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Numerical analysis of ecological survey data

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Numerical analysis of ecological survey data

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Content NUMERICAL ANALYSIS OF ECOLOGICAL SURVEY DATA by R obert W illiam Smith A D issertatio n P re se n te d to the FACULTY OF THE GRADUATE SCHOOL UNIVERSITY OF SOUTHERN CALIFORNIA In P a r tia l F ulfillm ent of the R eq u irem en ts for the D egree DOCTOR OF PHILOSOPHY (Biology) Jan u ary 1976 Copyright (Robert W illiam Smith) 19 76 UNIVERSITY OF SOUTH ERN CALIFORNIA T H E GRADUATE SC H O O L UN IV ER SITY PARK LOS A N G ELE S. C A L IF O R N IA 9 0 0 0 7 PK .P. '76 S 65& This dissertation, written by Robert William Smith under the direction of hi,^.... Dissertation Com mittee, and approved by all its members, has been presented to and accepted by The Graduate School, in partial fulfillment of requirements of the degree of D O C T O R OF P H I L O S O P H Y 9 / / f - Dean Date. DISSERTATION COMMITTEE Chdkman ACKNOWLEDGMENTS I would like to thank m y com m ittee ch airm an . Dr. K. Fauchald for his encouragem ent, advice, and co n sid erab le a s s i s tan ce w ith the m a n u sc rip t. D r. I. Straughan, Dr. G. Bakus, D r. D. J. R eish, and Dr. B. Abbott also m ade valuable suggestions concerning the m an u sc rip t. D r. Straughan in itially in tro d u ced m e to th e se types of analy ses and has since been a so u rce of en co u rag e m ent and valuable d isc u ssio n s. D r. W. Stephenson, during his stay at USC, m ade his c o n sid erab le experience with such m ethods available, and in effect g rea tly a c c e le ra te d m y p ro g re s s . D r. D. S. G orsline, b esid es teaching m e all about sedim ent an aly sis, provided m e w ith se d im e n t-s iz e d istrib u tio n form u lae, w hich led to id ea s on how to apply w eights in the w eighted d iscrim in an t an aly sis m ethod. Dr. C. G reene of SCCWRP provided m any useful d isc u ssio n s on m ethods, along with m uch im p o rtan t inform ation co ncernin g the data u sed in the study. D r. P e te r Ju m a rs and s e v e ra l of m y fellow g rad u ate students at USC also have provided m any stim u latin g d is c u ssio n s. 1 1 A d isse rta tio n of this scope could not have been w ritte n without the extensive financial support from the H arbor P ro je c t of the Allan Hancock Foundation (Dr. D. Soule and M. Oguri, D ire c to rs). T his funding orig in ated from g ran ts by Sea G rant, the Los A ngeles H arb o r D epartm ent, the Tuna R e se a rc h Foundation, the Southern C aliforn ia Gas C o ., and the U. S. A rm y C orps of E n g in eers. F in a n cial supp ort for the use of th e com puter was provided by the Biology D epartm ent at USC, the H arb o r P ro je c t, and the Southern C alifornia C o astal W ater R e se a rc h P ro je c t (SCCWRP). Nick Condap, Lou F rid k is, John E llis, along with m any of the consultants at the U niv ersity C om puter C enter, provided valuable a ssista n c e with the extensive com puter p ro g ram m in g re q u ire d in such a study. Nick Condap w rote the p ro g ram for plotting the d e n d ro g ra m s on the line p lo tte r. Cathy Link is resp o n sib le for the draw ing of the fig u res. The final typing w as done by O 'D etta Hawkins. L ast but not le a st, thanks to m y wife. Posy, who w as a constant so u rc e of encouragem ent and joy throughout the grueling and so m e tim es d ish earten in g p ro c e ss of w ritin g a d isse rta tio n . Ill TABLE OF CONTENTS Page AC KNOW LE DGME NTS ii TABLE OF CONTENTS iv LIST OF TABLES vii LIST OF FIGURES X LIST OF APPENDICES xvii C hapter I INTRODUCTION 1 II GENERAL METHODS OF ANALYSIS 8 A. Introduction 8 B. The C alculation of the In te r-e n tity D istances 9 C. T ran sfo rm atio n and Standardization of the Biotic Data 15 D. A gglom erative H ie ra rc h ic a l C lassifi- c at ion 26 E . O rdination 31 F . M u ltip le-d iscrim in an t A nalysis 36 G. M u ltip le-lin ear R e g re ssio n 39 H. C om puter P ro g ra m s 42 III GENERAL DESCRIPTION OF THE DATA 43 A. Introduction 43 B.. Methods 43 C. P re lim in a ry Data S um m arization 47 iv C hapter Page IV A STUDY AND EVALUATION OF THE E F F E C T S OF THE VARIOUS PRELIM INARY DATA MA NIPULATIONS IN THE NORMAL ANALYSIS USING THE BRAY-CURTIS INDEX A. Introduction B. M ethods C. R esu lts D. D iscussion 56 56 58 63 99 V DATA REDUCTION AND THE RELATIVE IM PORTANCE OF THE SPECIES IN THE NORMAL ANALYSIS A. Introduction B. M ethods C. R esu lts and D iscussion 155 155 158 161 VI RELATIONSHIPS BETW EEN THE SPECIES AND THE RESULTS OF THE NORMAL ANALYSIS A. Introduction B. Methods 171 171 172 VII ANALYSIS OF THE SPECIES USING CLASSI FICATION AND WEIGHTED DISCRIMINANT ANALYSIS A. Introduction B. Methods 183 183 184 VIII A DETAILED ANALYSIS OF THE DATA A. Introduction B. M ethods C. R esu lts 232 232 232 235 IX FURTHER CONSIDERATIONS OF THE SITE-W EIGHTING PROBLEM A. Introduction B. Methods C. R esu lts and D iscussion 327 327 331 344 X DISCUSSION A. What Is Being M easu red B. P red ictio n 355 355 361 V C hapter Page X (Continued) C. A nalysis of Data G athered O ver T im e 362 D. The P ro b lem of Uneven H abitat Sam pling 364 E. Site v s. Species S tandardization of the Data in the N orm al A nalysis 365 XI SUMMARY AND CONCLUSIONS 367 A. P a tte rn s in the B iological Data 367 B. R elationships betw een the Biotic P a tte rn s and the E nvironm ent 369 C. Data Reduction 370 D. E conom ical D isplay of the R elatio n ships betw een the Site and Species P a tte rn s 370 E. The C o rrec tiv e P ro c e d u re for Uneven H abitat Sam pling 370 F. U sefulness of the Methods 371 REFER EN C ES 372 APPENDICES A M U LTIPLE DISCRIMINANT ANALYSIS- CALCURATIONS AND IN TERPRETA TIO N OF RESULTS 384 A. In itial C alculations 384 B. W eighting C onsiderations 387 C. C alculation of the D iscrim inant Scores 389 D. R elationships betw een V ariables and the D iscrim in an t Space 390 E. A ssum ptions of the Method 393 F. Effect of O utlier Groups 394 G. L im its on the Num ber of V ariables Used 395 B LIST OF THE 94 MOST FREQUENTLY OCCURRING SPECIES 397 vi LIST OF TABLES Table P age 2- 1 F o rm u lae for the stan d ard izatio n s to be con sid e re d in the study. 19 3- 1 In te rc o rre la tio n s (xlOO) betw een the m e a su re d abiotic v a ria b le s. 54 4- 1 The in ter c o rre la tio n s (xlOO) betw een the calculated population p a ra m e te rs . 64 4- 2 C o rrela tio n s (xlOO) betw een sp e c ie s-d ista n c e contributions and population p a ra m e te rs for the v ario u s data types. 66 4- 3 S tandardized m u ltip le -re g re s s io n coefficients (ex p re ssed as percen tag e of to tal of absolute values for each axis), indicating the environ m en tal c o rre la tio n s with the ordination axes g en erated by each d a ta ty p e . 87 4- 4 C oefficients of se p a ra te determ in atio n from the d isc rim in an t analysis of the 11 groups defined by the various data types. 92 4- 5 C oefficients of se p a ra te determ in atio n from the d isc rim in an t analy sis of six groups defined by the v ario u s data types. 4- 6 C oefficients of se p a ra te d eterm in atio n from the d isc rim in an t analy sis of th re e groups as defined by the v arious data types. 98 4- 7 C o m p ariso n of site s on opposite sides of the abundance peak in Fig. 4-10. 104 vii Table Page 4- 8 C o m parison of som e site s se p a ra te d by an abundance count of th re e in Fig. 4-10. 106 4- 9 C o m parison of site s on opposite sides of the abundance peak, w ith the th re e sp ecies in Fig. 4-11. 111 4- 10 C o m pariso n of site s on opposite sid e s of the abundance peak, w ith s p e c ie s-m e a n stan d ard ized values for the th re e sp e cie s in Fig. 4-11. 115 4- 11 D em onstration of the effect of z e ro counts on the r e s u lts of a sp ecies-S D standard ization. 128 5- 1 T otal in form ation loss a sso c ia te d with each sp e c ie s-e lim in a tio n c rite rio n . 162 6- 1 Two cells in a hypothetical tw o-w ay table and a sso c ia te d cell su m m ariza tio n m e a s u re s . 177 7- 1 C o rre la tio n s (xlOO) betw een som e of the species c u rv e s in Fig. 7-2. 191 7- 2 In te r-s p e c ie s overlap in d ices to be com pared. 200 8- 1 C oefficients of se p a ra te d eterm in atio n from the d isc rim in an t an aly sis of sev en site groups. 260 8- 2 C oefficients of se p a ra te d eterm in atio n for the se p a ra te analyses of group p a irs not w ell se p a ra te d by the f ir s t two axes in the 11-group d isc rim in a n t an alysis. 266 8- 3 C om p ariso n of the m o re im p o rtan t v a ria b le s se p a ra tin g groups F and G. 267 8- 4 C o m p ariso n of the m o re im p o rtan t v ariab les se p a ra tin g groups H and I . 268 8- 5 C o m parison of the m o re im p o rtan t v a ria b le s se p a ra tin g groups J and K . 270 viii Table P age 8- 6 C om pariso n of the potentially im p o rtan t v a ria b le s se p a ra tin g groups D and E. 271 8- 7 W eighted m eans of the m o re im p o rtan t v a ria b le s for the sp e cie s in sp ecies group 1. 291 8- 8 W eighted m eans of the im p o rtan t v a ria b le s fo r the sp ecies in the indicated sp e cie s groups. 295 8- 9 S tandardized m ultiple re g re s s io n coefficients (xlOO). 315 IX LIST OF FIGURES F ig u re Page 2- 1 G eom etric re p re se n ta tio n of the sp e c ie s-to ta l and sp e c ie s-n o rm p a ra m e te rs . 20 2- 2 The p ro c e ss of building a dendrogram using flexible so rtin g stra te g y . 27 2- 3 Explanation of the flexible so rtin g stra te g y . 29 2- 4 The purpose of o rdination illu stra te d by com p a riso n of the efficiency of selected point projection s. 32 2- 5 D em onstration of the im po rtance of using m u ltiv ariate techniques when considering group d ifferen ces. 40 3- 1 P o sitio ns of the sam pling site s (after Sm ith and G reene, 1976). 44 3- 2 G eneral p attern s of sp e c ie s d iv ersity and abundance in the sam pling a re a (from Sm ith and G reene, 1976). 48 3- 3 P a tte rn s of the abiotic v a ria b le s in the sam pling a re a . 51 4- 1 PCA of the data types with the elem ents of the distance m a tric e s as a ttrib u te s. Axes I and II shown. 70 4- 2 PCA of the data types with the elem ents of the distance m a tric e s as a ttrib u te s. Axes I and III shown. 72 X F ig u re Page 4- 3 PCA space of Fig. 4- 1 w ith the c o rre la tio n s (xlOO) of the re sp e c tiv e data types w ith the total abundance indicated. 74 4- 4 PCA space of Fig. 4- 1 w ith the c o rre la tio n s of the re sp e c tiv e data types w ith the num ber of o c c u rre n c e s indicated. 76 4- 5 PCA space of Fig. 4-2 with the m axim um value for the re sp e c tiv e data types indicated. 79 4- 6 PCA of the data types w ith the m ultiple r e g re ssio n coefficients as a ttrib u te s. Axes I and II a re shown. 82 4- 7 PCA of the data types using the m ultiple r e g re s sio n coefficients as a ttrib u te s. Axes I and III a re shown. 84 4- 8 PCA of the coefficients of se p a ra te d e te rm in a tion gen erated by d iscrim in an t analy sis of the 11 groups defined by the v ario u s data types. 90 4- 9 PCA of the coefficients of se p a ra te d e te rm in a tion gen erated by d isc rim in an t analysis of the six groups defined by the vario u s data types. 94 4- 10 B asic m odel of rela tio n sh ip betw een sp ecies abundance (fitness) and p ertinent en v iro n m en tal variation. 100 4- 11 T h ree hypothetical sp ecies c u rv es along an environm ental gradient. 109 4- 12 S p e c ie s-m ea n stan d ard ized cu rv es from the data in Fig. 4- 11. 113 4- 13 D em o nstration of the rela tio n sh ip betw een k u rto sis and the re lia b ility of peak abundance values. 122 xi F ig u r e P age 4-14 The effect of the v arious sp ecies sta n d a rd izations on five hypothetical sp ecies c u rv e s. 125 4-15 H ypothetical environm ental gradient with s e v e ra l sp e cie s c u rv e s. 132 4-16 H ypothetical sp e cie s curve d em o n stratin g the effect of cu rv e tru n catio n on environ m en tal c o rre la tio n s . 13 9 4-17 D em o nstration of the relatio n sh ip betw een the num ber of groups se p a ra te d by an environ m en tal facto r and the co rresp o n d in g lev el on the d en d ro g ram . 143 4-18 D em onstration of the effect of the site -to ta l stan d ard izatio n on the data in Fig. 4-11. 150 4-19 D em onstration of a problem with site sta n dard izatio n s using the geo m etric in te rp r e ta tion of the site -n o rm standardization. 152 5- 1 The b est in fo rm a tio n -lo ss profile for each data type. 165 5- 2 The info rm atio n lo ss asso ciated with different n u m b ers of sam ple re p lic a te s at each site. 168 6- 1 The co n stru ctio n of a tw o-w ay coincidence table. 173 7- 1 H ypothetical c u rv e s dem o n stratin g the quali tative and quantitative com ponents of in te r sp ecies ov erlap . 185 7- 2 D em onstration of the instability of the c o r r e lation coefficient as a m e a su re of in te r-s p e c ie s overlap. 189 7- 3 R elationship betw een raw abundance counts and calculated in te r-s p e c ie s overlap. 194 Xll F ig u re Page 7- .4 R elationship betw een s p e c ie s -to ta l sta n d ard iz ed data and calcu lated in te r-s p e c ie s overlap. 196 7- 5 PCA of the v ario u s m ethods of calculating the in te r-s p e c ie s overlap. 201 7- 6 The problem of sam ple spacing in relatio n to the m ea su re m e n t of in te r-s p e c ie s overlap. 205 7- 7 C o m parison of stan d ard ized c u rv es using sp e c ie s-to ta l and sp e cie s-m ax im u m sta n d a rd izatio n s. F o r the raw data, the cu rv e shapes a re constant, but the widths vary . 209 7- 8 C om parison of stan d ard ized c u rv e s using the s p e c ie s -to ta l and sp e cie s-m ax im u m sta n d a rd izatio n s. The o rig in a l c u rv es v a ry in shape but a re constant in width. 212 7- 9 T h ree hypothetical sp e cie s c u rv e s used to d em o n strate re a so n s for using the B ra y -C u rtis index in the in v e rse an aly sis. 217 7-10 The application of site w eights to the B ray- C u rtis index calculations in the in v e rse analysis. 220 7-11 In fo rm a tio n -lo ss profile from random elim ina tion of sp e cie s. 222 7-12 F o rm a tio n of groups of sites for studying the e nvironm ental rela tio n sh ip s of the sp ecies with m ultiple d iscrim in an t analysis. 225 7-13 Two hypothetical sp ecies c u rv e s used to dem on s tra te the usefulness of w eighted calculations in m ultiple d isc rim in an t an aly sis. 227 8- 1 C lassificatio n of the 40 s ite s . 236 8- 2 P o sitio n s of the sites in the 11 site groups de fined by the n o rm a l c la ssific a tio n analysis. 238 xiii F ig u re Page 8- 3 D istribution of the site w eightings. 241 8- 4 C lassificatio n of the sp e cie s. 243 8- 5 S pecies-group p a ttern s in the sam pling a re a . 245 8- 6 P a tte rn s of the subgroups of sp e cie s group 1. 248 8- 7 Two-way coincidence table. 250 8- 8 S u m m aries of the cells of the tw o-w ay table. 252 8- 9 The sev en m ain groups of site s which a re se p a ra te d by the firs t two axes of the 11- group d isc rim in an t analysis. 255 8-10 P o sitio n s of the site groups at the 7 -group level. 258 8-11 D iscrim in an t analysis of site s at the 7 -group level. The firs t two axes a re shown. 261 8-12 D iscrim inant analy sis of site s at the 7 -group level. The firs t and th ird axes a re shown. 263 8-13 The seven m ain sp e cie s groups in the w eighted d iscrim in an t space. 274 8-14 The four subgroups of sp ecies group 1 in the w eighted d isc rim in an t space. 276 8-15 P a tte rn s of the sp e cie s abundances in the sam pling a re a . 279 8-16 W eighted d isc rim in an t analysis of the sp e cie s in the four subgroups of sp ecies group 1 . 289 8-17 W eighted d isc rim in an t analysis of the sp ecies in four of the sp e cie s groups. 293 xiv F ig u re Page 8- 18 The distrib u tio n of the sp e cie s abundances in the d isc rim in an t space for the sp e cie s in sp e cie s group 7 . 300 8-19 P lots of the ordination s c o re s on the firs t th re e axes. 303 8-20 The p a tte rn of site sc o re s on ordination Axis I . 306 8-21 The p a ttern of site s c o re s on ordination Axis II . 308 8-22 The p a ttern of site sc o re s on ordination Axis I I I . 310 8-23 C o m p ariso n of the sulfide p a tte rn with the p a tte rn of s c o re s on ordination Axis I I I . 313 8-24 O rdination tw o-w ay table for Axis I . 317 8-25 O rdination tw o-w ay table for Axis II . 319 8-26 O rdination tw o-w ay table for Axis III . 321 8-27 Plot of the w eig h ted -av erag e s c o re s for each sp e cie s in the ordination space of the firs t two axes. 324 9- 1 D em onstration of the effect of uneven habitat sam p ling on the sp e c ie s-m e a n stan d ard izatio n . 329 9- 2 The effect of weighting on the colum n to tals of the distance m a trix . 332 9- 3 Sam ple calculations for the application of two different sc a le s to the distance m a trix in Fig. 9-2. 337 XV F ig u r e P age 9- 4 Standard deviations of the m ean d istan ces (SD^ ) of the s ite s v s. the scale of the w eights. 339 9- 5 Plot of SD^ vs. the sc a le of the w eights using the 4 0 -site data. 345 9- 6 The p a ttern s of site w eights for the 4 0 -site data, calcu lated by the two altern ativ e m ethods. 349 9- 7 The p a ttern s of site w eights calculated by the two altern ativ e m ethods when using sim u lated data. 3 51 X V I U S T OF APPENDICES Appendix Page A M U LT IPL E DISCRIMINANT ANALYSIS- CALCULATIONS AND IN TER PRETA TIO N OF RESULTS 384 B U S T OF THE 94 MOST FREQUENTLY OCCURRING SPECIES 397 xvu CHAPTER I INTRODUCTION . . . fruitful m odels and hypotheses fo r s ta tis tic a l ecology a re likely to be b ased on biological r a th e r than purely m a th e m a tic a l p re m is e s . In s ta tis tic a l ecology in the p ast, m a th e m a tic a l and s ta tis tic a l c o n sid eratio n s have too often played the ro le of m a s te r r a th e r than serv an t; the subject is likely to m a r k tim e u n til its biological a sp ec ts re s u m e th e ir rightful place. -- D. W. Goodall (1970:122) The above quotation im p lies that in som e types of sta tis tic a l ecological stu d ies, the biological a sp ects have not always been given sufficient c o n sid eratio n . The m ain purpose of th is d isse rta tio n is to exam ine som e of the biological co n sid era tio n s w hich a re involved in the study of e co lo g ic a l-su rv e y data w ith m u ltiv a ria te s ta tis tic a l tech niqu es. T his leads to suggested m ethodological im p ro v em en ts in o rd e r that the m ethods m ay be c o n sisten t w ith the m o st frequently ob serv ed biological re a litie s . T his should in c re a s e the a ccu racy and pow er of th ese techniques. As u sed in th is study, the m ain intent of the m ethods is to study the rela tio n sh ip s betw een the sam p led o rg a n ism s and th e ir environm ent. C lassificatio n and ordination m ethods a re u sed to find p a tte rn s in the biolog ical data. T h ese m ethods take advantage of th e frequent high levels of redundancy in biotic p a tte rn s, i. e . , s e v e ra l sp ecies can resp o n d to the sam e en v iro n m en tal g radients o r discontinuities in the sam pling a re a . The sp e c ie s, acting in co n ce rt, w ill cau se stro n g p a tte rn s of change in the sp e c ie s abun dance (or b io m ass, etc. ) data. The n u m e ric a l m ethods to be used should detect th is since all sp ecies are c o n sid ered sim ultaneou sly. Such analysis is also econom ical because it is u n n e c e ssa ry to analyze each sp ecies se p a ra te ly . M ultiple r e g r e s s io n and m u ltip le -d isc rim in a n t analysis a re u se d to re la te the ordination and c la ssifica tio n r e s u lts to the m e a s u re d en v iro n m en tal (physical) p a ttern s. Since the re la tio n ships betw een the biota and the environm ent a re often m u ltiv a ria te (A lderdice, 1972), m u ltiv a ria te m ethods such as th e se a re e sse n tia l to avoid loss of im p o rtan t b io tic -e n v iro n m e n ta l c o rre la tio n s . It should be em p h asized th at th ese m ethods a re a m ean s of organizing com plex data into a form in which the p a tte rn s in the data a re evident. The in itia l re s u lt re p re s e n ts only a beginning; it is up to the ecologist to use this and other p ertin en t scientific info rm atio n to propose in tellig en t hypotheses about the e co sy stem . As the opening quote also im p h e s , hypothesis g en eratio n involves a c e rta in amount of c re a tiv ity - a situation^which is not alw ays co m patible w ith inflexible m a th e m a tic a l p re m is e s . Put another way. m a th e m a tic a l exactness cannot su b stitu te for intelligence and good ecolo gical se n se . The relatio n sh ip of th ese m ethods to ecology is so m e tim es m isu n d ersto o d . F o r exam ple, in a review of W h ittak e r's (1973) book on c la ssifica tio n and ordination, Johnson (1974a:688) sta te s, ". . . and m any w ill point out that ordination and c la ssifica tio n a re c o n sid era b ly behind the cutting edge of m o d ern com m unity stu d ies. ' Such an attitude com pletely m is s e s the point. The m ethods a re tools to be used by ecolo gists, and not ends in th e m se lv e s. They can be applied to a whole sp e c tru m of ecological stu dies from the th e o re tic a l (W hittaker, 1967; R ickiefs, 1972; Cody, 1974; G reen, 1974) to the p ra c tic a l problem of hypothesizing the environm ental im p a c t of hum an activ ities (M oore, 1973; C ro ssm a n et a l . , 1974; L ittle r and M u rray , 1975; Sm ith and G reene, 1976). Many asp ects of ecological sy ste m s cannot be studied e asily by co n tro lled e x p erim e n tal approach es but, instead, re q u ire that the ecologist o b serv e the eco sy stem as it n a tu ra lly o c cu rs. Cody (1974) calls th is approach a "n a tu ra l ex p erim en t. " This lim itation is e sp ecially im p o rtan t in studying environm ents which a re r e l a tively in a c c e ssib le . In addition, other eco sy stem s w ill be on a sc ale which is too sm a ll or the o rg an ism s w ill be too cry p tic for m any types of detailed study (e. g . , infaunal com m unities in m a rin e subtidal sed im en ts). It is not s u rp ris in g that the m o st com plete ecological studies have involved fo re sts, b ird s, in te rtid a l m a c ro fauna, etc. which contain re la tiv e ly larg e o rg a n ism s in habitats re a d ily a cc essib le to the ecologist. In the absence of co n tro lled studies, the ecologist m ust w o rk with c o rre la tio n a l data which th e o re tic a lly is insufficient to prove cause and effect. In the hands of an im aginative ecologist, how ever, such info rm atio n can lead to in te re s tin g and useful hypotheses which can p o ssib ly be te s te d m o re rig o ro u sly at a la te r tim e, or future "n atu ra l e x p erim e n ts" can be suggested. This point should be kept in m ind le st data be gath ered just for the sake of gathering data. The danger h e re is that the m any detailed c o r r e la tions o b serv ed m ay not be of any in te re s t to anyone (Pielou, 1975:3). As the com plexity of the data in c re a s e s , the m ethods used becom e m o re im p o rtan t. At the p re se n t sta te of the a rt, the ecologist m ust m ake s e v e ra l decisio n s. F o r exam ple, to find the p a tte rn s in the biotic data, th e re a re m any different m ethods of ordination and c la ssific a tio n to choose from . In addition, m any options concerning biological s im ila rity (or distance) m e a su re s, data tra n sfo rm a tio n , data standardization, and data reduction m ust be co n sid ered . The choices can affect significantly the final r e su lts. This m akes it im p o rtan t for the ecologist to u n d erstan d the biological im plications and assum ptio ns a sso c iated with each choice. Many ecologists u sing the m ethods seem to be unaw are of m any of th e se choices. The av ailab ility of co m puter p ro g ra m s is probably a facto r h e re . The m o st acc essib le and w id e sp rea d p ro g ra m packages a re w ritte n m ain ly for biom edical, so c ial- scien ce, geological, or taxonom ic an aly ses. The assu m p tio n s and techniques involved in th e s e a re a s of analysis m ay be m ea n in g le ss o r m isle ad in g when applied to ecological data. T hese p ro g ra m s n e v e rth e le s s a re applied without m odification to ecological an aly ses. In fact, the m ost ecologically-m eaningful options a re not even available in m o st of th e se co m p u ter packages. On the o th er hand, som e m ethods have been developed by eco lo g ists and, as such, the assu m p tio n s and m echanics of these techniques a re b e tte r und ersto o d by th e ir u s e r s . E xam ples a re the r e c u rre n t-g ro u p s analysis (E ag er, 1957), polar ordination (B ray and C u rtis, 1957), and d ire c t gradient analysis (W hittaker, 1967). T hese m ethods tend to be s im p le r than som e of the m o re com m only- u se d and so p h isticated techniques ( e .g ., p rin cip al com ponents an aly sis). T his does not im p ly th at the sim p le r m ethods a re un d e sira b le since often the sim p le m ethod can be shown to be p r e f e r able (W hittaker and Gauch, 1973). P a p e rs re p o rtin g the re s u lts of n u m erica l an aly ses often fo rc e the r e a d e r s to accept m uch of th e ir in te rp re ta tio n on faith. The r e a d e r should not have to re p e a t the sam e analysis to be able to evaluate the validity of the r e s u lts . T h ere a re som e m ethods available (such as tw o-w ay coincidence ta b le s, se e C hapter VI) that w ill display econom ically a m axim um am ount of inform ation about the an aly sis, and such techniques should be included in the re s u lts w henever p ra c tic a l. No detailed su rv e y of the applications of n u m e ric a l m ethods is given h e re sin ce th is has been done adequately elsew h ere (Noy_ M eir, 1970; W hittaker, 1973; Clifford and Stephenson, 1975). C hapter II gives a g e n e ra l d e scrip tio n of the m ain analytical m ethods to be c o n sid ere d . C hapter III d e sc rib e s the data which a re used to study the p a tte rn s of re s u lts when different m ethods a re applied. T hese data also are used in a d em o n stratio n of re s u lts w ith the p re fe r re d m ethods (Chapter VIII). C hapter IV is a study of the v ario u s c o n sid era tio n s involved in the an alysis of sa m p lin g -site (or spatial) p a tte rn s of the biotic data. C hapter V explores m ethods of deciding which, if any, sp e c ie s can be elim in ated in the analysis of the sa m p lin g -site p a tte rn without undue d isto rtio n of re s u lts . T his is im p o rtan t in c o n sid erin g the econom y of tim e, effort, ex pense, and display of r e s u lts . C hapter VI d isc u sse s techniques of re la tin g the sp e cie s data to the sa m p lin g -site p a tte rn s. C hapter VII involves the d ire c t an aly sis of the sp e cie s and groups of species and th e ir re la tio n sh ip s to the environm ent. C hapter VIII is a dem o n stratio n of the techniques d isc u sse d in the previous c h ap ters. C hapter IX is included as an afterthought. Some p ro b lem s re la te d to uneven sam pling of habitats a re d isc u sse d . O ther a sp ec ts of this topic a re introduced in C hapter Vll. CHAPTER II GENERAL METHODS OF ANALYSIS A. Introduction This c h ap ter contains a g e n e ra l d isc u ssio n of c lassificatio n , ordination, and m u ltip le -d isc rim in a n t a n aly ses. The c la ssifica tio n and ordination techniques a re used m ainly to find p a ttern s in the biotic data, and the d isc rim in an t analysis re la te s th ese p a ttern s to the enviro n m en tal data. The c la ssific a tio n and ordination m ethods to be used re q u ire input in the form of an in te r-e n tity distance m a trix . An entity is the unit to be analyzed. In te r-e n tity distan ces a re calculated from c o m p a riso n s of a ttrib u te s m e a su re d at or on the e n titie s. E ntities with s im ila r a ttrib u te m e a su re m e n ts w ill be se p a ra te d by only a sm a ll distan ce, and those with m o re d is s im ila r m e a su re m e n ts w ill be re la tiv e ly m o re distan t. The distance m a trix includes the calcu lated d istan ces betw een all possible p a irs of entities. In re la tio n to c la ssifica tio n and ordin ation of the biotic data, two m ain types of analysis are p u rsu ed in this study. A n o rm al analysis is an an aly sis w here the sam pling site s a re the entities and the sp e cie s counts a re the a ttrib u te s. An in v e rse analysis 8 entails using the sp e cie s as en tities and the sp e c ie s counts in the site s as the a ttrib u te s. When biotic data a re used, th e d istances so m e tim es a re r e f e r r e d to as "ecological" d istan c es (W hittaker, 1967). The calcu lation of the in te r-e n tity d istan ces is d isc u sse d f ir s t. B efore the d istan ces can be calculated, how ever, the raw sp e cie s data a re u su ally tra n s fo rm e d or stan d ard ize d , or both, and som e asp ects of th is topic a re d isc u sse d next. F inally, the g en eral p ro c e s s e s involved in th e c la ssifica tio n , ordination, and m ultiple- d isc rim in an t m ethods a re p resen te d . B. The C alculation of the In te r-e n tity D istances When quantitative data a re used, th e re a re th re e basic types of distance indices that w ill be co n sid ered and d isc u sse d (although not n e c e s s a rily used) in th is study. 1. The E u clid e a n -d ista n ce index (Clifford and Stephenson, 1975) is Dij = V ' Z - Xkj)^ . (2-1) k=l w h ere D^j is the E uclid ean distance betw een en tities i and j , Xj^ and X]^j a re the v alu es of attrib u te k in en tities i and j , resp ec tiv e ly , and n is the num ber of a ttrib u te s. T his m e a s u re is equivalent to the d istan ce betw een two points in a C a rte sia n coordinate sy stem (B e rs, 1969:629). In the n o rm a l an aly sis, each 9 dim ension of the coordinate sy ste m would c o rre sp o n d to a sp ecies and the points in the space would r e p re s e n t the site s . In the in v e rs e an aly sis, the dim ensions and points a re the sites and sp e cie s, re sp e c tiv e ly . Such a m e a s u re is sen sitiv e to the m agnitudes of the d ifferen ces betw een the a ttrib u te s. The la rg e r d ifferences n a tu ra lly w ill contribute a re la tiv e ly g re a te r amount to the sum of the sq u a re s of the d ifferen ces. U sually, attrib u tes on a la rg e r scale than o th er a ttrib u te s w ill have an in c re a s e d potential for g r e a te r d ifferen c e s. Insofar as this potential is rea liz e d , th is m e a s u re w ill be se n sitiv e also to the sc a le of the a ttrib u te s. To prevent e x ce ssiv e dom inance by attrib u tes w ith larg e d ifferences am ong the e n titie s, data tra n sfo rm a tio n s a n d /o r stan d ard izatio n s u su ally a re re q u ire d before calculation of the d istan ces. T h ere a re som e se rio u s p ro b lem s when this index is applied to ecological data. In the n o rm a l an aly sis, the sites with re la tiv e ly low er abundance counts a n d /o r few er sp e cie s w ill tend to be found n e a r the o rig in of the coordinate sy ste m . This is tru e re g a rd le s s of the qualitative sp e c ie s com position of the site s in question. Some of the u n d e sirab le re s u lts of such a situation a re shown by B an n ister (1968), F ie ld (1969), Gauch and W hittaker (1972), W hittaker and Gauch (1973), and C lifford and Stephenson (1975). S parcely populated site s w ith no or few sp e cie s in com m on can end up with re la tiv e ly sh o rt distances of se p ara tio n . Species can be 10 r a r e o r absent at a site for any num ber of com pletely different re a s o n s as far as the environm ent is concerned. A ccordingly, two s ite s re la tiv e ly lacking in the sam e sp ecies a re not n e c e s s a rily environm entally o r biotic ally s im ila r (Field, 1969). C o rre c tiv e m e a s u re s to o vercom e th is problem a re d isc u sse d la te r in the se ctio n on data sta n d ard iza tio n s and in C hapter IV. T his is h en ce fo rth r e f e r r e d to as the "d o u b le-z ero " problem b ecau se site s lacking m any of the sa m e sp e c ie s w ill tend to be found in the sam e v icinity in E uclidean sp ace. In the in v e rs e an aly sis, the index would be used as a m e a s u re of in te r-s p e c ie s o v e rla p . A d o u b le-zero c o m p a riso n at a site tends to in c re a s e the re la tiv e s im ila rity (m ea su red overlap) even though th e re is no o v erlap at the site of co m p ariso n . The r e lationship betw een d istan ce, overlap , and sim ila rity is d isc u sse d la te r in C hapter VII. The E uclidean d istance betw een two en tities is d ire c tly r e lated to the c o rre la tio n and co v arian ce betw een the sam e two en tities (A nderson, 1971). M ethods utilizing th e se two m e a s u re s (e. g . , PCA) w ith sp e cie s data w ill suffer the sam e d eficien cies as p o ss e s se d by the E u clid e a n -d ista n ce m e a su re . 11 2. The B ra y -C u rtis index is n B. . . n (2- 2 ) w h ere D^j is the d istan ce betw een entities i and j , Xj^ and X^. a re the values of a ttrib u te k in entities i and j , resp ec tiv e ly , and n is the num ber of a ttrib u te s. T his fo rm u la is equivalent to the com plem ent of the sim ila rity index u sed by B ray and C u rtis (1957), _i._e. , Dj^ = 1 - , w here Sj^ is the s im ila rity m e a s u re . The b in a ry c o u n te rp a rt of the B ra y -C u rtis index is called the CzekanowskL coefficient (Czekanow ski, 1913) o r the Dice index (Dice, 1945). However, when u se d w ith quantitative data, it is so m e tim es r e f e r r e d to as the Czekanowski m e a s u re (Hall, 1969, 1970; F ield , 1971; Goodall, 1973). Some of the im p o rtan t p ro p e rtie s of th is index are as follows: a) The c alcu late d distance has an u p p er lim it of one. T his is accom plished by the scalin g of the attrib u te c o m p ariso n s by the sum of the attrib u te valu es, b) D o u b le-zero co m p ariso n s have no effect on the d istan c es. F o r re a so n s d isc u sse d above, this is d e sira b le when sp e c ie s data a re used, c) Since the index is a single quotient, the d istan ce calcu lations a re se n sitiv e to the sc ale of the n u m b ers used. L arg e a ttrib u te d ifferen ces can influence 12 significantly the n u m e ra to r of the quotient, but la rg e n u m b ers w ill re c e iv e m o re w eight w h ether or not th ese n u m b ers re s u lt in larg e d ifferen ces when the a ttrib u te s a re co m p ared . F o r exam ple, if two entities have the sam e larg e value for a c e rta in attribute, the contribution of th is a ttrib u te to the final d istan ce w ill be z e ro in the n u m e ra to r of the above form ula, but w ill be the sum of the two larg e nu m b ers in the denom inator. This sum in the denom inator w ill have a significant effect on the final calcu lated distance. This situ ation is in c o n tra s t to the E uclidean distance m e a s u re , w here such a co m p a riso n would have the sam e effect w hether the dif fere n ce w as g e n erate d from a co m p ariso n of two la rg e n u m b ers o r two s m a ll n u m b e rs. H all (1969) d e m o n stra te s the effect of the sc a le of the n u m b ers on the B ra y -C u rtis d ista n c e s, and views this t r a it as being ecologically u n d esirab le. As w ill be shown la te r, it is ju st th is tr a it of the index that is u sed to advantage in both the n o rm a l and in v e rs e a n aly se s. Specifically, the influence of the v a rio u s a ttrib u te s can be se t at som e d e sire d lev e l by controlling th e ir sc ale and d istrib u tio n . As w ill be shown, th is c o n tro l is achieved by p re lim in a ry tra n s fo rm a tio n or sta n d ard iza tio n of the data. The fa ilu re to c o n sid er the scale of th e sp e c ie s counts can lead to se rio u s p ro b lem s sin ce a few larg e v alu es can end up d isp ro p o rtio n ately dom inating the an aly sis. 13 3. The C a n b e rra -m e tric d istance index (Boesch, 1973; C lifford and Stephenson, 1975) is n l^ki ' ^kil D = J _ \ (2-3) (Xki + X kj) ' k=l w h ere m is the num ber of n o n -d o u b le-zero a ttrib u te co m p ariso n s at site s i and j , and the o th er sym bols a re the sa m e as fo r the B ra y -C u rtis index. Only the frac tio n s involving n o n -d o u b le-zero c o m p a riso n s a re u se d in the sum and, accordingly, the sum is divided by m in ste a d of n. The index is the av erag e of m se p a ra te frac tio n s, and as such is not as se n sitiv e to the sc ale of the n u m b ers as is the B ra y - C u rtis in d e x .. T his is due to the fact that an a ttrib u te w ith a re la tiv e ly la rg e sc a le can only affect one of the s e v e ra l frac tio n s used to calcu late th is av erag e. T h e re is a problem of in se n sitiv ity w ith th is index when a ttrib u te values of so m e nu m ber g re a te r than z e ro a re com pared w ith a z e ro value. F o r exam ple, a co m p ariso n betw een 0 and 1 would re s u lt in the sa m e fractio n as a c o m p a riso n betw een 0 and 1000 ( l / l and lOOO/lOOO, resp ec tiv e ly ). To o v erco m e th is, a sm a ll fac to r u su ally about one-fifth of the sm a lle s t n u m b er in the data m a trix (C lifford and Stephenson, 1975) is su b tra c te d from the n u m e ra to r and added to the denom inator of each frac tio n involving 14 su ch a co m p ariso n . F o r the above exam ple (using a facto r of .2), the above fra c tio n s would becom e (1-. 2 )/(l+ . 2) = . 6667 and (1000-. 2)/(1000+. 2) = .9996. T hese r e s u lts a re m o re co m m en su ra te with the differen ces betw een 0 and 1 and 0 and 1000, re sp e c tiv e ly . The calcu late d d istan ces w ill v a ry betw een 0 and 1, as w ith the B ra y -C u rtis index. In su m m ary , the E uclidean distance m e a s u re is sen sitiv e to th e sc ale of the a ttrib u te differen ces and is affected by the d o u b le-z ero c o m p a riso n s in the calcu latio n s. The B ra y -C u rtis index is sen sitiv e to both the sc a le of the attrib u te d ifferen ces and the sc a le of the a ttrib u te s th e m se lv e s. The C a n b e rra -m e tric index, when co m p ared to the E u clid ean -d istan ce and B ra y -C u rtis m e a s u re s , is m uch le s s se n sitiv e to both the sc ale of the a ttrib u te s and the sc a le of the a ttrib u te d ifferen ces. N either the C a n b e rra - m e tric n or B ra y -C u rtis in d ices is affected by the d o u b le-z ero c o m p a riso n s, and both v a ry betw een 0 and 1. C. T ra n sfo rm a tio n and S tandardization of the Biotic D ata The sc ale and d istrib u tio n a l c h a ra c te ris tic s of the sp e c ie s- abundance data can be co n tro lled som ew hat by p re lim in a ry data tra n s fo rm a tio n s o r sta n d ard iza tio n s or both. A tra n s fo rm a tio n is a m a th e m a tic a l o p eratio n w hich is applied to each of a se t of n u m b e rs. The effect of a tra n s fo rm a tio n depends on the type of 15 tra n s fo rm a tio n and m agnitude of the values being tra n s fo rm e d . F o r in stan c e, a tra n s fo rm a tio n such as a log o r a cube ro o t w ill red u ce g re a tly the la rg e n u m b ers while having le ss effect on the s m a lle r n u m b e rs. In th is case , the tra n s fo rm e d data w ill s till be m onotonie w ith the raw data, i. e . , the la r g e r n u m b ers w ill s till be la rg e r, but the difference betw een the la rg e and sm a ll n u m b ers w ill be red u ced . A data sta n d ard iza tio n is a sc a le adjustm ent o r ra tio in volving division by som e p a ra m e te r of the site o r sp e cie s a sso c ia te d w ith the abundance count being stan d ard ized . The r e s u lts of a sta n d ard iza tio n a re not n e c e s s a rily m onotonie with the o rig in a l d ata and a re not alw ays com pletely p red ictab le without re fe re n c e to the data m a trix . 1, Data tra n sfo rm a tio n s D ata tra n s fo rm a tio n s often a re applied to data for the p u rp o se s of m eeting the assu m p tio n s of som e s ta tis tic a l p ro ce d u re to be u se d (B arn es, 1952; C a ssie and M ichael, 1968). In th is study, biolo gical data are u sed to calcu late ecological d istance betw een all p a irs of sam pling site s, or as a m eans of m ea su rin g in te r -s p e c ie s overlap. The distan ce index u sed (B ra y -C u rtis) is not a s ta tis tic a l m e a s u re w ith any a sso c ia te d p ro b ab ility distribution. N ev erth eless, w ith the use of th is and o th er sc a le - influe need in dices, th e re a re 16 som e im p o rtan t c o n sid era tio n s in re la tio n to data tra n sfo rm a tio n . a) The end re s u lt of a tra n s fo rm a tio n can be the convergence of the sc a le s of the data values for each sp e c ie s. The d eg ree of convergence can be controlled by the stre n g th of the tra n sfo rm a tio n . A sq u a re ro o t is usually a re la tiv e ly m ild tr a n s fo rm atio n and a fifth ro o t o r a log would be c o n sid ere d m o re se v e re . When using a s c a le -s e n s itiv e index and no subsequent data sta n d a rd i zation is to be applied, a tra n s fo rm a tio n would keep the abundant sp e cie s from com p letely dom inating the analy sis (Field, 1971). b) If a subsequent data sta n d ard iza tio n is to be ap plied, the effects of in te ra c tio n betw een the tra n s fo rm a tio n and stan d ard iza tio n should be c o n sid ere d . This topic is p u rsu ed la te r in th is section, c) E cological in te rp re ta tio n of r e s u lts of the analysis of biological data u su a lly en tails fu rth e r an aly sis to find c o r r e la tions with e n v iro n m en tal data. At le a st as a f ir s t approxim ation, th is n o rm ally involves m ethods w hich assu m e lin e a r rela tio n sh ip s betw een biotic and abiotic m e a s u re s . Since the rela tio n sh ip betw een sp e cie s abundances and environm ental g radien ts often a re exponen tia l (C assie and M ichael, 1968), a tra n s fo rm a tio n m ay be n e c e s s a ry so that assum ptions of lin e a rity can be justified. 17 2. Data sta n d ard iza tio n s A sta n d ard iza tio n can be applied to e ith er s ite s o r sp e c ie s, or both. The sta n d ard iza tio n p a ra m e te rs to be c o n sid ere d a re shown in T able 2-1. A dditional re fe re n c e s and d isc u ssio n on data sta n d ard iza tio n s a re found in A ustin and G reig-S m ith (1968), N oy-M eir (1970), C lifford and Stephenson (1975), and N oy-M eir, et al. (19 75). a) Species sta n d ard iza tio n s. Species stan d ard izatio n s attem pt to equalize the contribution of the different sp e cie s by con v e rtin g the sp e c ie s abundances to a ra tio of som e m e a su re of abundance o r v a ria b ility of the c o rre sp o n d in g sp e cie s. Species norm and to ta l a re m e a s u re s of the o v e ra ll abundance of a sp e cie s throughout the sam pling s ite s . The to ta l is sim p ly the sum of the abundances, and the norm is the distance of a point from the o rigin in a m u ltid im en sio n al space. H ere each dim ension of the sp ace r e p re s e n ts one site and the point in question r e p r e s e n ts a sp e c ie s. The distance of the point into a dim ension depends on the abundance of the sp ecies in the co rresp o n d in g site. The difference betw een the to ta l and the norm is shown in Fig. 2-1. E ach added o c c u rre n c e of a sp ecies w ill in c re a s e both the to ta l and the norm for that sp e cie s. Since, in the stan d ard izatio n , abundances a re to be divided by the co rre sp o n d in g norm or total, w id esp read sp e c ie s w ill show s m a lle r stan d ard ized values when co m p ared with 18 g o p g 3 W & 0 c . M . . H C Q 1 1 O ) s: G e n C Q ( Q O ) k r D C Q 0 1 U ■ M C Q o > ü o > G G P S - l O ) Si C Q o > C Q 0 1 O G U u o > a U o > a O o > a C Q T 3 a T 3 a • M a § C Q C Q C Q " S § 0 0 1 C Q C Q C Q C Q " G C N c < - ( « H G O O O 'ë - Æ y 0 S h O u O G C Q O G 1 C Q -g G - S ■ a 3 o r t 0 ) o > G o > G G o ^ o ^ O C Q " ë 1 Si a ja a X i a u o > X 3 a o > S 0 ) O o > “ i ü % 0 ) “ j 3 C 0 1 2 g g a g • M a a X î G x; G x; G C Q C Q G G G C Q G C Q C Q - W ' M -u - r - t - r - l I I • r ? G . X z H C O ! X " x" X >r % 0 1 II û z < H § H c o < H N U 3 H O h < W Q g ; g w X X X X c o =W E X Z I < H X IM-^ X ca*H I T3 O ) L , X X X X J g H X & X k E c o O ) c ■ I I s 1 O ) S I 0 ) H 19 F ig . 2-1. G eom etric re p re s e n ta tio n of the s p e c ie s -to ta l and sp e c ie s -n o rm p a ra m e te rs . 20 site 2 B site 1 O OB = abundance of sp e c ie s S in site 2 OA = abundance of sp e c ie s S in site 1 T otal = OB + OA N orm = OS = V OA^ + OB^ 21 m o re r e s tr ic te d sp e cie s of the sam e o r low er abundance lev e ls. W idespread sp e c ie s often a re abundant also and m ay display s m a lle r sta n d ard ize d v alu es than do the r a r e r , le s s abundant sp e cie s. All the values for th e se two sta n d ard iza tio n s w ill fall som ew here b e tw een 0 and 1. Species m axim um and th e sp e cie s m ean a re m e a s u re s of the g e n e ra l abundance level of a sp e cie s when it does o ccu r. Species m axim um sta n d ard iza tio n w ill put each sp e cie s on a sc a le w ith an id en tical upper lim it of one. Species m ean sta n d a rd ization has no se t upper lim it. It is im p o rtan t to em p h asize that the calculations involved in the m ean sta n d ard iza tio n only include data values g r e a te r than z e ro , i. e . , it re p r e s e n ts the m ean of the o c c u rre n c e s, not the m ean of the s ite s . The m ean of the site s is functionally equivalent to a to ta l stan d ard izatio n , since th is m ean is sim ply the to ta l divided by a num ber (num ber of site s ) w hich is a constant for all sp e c ie s. Species sta n d ard deviation is a m e a su re of the v a ria b ility of the sp e c ie s counts o v e r all the site s . Since th e re is n o rm a lly a high positive c o rre la tio n betw een the abundance and the sta n d a rd deviation of a sp e cie s, the sta n d a rd deviation also can be an in d ire c t m e a s u re of g e n e ra l abundance. T his stan d ard izatio n is u sed m o stly w ith m ethods which u tilize a c o rre la tio n m a trix , 22 sin ce such a sta n d ard iza tio n is p a rt of the calculation of a c o r r e la tion coefficient. E xam ples of such m ethods a re p rin c ip a l com ponent an aly sis (C assie and M ichael, 1968; Hughes and T hom as, 1971; M oore, 1974) and c la ssific a tio n s using c o rre la tio n s as a m e a s u re of s im ila rity (Ebbhng et a l . , 1970). b) Site sta n d ard iza tio n s. The site sta n d ard iza tio n s em p h asize the m an n e r in w hich the sp e cie s counts a re proportioned at a site . N um erically dom inant o rg a n ism s at a site w ill re c e iv e the h ig h er sta n d ard ize d values r e g a r d le s s of the level of the sp ecies counts at other s ite s . The sc a le c h a ra c te r is tic s of th ese sta n d a rd i zatio n s a re s im ila r to th o se of the c o rresp o n d in g sp e c ie s sta n d a rd i zatio n s. The site n o rm is the distance of a point ( r e p r e senting a site) fro m th e o rig in in a sp ace w here the dim ensions c o rre sp o n d to th e sp e c ie s. Or loci (1967a) reco m m en d s th is sta n d a rd izatio n as a m eans of overco m ing p ro b lem s with m ethods re la te d to E uclidean distance. N oy-M eir (1970) and N oy-M eir et al. (197 5), w orking with E uclidean distance, favored th is stan d ard iza tio n as producing the m o st "balanced" site pattern, since it equ alizes the contrib ution of each site in the an aly sis. The site to ta l and norm , stan d ard izatio n s w ill both r e s u lt in g en erally low er sta n d a rd iz e d values for sp e c ie s counts 23 from the m o re d iv e rs e and ric h s ite s , since each additional sp ecies w ill in c re a s e the to ta l o r norm by w hich the abundance values w ill be divided. Site m ean and m axim um sta n d ard iza tio n s, on the other hand, w ill allow the m o re d iv e rse site s to contain som e of the higher sta n d a rd iz e d v a lu e s . Site sta n d ard iza tio n s can be u sefu l when the sam pling in te n sitie s at all site s in the an aly sis are unequal. Under such c irc u m s ta n c e s, the re la tiv e p ro p o rtio n s of each sp ecies at a site w ill b e tte r r e p re s e n t the site than w ill the sp e c ie s counts or s p e c ie s -s ta n d a rd iz e d values, since raw abundances w ill not be m eaningful beyond a p a rtic u la r site. c) Double stan d ard izatio n s. T his type of sta n d a rd iz a tion involves the application of both a sp e c ie s and a site sta n d a rd iz a tion, e ith er in sequence o r sim ultaneo usly. Double stan d ard izatio n s would be expected to p o ss e s s som e of the c h a ra c te r is tic s of both the site and sp e cie s sta n d ard iza tio n s. 3. A pplication of a tra n sfo rm a tio n p rio r to sta n d ard iza tio n Since the rela tio n sh ip betw een sp e c ie s abundance counts and an en v iro n m en ta l g rad ien t can be exponential, one would expect to find one or a few values which a re c o n sid erab ly la rg e r than m o st other v alu es in biological abundance data. These ex trem e values can have a d isp ro p o rtio n ate effect on the p a ra m e te r of 24 sta n d ard iza tio n . The consequences of th is w ill v a ry with the type of sta n d a rd iz a tio n u sed . The norm or to ta l can becom e so la rg e that a ll o r n e a rly all the n u m b ers being stan d ard ized w ill be red u ced to unusually low values and re c e iv e little weight in the an aly sis. W ith the m axim um stan d ard izatio n , an e x tre m e m axim um n um ber can red u c e the rem a in in g n u m b ers to an insignificant frac tio n . The effect on the m ean sta n d ard iza tio n can be twofold. F ir s t, the peak sta n d a rd iz e d value can becom e v e ry larg e and the site or sp e cie s p o sse ssin g th is value can re c e iv e u n reaso n ab ly high w eight. Also, s m a lle r n u m b e rs can be d isp ro p o rtio n a te ly red u ced by an u n re a so n ably high m ean. In the situ atio n s noted above, the re s u ltin g m agnitude of the affected sta n d ard ize d values is not re la te d to any ecologically im p o rtan t c rite rio n , but only to the p re se n c e on one or a few ex tr e m e valu es in the calculation s of the p a ra m e te r of s ta n d a rd iz a tion. The problem caused by the e x tre m e counts can be a m e lio ra te d by a tra n s fo rm a tio n p rio r to the stan d ard izatio n . When such an adjustm ent is re q u ire d , the m o st ap p ro p ria te stra ta g e m is to apply the sa m e tra n s fo rm a tio n to the e n tire data m a trix ; to do otherw ise would involve subjective decisions w hich m ay be difficult to justify. In so m e e x tre m e c a s e s , how ever, c e rta in few sp e cie s w ill re q u ire a stro n g e r tra n s fo rm a tio n th an w ill the rem a in in g sp e c ie s. 25 D. A gglom erative H ie ra rc h ic a l C lassific atio n H ere the rela tio n sh ip s betw een the en tities are displayed in a tw o -d im en sio n al, h ie ra rc h ic a l s tru c tu re called a d en d ro g ram . The p ro c e s s of c o n stru ctin g the d en d ro g ram is explained with an exam ple. The sta rtin g point is the distance m a trix shown in Fig. 2 - 2a. The s h o rte s t distance in the m a trix is chosen and the two c o rre sp o n d in g en tities a re fused into a single group. This is re p re s e n te d on the d en d ro g ram by joining the entities with lines as shown on the rig h t side of the figure. The distance betw een the new entity (re p re se n tin g the fused en tities 1 and 3) and all other en tities m u st now be calculated. The m ethod of calculation u sed h e re is called the flexible so rtin g o r c lu s te rin g stra te g y (Lance and W illiam s, 1967; Sneath and Sokal, 1973; C lifford and Stephenson, 1975). Flexible s tra te g y (with the v a ria b le coefficient = -. 25) tends to re s u lt in m o re d istin ct and e q u a l-siz e d groupings on the d en d ro g ra m . T his fac ilitates the subsequent in te rp re ta tio n . T his m ethod is explained in Fig. 2-3. F ro m th e s e calculations, a new distance m a trix is c o n stru cted as in Fig. 2 - 2b. This p ro c e ss of entity (or entity-group) fusion and distance re c a lc u la tio n s is r e peated un til the den drogram is com pleted and all entities a re in one la rg e group. The ecologist can choose to delim it groups for in te rp re tiv e p u rp o ses at any lev el o r lev els on the final den drogram 26 F ig. 2-2. The p ro c e ss of building a dendrogram u sin g flexible so rtin g s tra te g y (see F ig. 2-3). E ntities c o rre sp o n d in g to the s h o rte s t d istan ce (c irc le d n u m b ers) a re fused and d istan ces to the new group a re calculated. The p ro c e ss is re p e a te d u n til a ll e n titie s and groups a re fused. 27 D istan ce M atrix Sam ple C alculations D endrogram E n tities 3 4 1 .60 .20 . 50 ,30 £ 2 .40 . 55 .15 .20 1 3 .25 . 52 n ■ . 10 0 4 . 56 1 3 1-3 2 4 .60 .26 . 55 61 0 . 56 ^1-3, 2 “ • G25 (Di2 + Dgg) - . 25 D^s = .625 (.60 + .40) - .25(. 10) = . 60 n n 1 s 4. 30 . 20 . 10 0 c . 1-3 2-5 d. 2-5 1-3-4 .80 .7 2 . 66 2-5 D i - 3. 2-5 = 625 (Di-3^2 + 5> ' • 25 Dg, 5 = .625 (. 60 + .61) - .25 (. 15) = . 72 n 1 2 5 4 1 3 D i-3 -4 ,2 -5 = 625 (D g. 5 , 1.3 + D2 . 5 . 4 ) " - .625 (.72 + . 66) - .25 (.26) = . 80 2 5 4 1 3 .30 .20 . 10 0 . 80 .70 . 60 . 50 .40 .30 .20 . 10 0 28 Fig. 2-3. E xplanation of the flexible so rtin g stra te g y . 29 a. B asic fo rm u la for d istance calculations follow ing fusions. %k = Dij 2 « = 1 -1 < ^ < 1 b. Value of p s e t at - .2 5 (C lifford and Stephenson, 1975). ^ h k “ .625 (Dj^ + D]^j) - .2 5 D^j c. G raphic re p re s e n ta tio n and explanation of the d istan ces (D). h J 1) i and j a re the entities or groups ju st fused. 2) h is the entity o r group to w hich the new distance is being calculated. 3) k is the position of the fused group (from i and j ) following the calcu latio n s. 30 T w o-w ay coincidence ta b le s a re e sp e c ia lly helpful in the decision as to what groups to c o n sid er (C hapter VI). E. O rdination O rdination m ethods also can s ta r t w ith the in te r-e n tity distance m a trix . E ach entity is im agined as a point in a m u lti d im en sio n al space, and the d istan c es betw een the points a re the c o rre sp o n d in g in te r-e n tity d ista n c e s. The points can form p a tte rn s w hich su m m a riz e re la tio n sh ip s betw een the en tities, but due to hum an lim itatio n s, th is space can be viewed in only a few dim en sio n s at a tim e . The an aly sis w ill be m o st efficient if the d im en sions u sed give a m axim um am ount of info rm atio n about the p a tte rn of points in the space. C onsider Fig. 2-4. M ost of the in fo rm atio n about the p a tte rn of points is re p re s e n te d c le a rly in the p ro jectio n s in F ig . 2-4d (line 3), since the sum of the in te r-p o in t d istan c es is c lo s e s t to the actual sum of the d ifferen c e s. F ro m th is it can be se e n th at the m ost efficient dim ension or axis w ill c o rre sp o n d to a line in the sp ace in a d irec tio n w h ere the points v a ry m o st, or w here the v a ria n c e of the p ro jec te d d istan c es is m ax im al (as on Une 3). The p rojections of the points onto an axis a re called s c o re s . The o rig in of th is sy stem is set at the c en ter of all points and the d istan ce fro m the o rig in to the p ro jectio n of a point is the m agnitude of the s c o re for that point or entity. S everal such axes can be se t 31 F ig. 2-4. The purp ose of ordination illu s tra te d by co m p ariso n of the efficiency of se le c te d point p ro jec tio n s. Note that the p ro jec tio n s onto line 3 re p r e s e n t alm o st a ll the d istan ces (92%) in the tw o -d im e n sio n al sp ace in (a). The points a re m ax im ally sp re a d out in the d ire c tio n of this Une. Symbols:% D = sum of the d istan c es (xlOO); %D^ot “ % of to ta l tw o -d im en sio n al distan ce. 32 85 10 a 30 41 65 80 b 70 39 c 29 36 d 26 e ID 778 a. D ista n c e m a t r ix (xlOO) and r e p r e s e n ta tio n in tw o -d im e n s io n a l sp a c e . b d f c e 08 a 29 02 25 05 b 26 02 22 c 24 04 d 20 e O V lin e 1 b. P r o je c te d d is ta n c e s (xlOO) on lin e 1. f b d c e 65 a b 30 30 60 00 30 28 c 30 28 d 02 e c. P r o je c te d d is ta n c e s (xlOO) on lin e 2. d f b e c 82 100 32 a 42 61 b 30 c 37 d e d. P r o je c te d d is ta n c e s (xlOO) on line 3. . c lin e 2 -# lin e 3 ED %D tot = 218 = 28 I D = 499 % Dtot ~ 64 I D = 712 = 92 33 up w ith the r e s tric tio n that all axes a re at rig h t angles to one another and m ath em a tica lly independent, i . e . , the s c o re s on different axes have an in te r - c o r r e la tio n of z e ro . It can be e a s ie r to in te rp re t and d iscu ss re s u lts when th e axis s c o re s a re independent since, th e o re tic a lly , each axis can then be c o n sid ere d as a unit without re fe re n c e to o th er axes. T his, how ever, is not alw ays the c ase in p ra c tic e . Subject to the above r e s tric tio n s , each su c c e ssiv e axis d is plays a m axim um am ount of the rem a in in g in te r-p o in t distance. The axes a re u su ally c o n sid ere d in o rd e r of the percen tag e of the to ta l distance o r v a ria n c e accounted for by the c o rre sp o n d in g s c o r e s . Im portan t e x te rn a l fa c to rs affecting the levels of the a ttrib u te s m e a s u re d at o r on each entity w ill often be c o rre la te d w ith the s c o re s on the m o re im p o rtan t o rdination axes. This is becau se th e se fa c to rs w ill cau se v a ria b ility in the a ttrib u te m e a s u re m e n ts. T his, in tu rn , leads to g re a te r in te r-e n tity d istan ces which u ltim ately w ill influence th e positions of the axes. P rin c ip a l com ponent an aly sis (PCA) is an ordination m ethod b ased on E uclidean d istan c es betw een the en tities. In ste ad of a d istan ce m a trix , in itia l data input u su ally is a m a trix of v a ria n c e s and c o v arian c es o r in te r -c o r re la tio n s betw een the a ttrib u te s (for o ther p o ssib ilitie s, se e Or loci, 1967b). M ore detailed d escrip tio n s 34 of PC A a re found in S eal (1964), M o rriso n (1967), Hope (1969), B lackith and R eym ent (1971), Cooley and Lohnes (1971), and Davis (1973). A nother o rd in atio n m ethod, called p rin c ip a l co o rd in ates a n aly sis (Gower, 1966, 1967; B lackith and R eym ent, 1971; Sneath and Sokal, 1973), s ta r ts w ith a d istan ce m a trix . In th is case , the d istan c es can be calcu late d from the B ra y -C u rtis , C a n b e rra - m e tric , o r any o th er re a so n a b le distance index. If E uclidean d istan c es a re used, the r e s u lts w ill be id en tical to a PCA. In th is study, both PCA and p rin cip al c o o rd in a te s analysis a re used. F o r data involving sp e cie s counts, p rin c ip a l co o rd in ates an aly sis is u sed sin ce d istan ce in d ices m o re a p p ro p ria te for this type of data can be u tilized . O rdination an aly se s not d ire c tly in volving sp e cie s counts w ill u se PCA. As m entioned, the PCA can s ta r t w ith e ith e r the in te r-a ttrib u te v a ria n c e -c o v a ria n c e o r c o r re la tio n m a trix . When the c o rre la tio n m a trix is used, the attrib u tes a re sta n d ard ize d by the a ttrib u te sta n d ard deviation. This equalizes the v a ria n ce and p re su m a b ly the w eight of the a ttrib u te s in the a n aly sis. Except w hen sta te d otherw ise, the c o rre la tio n m a trix is u se d in th is study in conjunction with PCA. 35 F . M u ltip le -D iscrim in a n t A nalysis T his m ethod is also calle d canonical v a riâ te s analysis (Seal, 1964) o r canonical a n aly sis of d isc rim in an c e (Hope, 196 9). The m ethod can be u sed to study the re la tio n sh ip s betw een p r e d e te rm in e d groups of en tities and a given set of a ttrib u te s o r v a ria b le s m e a s u re d on or at the en tities. E ach entity is im agined as a point in a m u ltid im en sio n al space in which each dim ension r e p r e s e n ts one of the a ttrib u te s. The distan ce of a point into a dim ension from the o rig in is d ire c tly re la te d to the m agnitude for the co rre sp o n d in g a ttrib u te . If the group form ation ( i . e . , the p re d e te rm in e d groups) is som ehow re la te d to som e of the m e a s u re d a ttrib u te s, then points re p re s e n tin g the entities in each group w ill c lu s te r in the dim ension or dim ensions c o rresp o n d in g to v a ria b le s v a ry in g m in im ally w ithin th e se g roups. A lso, the c lu s te rs for the differen t groups w ill se p a ra te from one another along a dim ension o r d im en sions c o rre sp o n d in g to v a ria b le s which v a ry m ax im ally betw een the groups. Point p ro jectio n s onto such d im ensio ns w ill be c o rre la te d w ith p ro jectio n s onto axes w hich a re positioned to display a m ax im u m amount of in tra -g ro u p c lu ste rin g along w ith a m axim um am ount of in te r-g ro u p se p a ra tio n . T his is a w ay to se e k out the v a ria b le s which c o rre sp o n d to the p a tte rn of entities in the p r e d e te rm in e d g roups. F o r in stan c e, in th is study, groups of 36 biotic ally s im ila r sam p lin g site s a re d elim ited by a c la ssific a tio n a n a ly sis. T h ese groups of site s (entities) can then be u sed in a m u ltip le -d isc rim in a n t a n aly sis w ith the environm ental v a ria b le s m e a s u re d at the site s as a ttrib u te s. T his w ill show which, if any, of the en vironm ental v a ria b le s c o rre sp o n d to the groups defined by the biota. Such r e s u lts can lead to hypotheses as to w hich en v iro n m e n ta l fa c to rs a re m ost im p o rtan t in d eterm in in g the co m position of the biota. Dahl et al. (1967) u se d d isc rim in a n t an aly sis to study r e l a tio n sh ip s betw een so il v a ria b le s and the locations of eight v e g eta tion al ty p es in conifer fo re s ts . A s e p a ra te an aly sis w as p e rfo rm e d to c o m p a re each p a ir of v eg etatio n al types - a p ro c e ss which r e q u ired 28 a n aly se s. If m o re than ju st a few groups a re involved, the r e s u lts from this approach can becom e r a th e r com plex and c u m b e rso m e . In the p re se n t study, all o r m o st of the groups a re analyzed sim ultaneo usly, leading to a m uch s im p le r and m o re in te g ra te d an aly sis. A group of site s also can be delim ited according to the p re s e n c e of a sp e cie s o r a sp e cie s group. Such groups also can be re la te d to the environm ent w ith m u ltip le -d isc rim in a n t analysis (G reen, 1971). In C hapter VII a ratio n ale fo r m odifying the u su al m u ltip le -d isc rim in a n t calcu latio n s is given, and the m odifications a re shown in Appendix A. 37 As in th e o rdination m ethods, s e v e r a l axes can be se t in the space. E ach axis is at rig h t angles to all o ther axes, and the axis s c o re s a re u n c o rre la te d from axis to axis w ithin and betw een the groups of en tities : (Hope, 1969:121). Subject to th e s e re s tric tio n s , each s u c c e ss iv e axis m ax im iz es the re m a in in g se p a ra tio n betw een the groups (as fa r as the p ro jec te d s c o re s a re concerned) while m in im izin g the re m a in in g se p a ra tio n s w ithin the groups. To de te rm in e w hich of the v a ria b le s a re re la te d to each axis, the c o efficients of se p a ra te d eterm in atio n a re u sed . See Appendix A for fu rth e r d isc u ssio n of the m u ltip le -d isc rim in a n t calculatio ns and in te rp re ta tio n . In re la tio n to the an aly sis of groups of s ite s , it could be arg u ed that a co m p licated m u ltiv a ria te m ethod such as this one is not n e c e s s a ry to find the im p o rtan t en v iro n m en tal v a ria b le s. The m ea n s and sta n d a rd deviations of the en v iro n m en tal m e a su re m e n ts at the s ite s in each group could be c o m p ared . V ariab les showing m in im a l overlap in m e a s u re m e n ts o r w idely differing m eans betw een the groups could be hypothesized as potentially im p o rtan t to the biota (F ield and M c F a rla n e , 1968; F ield , 1971; M oore, 1973; Lie, 1974). A ltern ativ ely , a one-w ay an aly sis of v a ria n ce could be p e r fo rm ed on each v a ria b le se p a ra te ly to te s t for n o n -rand om dif fe re n c e s in the m ean values of the groups (Dahl et al. , 1967). 38 T hese ap p ro ach es can be v e ry useful, but im p o rta n t in fo rm atio n can be lost since com binations of v a ria b le s (m u ltiv ariate rela tio n sh ip s) so m e tim es w ill b est explain group d ifferen ces (Cooley and Lohnes, 1962). Fig. 2-5 d e m o n s tra te s th is point. The c lu s te rs of points for the two groups a re c le a rly se p a ra te d in this space, but the se p a ra tio n is not along any one of the d im ensions c o rre sp o n d in g to a single v a ria b le . In fact, the im p o rtan ce of v a ria b le B would be com pletely m is s e d in a u n iv ariate an aly sis since the m ean s of both groups for th is v a ria b le a re n e a rly the sa m e . The b e st se p a ra tio n of the groups (along w ith a m a x im a l c lu s te rin g of the points within the groups) would be obtained by p ro jectio n s onto the dashed line called axis I. This axis is re la te d to both the dim ensions c o r resp o n d in g to the v a ria b le s and the re s u lts of the d isc rim in a n t an aly sis would indicate th is. In the ex p erien ce of the author, such m u ltiv a ria te re la tio n sh ip s are not uncom m on in ecological data. F u rth e rm o re , s e v e ra l en v iro n m en tal p a ra m e te r s often w ill v a ry significantly betw een groups, but the d isc rim in a n t analysis w ill em p h asize the m o re c o n sisten t of th ese p a ra m e te r s . T his g rea tly fa c ilita te s in te rp re ta tio n . G. M u ltip le -lin e a r R e g re ss io n T his m ethod is u se d to exam ine the re la tio n sh ip s betw een the m e a s u re d en v iro n m en ta l v a ria b le s (the independent v a ria b le s) 39 F ig. 2-5. D em onstration of the im p o rtan c e of using m u ltiv a ria te techniques when c o n sid erin g group d ifferen ces. A plot of v a ria b le m e a s u re m e n ts fo r two hypothetical site groups is shown. Note that the groups a re com pletely se p a ra te d in the dim ension re p re s e n te d by Axis I , and not by e ith er of the o rig in a l dim ensions re p re s e n tin g the v a ria b le s. 40 X P o in ts for site s in group 1 o P o in ts for site s in group 2 \ \ \ \ 20 20 V ariab le A \ Axis I \ \ 41 and the ordination axis s c o re s (dependent v a ria b le s). A s im ila r p ro c e d u re w as followed by C a ssie and M ichael (1968). This is a com m on sta tis tic a l p ro c e d u re and no detailed d e scrip tio n is given h e re . F o r m o re in fo rm atio n on the m ethod se e Seal (1964), D rap er and Sm ith (1966), Dixon (1970), and Cooley and Lohnes (1971). B efore u se in the an aly ses, the r e g re s s io n coefficients a re s ta n d a rd iz ed as in Steele and T o r r ie (1960:284, 299). T his puts all co efficients in sta n d a rd -d e v ia tio n units and the m agnitudes of the coefficients can be co m p ared d irec tly . H. C om puter P ro g ra m s A ll p ro g ra m s u sed in the p re se n t study w e re w ritte n by the author in P L / 1 co m p u ter language, and the com putations w ere p e rfo rm e d on the IBM 370/158 co m p u ter at the U n iv ersity of S outhern C alifornia C om puter C en ter. 42 CH A PTER III GENERAL DESCRIPTION OF THE DATA A. Introduction Some g e n e ra l c h a ra c te r is tic s of the data to be u sed in the study a re d e sc rib e d in th is ch ap ter. The data a re u sed in C h ap ters IV, V, and VII in conjunction w ith stu d ies of m ethods of an aly sis. In C h ap ter VIII, th e re is a detailed an aly sis using the p r e fe rr e d m ethods d isc u sse d in the p revious c h a p te rs. The data w e re co llected by the Los Angeles County Sanitation D istric t a b o ard the "S ea-S -D ee. " This sa m e group w as re sp o n sib le fo r the identification of the fauna and th e m e a su re m e n t of a ll p h y si c al p a ra m e te rs except m e r c u r y (m e a su re d by the Southern C alifo rn ia C o a stal W ater R e s e a rc h P ro je c t, E l Segundo, C alifornia) and se d i m en t d istrib u tio n (m ea su red by D. S. G orsline, D ep artm en t of Geology, U n iv ersity of S outhern C alifornia). B. M ethods 1. T im e and location of the sam pling F o rty site s off the P alo s V erdes P en in su la in Southern C alifo rn ia w e re sam p led in late August and e a rly S eptem ber of 43 o> C O C O CO C O 1 1 1 o C N LU C O C O L O C O CN CD f C N a. CO f CL C O L U CN L U (N " C N C N O ID C N CL 2 # / 00 CN o C O o m ^ o C D CO Û Û LU ^ Q < m O z C O Lb C O o .. C O o C O ^ 45 1973. Two m a jo r outlets d isc h arg e ap p ro x im ately 1.34 x lO ^ m ^ / day of dom estic and in d u stria l w a ste s in the a re a . The sam p lin g and outfall locations a re indicated in Fig. 3-1. U ntil June^ 1972 the out let to w ard the n o rth w est d isc h arg e d a d isp ro p o rtio n a te am ount of the h e a v ie r p a rtic u la te m a tte r. A fter th is tim e th e outflow of the h e a v ie r m a tte r h a s been m o re equally d istrib u te d betw een both pipes. Studies by SCCWRP (1973)1 indie ate that the c u rre n ts g e n erally flow to w ard the northw est. 2. Sam pling p ro ce d u re p At each site , four . 04m re p lic a te sa m p le s w ere taken w ith a Shipek grab. The sam p les w e re w ashed through a 1. 0 m m s ta in le s s - s te e l sc re e n , and the m a te ria l re m a in in g on the s c r e e n w as p re s e rv e d in 10 per cent b uffered fo rm alin. The sa m p le s w e re s o rte d and the o rg a n ism s identified to the sp ecies level w henever p ossible. 3. P h y s ic a l-c h e m ic a l data Sam ples and o b se rv atio n s for the abiotic (p h y sic a l/c h e m ical) data w e re obtained in s e v e ra l w ays. Sulfide potential (Eg in m V) and red o x po tential (E y in mV) m e a su re m e n ts w e re m ade on pore w a te r u sin g the p ro c e d u re of K alil (1973) and d e sc rip tiv e sed im en t c o a rs e n e s s (DSC) w as d e te rm in e d by touch for each biological sam p le. T hese data have been av erag e d at each statio n . Subsam ples from each 46 biological sa m p le w e re pooled at each statio n and analyzed for to ta l DDT (ppm, d ry weight) using gas chrom atography, and organic n itro g e n w as d e te rm in e d (per cent, d ry w eight) by the K jeldahl m ethod. A fifth Shipek sam p le w as tak en at each station and su b sam p les re ta in e d for the d e te rm in a tio n of to ta l m e rc u ry (ppm, d ry w eight, using the cold vapor technique) and sta n d ard sed im en t p a rtic le -s iz e g ra d e s. The siz e g ra d e s (W entworth scale in m m ) w e re com bined into p e r cent g rav el, sand, silt, and clay. M ean g ra in siz e and sta n d a rd deviation (in Phi units) also w ere d e te r m ined. C. P re lim in a ry Data S u m m arizatio n 1. Biotic data A to ta l of 54, 524 individual o rg a n ism s w ere sam p led and iden tified as belonging to 254 c a te g o rie s . Of th e se c a te g o rie s , 222 a re c o n sid e re d to re p re s e n t single sp e c ie s, although som e a re not given a sp e c ie s n am e. See Appendix B fo r a list of the 94 m o st freq u en tly o c c u rrin g sp e c ie s. The data from the four re p lic a te s at each site w ere pooled and the to ta l count for each sp e c ie s w as u sed in subsequent a n aly se s. The m ea n n um ber of individuals and sp e cie s, and the m e a n sp e c ie s d iv e rsity (B rillouin index, Pielou, 1969) at the s ite s a re su m m a riz e d in F ig. 3-2. 47 F ig. 3-2. G eneral p a tte rn s of sp e c ie s d iv e rsity and abundance in the sam pling a r e a (from Sm ith and G reene, 1976). 48 20 30 30 50 20 20 A. MEAN NO. OF SPECIES/SAMPLING SITE m B. MEAN NO. OF INDIVIDUALS/SQUARE METER X 1000 2.0 2.0 2.5 1.0 1.5 1.0 1.0 2.0 ® 1.5 1.5 C. MEAN DIVERSITY/SAMPLING SITE (BRILLOUIN) 49 2. Abiotic data The d istrib u tio n s of the m e a s u re d v a ria b le s in the sam pling a re a a re shown in Fig. 3-3. The in te rc o rre la tio n s of the v a ria b le s a re show n in T able 3 -1 . P r i o r to the c o rre la tio n c a lc u lations, som e of th e v a ria b le s w e re tra n s fo rm e d as ind icated in C hapter VIII. Due to the v e ry high in te rc o rre la tio n s of som e of the sed im en t v a ria b le s, the m ean g rain siz e and p e r cent silt a re not u sed in the subsequent an aly se s. T his avoids redundancy and fa c ilita te s in te rp re ta tio n . The N, Hg, DDT, sulfide, and eH v a ria b le s a re assu m e d to be re la te d to the d isc h arg e from the outfall. The la tte r two v a ria b le s a re re la te d to the lo cal lev els of d issolved oxygen, with la rg e negative potentials of each indicating low oxygen condi tions (P e rk in s, 1957; W hitfield, 1969; F e n c h el and Riedl, 1970). In the subsequent a n aly ses, the sc a le of the sulfide potential m e a su re m e n t is r e v e r s e d so that a h igher value actually in d icates a h ig h er lev el of d isso lv ed sulfide. Finally, the sedim ent p a ra m e te rs also a re som ew hat re la te d to outfall d isc h arg e , since finer p a rtic le s in the s ilt and clay ran g e a re r e le a s e d at the outfall. T h e re seem to be th re e m ain abiotic tre n d s in the sam pling a re a . The f ir s t is the se d im e n t-s iz e p a ttern , typified by 50 P a tte rn s of the abiotic v a ria b le s in the sam p lin g a re a . o o o o CL C l W ) W ) Kl (f) O SB. T ^ 53 m ü) § 5 ■ 5 % (D (D I I m CD co 0 0 In " o s k ü Q ni • • > > m • i p • I - ) C L r Q S u c d CD CD T d C L CD c d Î H 0 U ai ai c d CD d CD a •r4 m ;=! T 3 u CD CD CD ■y T 3 c d -M Ü d 0 CD CD d CD • iH r o 01 ■ J t? c d CD rO 0 13 CD 0 0 S 0 -M c , T -H r-H 0 % c d S + H p ! m 01 c r * d CD c d Ü 0 U •rH 0 " c d CD 1 — 4 G CD U u s i -M CQ CD 0 1 —4 0 CD r O u ■ H Cd CD c d •1-4 CD h Ü S h c d 1 — 1 bo > I 0 0 (D T — 4 H U0§oj:q.Tjsi otubS j o % ss3uasj[B03 !|.uaraip3s m < î) •A3Q *p;s (T^d) 3ZTS uveno UB3]M <N i a a i ^ ; o i (= S g )T B n u 3 ;o a o Œ) mdscL 0 0 CD lO % ^ n îS % m pUBg 0^ M X3ABJO % ^ M C D I — I •iH I I co îo o 1 > - 1 0 0 I T-l tD ID l O i D 0 0 I I lO 0)1 T -Il l> - T h D-|co|(N I I CM 0 0 CD C O ) CO O 1 —I 1 — I CD I I CD CD CO LO rH CM T f CD (O LO "M 4 r-l I l I I I 0 0 CM CO CM CM LO O ) O ' 1 T — I 1 1 CM T — I < 0 I I O ) 0 | (0) lO tH CM 0 0 CO CM CD o CD rH I I 0 ) I C 0 O L 0 L 0 r H ( 0 ) ' ^ C D Oo| CD LO 1 —1 LO t —I LO \1 * 1 —1 I I C M C M | l > - C D a ) O C D C O O ) C O M^D-I o c d M ^ lO i-HCOCMCM §1 M 1 0 c o l I>- rH | 0 ) | CM D -| CO I>-| I I I I 1 0 CO T — I o CD lO r- I I 1 ■ ^ C M B ^ i O - M ^ O M ^ C O l O C D C D O O O t- I O O M ^ i- I C M O O O O O I l I I I I g <D £ M m m ai <ü n < D m o O G (D tu O O S o § tu O , u 8 - 0 Q ^ 1 1 co H I I fL ^ (D ^ "O I I c ^ H M 5 1 — 1 c d • r - l "S CD O P h c d g 'V O CD H p 4 54 th e d istrib u tio n of the sed im en t c o a rs e n e s s v a ria b le . The second is the som ew hat s im ila r p a tte rn of the N, Hg, and DDT m e a s u r e m e n ts, and the th ird is th at shown by the sulfide and eH m e a s u r e m e n ts. T hese th re e g e n e ra l p a tte rn s , although not independent, a re sufficiently different so they would be expected to show varying c o rre la tio n s w ith the biotic data. At som e s ite s , th e r e a p p ea rs to be a d isc re p a n c y betw een the m ec h an ica lly m e a s u re d se d im e n t-s iz e p a ra m e te rs (per cent g rav el, sand, etc. ) and the d e sc rip tiv e m ethod (sedim ent c o a rs e n e s s ). The la tte r m ethod m ay be m o re re lia b le since it was e stim a te d on each re p lic a te . T his is in c o n tra st to the fo rm e r m ethod w hich is b ased on a single sam p le. Also, the m ec h an ica l p ro c e ss in g m ay d e stro y som e of the p ro p e rtie s of the sed im en t w hich the la tte r m ethod m ay detect (Johnson, 1974b). T his would be a r e s u lt of the d e stru c tio n of th e organic m a tte r p r io r to the m e c h a n ic a l an aly sis. 55 CHA PTER IV A STUDY AND EVALUATION OF THE E F F E C T S OF THE VARIOUS PRELTMINARY DATA MANIPULATIONS IN THE NORMAL ANALYSIS USING THE BRAY-CURTIS INDEX A. Introduction The p u rp o se of this ch ap ter is to exam ine a set of data types (with r e s p e c t to tra n s fo rm a tio n s , sta n d a rd iz a tio n s, etc. ) in o rd e r to illu s tra te and ev aluate som e of the p ro p e rtie s of each, and also to d e te rm in e if and how in te rp re ta tio n s of the final re s u lts of the n o rm a l an aly sis w ill be affected by the choice of data type. L ittle attention w as given to this topic u n til re c e n tly when a detailed a n aly sis w as re p o rte d by S m artt et al. (1974). T hese au thors exam ined the p ro p e rtie s of s e v e r a l ty p es of t e r r e s t r i a l veget ational m e a s u re m e n ts and c o m p a red them at th re e levels: (1) at the lev e l of the data m a trix , (2) at the le v e l of the c la s s ific a tion r e s u lts , and (3) at the lev el of the e n v iro n m en ta l c o rre la tio n s . The f ir s t appro ach involved exam ination of c o rre la tio n s betw een the values of each m e a s u re m e n t type, the second included c o m p a riso n of d e n d ro g ra m s g e n e ra te d by each type, and the la s t involved c o m p a riso n of each d en d ro g ram with a su bjectiv e c la ssific a tio n based 56 on the a u th o rs' knowledge of the physical and biotic environm ent. F inally, to c o m p a re the different types on an o v e ra ll b a sis, c o r r e lations betw een each data type at all lev els of c o m p a riso n w e re u sed as a ttrib u te s in a PCA. (The d a ta ty p e s w e re the e n titie s. ) The data types u sed in the study w e re m o re applicable to t e r r e s t r i a l plant data and did not include any sp e cie s sta n d a rd iz a tio n s. In th is sectio n , d ata types w ill be exam ined at the level of the distance m a trix and at the level of e n v iro n m en tal c o rre la tio n s . B e n th ic -m a rin e data of the type u se d h e re cannot be u se d su b je c tiv e ly to define an "o p tim al" p a tte rn (as in S m a rtt et a l., 1974) for c o m p a riso n w ith the r e s u lts of the different data ty p es. E valuation w ill have to be b ased on an u n d erstan d in g of the p ro p e rtie s of the different data ty p es and the effect of th ese p ro p e rtie s on the goals of the an aly sis. The d istan ce calcu latio n s a re b ased on th e B ra y -C u rtis index,w hich is u se d b e c a u se (1) it is im p ra c tic a l at th is stag e to te s t s e v e ra l in d ic es. As w ill be seen, extensive calcu latio n s a re involved in th is type of study; and (2) when u se d in conjunction w ith a p ro p e r sta n d ard iza tio n , th is index has so m e d e sira b le p ro p e rtie s w hich a re d isc u sse d la te r in the ch ap ter. 57 B. M ethods 1. G eneral app ro ach The p ro c e d u re is as follows: a) Some g e n e ra l population c h a r a c te ris tic s of each sp e c ie s a re d e te rm in e d . b) The in te r c o rre la tio n s of th ese population p a ra m e te r s a re calcu lated and d isc u sse d . c) A sta n d a rd iz e d m e a s u re of the re la tiv e am ount of d istan c e which each sp e c ie s c o n trib u te s to the final d istan ce m a trix is c alcu late d for each d a ta type. d) The c o rre la tio n s betw een the population p a ra m e te rs ^from a)] and the sta n d a rd iz e d d istan ce contributio ns [from c)] a re c alcu late d for each d ata type. T his should show som e of the c h a r a c te r is tic s of the sp e c ie s which re c e iv e higher w eightings in the d istan ce calcu latio n s w ith the v a rio u s d a ta ty p e s . e) An i n te r - s i te d istance m a trix is calcu late d for each d a ta type. f) The d a ta ty p e s (as en tities) a re c o m p a re d using PC A w ith the elem en ts of the d istan ce m a tric e s as a ttrib u te s. The p a tte rn s shown h e re a re then re la te d to the c o rre la tio n s w ith the population p a ra m e te r s . 58 g) E ach distan ce m a trix is u se d in a p rin cip al co o rd in ate a n aly sis, and th e c o rre la tio n s betw een the f ir s t th re e axes thus defined and th e m e a s u re d e n v iro n m en tal v a ria b le s w ill be d e te rm in e d w ith m u ltip le lin e a r re g re s s io n . h) The sta n d a rd iz e d r e g r e s s io n coefficients [from g)] for each data type w ill be u sed as a ttrib u te s in a PCA. This w ill s e rv e to c o m p a re the r e s u lts of the different data types at the in te r p re ta tiv e lev e l of the an aly sis, a ssu m in g that the pu rp o se of the an aly sis is to identify p o ssib le re la tio n sh ip s betw een the biota and the environm ent. i) E ach d istan ce m a trix w ill be u se d in a c la ssific a tio n of s ite s . F o r each re s u lt, groups w ill be d elim ited at th re e lev els of the d e n d ro g ra m , and the en v iro n m en tal c o rre la tio n s c o r r e sponding to th e se groups w ill be d e te rm in e d w ith m u ltip le- d isc rim in a n t a n a ly sis. The coefficients of s e p a ra te d eterm in atio n (which indicate the w eight of the v a rio u s v a ria b le s in group s e p a r a tion) w ill be u sed in a PC A to co m p are data ty p es at the in te r p re ta tiv e lev el as in h). 2. D ata types u sed F ifteen data types a re co n sid ere d . T hey include site and sp e c ie s sta n d a rd iz a tio n s u sin g the m ean, n o rm , total, and m axim um , along w ith sp e c ie s sta n d a rd -d e v ia tio n and 59 s im u lta n e o u s -double sta n d a rd iz a tio n s. T hese ten data sta n d a rd iz a tions a re applied following a tra n s fo rm a tio n by a sq u a re ro o t. Due to sp e c ia l in te re s t in the p ro p e rtie s of the s p e c ie s -m e a n sta n d a rd i zation, two additional such sta n d ard iza tio n s a re included. One is applied to raw data and the o th e r to c u b e-ro o ted data. F inally, p re s e n c e -a b s e n c e data (henceforth called b in ary data) and data tra n s fo rm e d by a sq u a re ro o t and by a cube ro o t (without any stan d ard izatio n ) a re included. R eferen ce in the text to the sp e c ie s- m e a n sta n d ard iza tio n w ill r e f e r to the one p reced ed by the sq u a re - ro o t tra n s fo rm a tio n u n le ss sta te d o th erw ise. 3. Data red u ctio n To avoid e x c e ssiv e com putational tim e , the sp e c ie s d ata a re red u ced . A ll sp e c ie s which o ccur at four o r m o re site s a re included. The 94 sp e c ie s in th is categ o ry a re u sed in the su b sequent a n aly ses. The effect of such data red u ctio n on the distance calcu latio n s is d isc u sse d in C hapter V. 4. C alculation of the sta n d a rd iz e d sp e c ie s -d is ta n c e contributions When u sing the B ra y -C u rtis m e a s u re , the d istance c o n trib u ted by each sp e c ie s (to the final ecological d istan ce betw een two site s ) is not additive. T his m ean s that the sum of the d istan ce co n trib u tio n s of each sp e c ie s does not equal the c alcu lated d istan ce betw een the two s ite s . T his is due to the fact that th is m e a s u re is 60 a single quotient and the contrib ution of a sp e c ie s in a c o m p a riso n is p a rtia lly dependent on the values fo r the o th er sp e c ie s at th e s ite s being co m p a red . In o rd e r to c alcu late the d istan ce a ttrib u t able to a single sp e c ie s in a single c o m p a riso n , it is n e c e s s a ry to calcu late the d istan ce w ith the sp e c ie s included and then re c a lc u la te the d istan ce w ith the sp e c ie s excluded. The absolute value of the d ifferen ce betw een th e s e two values w ill give a m e a s u re of the d istan ce co n trib u tio n of th at sp e c ie s. F o r each sp e c ie s, the sum of the d istan c e contrib u tio n s over all c o m p a riso n s is then calculated. Since d o u b le-z ero c o m p a riso n s a re not c o n sid ered w ith the B ra y -C u rtis index, each sp e c ie s w ill not p a rtic ip a te in the sa m e nu m b er of c o m p a riso n s. Species involved in m o re co m p a riso n s w ill n a tu ra lly have a h ig h er po tential sum of distance c o n trib u tio n s. To elim in ate this effect the to ta l d istan ce c o n trib u tion for each sp e c ie s is divided by the n u m b er of co m p a riso n s (i. e . , the n u m b er of n o n -d o u b le -z e ro m atch e s) in w hich the sp e cie s w as involved. T his sta n d a rd iz a tio n w ill give the d istan c e p e r c o m p a ri son. 5. P opulation p a ra m e te rs F o r each sp e c ie s, the following a re en um erated: a) The nu m b er of site s in w hich the sp e cie s o c c u rs. T his w ill be r e f e r r e d to as the n u m b er of o c c u rre n c e s . 61 b) The abundance p e r o c c u rre n c e ; c) the to ta l abundance; d) the m axim um abundance value; e) the sta n d a rd deviation over all s ite s ; and f) a rough m e a s u re of the am ount of a sso c ia te d sam p lin g e r r o r . F o r th is p urpose, a m odification of the coefficient of v a ria tio n (Sokal and Rohlf, 1969) is u sed . In g e n eral, the coefficient of v a ria tio n is sim p ly the sta n d a rd deviation of a sam p le divided by the m e a n of the sa m p le . The d ivision by the m ea n can c els out the effect of the sc a le of the n u m b ers in the sa m p le , and consequently sa m p le s w ith different sc a le s can be co m p a red fo r v a ria b ility . H ere, each sp e c ie s is c o n sid ere d a sam p le, and the sta n d a rd deviation of the sp e c ie s is the to ta l w ith in -site s ta n d a rd deviation. To calculate th is, it is n e c e s s a r y to use th e o rig in a l data m a trix befo re pooling of the four re p lic a te s at each site has tak e n place. A sum of s q u a re s (sq u a red deviations from the site m ean) is accum ulated o v e r a ll s ite s w h e re the sp e cie s o c c u rs. D ivision of the to ta l sum of s q u a re s by 4 x N - N (w here N = the n u m b er of site s in which the sp e c ie s o c c u rs) gives the in tr a - s ite v a ria n c e and the sq u a re ro o t of th is gives the sta n d a rd deviation. F inally , the sta n d a rd deviation is divided by the m e a n abundance in the N s ite s of o c c u rre n c e . 62 T h is m e a s u re w ill so m e tim e s u n d e re stim a te the sa m p lin g e r r o r involved w ith the r a r e r sp e c ie s, sin ce th e re is a h ig h er p ro b ab ility th at in the sam p lin g th ey w ill be m is s e d e n tire ly at som e site s at w hich th ey a re actu ally p re s e n t. Sites devoid of a sp e c ie s do not e n te r into the c alcu latio n s (for that sp ecies) and th is additional v a ria b ility w ill not be d etected . The m o st a p p ro p ria te s tra te g y is to allow for th is so u rc e of e r r o r in the in te rp re ta tio n r a th e r than t r y to guess w here a sp e c ie s could be p re s e n t but not sam pled , or to a ssu m e th at all sp e c ie s a re potentially p re s e n t at a ll s ite s . It is to be expected th at h ig h er sam pling e r r o r is a sso c ia te d with the r a r e r sp e c ie s (Lloyd, 1967). The coefficient of v a ria tio n is included h e re b e ca u se it obviously is u n d e sira b le if an analysis gives ex cessiv e w eight to the m o re u n c e rta in sp e c ie s counts, and th is would be d etected by o b serv in g c o rre la tio n s w ith th is p a ra m e te r. C. R e su lts 1. P opulation p a ra m e te rs T able 4-1 show s the c o rre la tio n s betw een the p a ra m e te r s . It is apparent that : a) A ll the m e a s u r e s of abundance (num ber p e r o c c u r re n c e , to ta l abundance, and m axim um ) a re highly in te r c o rre la te d . 63 1 ,-1 1 4-4 o c u 'o ia o > o U m e u ü e u u u o 6 d ) o g O O = tte o 0 § T J d 1 o H 1 s g % c u Q & ’ -a s & Q # O c c u rre n c e s X 51 52 48 -47 50 #1 O c c u rre n c e 51 X 97 98 -23 99 T o tal Abundance 52 97 X 97 -27 98 M axim um 48 98 97 X -19 99 C oefficient of V a riatio n -47 -2 3 -27 -19 X -22 S tandard D eviation 50 99 98 99 -22 X T able 4-1. The in te r c o rre la tio n s (XlOO) betw een th e c alcu late d population p a ra m e te rs . 64 b) T h ere a re m o d e ra te ly high c o rre la tio n s (.48 to .75) betw een the n u m b er of o c c u rre n c e s and the abundance m e a s u re s . T his show s th at the m o re w id esp read sp e c ie s a lso tend to be m o re abundant. c) The coefficient of v a ria tio n is c o rre la te d nega tiv ely w ith all other p a ra m e te rs , with the low est negative c o r r e lation being w ith the num ber of o c c u rre n c e s . E vidently sp e cie s w id e sp re a d over the sam p lin g a re a tend also to be w id e sp rea d at each sam p ling site, re s u ltin g in re la tiv e ly m o re co n sisten t sam ple counts. d) The sta n d a rd deviation is v e ry highly c o rre la te d (.9 8 to .99) w ith all abundance m e a s u r e s . It has slightly h ig h er c o rre la tio n s w ith the n u m b e r/o c c u rre n c e and the m axim um , both of w hich a re m e a s u r e s of g e n e ra l abundance w h ere the sp e cie s o c c u rs r a th e r th an w ith the to ta l abundance. 2. C o m p ariso n of data types at the level of the distance m a trix a) R elatio n sh ip s betw een sta n d a rd iz e d distan ces and population p a ra m e te rs F o r each data type, c o rre la tio n s betw een the s p e c ie s -d is ta n c e contribu tions and th e population p a ra m e te rs a re show n in T able 4-2, The purpose h e re is to show som e of the c h a r a c te r is tic s of the m o re heavily w eighted sp e c ie s w ith each data type. 65 C O o g o u u ^ 3 o 6 4k C D q C D q q o o O 4 k C D O 1 "S 1 s 1 4 H I I 4 H - r H C D u O c d u > 1 1 X J c t S C i r - 4 rt ^ < p U2Q Species M ean (raw) 65 17 23 13 -31 16 Species M ean (sq. r t. ) 75 23 28 20 -32 22 Species M ean (cube r t. ) 80 29 33 26 -34 28 Species M axim um 21 -16 -07 -17 -09 -15 Species Norm -02 -18 -10 -18 06 -17 Species T otal -42 -22 -18 -21 29 -22 Species Standard D eviation 24 -10 -01 -11 -07 -09 Square Root 73 89 89 84 -40 86 Cube Root 82 81 82 76 -44 78 Site M ean 72 89 90 84 -41 86 Site M axim um 64 92 95 89 -38 91 Site Norm 67 92 94 89 -40 91 Site T otal 68 92 94 88 -39 90 Sim ultaneous Double 49 58 60 53 -28 55 B in ary 90 53 55 50 -39 52 T able 4-2. C o rre la tio n s (XlOO) betw een sp e c ie s -d is ta n c e c o n tri butions and population p a ra m e te r s for the v a rio u s data ty p es. 66 It can be seen that: 1) The s ite -s ta n d a rd iz e d and u n stan d ard ized , but tra n s fo rm e d data a re a ll highly c o rr e la te d w ith the abundance p a ra m e te r s . Of th e se , the site m ean, s q u a re -ro o t and c u b e -ro o t d ata have the low est (although s till re la tiv e ly high) c o rre la tio n s w ith the abundance m e a s u re s and the highest c o rre la tio n s w ith the n u m b er of o c c u rre n c e s . T hese th re e data types a re alike in that th ey a re not c o n stra in e d , i. e. , th e re is no se t u p p er lim it to th e ir p o tential v alu es. 2) Species m ax im u m , n o rm , total, and sta n d a rd - deviation sta n d a rd iz e d data a re n egatively c o rre la te d w ith the abundance m e a s u r e s . The c o rre la tio n s w ith the n um b er of o c c u r re n c e s also a re low -positive o r negative, as in the c ase of the n o rm and to ta l. T hus, th ese m e a s u re s give m o re w eight to the le s s-a b u n d a n t and, in the c a se of at le a s t one of th ese, to the le s s - w id e sp re a d sp e c ie s. T his m a y b e an u n d e sira b le situatio n since the r a r e r sp e c ie s u su a lly w ill have h ig h er sam p lin g e r r o r (see Table 4-1). T his is co n firm ed by o b se rv in g the c o rre la tio n s w ith the coefficient of v a ria tio n (Table 4-2). T hese four sp e cie s sta n d a rd iz a tio n s show positive (in th e c a se of sp e cie s total) or n e a r z e ro c o rre la tio n s w ith the coefficient of v a ria tio n , indicating that g r e a te r o r n e u tra l w eight is being given to the m o re u n c ertain 67 sp e c ie s counts. A ll the re m a in in g data types show m uch b e tte r negative c o rre la tio n s w ith the coefficient of v a ria tio n , indicating a m o re d e sira b le situ atio n at le a st w ith re s p e c t to sam p lin g e r r o r . 3) Am ong the sp e c ie s sta n d ard iza tio n s, sp e c ie s- m e a n data a re unique, w ith m o d e ra te to low positive c o rre la tio n s w ith the abundance m e a s u re s , re la tiv e ly high c o rre la tio n s w ith the n u m b er of o c c u rre n c e s , and re la tiv e ly favo rable negative c o r r e lations w ith the coefficient of v a ria tio n . 4) The b in a ry data a re the m o st highly c o rre la te d w ith the n u m b er of o c c u rre n c e s , and sin ce (as shown in T able 4-1) the n u m b er of o c c u rre n c e s is p o sitiv ely c o rre la te d w ith the abundance p a ra m e te rs , m o d e ra te ly high c o rre la tio n s w ith th e se m e a s u r e s a re re a liz e d also. 5) The sim u lta n eo u s-d o u b le sta n d a rd iz a tio n is m o d e ra te in all its c o rre la tio n s . T his is probably due to the balan cing effect of the two opposite ty p e s of sta n d ard iza tio n in volved h e re . b) C o m p ariso n of the d istance m a tric e s P r e lim in a r y an aly sis indicated that the m ain tre n d am ong the d istan ce m a tr ic e s w as re la te d to the g e n e ra l lev el of the d ista n c e s in each m a trix . T his fac to r would not be expected to affect the re s u lts of an a n aly sis sin c e it is the re la tiv e m agnitudes 68 of the d istan c es w ithin a m a trix th at a re im p o rta n t. F o r exam ple, if a constant am ount w e re added to each elem ent of a distance m a trix , the p a tte rn show n by the re s u ltin g c la ssific a tio n or o rd in a tion would be id e n tic a l w ith or without the addition of the constant. W ith th is in m ind, each distance m a trix w as sta n d a rd iz e d by the to ta l of all its e le m e n ts. T his would re s u lt in an id en tica l m ean d istance for each d istan ce m a trix , and thus the siz e effect would be re m oved. The r e s u lts of the PC A of the data typ es w ith the elem en ts of the d istan ce m a tr ic e s as a ttrib u te s a re shown in F ig s. 4-1 and 4-2. Axis I s e p a ra te s the sp e c ie s sta n d ard iza tio n s from the u n sta n d ard ize d and s ite -s ta n d a rd iz e d data. T hese distinctions a re re la te d to the re la tiv e weight given to the m o re abundant sp e c ie s. F ig u re 4-3 is the sam e as Fig. 4-1 but the c o rre la tio n s of each data type w ith to ta l abundance (from T able 4-2) a re included. The sp e cie s sta n d a rd iz a tio n s, situ ated m o re to w ard the negative end of Axis I, show low er c o rre la tio n s and th e u n sta n d ard ize d and s ite -s ta n d a rd iz e d data display the h igher c o rre la tio n s . A xis II s e e m s to be re la te d to an in te ra c tio n b e tw een data type and th e w eight given to the m o re w id e sp rea d sp ecies Included in Fig. 4-4 a re the c o rre la tio n s of each data type w ith the n u m b er of o c c u rre n c e s (from Table 4-2). W ithin the v a rio u s 69 F ig . 4-1 PC A of the data typ es w ith the elem ents of the d istan ce m a tr ic e s as a ttrib u te s. Axes I and II shown. The % of the to ta l v a ria n c e accounted for by each axis is in cluded in p a re n th e s e s. Sym bols: O sp e cie s sta n d a rd i zation; # site sta n d ard iza tio n ; x other; MAX = m a x i m um ; CBRT = cube ro o t; SQRT = sq u a re root; SIMDBL = sim u ltan eo u s double. 70 AXIS H (34%) X BINARY O M EAN (C B R T) M AX O o O SD NORM o m e a n (SQRT) X SIM D BL O T O T A L O MEAN (RAW) C B R T X SQRT ___X ___ M EAN ( 43%) AXIS I T O T A L • NORM M AX 71 F ig . 4-2. PC A of the data types w ith the e lem en ts of the d istance m a tr ic e s as a ttrib u te s. Axes I and III shown. Sym bols a re as in F ig . 4-1. 72 A X IS III (9%) • T O T A L S IM D B L M A X O • • T O T A L NORM B IN A R Y N O R M ^ O M A X SD O O • ...... A X IS I M E A N (43% ) O M E A N (C B R T ) O M E A N (SQ R T ) O M E A N (R A W ) X C B R T X SQ R T 73 F ig . 4 -3. PCA space of F ig. 4-1 w ith the c o rre la tio n s (xlOO) of the re s p e c tiv e data types w ith the to ta l abundance ind i cated. Note tre n d of h ig h er values tow ard th e positive end of Axis I. 74 82 33 90 89 AXIS I 94 - 0 7 60 -10 -18 23 75 F ig . 4-4. PCA sp a ce of Fig. 4-1 w ith the c o rre la tio n s of the re s p e c tiv e d ata types w ith the num b er of o c c u rre n c e s indicated. Note that w ithin the v ario u s c a te g o rie s of data typ es th e re is a tre n d of low er values to w ard the negative end of Axis II. 76 B in a r y AXIS 1 1 90 U n s ta n d a r d iz e d S p e c ie s M e a n S td s . '80 S ite S td s . 72 A X IS I ,68 • 67 O th er S p e c ie s S t d s . / SIM D B L • 64 49 24 - 0 2 - 4 2 65 77 c a te g o rie s d elim ited in the figure, the types w ith the higher c o r r e lations tend to be situ ated to w ard the positive end of the axis. A xis III a p p ea rs to be due to an in te ra c tio n betw een the data type and th e am ount of c o n stra in t on the data v a lu e s. In Fig. 4-5, the m axim um value for each data type is shown. W ithin a c a te g o ry th e re is a tre n d of h ig h er values tow ard the negative end of th e axis. In su m m a ry , it a p p ea rs th at th e re a re th re e m ain p ro p e rtie s of the data type w hich affect p e rtin en t a sp ec ts of the d istan ce c alc u la tio n s. The w eight given to abundance is of p rim a ry im p o rtan c e . U sing th is c rite rio n , the data can be divided up into four m ain types - site sta n d a rd iz a tio n s, u n sta n d ard ize d , sp e c ie s- m ea n sta n d a rd iz a tio n s, and other sp e cie s sta n d a rd iz a tio n s. T hese a re liste d in o rd e r of d e c re a sin g e m p h a sis on abundance. In addition, th e re a re the sim u ltan eo u s-d o u b le and b in a ry data. Since it has p ro p e rtie s of both, the sim u ltan eo u s-d o u b le data a re in te r m ed iate betw een the site and sp e c ie s sta n d a rd iz a tio n s. The b in ary data a re unaffected by abundance but a re p o sitiv ely c o rre la te d with it only b ecau se the n u m b e r of o c c u rre n c e s and abundance a re c o rre la te d (see T able 4-1). The lack of em p h a sis on abundance m ak es b in ary data m o re s im ila r to the sp e c ie s sta n d ard iza tio n s. 78 F ig . 4 -5. PC A sp ace of Fig. 4-2 w ith the m axim um value fo r the re s p e c tiv e data ty p es indicated. Note the tre n d of h ig h er values (redu ced c o n stra in t) to w ard the negative end of Axis III w ithin the c a te g o rie s . 79 SIM D B L S ite S td s . .3 2 . 57 . 39 O th er \ S p e c ie s S td s . B in a r y . 95 11. 89 616 A X IS I 4 . 02 S p e c ie s M ea n S td s . 6. 71 1 5 . 73 14. 0 U n s ta n d a r d iz e d 52.74 80 W ithin th ese c a te g o rie s of data type (based on the w eight given to abundance), the p ro p e rtie s of im p o rta n c e ap p ear to be the w eight given to the nu m b er of o c c u rre n c e s and the re a liz e d c o n stra in t on the data v a lu e s. 3. C o m p ariso n of the data typ es at the lev el of en v iro n m en ta l c o rre la tio n s a) O rdination P re lim in a r y PCA of the data types w ith the sta n d a rd iz e d m u ltip le -re g re s s io n coefficients as a ttrib u te s indicated the following: 1} The sign of the coefficients cannot be u se d since the o rie n ta tio n of the positive and negative ends of the b io tic - o rd in atio n axes is a rb itr a r y ; 2} the in te rp re ta tio n is b a sed m ain ly on the la r g e r (and u su a lly m o re significant) coefficients and, as such, it is d e sira b le th at they re c e iv e g re a te r w eight in the a n aly sis. The f ir s t p roblem is e asily solved by u sin g only the absolute value of the r e g r e s s io n coefficien ts. In re la tio n to the second p ro b lem , g r e a te r w eight can be given to the la r g e r coeffi cie n ts by using the v a ria n c e -c o v a ria n c e m a trix in stea d of the c o rre la tio n m a trix in the PCA. The r e s u lts of the PCA w ith th e se m odifications a re shown in F ig s. 4-6 and 4-7. T h e re is a fa irly good r e s e m blance betw een th e se r e s u lts and the r e s u lts of the PCA of the d istan c e m a tr ic e s . The g e n e ra l p a tte rn of F ig. 4-6 i s s im ila r to F ig . 4-6. PC A of the data types w ith the m u ltip le re g r e s s io n co efficien ts as a ttrib u te s. Axes I and II a re shown. 82 AXIS n (21 %) % S IM D B L T O T A L O BINARY X NORM O O M A X O SD O M E A N (C B R T ) O M E A N (SQ R T ) X C B R T MAX T O T A L NORM A X IS I (61%) M E A N O M E A N (R A W ) SQ R T X 83 Fig. 4-7. PC A of the data ty p es u sin g the m ultiple r e g re s s io n coefficients as a ttrib u te s. Axes I and III a re shown. 84 AXIS m (9%) X B IN A R Y O M E A N XCBRT), , C B R T X TOTAL 3^ NORIVP M A X o M E A N (SQ R T ) M E A N SQ R T X M A X * NORM T O T A L X SIM D B L A X IS I ( 6 1%) O M E A N (R A W ) 85 the p a tte rn shown in F ig. 4-2, and Fig. 4-7 r e s e m b le s Fig. 4-1. Thus it a p p e a rs that the c h a r a c te r is tic s w hich affect the d istance c alcu latio n s also affect (in a fa irly sy ste m a tic m a n n e r) the p a tte rn of en v iro n m en tal c o rre la tio n s shown in the r e s u lts of the ordination a n a ly s is . T able 4-3 contains the m o re im p o rta n t m odified r e g r e s s io n co efficients for each d a ta ty p e . The colum ns a re a r ran g e d in the sa m e o r d e r as the re s p e c tiv e data ty p es on Axis 1 (F ig s. 4-6 and 4-7). The v a ria b le s which have no m o re than one r e g r e s s io n coefficient significant at the 20 p er cent lev el a re ex cluded. Some g e n era liz a tio n s can be m ade from exam ination of T able 4-3. 1) F o r a ll d ata ty pes, th e re a re significant c o rre la tio n s for depth I, sulfide 11, eH II, and sulfide III. F o r all but a few data ty p es, the sig n ifican ce is at the five p e r cent level. 2) The s p e c ie s - to ta l and b in a ry data a re unique in th at (am ong o th e r things) each lacks a sig nificant coefficient for a v a ria b le singled out as being significant using all o th er data types. 3) The data types tow ard the negative end of Axis 1 (left side of T able 4-3) em phasize n itro g e n II, sed c 111, and sulfide 111, w hile d e em p h asizin g n itro g en I, depth II, DDT III, and eH III. 86 u ia o u 9 IT S X B U I S IT S ’-H i CO CM O I t o .'-4 T f r -l --t o o c- rH O ( M - G a ) t T B a t U t o i t o i sa io a d g ajbs I T B a t U lO c o j c o l aiTg o o ! io| T qpuiT S IJqo O i l CO o ir a (5Jbs)u-eatu sa io a d g CO ^ 1 O I O T - t to h ! i r a I xism sa io a d g sa io a 9 g u ia o u sa io a d g (^.xtop-eaui O il ir a i sa io a d g leiq: c u 3 | sa io a d g o ir a | §181 0 0 CO o o O i C - CO O I O O O CM CO O i CO O I O O O CM C O O i o o t o i CO c o l CO 11 O * o t o O c - CO T —1 I T —1 I O «I g ml o 1 —1 t o i C - lOi 1 - 1 1 - 1 O C M t o CO IH CO CM I CD 1-1 ' T f l T f t ^ 1 —1| O i 1 —11 CM O CM ira i i-H O i l o c o l CO -4^ o l c - | i-H CM I 1 —t 1 —11 1 —1 1 —1 O 1 —1 CM 1 —1 1 1 —1 CO Tfi CMI O i - l l 1-1 1-1 CM 1-1 CM O I CO O I CM O i l CM 1-1 CM 1-1 1-1 O i l T f O CM O il O il ira O i l CM c o l t o j t o O I CM O il i r a i t — I c — j T f i t o j -c f i| CM CM I-H I f t o c o j CO "41 O i I c o j i r a CO O i l 1-1 CJij CO CD] C - | O i l i-H O 1 —1| 1-H i-H l§ I 1 . a iw b e o S d .S c C T J C O m < v S ' s r t te- CO . r t c u V T 3 il O CO CJ 0 ) 'O IJ ( U L , CL CU CO " O S § ! CO C D U k o c u CO c u e q 0 ) 0 ) 5 ^ rt 'o I h fi! c u Si c u ( U c S i m cci < U -u Si 3 S S i Ü m" - C I i r c (U ■ o s CO . C L ” I -o I I I ?■= ^<3 S s e g k œ « 2 (g I T 5 H 4 1li siiii i t h is c u a c u 7 3 a 5 b O a C Oa c u : s 5 ! g n i l s I CU 87 Although th e re a re m any s im ila r itie s in the p a tte rn s of e n v iro n m en ta l c o rre la tio n s using the d ifferen t data types, th e re also a re so m e im p o rtan t d ifferen c e s, e sp ec ially on the th ird axis. The a r e a su rv e y ed contains so m e v e ry stro n g g ra d ie n ts and all the data types seem to d etect at le a s t som e aspect of th e se tre n d s . O ther p o ssib le explanations for the s im ila r itie s in the r e s u lts a re d isc u sse d la te r in the c h a p te r, b) C lassific atio n The d e n d ro g ra m s (not included) a re exam ined at the 11-, s i x - , and th re e -g ro u p lev els. H ere the coefficients of se p a ra te d e te rm in a tio n from the d isc rim in a n t a n aly se s of the groups fro m each data type a re u se d as a ttrib u te s in a PCA. The v a ria n c e - c o v arian c e m a trix again is u se d to give g r e a te r w eight to the la r g e r co efficien ts. Id en tica l groups can re s u lt at any one lev el for s e v e r a l different data ty p es. T his is without re f e re n c e to the h ie ra rc h y by w hich th e s e groups a re connected. H ow ever, when id e n tic a l groups do have d ifferen t h ie ra r c h ic a l a rra n g e m e n ts , this can be d etected w ith an an aly sis at a h igher lev e l of the d endro g ra m . F o r exam ple, th e re a re a few data ty p es w hich re s u lt in the s a m e groups at th e 1 1 -group level, but tho se w ith different h ie r a rc h ie s w ill be d istin g u ish ed in the a n aly sis at the six -g ro u p level. 88 1) The 1 1 -group lev el H ere all but the s p e c ie s -to ta l (12 groups) and the s ite -m e a n (10 groups) a re analyzed at the 1 1 -group level. T hese two data ty pes did not lend th e m se lv e s e a sily to an 1 1 -group division. It w as felt that it would be b e tte r to u se a d ifferen t n u m b er of groups ra th e r than include u n u su al fusions or sp lits . The r e s u lts of the PC A using the coefficients fro m the f ir s t th re e d isc rim in a n t axes a re show n in Fig. 4-8. Both axes s e p a ra te the site and sp e c ie s sta n d a rd iz a tio n s . The double sta n d a rd iz e d and c u b e -ro o t data c lu s te r w ith the sp e cie s s ta n d a rd i zatio n s, w h e re a s s q u a re -ro o t d ata is m o re a sso c ia te d w ith the site sta n d a rd iz a tio n s. M ost of the d istan ce in both dim ensions, how ever, is due to the p e rip h e ra l p o sitio n s of the bin ary , sq u a re - ro o t, and s ite - to ta l data ty p es. The sp e c ie s-m a x im u m data also are s e p a ra te d som ew hat fro m the o th er sp e c ie s sta n d a rd iz a tio n s. T able 4 -4 show s the m o re im p o rtan t coeffi cie n ts of s e p a ra te d e te rm in a tio n fro m the d isc rim in a n t a n aly sis of each data type. All the data ty p es show high coefficients for depth and se d im e n t c o a rs e n e s s on the f ir s t d isc rim in a n t axis. T his r e se m b le s the r e s u lts of sta n d a rd iz e d r e g r e s s io n coefficients for the f ir s t o rd in atio n axis (Table 4-3). On th e 's e c o n d d isc rim in a n t axis, a ll data types but the s q u a re -ro o t, s ite -to ta l, and b in ary d ata a re 89 F ig . 4-8. PCA of the coefficients of se p a ra te d e te rm in a tio n g e n e ra te d by d isc rim in a n t an aly sis of the 11 groups defined by the v a rio u s data ty p es. 90 SQ R T A X IS n (34%) * T O T A L X B IN A R Y M E A N • . M A X ^ N O R M AXIS I (49%) T O T A L O O O M E A N M A X (R A W ) O M E A N / M E A N (C B R T ) (S Q R T ) NORM & SD C B R T ^ S IM D B L 91 c d O Û uim s I s I CD g I CD CD c d O CQ CD •rH O CD C L m a s ( D CQ CD •p H o CD Q . m m CD •rH o CD a m u cr CQ ^ s I CD a CQ (D •rH o (D CL m o Sed C I 20 13 26 22 23 22 10 23 23 Depth I 73 68 57 69 59 67 80 67 66 Clay II 20 13 18 15 16 20 18 20 Sed C II 19 26 24 27 18 32 32 N itrogen II 42 12 Hg II 15 DDT II 31 19 20 20 11 19 22 22 15 eH II 23 19 15 22 14 19 12 13 Clay III 13 SD m 18 11 11 8 14 11 8 Sed C III 53 22 23 15 11 9 10 Depth III 19 10 23 25 25 21 19 20 Sulfide III 49 33 11 17 DDT III 12 eH III 10 18 17 23 25 37 29 39 T able 4-4. C oefficients of se p a ra te d e te rm in a tio n fro m the d is c rim in a n t a n aly sis of the 11 groups defined by the v a rio u s d a ta ty p e s . Only the la r g e r v alu es a re listed. The colum ns a re in the s a m e .o r d e r as the d a ta ty p e s on A xis II in Pig. 4-8. 92 fa irly s im ila r. On the th ir d d isc rim in a n t axis, the s p e c ie s - m axim um and b in a ry d ata a re unlike a ll o th er ty p es. On the whole. Fig. 4-8 and T able 4-4 show a g e n e ra l s im ila rity of in te rp re ta tio n using m o st data tÿ p es. M ost of the v a ria tio n is provided by th re e p e rip h e ra l data ty p es w hich a re e x tre m e by one c r ite r io n o r an o th er. The s q u a re - ro o t d ata a re the le a st c o n stra in e d data type, the s ite -to ta l is the m o st c o n stra in e d site sta n d ard iza tio n , and th e b in a ry data are the only type without a quantitative com ponent. 2) The six -g ro u p level The r e s u lts of the PCA a re in Fig. 4-9. At th is level, all the s ite -s ta n d a rd iz e d , u n sta n d ard ize d , and double sta n d a rd iz e d d ata show id e n tic a l groups. The b in a ry data a re stro n g ly p e rip h e ra l, and the sp e c ie s and site sta n d a rd iz a tio n s a re w ell se p a ra te d . The sp e c ie s-m a x im u m sta n d a rd iz e d data a re s till som ew hat se p a ra te d from the o th er sp e cie s sta n d a rd iz a tio n s on PCA Axis II. T able 4-5 contains the coefficients of s e p a ra te d e te rm in a tio n for the f ir s t th re e d isc rim in an t ax es. The f ir s t d is c rim in a n t axis is s till m o stly re la te d to depth and se d im e n t c o a r s e n e ss, although the sp e c ie s-m a x im u m and b in ary data a re e sp ec ially heav ily w eighted for depth. On d isc rim in a n t Axis 11, the sp e cie s 93 F ig . 4-9. PCA of th e coefficients of s e p a ra te d e te rm in a tio n g e n e ra te d by d isc rim in a n t a n aly sis of the six groups defined by the v a rio u s data ty p es. 94 A X IS n (32% ) O M A X X B IN A R Y M E A N (R A W ) (S Q R T ) (C B R T ) O — --------------- ------------ -------------------- A X IS I N O R M <56%) SD O T O T A L N O R M M A X M E A N T O T A L * SQ R T C B R T S IM D B L 95 0 0 k C Q C Q 0 0 • r H tH a a 0 0 C L a, mm c d § 0 s C Q 0 • r H 0 0 CL m " Id H-) O H - J C Q 0 • r H 0 0 a m ; 1 •s C Q 0 • r H 0 0 C» a 0 0 0 0 ^ q 4 -j H - j H - > p q 3 m m m cn U Sed C I 30 30 35 11 35 9 Depth I 49 53 42 76 42 70 eH I 9 C lay II 15 18 13 24 10 7 Sed C II 28 27 22 21 17 Depth II 19 14 23 N itrogen II 16 Sulfide II 23 25 Hg II 13 DDT II 17 11 17 18 eH II 12 18 32 30 Sand III 11 SD III 14 25 Sed C n i 10 11 9 14 Depth III 12 22 Sulfide III 38 33 29 48 30 27 eH III 36 43 29 30 9 T able 4-5. C oefficients of s e p a r a te d e te rm in a tio n from the d is- c rim in a n t an aly sis of six groups defined by th e v a rio u s data types The colum ns a re in the sa m e o rd e r as the data types on Axis I in Fig. 4-9. 96 sta n d a rd iz a tio n s would be in te r p re te d quite differently from the s ite -s ta n d a rd iz e d (etc. ) and the b in a ry data in that the la tte r two a re m o re re la te d to sulfide and eH r a th e r than se d im e n t and depth. On the th ird d isc rim in a n t axis, th ese two g e n e ra l data types a re again m o st unlike the re m a in in g ty p es. 3) The th re e -g ro u p level At this level, th e re is a convergence of the r e s u lts of m o st of the data ty p es. In fact, 12 different ty p es r e su lted in id en tical grou ps on the d e n d ro g ra m . The coefficien ts of se p a ra te d e te rm in a tio n for the four d ifferen t c la ssific a tio n r e s u lts a re show n in T able 4 -6. The f ir s t d isc rim in a n t axis d iffers from the f ir s t axis at the 11- and six -g ro u p lev e ls. Two of the re s u lts em p h a siz e depth and sulfide, and the o th er two em p h asize depth and sed im en t p a ra m e te r s , w ith h e a v ie r w eight on the se d im e n t. 4) P a tte r n of the r e s u lts In c o m p a rin g the re s u lts at the th re e different le v e ls, a p a tte rn se e m s to e m e rg e . At the 11-group level, th e re a re a few o u tlie rs , but in g e n e ra l m o st of the site and sp e c ie s sta n d a rd iz a tio n s a re only w eakly se p a ra te d . At the th re e -g ro u p lev el th e re is even a stro n g e r con v erg en ce of the r e s u lts of a ll but th re e data ty p es. At the six -g ro u p level, how ever, the sp e c ie s s ta n d a rd i z atio n s a re stro n g ly se p a ra te d from the s ite -s ta n d a rd iz e d , un sta n d a rd iz e d , and d o u b le -sta n d a rd iz e d d a ta ty p e s . This s e e m s 97 Clay I Sed C I Depth I Sulfide I Clay II Sed C II Depth II N itrogen II Sulfide II Hg II DDT II eH II C 66 16 26 14 30 16 0 2 0 ’o 0 a t i o . “ S' 0'-^ II 64 10 13 8 10 I § 0 0 2 0 •rH o 0 CL m 13 47 19 12 10 a j O E h 02 0 •iH 0 0 CL m 14 44 20 40 46 12 19 11 33 T able 4-6. C oefficients of se p a ra te d e te rm in a tio n from the d isc rim in a n t an aly sis of th re e groups as defined by the v a rio u s d ata types. 98 to in d ic ate (with the p re s e n t data) that the effects of the g e n e ra l d a ta type u sed in the c la ss ific a tio n a n aly sis w ill be m o st evident at the m iddle lev els r a th e r than at the v e ry top or bottom of the d e n d ro g ra m . The p o ssib le re a s o n s for th is p a tte rn a r e exp lo red la te r in the c h ap ter. D. D iscu ssio n 1. The choice of data type F re q u en tly the pu rp o se of the n o rm a l an aly sis is the definition of a p a tte rn of s ite s w hich w ill c o rre sp o n d to the m ain biotic a lly -im p o rta n t e n v iro n m en ta l fa c to rs in the a re a u n d er study (A ustin, 1968; C a ssie and M ichael, 1968; Field, 1971; A ustin et al. , 1972; M oore, 1973; N oy-M eir et al. , 1973). As d e m o n stra te d in th is c h a p te r, the p a tte rn of d ista n c e s (B ra y -C u rtis index) betw een the s ite s can be re fle c te d in the fin al e n v iro n m en tal c o rre la tio n s . T h is lead s to the question of w hich data type w ill r e s u lt in d istan c es w hich w ill give the b e st c o rre sp o n d e n c e to changes in the m o re im p o rta n t e n v iro n m en ta l fa c to rs . To study th is question it is n e c e s s a r y to r e v e r t to the b a sic m od el of the re la tio n sh ip betw een the fitn e ss of a sp e c ie s and p e rtin e n t changes in the lo cal e n v iro n m en t. F ig u re 4-10 show s the c la s s ic b e ll-sh a p e d cu rv e w ith the peak at a p a rt of the en v iro n m en t w h e re the o rg an ism in questio n is m o st fit (co n sid erin g 99 F ig. 4-10. B asic m odel of re la tio n sh ip betw een sp e c ie s abundance (fitness) and p e rtin en t e n v iro n m en tal v a ria tio n . 100 D EFG HI J KLM NO P R S T S it e s 0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 E n v ir o n m e n t a l Sc a le E n v ir o n m e n t a l C h a n g e DEFGHI J KL MN OP Q R S T 0 1 2 4 6 91 21 41 51 41 29 6 4 2 1 0 D a ta M a tr ix 101 bio lo g ical and p h y sical fa c to rs ), and w ith d e c re a s in g fitn ess in e ith e r d irectio n away fro m the peak. The en v iro n m en ta l change along the h o riz o n ta l axis is a ssu m e d to be continuous and uni d ire c tio n a l. F o r th e o re tic a l d isc u ssio n s of th is concept see W hittaker (1967), L evins (1968), and P ian k a (1974); fo r exam ples o b se rv e d in n a tu re see T e rb o rg h (1971), A ustin (1972), Johnson and R is s e r (1972), C opeland and B ech tel (1974), and N oy-M eir (1974). In th e follow ing d isc u ssio n , the abundance o r re la tiv e abundance of a sp e c ie s is u se d as a m e a s u re of the fitn e ss of a sp e c ie s in a p a rtic u la r en v iro n m en t. F o r the sake of sim p licity , the m o d el only in clu d es a single e n v iro n m en ta l g rad ien t, but the sa m e arg u m e n ts can be applied in the c a se of additional en v iro n m e n ta l g ra d ie n ts. U nless sp ecified o th erw ise, the w ord ’’d istan c e" w ill r e f e r to the c alcu late d B ra y -C u rtis d istan ce betw een s ite s . In th is m odel, the abundance of an o rg an ism changes in a n e a rly continuous fashion in re sp o n se to a continuous change in the environm ent. T h e re fo re , the id eal d ista n c e s betw een site s should c o rre sp o n d d ire c tly w ith the e n v iro n m en tal change. In Fig. 4-10 site s D and T a re at the two e x tre m e ends of the env iro n m e n ta l g rad ie n t and also a re s e p a ra te d by a d istan ce of one since they have no sp e c ie s counts g r e a te r than z e ro in com m on (assu m in g only the single sp e c ie s is p re se n t). Since th e re a re 16 102 u n its of e n v iro n m en ta l change betw een site s D and T , the id ea l distance calcu latio n s would change 1/16 (.0625) for each unit of e n v iro n m en ta l d ifferen ce betw een any two s ite s being co m p ared , i . e . , the id eal d istan ce betw een site s A and B = . 0625 x E^-g , w h e re E^-g = units of e n v iro n m en ta l change betw een site s A and B. The d istan ce calcu latio n s w ith the single sp e c ie s in F ig. 4-10 a re c o n sid e re d f ir s t. Table 4-7 in clu d es c o m p a riso n of s ite s equidistant fro m , but on opposite sid e s of the site w ith the p eak abundance count (site L ). Since id en tica l abundance counts a re found at th e se points, each d istan ce w ill be z e ro (with any d istan ce index). Note th at the c alcu late d d istan c es show a b e tte r c o rre sp o n d e n c e w ith the id ea l distance as the abundance counts of the site s being c o m p a re d a re c lo s e r to the peak abundance count. F o r exam ple, s ite s F and R contain sp e c ie s counts of two (which is 2 /1 5 of the peak value) and the calcu lated d istan ce d iffers from the id e a l by a re la tiv e ly la rg e am ount (. 75). In c o m p a riso n , site s K and M each contain counts w hich a re 14/15 of the peak value, and the c alcu lated d istan c e v a rie s from th e id e a l by . 12 5. The r e a s o n for th is effect c le a rly is due to the fact th at the cu rv e in te r s e c ts the sa m e abundance count at two points (Austin, 1972; B eals, 1973). Although the site s c o rre sp o n d in g to 103 Site C o m p a riso n s F - R G-Q H -P I-O J - N K-M U nits of E n v iro n m e n ta l Change 12 10 8 6 4 2 Id e a l D ista n ce .7 5 0 .6 2 5 .500 .3 7 5 ,2 5 0 . 125 D ista n c e 0 0 0 0 0 0 D iffere n ce .7 5 0 .6 2 5 .5 0 0 .3 7 5 .2 5 0 .1 2 5 T ab le 4 -7 . C o m p a riso n of s ite s on opposite sid e s of the abundance p eak in F ig . 4-10. 104 th e s e points a re id e n tic a l biotic ally, they a re not id en tica l e n v iro n m en tally . The d isc re p a n c y betw een the biotic and e n v iro n m en tal change w ill depend on th e shape of the c u rv e and the m agnitude of the abundance values in question. Since the two halves of the c u rv e co n v erg e at the peak (at one point in th e environm ent), the e n v iro n m e n ta l change d e c re a s e s as points c lo s e r to the peak (but on opposite sid e s) a re being c o m p a red . T able 4-8 show s som e c o m p a riso n s of s ite s con tain in g abundance counts w hich differ by so m e constant am ount g r e a te r than z e ro . Site 1 d iffers from s ite s H and P by a count of th re e , and a count of th re e lik ew ise s e p a ra te s site L from s ite s J and N (Fig. 4-10). Of the c o m p a riso n s w ith site 1, the one m o st in a g re e m e n t w ith th e id e a l is the H-1 c o m p a riso n . T his is due to the fact that both s ite s a re c lo se to g eth er on the sa m e sid e of th e abundance peak, w h e re the s m a ll change in abundance c o rre sp o n d s to a s m a ll change in th e e n v iro n m en t. The 1 -P c o m p a riso n , w hich is fu rth e r fro m the ideal, involves a site w hich is on the opposite side of the abundance peak, and the s m a ll change in the sp e c ie s count does not c o rre s p o n d as w ell to the la r g e r en v iro n m e n ta l d ifferen ce betw een th ese two s ite s . C o m p ariso n s w ith site L (the peak site) a re not affected by th is problem sin ce c o m p a ris o n w ith a site on any o th er p a rt of the c u rv e w ill involve 105 Site C o m p ariso n s H-I I - P J - L L -N U nits of E n v iro n m e n ta l Change 1 7 2 2 Id ea l D istan ce .0 6 2 5 .43 7 5 .12 50 .1250 In sta n c e .2 00 .200 .110 .110 D ifferen ce .1375 .2375 .0 1 5 0 .0150 T able 4 -8 . C o m p a riso n of so m e s ite s s e p a ra te d by an abundance count of th re e in F ig . 4-10. 106 c o m m e n s u ra te biotic and e n v iro n m en ta l change. T his is re fle c te d in the fact th at both c o m p a riso n s w ith L a re equally s a tis fa c to ry and both a re c lo s e r to the id e a l than the I - P c o m p a riso n . The im p o rta n t point h e re is th at w hen at le a s t one of th e s ite s in a c o m p a riso n co ntains a peak or n e a r-p e a k abundance count for a sp e c ie s, the in fo rm a tio n c o n trib u ted to the d istan ce calcu latio n s by th at sp e c ie s should a g re e fa irly w ell w ith the en v iro n m e n t. When both counts of a sp e c ie s a re fu rth e r from the peak v a lu e s, the c o rre sp o n d e n c e w ith th e environm ent can be le s s re lia b le . As in the f ir s t exam ple, th is effect is due to the dupli cate abundance counts on opposite sid e s of the peak abundance. The fact th at c o m p a riso n H-I deviates fu rth e r from the id ea l than does J - L or L -N is due to the v ary in g slope of the c u rv e and also to a c h a r a c te r is tic of the d istan ce m e a s u re . As m en tio n ed in C hapter II on the B ra y -C u rtis index the am ount of d istan c e re s u ltin g from a d ifferen ce in sp e cie s counts w ill depend on the g e n e ra l m agnitude of the n u m b e rs from w hich the differen ce w as calcu late d (Hall, 1969). A d ifference betw een two s m a lle r n u m b e rs w ill r e s u lt in a g r e a te r d istance th an w ill the sa m e d iffe r ence w hen c alcu lated fro m two la r g e r n u m b e rs. Thus a d ifference of th re e low er on the c u rv e (as in c o m p a riso n H -I) w ill lead to g r e a te r d istan ce than w ill a d ifferen ce of th re e h igher on th e c u rv e 107 (as in c o m p a riso n J -L ). Since d istan ce H -I is g re a te r than the id eal, the additional d istan ce due to th is effect is re sp o n sib le p a rtia lly for the deviation from the ideal. In the c a se of c o m p a riso n I - P , how ever, the id e a l is la r g e r th an the c alc u la te d distance, and the la tte r would be even fu rth e r from the id ea l w ithout th is effect. This effect w ill be le s s d ra s tic when th e r e a re m o re sp e c ie s at the s ite s being c o m p a re d sin ce such s m a lle r n u m b e rs w ill be dom inated by the la r g e r n u m b e rs in the calcu latio n s. Two additional sp e cie s a re c o n sid e re d now along the sa m e e n v iro n m en ta l g rad ie n t (Fig. 4-11). T h ese sp e c ie s (called sp e c ie s B and C ) a re le s s abundant and le s s w id e sp re a d than is the o rig in a l sp e c ie s (now c alle d sp e c ie s A). T his is c o n sisten t w ith the c o rre la tio n s in T able 4- 1 w h ereb y the n u m b er of o c c u r re n c e s and the abundance p e r o c c u rre n c e a re p o sitiv ely c o rre la te d (r = . 51). The c o m p a riso n s in T able 4-7 a re re c a lc u la te d w ith the tw o new sp e c ie s included, and the r e s u lts a re shown in T able 4-9. The d ifferen c e s betw een th e id e a l d ista n c e s and the c alcu lated d istan c es e ith e r a re the sa m e as or m o re fav o ra b le than the d is ta n c e s w ith only one s p e c ie s . Thus the in c lu sio n of additional sp e c ie s can add u se fu l in fo rm a tio n and im p ro v e s the c o rre sp o n d e n ce w ith th e enviro n m en t. 108 F ig . 4-11. T h re e hypothetical sp e c ie s c u rv e s along an e n v iro n m e n ta l g rad ie n t. 109 T O C L ) S LO o H m c ç u o < D r— 4 d o T O G H to T -t LO 1 - 4 rt* f— I c r > T — t CM n 4 1 -4 r-4 O O ) 00 t> CO H4 lO S O ^ & 4 CM M 1 -4 Û o aou-epunqv 110 F -R G“ Q H -P I-O J - N K -M U nits of E Change 12 10 8 6 4 2 Id e a l D ista n ce . 750 . 625 . 500 .3 7 5 .250 . 125 D ista n ce .333 .333 . 400 . 182 . 077 . 000 D ifferen ce . 417 .292 . 100 . 193 . 173 . 125 T able 4 -9 . C o m p a riso n of s ite s on opposite sid e s of the abundance peak, w ith the th r e e s p e c ie s in F ig. 4-11. I l l In Fig. 4-11 it can be se e n that sp e c ie s A is the m o st heavily w eighted sp e c ie s at a ll s ite s since it is the m o st abundant at a ll s ite s . The m o st re lia b le in fo rm a tio n about sp e c ie s A , how ever, is n e a r its peak. Solely due to its re la tiv e ly high abundance and w id e sp re a d o c c u rre n c e , it is heavily w eighted both at its peak and fa r from its peak. F u rth e rm o r e , if sp e cie s B and C a re to re c e iv e the d e s ire d h igher w eighting to w ard th e ir r e s p e c tive peak counts, it is obvious that the overw helm ing effect of the abundance of sp e c ie s A m u st be elim in ated . T his can be acco m plished by a sp e c ie s sta n d ard iza tio n . The p ro p e rtie s of the sp e cie s sta n d a rd iz a tio n s a re d isc u sse d fu rth e r in the next sectio n. F ig u re 4-12 show s a plot of the data in Fig. 4-11 after a s p e c ie s -m e a n sta n d a rd iz a tio n . It is c le a r that now each sp e c ie s w ill be heav ily w eighted in the d istance c alcu latio n s involving site s w here the peak counts ap p ear. T able 4-10 contains the d istan c es calcu late d w ith the sta n d a rd iz e d v alu es. The o v e ra ll c o rre s p o n dence w ith the environm en t has been im p ro v e d g re a tly by the sta n d a rd iz a tio n . The sum of the deviations fro m the id eal w ith the u n sta n d a rd iz e d data (T able 4-9) w as 1.3 and the sum for the sta n d a rd iz e d data is . 59 5. T his r e p re s e n ts a significant 54 per cent im p ro v e m e n t of the c o rre sp o n d e n c e w ith the id ea l d istan c es. The sa m e am ount of im p ro v em en t could not have been ob tain ed by a site sta n d a rd iz a tio n o r a data tra n s fo rm a tio n since the 112 S p e c ie s-m e a n sta n d a rd iz e d c u rv e s from the data in 4-12. 4-11. 0 > " c ü o C Q aoTjepunqv p s z ip j ^ p iT B ^ g U B aui-saioadg C Q d > eg I - O * - O l-H Î X 3 CO T — 4 lO y-i 1 — t 0 0 T - 4 CM I T -4 1 — 4 O v 4 (Ji 0 0 r- CD lO 0 0 114 F - R G-Q H -P 1 - 0 J - N K -M Units of E Change 12 10 8 6 4 2 Id ea l D istan ce .75 0 . 625 . 500 . 375 . 250 . 125 D istance . 650 . 650 . 710 . 450 .240 . 000 D ifferen ce . 100 . 075 . 210 . 075 . 010 . 125 T able 4-10. C o m p a riso n of s ite s on opposite sid e s of the abundance peak, w ith s p e c ie s - m e a n s ta n d a rd iz e d v alu es fo r the th re e s p e c ie s in F ig . 4-11 (see F ig . 4-12). 115 v a lu e s for sp e c ie s A w ould s till be dom inant at a ll s ite s and the peaks of sp e c ie s B and C would not re c e iv e th e ir p ro p e r w eight. In fact, the site sta n d a rd iz a tio n s would often in c re a s e the dom inance of the la r g e r abundance counts (see Table 4-2 and Fig. 5-1). T his w ould d e c re a s e r a th e r th an in c r e a s e the c o rre sp o n d e n c e w ith the en v iro n m en t sin ce the situ atio n w ould becom e m o re like the sin g le sp e c ie s exam ple (Fig. 4-11 and T able 4-7). At th is point, the d isc u ssio n can be s u m m a riz e d as follow s. In a c o m p a riso n of two s ite s , the sp e c ie s w hich w ill con trib u te th e m o re re lia b le in fo rm a tio n (as fa r as the env iro n m en t is co n cern ed ) w ill be the ones w ith v alu es c lo s e r to th e ir peak v alu es in at le a s t one of the two s ite s being co m p a red . F o r th is re a s o n it is im p o rta n t that the data type u sed allow g r e a te r w eight for each sp e c ie s w hen it o c c u rs n e a r its peak abundance. A d istan ce index such as the B ra y -C u rtis index gives g r e a te r w eight to th e la r g e r n u m b e rs in the distance c a lcu latio n s. Thus when su ch an index is used , a sp e c ie s sta n d a rd iz a tio n is re c o m m e n d e d to give each sp e c ie s re la tiv e ly la rg e sta n d a rd iz e d v alu es in place of its peak abundance counts. T his is n e c e s s a r y sin c e the m agnitude of the peak abundance level w ill v a ry fro m sp e c ie s to sp e c ie s ; without the sta n d a rd iz a tio n the absolute abundance w ill be e m p h asized r a th e r th an the peak valu es of each sp e c ie s. T his com bination of 116 d ata type (sp ecies sta n d ard iza tio n ) and d istan c e index (B ra y -C u rtis) p ro v id es a sim p le and e a sily -im p le m e n te d m ea n s of s tr e s s in g the m o re re lia b le , e n v iro n m e n ta lly -re la te d in fo rm a tio n in the biotic data. Thus fa r the effect of the sta n d a rd iz a tio n a n d /o r tra n s fo rm a tio n is evalu ated only in re s p e c t to the B ra y -C u rtis index. With the C a n b e r r a - m e tr ic index, the la r g e r and s m a lle r sp e c ie s counts w ill re c e iv e m o re equal w eight. In th is c a se , the m o re re lia b le sp e c ie s c o m p a riso n s (those involving at le a st one n e a r - p e a k count) w ill be m o re equally w eighted w ith the le s s re lia b le c o m p a ris o n s (those w ith both counts fu rth e r from the peak counts). As long as m o st of the sp e c ie s c o m p a riso n s a re of the m o re re lia b le type, the re s u lts should be re a so n a b le . How ever, th e re is no a s s u ra n c e th at this w ill alw ays be the c a se . T h ere is c le a rly a need for fu rth e r study on the re la tio n sh ip betw een data sta n d a rd iz a tio n and the C a n b e r ra - m e tric index. It should be pointed out that the techniques u se d h e re a re not m ea n t to p ro v e th a t one p ro c e d u re is b e tte r th an another. The m ain in ten t of the c u rv e s and c alcu latio n s is to d e m o n stra te an idea. It would be p o ssib le to se t up a com bination of c u rv e s w hich would show that another p ro c e d u re is optim al. The r e a l te s t r e s t s in how w ell the situ atio n depicted fits actual data. 117 An im p o rta n t p a rt of the m o d el is the a ssu m p tio n th at th e r e w ill be sp e c ie s w hich a re both w id e sp re a d and re la tiv e ly abundant. T h ese sp e c ie s can s till have high abundance counts (when c o m p a re d to o ther sp e c ie s counts) in s ite s w hich a re distan t (along the en v iro n m en tal continuum ) from the s ite s w ith th e ir p eak co unts. It has been d e m o n s tra te d w ith the p re s e n t data th at the m o re com m on sp e c ie s tend to be m o re abundant (Table 4-1), and o th e rs have found th is to be the c a se also (H airston, 1959; M cN aughton and Wolf, 1970; Lane," 197 5). On the o th er hand, R ick iefs (1972), w o rking w ith b ird s , found no c o rre la tio n betw een abundance and b re a d th along the a ssu m e d en v iro n m en ta l g ra d ie n ts . H ow ever, it is not n e c e s s a r y to show such a c o rre la tio n fo r the m o d el to be valid. The p re s e n c e of s e v e ra l le s s-a b u n d a n t but w id e sp re a d sp e c ie s would b r e a k down the c o rre la tio n , but th is would not m ea n that abundant and w id e sp re a d sp e c ie s a re absent co m p letely . It could be arg u ed th at the sp e c ie s-a b u n d a n c e c u rv e s along e n v iro n m en ta l g ra d ie n ts w ill not alw ays be a p p ro x im ate ly b e ll-sh a p e d . T h ere could be s e v e r a l r e a s o n s fo r th is. a) If only one e n v iro n m en ta l fa c to r (one dim ension) is being c o n sid ere d , but the sp e c ie s also' a re resp o n d in g to o th er re la tiv e ly independent fa c to rs w hich change in the study a re a , the 118 effects of the d ifferen t fa c to rs could be confused. In th is c a se the re la tio n sh ip s betw een the abundance and the environ m ent would have to be c o n sid e re d in a la r g e r d im e n sio n al sy ste m w here the abundance data indeed m ay be com patible w ith the b e ll-c u rv e m o d el (P ianka, 1974:191). b) B iotic in te ra c tio n s m ay in te r f e r e w ith su ch a r e sponse to th e en v iro n m en t. T h e re a re th re e b a sic re s p o n s e s of sp e c ie s to co m p etitio n o r p red a tio n as fa r as th e ir re la tio n sh ip s to the p e rtin e n t en v iro n m en ta l g ra d ie n ts a re c o n cern ed (M ueller- D om bois and E llen b e rg , 1974). T hey m ay (1) n a rro w th e ir b rea d th of o c c u rre n c e ; (2) d isp lace the peak of abundance in one d irec tio n away fro m the co m p etitio n or p red atio n; o r (3) display two o r m o re p eak s, being d e p re s s e d in the in te rn a l p a rt of the c u rv e w h ere the com petition or p red a tio n is the g re a te s t. Only the th ird p o ssib ility is in co m p atib le w ith the a ssu m p tio n s of the m odel. F o rtu n ately , th is is probably the le a s t com m on re s p o n s e (Colw ell and F u en tes, 197 5). The secon d p o ssib ility s e e m s to be the a lte rn a tiv e m o st com m only en co u n tered . c) Sam pling e r r o r m ay in te r f e r e w ith a c c u ra te m e a s u re m e n t of the abundances and the a ssu m e d c u rv e s m ay not be re a liz e d . T his pro b lem c a n be solv ed by (1) adequate sam p lin g tech n iq u es, and (2) the u se of a n aly tic al m eth o d s w hich do not 119 e m p h a siz e the sp e c ie s w hich a p p ea r to be m o re su sce p tib le to sa m p lin g e r r o r . As fa r as (2) is con cern ed , the re la tio n sh ip b e tw een the d ifferen t data ty p es and the sam p lin g e r r o r (as m e a s u re d by th e coefficient of v a ria tio n ) is exam in ed e a r lie r in the c h a p te r. d) Some sp e c ie s m ay have different ecotypes r e p r e se n ted in the data. T his m ay c a u se peak abundances for the sa m e s p e c ie s in quite different e n v iro n m en ts. This could be a se rio u s p ro b lem but if th e re is so m e w ay to identify the d ifferen t ecotypes, th e y should be c o n sid e re d as if th ey w e re se p a ra te sp e c ie s in the a n a ly sis. O th erw ise, if w idely d iv erg en t but unidentifiable ecotypes for a sp e c ie s a re known to ex ist in the study a re a , such s p e c ie s could be e lim in a te d from the data. e) The sc a le of the sam p lin g could be e ith e r too sm a ll o r too la rg e to include b e ll c u rv e s . T his is d isc u sse d la te r in the se ctio n . 2. F u rth e r c o n sid e ra tio n of sp e c ie s sta n d a rd iz a tio n s Given th at a sp e c ie s sta n d a rd iz a tio n m ay allow p ro p e r w eighting for the m o re e n v iro n m en ta lly in fo rm a tiv e sp e c ie s counts, fu rth e r ex p lo ratio n of th e p ro p e rtie s of the sp e c ie s sta n d a rd iz a tio n s is in o rd e r . In th e follow ing d isc u ssio n it is a ssu m e d that the d is tan c e m e a s u r e involved is se n sitiv e to the sc ale of the sta n d a rd iz e d v a lu e s, as is th e B ra y -C u rtis index. 120 U sing the m o d el of the b e ll c u rv e s, a sa tis fa c to ry sp e c ie s sta n d a rd iz a tio n should m e e t th re e re q u ire m e n ts : a) The shape of the c u rv e is im p o rta n t. In Fig. 4 - 13a the abundances at s ite s A and B a re both peak values yet th ey a re d is s im ila r en v iro n m en tally . With such a cu rv e, th e p eak values w ill not be as re lia b le an in d ic a to r of the en viro nm ent as would the peak v alu es in a c u rv e such as th a t shown in Fig. 4- 13b. Thus it would be d e s ira b le if th e peak sta n d a rd iz e d values fo r the sp e c ie s w ith "flat" (platykurtic) c u rv e s w e re of a low er m agnitude th a n for th o se of sp e c ie s w ith m o re "peaked" (leptokurtic) c u rv e s. b) Holding the shape of the c u rv e constant, it is d e s ira b le that the peak sta n d a rd iz e d v alu es of the r a r e r , le ss abundant sp e c ie s would not be g r e a te r than th o se of th e m o re abundant, w id e sp re a d sp e c ie s, sin c e th e la tte r sp e c ie s contain in fo rm a tio n c o n cern in g a la r g e r p o rtio n of the sam p lin g a re a and u su a lly a re a sso c ia te d w ith low er sa m p lin g e r r o r (T able 4-1). c) The sta n d a rd iz a tio n should be "sta b le " in the se n se that the s ta n d a rd iz e d v alu es for a sp e c ie s should not be affected by the in clu sio n of s ite s w hich contain no o c c u rre n c e s of the sp e c ie s in question. T his re q u ire m e n t is included sin ce th e site s at w hich a sp e c ie s is absent contain no additional in fo rm a tio n in re la tio n to the shape or height of the c u rv e for th at sp e c ie s. 121 F ig . 4 -13. D e m o n stratio n of the re la tio n sh ip betw een k u rto sis and th e re lia b ility of peak abundance v a lu e s. See tex t for explanation. 122 a. A B 123 To evaluate the sp e c ie s sta n d a rd iz a tio n s in light of th e se th re e c r ite r ia , the sta n d a rd iz e d values of each, as applied to five hyp othetical sp e c ie s c u rv e s, a re exam ined (Fig. 4-14). The f ir s t four sp e c ie s v a ry in the am ount of k u rto sis (" fla tn e ss" or "p eak ed n ess") of th e ir c u rv e s, w ith the peak abundance value (24) and th e n u m b er of o c c u rre n c e s (11) held c o n stan t. The fifth curve is ap p ro x im ate ly th e sa m e shape as the th ir d c u rv e but the sp e c ie s is le s s abundant (peak = 12) and le s s w id e sp re a d (five o c c u rre n c e s ). T his la s t sp e c ie s is in clu d ed to te s t c rite rio n b) w hich w ill be c o n sid e re d sa tis fie d if th e peak values of this sp e c ie s a re not g r e a te r th an th o se of the f ir s t two sp e c ie s (which a re m o re lepto k u rtic , abundant, and w id esp read ). The raw d ata a re shown in the top se t of c u rv e s in Fig. 4-14. A ll the sp e c ie s sta n d a rd iz a tio n s except the sta n d a rd deviation s ta n d a rd iz a tio n (SD) fulfill re q u ire m e n t c) A dditional z e ro v alu es fo r a sp e c ie s w ill not a lte r th e m ax im u m , m ean, n o rm , o r to ta l by w hich th e abundance counts a re divided. The c u rv e s for the sp e c ie s m ax im um sta n d a rd iz a tio n violate re q u ire m e n t a) sin ce the lep to k u rtic and p laty k u rtic c u rv e s all have id e n tic a l peak v a lu e s. The c u rv e s fo r the s p e c ie s -m e a n sta n d a rd iz a tio n fulfill a ll th re e c r ite r ia . ' The peak sta n d a rd iz e d values v a ry d ire c tly w ith the k u rto sis of a ll th e c u rv e s, and the 124 F ig . 4-14. The effect of the v a rio u s sp e c ie s sta n d a rd iz a tio n s on five hypothetical sp e c ie s c u rv e s. The sp e c ie s s ta n d a rd iz atio n applied is in d ic ate d at the left of each se t of c u rv e s. E leven s ite s a r e included in the SD c a lc u lation s fo r sp e c ie s 5. 125 SPECIES 1 SPECIES 2 SPECIES 3 SPECIES 4 SPECIES 5 L H CM - 126 peak sta n d a rd iz e d value fo r sp e c ie s 5 is le s s th an th o se of sp e c ie s 1 and 2. The s p e c ie s -n o rm and s p e c ie s - to ta l s ta n d a rd iza tio n s both fulfill re q u ire m e n t a)_, but each fails to m e e t r e q u ire m e n t b), sin c e in each c a se sp e c ie s 5 has the highest s ta n d a rd iz e d value of a ll the sp e c ie s. The sp e c ie s SD s ta n d a rd iza tio n a p p e a rs to m e e t c r ite r io n b) but fails on c r ite r io n a) in th at sp e c ie s 4 and sp e c ie s 2 have h ig h er sta n d a rd iz e d values th a n sp e c ie s 1 , As m entioned, the sp e c ie s SD sta n d a rd iz a tio n vio la te s re q u ire m e n t c). The dashed line for sp e c ie s 4 (Fig. 4-14) show s the sta n d a rd iz e d v a lu e s w hen the two z e ro s for the end site s a r e included in the c a lc u la tio n s. W hen th is is done, th e c u rv e goes fro m the hig hest peak to the low est peak. T able 4-11 dem on s t r a te s fu rth e r the in sta b ility of th is sta n d ard iza tio n in re la tio n to the n u m b er of z e ro s in the data m a trix . A sp e c ie s w ith four o c c u rre n c e s of a count of one is show n w ith in c re a s in g n u m b ers of z e ro counts (i. e . , m o re s ite s a re included which do not contain any of th is sp e c ie s). W ith te n to ta l s ite s , all the s ta n d a rd iz e d values a re 1. 58 , w hile w ith 100 site s they a re a value of 5. As the sp e c ie s b eco m es re la tiv e ly r a r e r in re la tio n to the to ta l num ber of s ite s , it is actu ally becom ing m o re heavily w eighted. This situ atio n can lead to violations of c r ite r io n b). It should be noted also th at th is sp e c ie s has a co m p letely "flat" 127 T otal # Sites S tan d ard ized V alues 10 1.58 20 2. 24 40 3 .1 6 60 3.8 7 80 4. 47 100 5.00 T able 4-11. D enaonstration of the effect of z e ro counts on the r e s u lts of a sp e c ie s-S D s ta n d a rd iz a tion, The sp e c ie s depicted o c c u rs w ith a count of 1 at each of four s ite s . A dditional s ite s contain counts of 0. Note the in c re a s e in sta n d a rd iz e d values as m o re s ite s a re added. 128 c u rv e w hich would b eco m e h eav ily w eighted as m o re z e ro s a re added. T his, of c o u rs e , would v io late re q u ire m e n t a). It could be arg u ed that th e se p ro b le m s m ight be re c tifie d by c alcu latin g the sta n d a rd deviation w ithout in clu sio n of the z e ro s (as is done w ith th e s p e c ie s -m e a n ). T his is not the solution sin ce so m e tim e s a b e tte r r e s u lt is re a liz e d w ith the z e ro s included. F o r in sta n c e , s p e c ie s 4 is m o re c o n siste n t w ith c rite r io n a) w ith the z e r o s included. F u r th e rm o re , a sp e c ie s w hich has the sa m e abundance count w ith each o c c u rre n c e w ill have a sta n d a rd deviation of z e ro and sta n d a rd iz e d v alu es of infinity. The p ro p e rtie s of th e sp e c ie s sta n d a rd iz a tio n s show n in F ig . 4-14 and T able 4-11 a re c o n siste n t w ith the c o rre la tio n s show n in T able 4-2. The s p e c ie s -to ta l, n o rm , SD, and m axim um sta n d a rd iz a tio n s , w hich can r e s u lt in g r e a te r or equal w eight fo r th e r a r e r sp e c ie s, a ll show low er c o rre la tio n s w ith the abundance p a r a m e te r s and the n u m b er of o c c u rre n c e s and, subsequently, h ig h e r c o rre la tio n s w ith the coefficient of v ariatio n . E vidently, in the p re s e n t data the m o re w id e sp re a d sp e c ie s ten d to be som ew hat m o re lep to k u rtic in the d istrib u tio n of th e ir abundance v a lu e s. T his w as c o n firm e d by exam ination of the data m a tr ix . Since the s p e c ie s - m e a n sta n d a rd iz a tio n gives m o re w eight to the m o re lep to k u rtic s e ts of abundance counts, the c o rre la tio n s 129 w ith th is sta n d a rd iz a tio n would be c o n siste n t w ith the c o rre la tio n s w ith the n u m b e r of o c c u rre n c e s (a m e a s u re of how w id e sp re a d a sp e c ie s is). F o r exam ple, both the s p e c ie s -m e a n sta n d a rd iz a tio n (see T able 4-2) and the n u m b er of o c c u rre n c e s (see T able 4-1) a re po sitiv ely c o rr e la te d w ith the abundance m e a s u re s and n eg ativ ely c o rr e la te d w ith the coefficient of v a ria tio n . In su m m a ry , the s p e c ie s -m e a n sta n d a rd iz a tio n a p p e a rs to have the m o st d e sira b le p ro p e rtie s of all sp e c ie s s ta n d a rd iz a tions c o n sid e re d . To th e knowledge of the author, this s ta n d a rd iz a tion has not been u se d n o r te s te d in any p rev io u s study. The g e n e ra l re lu c ta n c e of m an y eco lo g ists to u se sp e c ie s sta n d a rd iz a tio n s is due to the d isp ro p o rtio n a te w eight given to the r a r e r sp e c ie s by th o se sta n d a rd iz a tio n s th at have been te s te d (N oy-M eir, 1970; F ield , 1971; G oldsm ith, 1973; C lifford and Stephenson, 1975). 3. P a tte rn of the c la ss ific a tio n r e su lts and the b e ll-c u rv e m o del W ith a few exceptions, the en v iro n m en ta l c o rre la tio n s w ith the c la s s ific a tio n r e s u lts fro m the d ifferen t data types g e n e ra lly co n v erg e at the v e ry low er (11 groups) and u p p e r (th ree groups) lev e ls of the d e n d ro g ra m . At the m iddle lev el of the d en d ro g ram (six g ro u p s), how ever, the r e s u lts of the s ite -s ta n d a rd iz e d and un sta n d a rd iz e d data d iv erg e fro m those of the s p e c ie s -s ta n d a rd iz e d data (F ig. 4-9). T his p a tte rn of r e s u lts m ay be re la te d to the b ell 130 c u rv e m odel. The d isc u ssio n h e re w ill c e n te r around F ig. 4-15, w hich depicts a sing le e n v iro n m e n ta l g rad ien t w ith s e v e r a l sp e c ie s c u rv e s of v ary in g sh a p e s, heights, and peak points. When c o m p a rin g s ite s clo se to g e th e r on the e n v iro n m e n ta l g rad ien t ( e .g ., s ite s A and B ), m o st sp e c ie s counts v a ry a p p ro x im ate ly continuously w ith the env iron m ent (as in c o m p a riso n H -I in T able 4-8). A ccordingly, the d ecisio n as to w hich sp e c ie s a re m o re heavily w eighted (by data sta n d ard iza tio n ) in th e d istan ce c alcu latio n s b eco m es le s s im p o rta n t, and the r e s u lts of th e dif fe re n t data types would be expected to converge. T his id e a is con- \ s is te n t w ith the c o n v erg en c e of c la ss ific a tio n r e s u lts at the 11- group lev el. At th is level, the c la ss ific a tio n is b a sed on the s h o rte r d ista n c e s in the d istan c e m a trix , and th e se s h o r te r d ista n c e s p re s u m a b ly w ill be betw een s ite s c lo se to g eth er on th e e n v iro n m e n ta l g ra d ie n ts as a re s ite s A and B in F ig . 4-15. When c o m p a rin g s ite s som ew hat fu rth e r a p a rt on the e n v iro n m en ta l g rad ien t (e.g., s ite s A and C ), sp e c ie s counts on both sid e s of th e peak of so m e of the c u rv e s w ill be encou ntered. W ith such sp e c ie s, a situ atio n s im ila r to the c o m p a riso n s in T able 4-7 o r c o m p a riso n I - P in T able 4-8 a ris e s , and th e changes in so m e abundance counts a re not c o m m e n s u ra te w ith the e n v iro n m e n ta l change. O ther sp e c ie s w ill not be as affected by th is 131 F ig . 4-15. H ypothetical en v iro n m en tal g rad ien t w ith s e v e ra l sp e c ie s c u rv e s . 132 T 3 133 p ro b lem and th e changes in abundance counts w ill be m o re in Une w ith the e n v iro n m en ta l c h an g es. In th is c a se , th e w eight given to the v a rio u s sp e c ie s in th e d istan c e c alcu latio n s (by data s ta n d ard izatio n ) b eco m es m o re im p o rta n t, sin ce a ll sp e c ie s a re not conveying the s a m e in fo rm a tio n in re la tio n to th e en v iro n m en t. T his is c o n sisten t w ith th e d iv erg en ce of th e c la ss ific a tio n r e s u lts at the 6 - group level, w h e re th e m o d e ra te d istan c es b eco m e im p o rtan t in the building of th e d e n d ro g ra m . Such d ista n c e s w ould be a s s o c ia te d w ith the lev e l of e n v iro n m en ta l change d isc u sse d h e re . F in ally , w hen c o m p a rin g s ite s v e ry fa r a p a rt on the e n v iro n m en ta l g rad ie n t ( e .g ., s ite s A and D ), few, if any, of the sa m e sp e c ie s w ill be p re s e n t in both s ite s . H ere n e a rly all s p e c ie s at th e two s ite s being c o m p a re d w ill add to the calcu late d d ista n c e and th e re w ill be m in im a l p ro b lem s w ith duplicate counts on both sid e s of the peak s of the c u rv e s. In o th er w o rd s, each sp e c ie s w ill convey s im ila r in fo rm a tio n in re la tio n to the e n v iro n m en t, and the w eight given to each sp e c ie s again b e c o m e s le s s im p o rta n t. T his w ould explain the c o n v erg en ce of c la ss ific a tio n r e s u lts at the 3-g ro u p level, sin ce the longer d ista n c e s (which c o rre sp o n d to th e g r e a te r am ounts of e n v iro n m en ta l d istan c e b e tw een site s ) a re m o re im p o rta n t at the top of the d e n d ro g ra m . 134 T h e re a re so m e exception s to the p a tte rn of c o n v e r gence and d iv erg en ce of the c la ss ific a tio n re s u lts w ith the different data ty p e s. At the 11- and 6 - group lev e ls, the r e s u lts w ith the b in a ry d a ta a re unique. T his w ould be expected sin ce a quantitative com ponent is lacking, and the b e ll-c u rv e m odel is highly dependent on d iffe re n c e s in sp e c ie s counts at s m a ll and m o d e ra te ly la rg e e n v iro n m en ta l d istan c es along the g rad ie n t. At the 3 - group level, how ever, m o st of the re le v a n t d ista n c e s a re betw een v e ry differen t s ite s (biotic ally and environm ent ally) w h e re m uch of the biotic v a ria tio n is of a q ualitative n a tu re , and the b in a ry r e s u lts co nverge w ith th o se of the o th e r data ty p e s. At the 11-group level, the r e s u lts w ith th e sq u a re - ro o t and s ite - to ta l data differ sig n ifican tly from those of the o th e r data ty p e s. As noted, each of th e se ty pes a re e x tre m e as fa r as c o n s tra in t is concerned, w ith the s q u a re - ro o t data the le a s t con s tra in e d and th e s ite - to ta l d ata the m o s t c o n stra in e d site s ta n d a rd i z atio n . T his m ay be a s s o c ia te d som ehow w ith th e unique r e s u lts obtained w ith th e se data ty p e s. E sp e c ia lly at the 6 - group level, the r e s u lts w ith the s p e c ie s -m a x im u m sta n d a rd iz a tio n ten d to diverge from th o se of the o th e r sp e c ie s sta n d a rd iz a tio n s. As show n in Fig. 4-14, th is s ta n d a rd iz a tio n is unique in that the peak sta n d a rd iz e d v alu es fo r a 135 s p e c ie s a re independent of the k u rto sis of the d istrib u tio n of sp e c ie s cou nts. As d isc u sse d , this c h a r a c te r is tic c an affect how w ell the ca lc u la te d d ista n c e s w ill c o rre sp o n d to the a sso c ia te d env iro n m e n ta l change betw een s ite s . T his is fu rth e r d isc u sse d in C h ap ter IX. It should be noted th at the sc a le of the sam p lin g would be im p o rta n t in th e application of th e s e id e a s. If the sam pling s ite s a re re la tiv e ly fa r a p a rt on the e n v iro n m e n ta l g rad ien t, then th e re m ay not be c o n v erg en ce at the bottom of the d e n d ro g ram of the type h y p o th esized fo r the 1 1-group level. If th e s ite s a re su f ficien tly different en v iro n m en ta lly and biotic ally, a convergence of the type h y p o th esized at the th re e -g ro u p lev e l m ay take place at the bottom of the d e n d ro g ra m . On the o th e r hand, if the site s all a re re la tiv e ly c lo se to g e th e r along the en v iro n m en ta l g rad ien t, th e re m ay not be m an y o r any full b ell c u rv e s in the data, and the r e s u lts for the d ifferen t data types would tend to c o n v erg e, p resu m ab ly , at all lev e ls of the d e n d ro g ra m . E ven in th is c a se sp e c ie s -m e a n s ta n d a rd iz e d o r tra n s fo rm e d data m a y be b e st sin c e a site s ta n d a rd iza tio n m ay u tilize only a s m a ll am ount of the sp e c ie s data (i. e. , m o stly the abundant sp e c ie s), and the o th er sp e c ie s s ta n d a rd iz a tions m ay o v e re m p h a siz e the im p o rta n c e of the le s s - r e lia b le sp e c ie s counts of the r a r e r sp e c ie s. T his effect is d e m o n stra te d 136 in C h a p te r V, in re la tio n to the effects of the elim ination of sp e c ie s fro m th e data. Some additional p ro b le m s w ith site sta n d a rd iz a tio n s a re d is c u s s e d below. 4. P a tte rn of the o rd in atio n r e su lts and the be 11-c u rv e m odel W ith o rd in atio n , the la r g e r d ista n c e s a re m o st im p o rta n t when the position of the f ir s t axis is d e te rm in e d (Rohlf, 1968). T his being the c a se , one m a y su sp e c t a g e n e ra l co n v erg en ce of r e s u lts w ith the d ifferen t data types on the f ir s t axis. T his w ould be s im ila r to the c o n v erg en ce of r e s u lts at the top of the d e n d ro g ra m (th re e -g ro u p level) in the c la ss ific a tio n r e s u lts , w h e re the lo n g er d istan c es a re lik ew ise m o re im p o rtan t. The in te r p r e ta tio n s of the f ir s t a x is, as f a r as en v iro n m en ta l c o rre la tio n s a re co n ce rn e d , a re indeed quite s im ila r in that alm o st all data typ es show significant co efficien ts for depth and sed im en t c o a r s e n e s s (T able 4-3). T he position of the second and th ird axes would be influenced by m o re m o d e ra te d ista n c e s, and som e d iv erg en ce of r e s u lts w ould be expected w ith the d ifferen t data ty p es. T h e re s e e m s to be a fa ir am ount of d iv erg en ce of r e s u lts on the th ird axis (T able 4-3). T his c o rre sp o n d s w ith the d iv erg en ce of the c la s s ific a tio n re s u lts at the six -g ro u p level. 137 5. R ange and efficiency of the a g g lo m e ra tiv e - c la ss ific a tio n , m u ltip le -d is c rim in a n t m ethod The re lia b ility of the calcu late d d ista n c e s s e e m s to depend on w hich c o m p a ris o n s a re being c o n sid e re d . D istances betw een s ite s w hich a re s e p a ra te d by a m o d e ra te am ount on the e n v iro n m en ta l g rad ie n t(s) m a y be affected by the b e ll-c u rv e p ro b lem , w hich can be a llev iated som ew hat by the s p e c ie s -m e a n sta n d a rd iz a tio n . The d ista n c e s betw een s ite s v e ry w idely s e p a ra te d on the en v iro n m en ta l g rad ie n t ( e .g . , site s A and D in Fig. 4-15) can be d isto rte d som ew hat by the tru n c a te d n a tu re of the b ell c u rv e s (Swan, 1970). T his concept is d isc u sse d in re la tio n to F ig. 4-16. In the c o m p a riso n of s ite s A and B , the con trib u tio n of th is sp e c ie s to the d istan c e calcu latio n s is due to the differen ce of 10 betw een the two s ite s . W hen c o m p a rin g s ite s A and C, the co n trib u tio n of the sp e c ie s is id e n tic a l even though th e se two s ite s a re m o re e n v iro n m en ta lly d is s im ila r than a re s ite s A and B. T hus, beyond the point w h e re a sp e c ie s c u rv e is tru n c a te d , the sp e c ie s counts for th at s p e c ie s c e a se to provide additional in fo rm a tion in re la tio n to e n v iro n m e n ta l change. The c alcu latio n s of the longer d istan c es betw een s ite s w hich a re s e p a ra te d by la rg e r e n v iro n m en ta l d iffe re n c e s u su a lly w ill involve m an y such tru n c a te d c u rv e s . T his m ea n s th at the re s u ltin g d ista n c e s betw een such site s 138 F ig . 4-1 6. H ypothetical sp e c ie s c u rv e d e m o n stra tin g the effect of c u rv e tru n c a tio n on e n v iro n m e n ta l c o rre la tio n s . 139 C Q (D to O U pq g • i ~ t ■ § b) " c ti (D S c 0 1 H 0otrepunqY 140 can be u n d e re s tim a te d when c o m p a re d w ith the a c tu a l d e g re e of en v iro n m en ta l change. The m o st r e lia b le d ista n c e s se em to involve c o m p a ri sons of site s w hich a re c lo se to each o th er on th e e n v iro n m en ta l g rad ie n t, sin ce m o st sp e c ie s counts u su a lly w ill change continu ously w ith the en v iro n m en t betw een such s ite s . M ost o rd in atio n m ethod s (PCA, p rin c ip a l co o rd in ate a n a ly sis, p o lar ordin atio n , etc. ) heavily w eight the g r e a te r d ista n c e s (Rohlf, 1968). As d e m o n stra te d above, how ever, th e se d ista n c e s a re not the m o st re lia b le d istan ces in the d istan c e m a trix as fa r as en v iro n m en ta l c o rre la tio n s a re co n ce rn e d . Some c o n se quences of this in o rd in atio n a re d isc u sse d by N oy-M eir and A ustin (1970), Swan (1970), A ustin and N oy-M eir (1971), A ustin (1972), Gauch and W h ittak er (1972), and B eals (1973). The p a tte rn of s ite s in the o rd in atio n sp a c e can becom e com plex and the in te r p re ta tio n m ay be m o re difficult. The s e v e rity of th is p ro b lem depends on the biotic p a tte rn s in the a re a su rv ey ed , th e tr a n s f o r m a tio n s and sta n d a rd iz a tio n s applied to the data, and the o rd in atio n m eth o d used . A g g lo m erativ e c la ssific a tio n , on the o th er hand, in itia lly u s e s the s h o r te s t d ista n c e s , and as the d e n d ro g ra m is built, the longer (and p o ssib ly le s s re lia b le ) d istan c es a re u tilize d . 141 If the s ite s connected at the bottom of the d en d ro g ram a re not too d is s im ila r en v iro n m en tally , th is p a rt of the d en d ro g ram w ill be b a se d on the m o st re lia b le d ista n c e s . F u rth e rm o re , as the d e n d ro g ram is built, e r r o r s m ay accu m u late and c au se additional un c e rta in ty at the u p p er lev e ls (Rohlf, 1968, 1970). T rad itio n a lly , the u p p e r le v e ls of the d en d ro g ra m have been em p h a size d . Since the u p p e r sp lits s e p a ra te m an y of the s ite s (to be d e m o n stra te d below in Fig. 4-17), it is a ssu m e d th at such sp lits w ill c o rre s p o n d to the m a jo r fa c to rs in the en v iro n m e n t. When m u ltip le d isc rim in a n t a n aly sis is u sed with agglom e r ative c la ssific a tio n , it is not n e c e s s a r y to u se the m o re u n c e rta in links on the u p p e r le v e ls of the d en d ro g ram in an attem p t to find the m o re im p o rta n t e n v iro n m en ta l fa c to rs . F ig u re 4 - 17a r e p r e s e n ts an id e a l d e n d ro g ra m in which each split c o rre s p o n d s to division of th e s ite s acco rd in g to som e e n v iro n m e n ta l fa c to r. F o r in sta n c e , fa c to r A m ay r e p r e s e n t depth, and the sh a llo w er s ite s would be on one side of the sp lit and the d e ep e r s ite s on the o th er. F ig u re 4 - 17b shows the fac to r by w hich each p a ir of groups on the d e n d ro g ra m is se p a ra te d . T his i s d e te rm in e d by tra c in g a path fro m one group being c o n sid e re d to th e o th e r, and th e fac to r a sso c ia te d w ith the node w hich is c o m p letely t r a v e r s e d is the fac to r s e p a ra tin g the two group s. It can be 142 F ig , 4-17. D em o n stratio n of the re la tio n sh ip betw een the n u m b er of groups s e p a ra te d by an e n v iro n m en ta l fa c to r and th e c o rre sp o n d in g lev el on the d e n d ro g ra m . 143 F a c to r A F a c to r C F a c to r B F a c .C F a a F F a a G 2 3 1 5 8 4 6 7 G roup # a. H y p o th etical id e a l d e n d ro g ra m . E n v iro n m e n ta l fa c to r s c o r r e sponding to e ac h sp lit in d ic a te d . = » te P o s 2 3 GROUP # 4 5 6 7 8 # G roup p a ir s e p a r a te d b] e ach fa c to r 1 D C C A A A A A 16 2 C C A A A A B 4 3 E A A A A C 4 4 A A A A D 1 5 F B B E 1 6 B B F 1 7 G G 1 b. F a c to r s s e p a r a tin g the g ro u p s in the h y p o th etical d e n d ro g ra m , 144 s e e n th at m o re groups a re s e p a ra te d by fa c to rs w hich c o rre sp o n d to the hig h er lev els of the d e n d ro g ra m . Sixteen p a ir s of groups a re s e p a ra te d by fa c to r A, four p a irs by fa c to rs B and C , and one p a ir each by th e re m a in in g fou r fa c to rs. T his is a c o n se rv a tiv e ta lly sin c e quite often the sa m e fa c to r w ill be a s s o c ia te d w ith su b d iv isio n at s e v e r a l le v e ls on th e d en d ro g ra m . F o r each ax is, the d isc rim in a n t a n aly sis m ethod e m p h a siz e s v a ria b le s w hich m a x im iz e the b etw een -g ro u p d ifferen ces and m in im iz e the w ith in -g ro u p d ifferen c e s (see C h ap ter II). If m e a s u re d , the en v iro n m en ta l v a ria b le s w hich c o rre s p o n d to the s e p a ra tio n of s e v e ra l groups at the low er levels of the d e n d ro g ra m (e, g. , fa c to r A in F ig. 4- 17) w ill be re sp o n sib le fo r a re la tiv e ly la rg e portion of th e b etw ee n -g ro u p d ifferen c e s. C onsequently, th e s e v a ria b le s w ill be h eav ily w eighted on the e a rly d isc rim in a n t ax es. The v a ria b le s ex clu siv ely se p a ra tin g adjacent groups on the lo w er lev els of the d e n d ro g ra m ( e .g ., facto r D in Fig. 4-17) w ill be heav ily w eighted on the la te r ax es. Thus, th e v a ria b le w eightings on the d isc rim in a n t axes convey a h ie ra rc h y of e n v iro n m e n ta l fa c to rs of th e ir own, independent of the actu al d en d ro g ram h ie r a rc h y above the le v e l of the groups chosen. 145 6. "R o b u stn e ss" of the e n v iro n m en ta l c o rre la tio n s in re la tio n to d ata type If the e n v iro n m en ta l c o rre la tio n s with the groups at th e low er p a rt of the d e n d ro g ra m tend to converg e w hen d ifferen t d ata ty p es a re u se d (as happened w ith th e 1 1 -group d isc rim in a n t a n a ly sis r e s u lts ), th is m ethod (a g g lo m e ra tiv e -c la ssific a tio n and m u ltip le -d is c rim in a n t a n aly sis) w ill be re la tiv e ly " ro b u st" ( i . e . , the d ata type used b eco m es le s s c ritic a l). F o r in sta n c e , if for so m e re a s o n the sa m p lin g in te n sity at each site is not equal and a site sta n d a rd iz a tio n b e co m es n e c e s s a ry , it is p o ssib le th at th is w ould not lead to a d r a s tic a lly d ifferen t r e s u lt w hen c o m p a re d to an a n a ly sis u sing a m o re efficient s p e c ie s -m e a n sta n d a rd iz a tio n (if the sa m p lin g in te n sity w e re equal). T his would only happen, h o w ev er, if the sc a le of sa m p lin g w e re conducive to the co n v erg en ce of r e s u lts , and the c o n v erg in g lev e ls of the d e n d ro g ra m w e re u sed in th e d isc rim in a n t a n a ly sis. 7. C o m p ariso n w ith d ivisive h i e r a rc h ic a l c la ss ific a tio n m ethods D ivisive c la s s ific a tio n techniques (W illiam s, 1971) build the d en d ro g ra m by s u c c e s s iv e sub div isions, s ta rtin g at the top. T his avoids the p ro b lem of e r r o r build-up to w ard the top of the d e n d ro g ra m as happens w ith agglom e r ative c la ssific a tio n . F o r th is re a s o n , divisive c la s s ific a tio n is appealing to e co lo g ists who 146 e m p h a size the u p p er le v e ls of the d en d ro g ra m in th e ir a n aly se s. The e r r o r w ith the d ivisive m ethods w ill accu m u late to w a rd the bottom of the d e n d ro g ra m and th e re w ill be le s s confidence in the groups s e p a ra te d out by the low er sp lits. Like the divisiv e c la ss ific a tio n m eth o d s, m o st o rd in atio n tech n iq u es a re designed to stud y efficien tly the m a jo r e n v iro n m en ta l fa c to rs at the expense of the m o re m in o r fa c to rs . In fact, m an y of the p o ly th e tic -d iv isiv e c la ss ific a tio n m ethods are b a sed on the s e p a ra tio n of s ite s along an o rd in atio n axis for each su b d iv isio n (L a m b e rt et al. , 1973; N o y -M eir, 1973; H ill et al. , 1975). In c o n tra s t, a g g lo m era tiv e c la ss ific a tio n w ith m ultiple d isc rim in a n t a n a ly sis can u su a lly provide re lia b le en v iro n m en tal in fo rm a tio n in re la tio n to both m a jo r and m in o r fa c to rs . If the sam p lin g sc a le is a p p ro p ria te and groups low er on the d en d ro g ram a re d elim ited , the m o st e n v iro n m en ta lly efficient d istan c es a re u se d (the s h o r te r ones) and a m in im um of e r r o r build-up is r e alized . Subsequent d isc rim in a n t an aly sis c an provide the in fo rm a tion on both the m a jo r and m in o r e n v iro n m en tal fa c to rs . It is in te re s tin g to note that the en v iro n m en tal c o rre la tio n s w ith the f ir s t o rd in atio n axis (T able 4-3) a re som ew hat s im ila r to the en v iro n m en ta l c o rre la tio n s w ith the f ir s t d isc rim in a n t axis 147 (T ables 4 -5 and 4-4) at the six - and 1 1-group le v e ls. T his in d ic ate s that in fo rm a tio n c o n cern in g the m a jo r e n v iro n m e n ta l fa c to rs indeed is available at the low er lev els of the d e n d ro g ra m ju st as it is av ailab le on the f ir s t o rd in atio n axis. It should be pointed out, how ever, th at the d is c r im i nate a n aly sis is dependent on th e m e a s u re m e n t of v a ria b le s a s s o c i ated w ith the re le v a n t e n v iro n m en ta l fa c to rs . In the absence of su ch data, the efficiency and u se fu ln ess of the m ethod w ill d e c re a s e . 8. Site sta n d a rd iz a tio n s and E u clid ean d istan ce Som e confusion m ay re s u lt if the conclusions re a c h e d by N o y -M eir (1970) and N oy-M eir et al. (1975) a re c o m p a re d w ith th e re c o m m e n d a tio n s in th is c h a p te r as fa r as d ata sta n d a rd iz a tio n s a re c o n ce rn e d . T hese au th o rs rec o m m e n d a site sta n d a rd iz a tio n in the n o rm a l an aly sis w hen u sing the E u c lid e a n -d ista n c e m odel. Such a co n clu sio n does not c o n tra d ic t the p re s e n t study since a d ifferen t d ista n c e m o d el (B ra y -C u rtis) is u se d h e re . It is not s u rp r is in g th at a site sta n d a rd iz a tio n would be b e n eficial w ith E u clid ean d istan ce in light of the fact that O r loci (1967a) re c o m m e n d e d a site -n o rm sta n d a rd iz a tio n (in conjunction w ith E u clid ean d istan c e) as a m e a n s of c o rre c tin g fo r the o th erw ise un accep tab le p ro p e rtie s of th is m o d el (the d o u b le -z e ro problem ; 148 C h ap ter II). U nfortunately, th e r e a re so m e p otential d raw backs w ith site sta n d a rd iz a tio n s . F o r exam ple, if the sp e c ie s data in Fig. 4-11 a re sta n d a rd iz e d by site to tal, the re s u ltin g sp e c ie s c u rv e s a re as in F ig s . 4 -18a, b. In F ig. 4 - 18c, the re s u ltin g E uclidean d ista n c e s betw een site 1 and the o th e r s ite s a re shown. The r e s u lts c le a rly a re n o n se n sic a l. Am ong o th e r p ro b le m s, the end s ite s a re id e n tic a l w ith the m iddle s ite s and the p a tte rn of site d istan c es is th e sa m e on e ith e r sid e of the m iddle s ite s . H ere the sa m e poor r e s u lts w ould be found w ith any o th e r d ista n c e index b e ca u se th e fault lie s w ith the s ite sta n d a rd iz a tio n . The re a s o n for su ch p ro b le m s can be illu s tr a te d by r e f e r e n c e to th e g e o m e tric in te r p re ta tio n of the site s ta n d a rd iz a tio n s . The sta n d a rd iz a tio n by site n o rm is equivalent to p ro jec tin g th e s ite points in the E u clid ean sp a ce onto the su rfa c e of a c irc le , sp h e re , or h y p e rs p h e re (depending on th e n u m b er of dim ensions involved) w ith a ra d iu s of unity (O rloci, 196 7a; N oy-M eir et a l . , 1975). A sta n d a rd iz a tio n by site to ta l is c o m p a rab le to p ro jec tin g the s ite points onto a unit hypotenuse, plane, o r h y p er plane (N oy-M eir et al. , 1975). In F ig. 4-19, it can be se en th at w ith the s ite - n o rm sta n d ard iza tio n , the p ro jec tio n s onto the unit c ir c le can be id e n tic a l or s im ila r even though th e points a re lo cated o rig in a lly 149 F ig . 4-18. D e m o n stratio n of the effect of the s ite - to ta l sta n d a rd iz a tio n on the data in Fig. 4-11. In (c) the d istance be tw een s ite s 1 and 15 is not due to the sta n d a rd iz a tio n sin ce the ra w data a re id e n tic a l at th e s e s ite s . The sa m e p ro b le m s also w ill a r is e w ith a ll the o th er site st an d ard iz a tio n s . 150 S p ecies A 1 S p ecies B S pecies C Sites a. S ta n d ard ize d s p e c ie s c u rv e s 1, 0 .67 .67 . 6 .82 .92 1 1 1 .92 .82 . 6 .67 .67 1.0 0 .33 .33 .4 .18 .08 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 .08 .18 . 4 .33 .33 0 ( D b. S ta n d a rd iz e d d a ta m a tr ix S ites 1 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 0 .4 7 . 47 . 57 .2 5 . 11 0 0 0 a i .25 .57 .47 .47 0 c. D ista n c e s b etw een s ite 1 and the o th e r s ite s . 151 F ig . 4-19. D e m o n stratio n of a p ro b lem w ith site sta n d a rd iz a tio n s using the g e o m e tric in te rp re ta tio n of the s ite -n o r m sta n d a rd iz a tio n . The s ite points a re p ro je c te d onto the su rfa c e of a c ir c le w ith a ra d iu s of unity. Note th at id e n tic a l sta n d a rd iz e d v a lu e s a re a re s u lt of p ro jec tio n s fro m d ifferen t lo cations in the space. 152 u O ) a. CO 3 • s • R 2 1 2 S p e c ie s X 153 in quite d ifferen t p a rts of the sp ace. F o r exam ple, p ro je c tio n s of points P - T o r L and M would r e s u lt in id e n tic a l sta n d a rd iz e d v a lu e s. A s im ila r situ a tio n would be found w ith the p ro je c tio n s c o rre sp o n d in g to the o th er site sta n d ard iza tio n . T hus, if the effective sam p lin g in te n sity at each site is eq u al and it is m ean ingful to d ire c tly co m p a re abundance counts at the d ifferen t s ite s , then a site sta n d a rd iz a tio n can lead to a loss of w hat m a y be im p o rta n t in fo rm a tio n in calcu latin g the i n te r - s ite d ista n c e s . F ro m th is p e rsp e c tiv e , it would ap p ear th at the site sta n d a rd iz a tio n s a re po ten tially tro u b le so m e data m an ip u latio n s w hich a re re q u ire d by th e u n eco lo g ical n a tu re of the E u clid ean - d istan ce m o del. In c o n tra s t, data an aly sis using the B ra y -C u rtis d istan c e m o d el does not r e q u ire n e c e s s a r ily a potentially u nde s ir a b le data sta n d a rd iz a tio n . 154 CH A PTER V DATA REDUCTION AND THE RELA TIV E IM PORTANCE OF THE SPEC IES IN THE NORMAL ANALYSIS A. In tro d u ctio n Not a ll sp e c ie s w ill co n trib u te equally to the p a tte rn s to be defined in the site a n a ly sis. In fact, the co n trib u tio n of s e v e ra l sp e c ie s w ill be so negligible th at th ey could be e lim in ated from the a n a ly sis w ithout a lte rin g the r e s u lts (Day et al. , 1971). The r e m o v al of a la rg e n u m b er of u n im p o rtan t s p e c ie s can lead to con sid e ra b le savings of c o m p u ter tim e and s to ra g e sp a ce along w ith allev ia tin g s e v e r a l of the lo g istic a l p ro b le m s involved in such an a n a ly sis. The im p o rta n c e of a sp e c ie s w ill v a ry w ith the m ethod of a n a ly sis, and even w ithin one m ethod the w eight of a sp e c ie s w ill be dependent upon the tra n s fo rm a tio n and sta n d a rd iz a tio n u sed . Thus if one w ish e s to elim in ate so m e sp e c ie s fro m the a n aly sis, it is im p o rta n t to have av ailab le an a p p ro p ria te c rite r io n for id en tificatio n of th e re la tiv e im p o rta n c e of a sp e c ie s. F ie ld (1971), w orking w ith benthic m a rin e data, u se d all sp e c ie s above an a r b itr a r ily ch o sen n u m b er of individuals (<10 155 in d iv id u als in 55 site s ). T his re d u c e d his n u m b er of sp e c ie s from 298 to 131, but only changed the n u m b er of individuals fro m 19, 667 to 19, 223. Stephenson et al. (1970), in itia lly e lim in a te d s p e c ie s o c c u rrin g in few er than eight of 400 dredge sta tio n s. W illiam s and Stephenson (1973), u sin g a c la ss ific a tio n m eth od involving a form of E u clid ean d istan c e, p ro p o sed a m eth od w h ereb y the to ta l d istan ce c o n trib u te d by each sp e c ie s (in the fin al d istan c e m a tr ix c o m p a rin g all p a irs of s ite s ) could be c alcu late d . The sp e c ie s then can be s o rte d in descending o r d e r a cc o rd in g to th e ir d istan c e co n trib u tio n s, and the bottom of the lis t can be elim inated. The c u t-o ff point r e q u ir e s an a r b i t r a r y decisio n. O r loci and M ukkattu (1973) p ro p o sed a m ethod fo r c a lc u lating the re la tiv e con trib u tio n of each sp e cie s to a c o v arian c e s tr u c tu r e . T his would c o rre s p o n d to the sp e c ie s im p o rta n c e in m ethods r e la te d to E u clid ean d ista n c e . The sp e c ie s w e re ran k e d in o r d e r of th e ir c a lc u la te d c o v a ria n c e contributions and th en d ifferen t n u m b e rs of s p e c ie s w e re e lim in ated from th e bottom of the lis t. The am ount of in fo rm a tio n lo ss (defined as " s t r e s s " ) w as c a lc u la te d a fte r each elim in atio n and plotted against the p e r centage of sp e c ie s elim in ated . The re s u ltin g c u rv e can show the eco lo g ist the am ount of in fo rm a tio n lo ss at the d ifferen t lev e ls of sp e c ie s red u ctio n , and in tu rn m ak e the decision of finding a good 156 c u t-o ff point m u ch e a s ie r . A m o d ificatio n of th is m ethod is u se d in th is se ctio n . The la s t two d a ta -re d u c tio n m etho ds m entioned a re a s s o c i ated w ith E u clid ean d istan c e in w hich th e sp e c ie s d istan ce c o n trib u tio n s a r e additive. As noted in C h ap ter IV, the B ra y -C u rtis m e a s u re is not additive and th e d ista n c e co n trib u tio n of a sp e c ie s in a site c o m p a riso n w ill depend p a rtia lly on the valu es for the o th er sp e c ie s involved in th e sa m e c o m p a riso n . C onsequently, the s p e c ie s -d is ta n c e c o n trib u tio n s c a lc u la te d by the m ethod u se d in C h ap ter IV (sectio n B. 4) w ill only apply to the data m a trix from w hich they a re c alcu lated . When sp e c ie s a re being s u c c e s s iv e ly elim in ated , th e data m a trix is changing c o n stan tly and the s p e c ie s - d ista n c e c o n trib u tio n s m u st be re c a lc u la te d w ith each sp e c ie s elim in atio n . W hen th e r e a re a la rg e n u m b er of sp e c ie s, as in th is study, the com putation tim e r e q u ir e d fo r th is p ro c e ss is fo rm id a b le . For th is re a s o n it is not re a so n a b le to attem p t to elim in ate sp e c ie s a cc o rd in g to th is p ro c e d u re . H ow ever, it is p o ssib le that the r e calcu latio n s a r e la rg e ly u n n e c e s s a ry each tim e a sp e c ie s is e lim in ated , sin c e th e ran k in g of d istan ce c o n trib u tio n s m ay not change even though th e d istan c e co n trib u tio n s m ay change so m e w hat. To te s t th is, the u n sta n d a rd iz e d s p e c ie s -d is ta n c e c o n trib u tions (henceforth c alle d to ta l d istan ce) fo r a ll sp e c ie s a re c alcu late d 157 and te s te d . In addition, e lim in atio n lis ts b a se d on so m e e asily - c alc u la te d population p a ra m e te r s a re c o n stru c te d . The p e r fo rm a n c e s of the v a rio u s lis ts a re c o m p a re d and ev alu ated using five re p re s e n ta tiv e d ata ty p es. The m ethods u se d in th is c h a p te r also can be u se d to show the am ount of in fo rm a tio n lo st by using few er re p lic a te s than the to ta l n u m b er tak en at each site . This technique is d e m o n stra te d below w ith the p re s e n t data. B. M ethods 1. P opulation p a ra m e te r s and d ata ty p es The d a ta ty p es to be te s te d a re the sp e c ie s m ean, sp e c ie s to ta l, site to tal, cube ro o t, and b in a ry . F o r each p a ra m e te r , a lis t is com piled w ith the sp e c ie s in ra n k o rd e r a c c o rd in g to that p a ra m e te r. The sp e c ie s show ing the h ig h est p a ra m e te r value is at th e top of the lis t. The population p a ra m e te rs u se d as c r i t e r i a fo r d a ta elim in atio n a re the n u m b er of o c c u rre n c e s , the to ta l s q u a re -ro o t abundance, and the s q u a r e - r o o t abundance p e r o c c u rre n c e . The s q u a r e - r o o t tra n s fo rm a tio n w as applied to th e data b efore c alcu latio n of th e se p a ra m e te r s sin ce the sta n d a rd i zatio n s being te s te d a r e p re c e d e d likew ise by a sq u a re ro o t. How e v e r, a few te s t tr ia ls show ed that the r e s u lts a re about the sam e 158 w ith or without the s q u a r e - r o o t tra n s fo rm a tio n . As a co n tro l, a lis t of sp e c ie s in ran d o m o rd e r also is u se d for data re d u c tio n and c o m p a re d w ith the o th er c r i t e r i a . As m en tio n ed above, a to ta l- d istance list a lso is te s te d . 2. C alculation of in fo rm a tio n lo ss The f ir s t ste p involves calcu latio n of the in te r - s ite d istan c e m a trix u sin g a ll 222 sp e c ie s. Given a lis t of sp e c ie s in o rd e r of elim in atio n (with the la s t to be e lim in a te d at the top of the list), a c e r ta in n u m b er of sp e c ie s c o rre sp o n d in g to the bottom p a rt of the lis t a re re m o v e d fro m the data m a trix and the d istance m a trix is re c a lc u la te d . The changes in the d istan c e m a trix (com p a re d to the o rig in a l d istan c e m a trix ) a re c o n sid e re d in fo rm atio n lo ss (or " s tr e s s " ) due to th e re m o v a l of the s p e c ie s . The p e r cent in fo rm a tio n lo ss is c a lc u la te d as 100 x (1 - r^ ), w h e re r is the p ro d u c t-m o m e n t c o rr e la tio n betw een th e o rig in a l and the s u b s e quent d istan ce m a trix (O rloci and M ukkattu, 1973). In th is c a se , r2 is a m e a s u re of the s im ila r ity betw een th e tw o d istan ce m a t r i c e s and th is value s u b tra c te d fro m one would r e p r e s e n t the am ount of change. See Steel and T o r r ie (1960:18 7) for an exp Ian- o at ion of the use of r in ste a d of r . T his p ro c e s s is continued w ith the elim in atio n of m o re sp e c ie s and the c alcu latio n of a new p e r cent in fo rm a tio n lo ss value w ith each elim in atio n . 159 F o r each lis t and each d ata type, 10 su c c e ss iv e d ata e lim in a tio n s a re p e rfo rm e d , and the n u m b e r and p e r cent of the sp e c ie s re m a in in g a fte r each elim in a tio n is in d icated in the r e s u lts (F ig. 5-1). 3. A s s e s s m e n t of the p e rfo rm a n c e of the c r i t e r i a for data elim in atio n F o r a given data type, the m o st d e sira b le seq u en ce for sp e c ie s elim in atio n would be one w hich is a sso c ia te d w ith the le a s t am ount of to ta l in fo rm a tio n lo ss. Such a sequence would allow a m ax im u m am ount of data re d u c tio n w ith m in im a l change in the r e s u lts . H ow ever, the to ta l in fo rm a tio n lo ss m ay be of in te re s t only o v e r a c e r ta in ra n g e of lo ss v a lu e s. F o r in stan c e, one m ay w ant only to c o m p a re p e r cent in fo rm a tio n -lo s s v alu es below the 20 p e r cent lev el, sin c e lo ss above th is value m ay.be c o n sid e re d too high to to le ra te in the a n a ly sis, and the p e rfo rm a n c e of the lis ts fo r e lim in a tio n would becom e im m a te r ia l above th is ra n g e . Since the in fo rm a tio n -lo s s p ro file s of the d ifferen t data ty p es v a ry c o n sid era b ly , the elim in atio n lis ts a re a s s e s s e d o v e r v a ry in g ra n g e s acco rd in g to the g e n e ra l c h a r a c te r is tic s of the r e s u lts fo r e ac h data type. The c rite r io n of a s s e s s m e n t is the to ta l p e r cent in fo rm a tio n lo ss over the ch o sen ran g e of elim in atio n (which is held co n stan t fo r each d ata type). T his is c alcu lated by su m m in g each lo ss value in the ran g e of in te re s t. 160 4. A s s e s s m e n t o f the effects on the d istan c e m a tr ix of using d ifferen t n u m b e rs of r e p lic a te s at each site In the p re s e n t c ase , four r e p lic a te s w e re tak en at each s ite . It is a ssu m e d th at a) the m axim um am ount of available in fo rm a tio n is obtained by u sin g all four r e p lic a te s in the d istance c a lc u la tio n s, and b) the re s u ltin g d istan ce m a tr ix is a sso c ia te d w ith z e ro in fo rm a tio n lo s s . The effect of u sin g only th re e of the four r e p lic a te s can be show n by c alcu latin g a d istan ce m a trix b a se d on th re e of the re p lic a te s , and then finding the c o rre la tio n betw een th is m a trix and one involving four r e p lic a te s . The in fo rm a tion lo ss is c a lc u la te d fro m th is c o rre la tio n as d isc u sse d above. T his p r o c e s s then is re p e a te d for two r e p lic a te s and fo r a single s a m p le . R andom se le c tio n is u se d to d e te rm in e w hich of the r e p li c a te s a re included in each set of c a lc u la tio n s . D istance m a tric e s a re u sed in both th e n o rm a l and in v e rs e a n a ly se s (see C hapter VII), and the a sso c ia te d in fo rm a tio n lo ss is c alc u la te d fo r both c a s e s . C. R e su lts and D isc u ssio n 1. The effects of sp e c ie s e lim in atio n The to ta l in fo rm a tio n lo ss fo r each data type and each elim in atio n c r ite r io n is shown in T able 5-1. F o r each data type, the ran g e c o n sid e re d is in d icated by the m ax im u m n u m b er of sp e c ie s e lim in ated , and the to ta l p e r cent in fo rm a tio n lo ss is 161 L O L O LO CO CD CO CO CO CO tu o r-H CM -a r4 CO I — I CD CO CO CM lO o: o CO r— I CO LO CO 0 3 = tt= 1 -4 o CO lO CM LO CO CO CO CD O CO CO CD CM CM CO CM CO CM CO CM O CO -a CM O CM CO CM T — ( oa rH + j nJ O lO o LO o CO o LO a oa + -> rH r— I rH rH LO 162 ra n k e d w ith the b e st p e rfo rm in g (low est to ta l lo ss) c rite rio n ran k e d f ir s t. The n u m b er of o c c u rre n c e s is the only p a ra m e te r te s te d for the b in a ry data sin ce h e re the to ta l abundance w ill equal the n u m b er of o c c u rre n c e s and the abundance p e r o c c u rre n c e w ill be unity fo r a ll sp e c ie s. The b e st c r ite r io n fo r e lim in a tio n v a rie s w ith the data type. W ith s ite - to ta l and s p e c ie s - m e a n data, the b est is the total-abundance m e a s u r e and w ith the cube ro o t it is the abundance p e r o c c u rre n c e . E xcept for the sp e c ie s to tal, the to ta l-d is ta n c e m e a s u re fa re s p o o rly w hen c o m p a re d w ith the o th er p a ra m e te r s . E vidently re c a lc u la tio n s a re re q u ire d w hen building the s p e c ie s - d istan ce e lim in a tio n lis t. O th erw ise, a d ire c t m e a s u re of the d istan ce su c h as th is would be expected to show the b e st p e r fo rm a n c e . W ith the s p e c ie s - to ta l data, the to ta l-d is ta n c e m e a s u re show s the b e st p e rfo rm a n c e . T his is not s u rp ris in g in light of the fact th a t th e sta n d a rd iz e d d ista n c e s for th is data type a re n eg ativ ely c o r r e la te d w ith th e o th er p a ra m e te r s u se d h e re (see T able 4-2). H ow ever, the s p e c ie s -d is ta n c e lis t s till would be suboptim al; it only show s the b e st p e rfo rm a n c e b e ca u se the o th e r p a ra m e te r s a re u n su itab le for th is type of data. 163 All th e p a ra m e te r s te s te d p e rfo rm b e tte r than the ran d o m lis t, although w ith the s p e c ie s - to ta l d a ta the im p ro v e m e n t is not v e ry im p re s s iv e . In F ig . 5-1, the b e st in fo rm a tio n -lo s s pro file for each d ata type is plotted. As can be seen , the n u m b er of sp e c ie s that can be e lim in a te d w ithout undue d isto rtio n of the r e s u lts w ill v a ry w ith th e d ata type u sed . If the am ount of in fo rm a tio n lo ss that could be to le ra te d w ithout a significant change in p a tte rn would be about five p e r cent, it c an be se e n from F ig. 5-1 th at the s ite - to ta l d ata w ould u tilize about 33 sp e c ie s, the c u b e -ro o t data about 67 sp e c ie s, the s p e c ie s -m e a n data about 78 s p e c ie s , the b in a ry data about 115 sp e c ie s, and the s p e c ie s -to ta l d ata about 135 sp e c ie s. W ith the s ite - to ta l and c u b e -ro o t data, the m o re abundant sp e c ie s dom inate the a n aly sis (see T able 4-2) and the r e m o v al of the m any le ss-a b u n d a n t sp e c ie s does not a lte r the p a tte rn of th e r e s u lts . T his is tru e e sp e c ia lly w ith the s ite - to ta l data w h ere the lo ss w ith only five sp e c ie s is s till le s s th an 20 p e r cent. The s p e c ie s -m e a n and b in a ry data a re le s s dom inated by the abundant sp e c ie s and thus m o re sp e c ie s w ill c o n trib u te to th e p a tte rn . The s p e c ie s -to ta l p a tte rn would be expected to be b ased upon the m o st sp e c ie s sin c e the r a r e sp e c ie s a re given re la tiv e ly g r e a te r w eight (see T ab le 4-2), and th e re a re m any r a r e sp e c ie s in the to ta l d ata m a trix . 164 F ig . 5-1. The b e st in fo rm a tio n -lo s s pro file for each d a ta ty p e . The c rite rio n for sp e c ie s elim in atio n is in d icated in p a re n th e s e s. 165 ■ d ( U Q . 0 ( h Q > x > 1 o o o o c o f f i w c o o co c o co 00 C 3 l O o co o o o o o 0 0 o o o 13 ( U I C m ( U ü ( U & OOT X ( g j - T) s SOI uo T i-em ao jtri % 166 2. The effects of sa m p le re p lic a tio n F ig u re 5-2 show s the in fo rm a tio n lo ss a sso c ia te d w ith the d ifferen t n u m b er of re p lic a te s in both the n o rm a l and in v e rs e a n a ly se s. In the n o rm a l a n a ly sis, about 70 p e r cent of the in fo rm a tio n contained in all four r e p lic a te s is contained in ju st one of the r e p lic a te s . E ach additional re p lic a te included adds an in c re a s in g ly s m a lle r p e rc e n ta g e of th e to ta l in fo rm a tio n contained in all four r e p lic a te s . F o r exam ple, only a five p e r cent gain in in fo rm a tio n is re a liz e d by u sin g four in ste a d of th re e r e p lic a te s . The in v e rs e a n aly sis a p p e a rs to be m o re se n sitiv e to the lev el of re p lic a tio n . A single r e p lic a te contains about 55 p e r cent of th e to ta l in fo rm a tio n in four re p lic a te s , and the fo u rth r e p lic a te s till adds about 10 p e r cent to th e to ta l in fo rm atio n . When c o n sid e rin g a n a ly se s b a se d on eco lo g ical d is ta n c e s , th e s e r e s u lts in d ic a te that: a) A la rg e am ount of th e to ta l in fo rm a tio n is co n tain ed in a single re p lic a te . T hus, in so m e c a s e s , r e s u lts b a sed on one or two r e p lic a te s m ay not differ ap p re cia b ly fro m r e s u lts b a se d on s e v e r a l re p lic a te s . b) If the a n a ly sis is to e m p h a size the study of sp e c ie s gro u p s (in v e rse a n aly sis), m o re re p lic a tio n m ay be n e c e s s a r y than w ould be the c a s e w ith the n o rm a l a n a ly sis. 167 F ig . 5-2. The in fo rm a tio n lo ss a s s o c ia te d w ith different n u m b ers of sa m p le re p lic a te s at each site . The d istance m a tric e s fo r the n o rm a l an aly sis a re b a se d on s p e c ie s -m e a n s ta n d a rd iz e d data w ith a p rio r s q u a r e -r o o t tra n s fo rm a tio n , and the d ista n c e s for the in v e rs e a n a ly se s a re b a sed on s p e c ie s-m a x im u m sta n d a rd iz e d d ata (see C hapter Vll) w ith a p r io r s q u a r e -r o o t tra n s fo rm a tio n . 168 0 0 C M O o o o o o o o o o o C Q •4-3 rt Ü d 0 ) = tf c O i CO CD lD rjH r o C S J 880% IX O T ^ B U ia O J U I % 169 It should be noted that the in fo rm a tio n lo ss as d is c u s s e d h e re is only a re la tiv e te r m . To d e te rm in e the absolute am ount of in fo rm a tio n lo ss involved in the p re s e n t data, m o re r e p lic a te s than four would be n e c e s s a r y . The n u m b er of re p lic a te s a s s o c ia te d w ith the point w h ere th e c u rv e beco m es n e a rly p a ra lle l to the a b s c is s a would be c o n sid e re d the m in im u m point of in fo rm a tion lo ss in an absolute se n se . T his technique can be valuable e sp e c ia lly in p r e lim in a ry su rv e y w ork. S e v e ra l re p lic a te s can be taken at each site of in te r e s t and the re s u ltin g in fo rm a tio n -lo s s c u rv e s (e. g . , F ig . 5-2) can be c o n stru c te d . The c o st and effort involved w ith each additional re p lic a te can then be b alan ced against the am ount of additional in fo rm a tio n to be re a liz e d . T his w ill allow the eco lo g ist to d e te rm in e the m o st e co n o m ic al lev e l of sa m p le re p lic a tio n in su b seq u en t su rv e y s of the sa m e a re a . 170 CHAPTER VI RELATIONSHIPS B ETW EEN THE SPEC IES AND THE RESU LTS OF THE NORMAL ANALYSIS A. In tro d u ctio n The re la tio n s h ip s betw een the sp e c ie s and s ite groups from the c la ss ific a tio n r e s u lts a re often disp lay ed by the u s e of tw o-w ay coinciden ce ta b le s (Kikkawa, 1968; Stephenson and W illiam s, 1971). T his sim p ly c o n s is ts of a r e a r r a n g e d data m a trix , w ith the colum ns (each re p r e s e n tin g a site) a rra n g e d in the sa m e o r d e r as they o c cu r on the d e n d ro g ra m of the n o rm a l a n a ly sis, and w ith the ro w s (each r e p r e s e n tin g a sp e cie s) in the sa m e o r d e r as they o ccu r on a d e n d ro g ra m of an in v e rs e a n aly sis (C h ap ter VII). Since the s im ila r s ite s and sp e c ie s w ill be grouped to g e th e r on the re s p e c tiv e d e n d ro g ra m s, the ro w s and colum ns of the tw o-w ay tab le w ill con s i s t of blocks of s im ila r s ite s and s p e c ie s . T his m a k e s it e a s ie r to se e the "co in cid en ce" of the site and sp e c ie s p a tte rn s . Since the tw o -w ay ta b le can becom e la rg e and c u m b e rso m e when n u m ero u s sp e c ie s and s ite s a re involved, a few m od ifications of the u su a l p ro c e d u re a re p ro p o sed in th is se ctio n . The firs t m o d ificatio n involves s u m m a riz a tio n of the c e lls of the tw o-w ay 171 tab le. The c e lls a re the gro u p s of e n trie s w h e re the site and sp e cie s group s co incide. The second p ro p o sa l involves the u se of sy m bols in s te a d of n u m b e rs in the body of the tw o-w ay tab le. T w o-w ay ta b le s a lso can be u sefu l in re la tin g the sp e c ie s to the o rd in atio n r e s u lts . C olum ns (sites) a re o rd e re d acc o rd in g to th e ir re s p e c tiv e s c o r e s on an axis of in te re s t, and row s (sp e cie s) a re o rd e r e d acc o rd in g to a w e ig h te d -a v e ra g e sc o re c a lc u la te d from the sp e c ie s abundances at each site and th e c o rre sp o n d in g site s c o re on the axis in question. T h ese w eighted a v e ra g e s fo r each sp e c ie s can be plotted also in o rd in atio n sp ace. The abundance of each sp e c ie s at each site also can be plotted in the o rd in atio n space (A ustin, 1968). T his can be v e ry in fo rm a tiv e , but has the d raw b ack th a t a s e p a ra te plot is u su a lly r e q u ir e d fo r each sp e c ie s. T his ap p ro ach w ould be im p ra c tic a l c o n sid e rin g the la rg e n u m b er of sp e c ie s in the p re s e n t study, and is not p u rsu e d fu rth e r. B. M ethods 1. C la ssific a tio n tw o-w ay ta b le s The c o n stru c tio n of a tw o-w ay tab le is b est illu s tra te d w ith a th e o r e tic a l exam ple. A site c la ss ific a tio n of the five site s in F ig . 6 - lc is shown in Fig. 6 - la , and a sp e c ie s c la ss ific a tio n of the sa m e d ata is shown in F ig. 6 - lb . By r e a rra n g in g the colum ns 172 The c o n stru c tio n of a tw o-w ay coincidence tab le. S ite g ro u p 2 S p e c ie s g ro u p 1 S p e c ie s g ro u p 2 2 3 4 1 5 $ * o a- B C A D S ites a. S ite c la s s if ic a tio n A B C D b. S p e c ie s c la s s if ic a tio n c. H y p o th e tic a l d a ta m a tr ix S ite s to 0 1 B C A D d. T w o -w ay ta b le 174 and row s of Pig. 6 - l c to c o rre sp o n d w ith th e se d e n d ro g ra m s, a tw o-w ay tab le su c h as that shown in Fig. ‘ 6 - Id r e s u l ts . It is now e a sy to s e e why the s ite s and sp e c ie s c lu s te re d the w ay they did. F o r exam ple, s ite s 2, 3, and 4 (site group 1) a re alike in that they con tain re la tiv e ly h ig h er counts of sp e c ie s B and C and low er counts of sp e c ie s A and D. C o n v ersely , sp e c ie s B and C c lu s te re d due to th e fact th at they occu r to g e th e r in h ig h er n u m b e rs in the f ir s t site group and in low er n u m b e rs in the second site group. The lin es a re draw n on the tw o-w ay tab le to delim it the ch o sen site and sp e c ie s group s. The e n trie s e n clo se d by the lin e s a re c o n sid e re d in a "cell, " w h ere a p a rtic u la r site and sp e c ie s group coincide. 2. C la ssific a tio n tw o-w ay table s u m m a r ie s The siz e and com plexity of the tw o-w ay tab le could be c o n sid e ra b ly red u c e d by su m m a riz in g each c e ll w ith a single n u m b e r w hich w ould r e p r e s e n t the g e n e ra l n a tu re of the c e ll con te n ts. One such m ethod, u sed by Webb et al. (1967), Kikkawa (1968), W illiam s et al. (1969), and M oore (1973), involves the use of the c e ll "density, " w hich is the p e rc en ta g e of the c e ll e n trie s w ith v alu es g r e a te r than z e ro . T his m ethod ig n o re s the quantita tive in fo rm a tio n in the d ata m a trix , but sin c e the n u m b er of o c c u rre n c e s often a re c o rre la te d w ith the g e n e ra l abundance of a 175 s p e c ie s (for exam ple, se e T ab le 4-1), the m ethod m ay give r e s u lts s im ila r to a s u m m a ry w hich is b a se d on abundances. A second m ethod, u se d by Stephenson and W illiam s (1971) and W illiam s and Stephenson (1973), entails the c alcu latio n of the m e a n abundance p e r ele m e n t in each cell. In m o st c a s e s , e ith e r one of th e s e m ethods w ill s u m m a riz e ad equately the tw o-w ay tab le but, in so m e c a s e s , m is le a d in g re s u lts m ay o c c u r. C o n sid e r, for exam ple, the c e lls show n in T able 6-1. In the c a se of both th e c e ll-d e n s ity and m ea n - abundance m e a s u re s , it a p p e a rs that sp e c ie s group 2 is b e tte r r e p r e s e n te d in site group 1 sin c e th e c e ll s u m m a rie s fo r sp e c ie s g roup 2 a re h ig h e r. The h ig h er values of sp e c ie s group 2, how ever, a re due only to the p re s e n c e of so m e sp e c ie s w ith z e ro e n trie s in s p e c ie s group 1. T his m a k e s it p e rilo u s to c o m p a re the re la tiv e am ount of the d ifferent s p e c ie s group s at a site group, sin c e the in clu sio n of r a r e o r in freq u en t sp e c ie s in a c e ll can c an c e l out the effect of the m o re abundant sp e c ie s in the sa m e cell. T his can tak e place even if the r a r e r sp e c ie s a re at th e ir m axim um o c c u r re n c e s and abundances in the cell. A solution to the p ro b lem is to u se the to ta l c e ll abundance. In th is c a se , how ever, it would be difficult to c o m p a re th e re la tiv e am ount of a sp e c ie s group in the v a rio u s site g roups. 176 CO CO CO CO LO to -M to I - H to I -H CM o o to I- -- -I lO o LO o C L CM CM CM tUO’—I O o CM CM CM CM m to a I — I m 177 sin c e th e re no doubt w ill be d ifferen t n u m b e rs of s ite s in m any of the s ite g ro u p s. C ells c o rre sp o n d in g to th e l a r g e r s ite groups can accu m u late po ten tially la r g e r abundance to ta ls . T his p ro b lem can be solved by sta n d a rd iz in g each c e ll to ta l abundance by the num ber of s ite s in the c o rre sp o n d in g site group. It m ay be b e s t to tr a n s fo r m a n d /o r sta n d a rd iz e the d ata m a trix b efo re s u m m a riz in g the c e lls in th is m a n n e r. With raw data, the la r g e r num bers m ay dom inate the c alcu latio n s c o m p letely . When the m ain in te re s t lie s in the sp e c ie s m a k e -u p of the v a rio u s site g ro u p s, the sa m e data type u se d in th e n o rm a l an aly sis could be u sed . T his would give each sp e c ie s the sa m e w eighting it re c e iv e d in the a n aly sis fro m w hich the site groups w e re g e n erate d . When the m ain in te r e s t is in the d istrib u tio n of each sp e c ie s group throughout the s ite s , the sa m e data type u sed in th e in v e rs e an aly sis m a y be m o st useful. When th e re is in te r e s t in both the s ite - and s p e c ie s -g ro u p p a tte rn s , it m a y be n e c e s s a r y to c o n stru c t two s u m m a riz a tio n ta b le s if d ifferen t d ata ty p es a re u se d in the n o rm a l and in v e rs e a n a ly se s. F inally, the su m m a riz a tio n tab les u su a lly w ill be e a s ie r to in te r p re t when th e v alu es a re co n v erted to p e rc e n ta g e s . The n u m b e rs in the s u m m a ry tab le em p h asizin g the s ite groups would be c o n v erted to th e p e rc e n ta g e of the re s p e c tiv e colum n 178 to ta ls , and e n trie s in th e tab le e m p h a sizin g the sp e c ie s groups would be e x p re s s e d as p e rc e n ta g e s of the re s p e c tiv e row to ta ls. T h is ap p ro ach of tw o -w ay tab le s u m m a riz a tio n is d isc u sse d and d e m o n s tra te d in Stephenson et al. (197 5). 3. Two-way ta b le s u sing s y m bols in s te a d of n u m b ers A nother m eth od of red u c in g the p h y sic al siz e and co m p lex ity of th e tw o-w ay ta b le s is th ro u g h th e u se of sym bols in the body of the ta b le in ste a d of n u m b e rs . A sin g le sy m b o l u su a lly w ill re q u ire le s s sp a ce than a n u m b er, and no sp acin g is re q u ire d betw een sy m b o ls as is the c a se w ith n u m b e rs . To u se th is m ethod, a ll sp e c ie s abundance counts m u s t be c o n v e rte d to a com m on sc a le w hich can be r e p re s e n te d by a re a so n a b ly s m a ll n u m b er of sy m b o ls. T his would re q u ire a sp e c ie s sta n d a rd iz a tio n such as a m ax im u m o r a m ea n sta n d a rd iz a tion p rio r to c o n v e rsio n to sy m b o ls. Both of th e se sta n d a rd iz a tio n s have an id en tifia b le and in te r p re ta b le point on th e ir sc a le w hich o c c u rs for e v e ry s p e c ie s . T his point is the sta n d a rd iz e d value of one, w hich w ill r e p r e s e n t th e m axim um o r m e a n value, resp ectiv ely , for each sp e c ie s. The ra n g e of the sta n d a rd iz e d v alu es is divided into in te rv a ls and a sp e cific sym bol i s a ssig n e d to r e p r e s e n t each su ch in te rv a l. If the ra n g e of sta n d a rd iz e d valu es is p red ic ta b le, it 179 b e c o m e s u n n e c e s s a ry to s e a r c h the sta n d a rd iz e d data m a trix fo r the ra n g e of v a lu e s. In stead , the sy m b o ls and the in te rv a ls they r e p r e s e n t can be se t up p rio r to the a n a ly sis. The ran g e of sp e c ie s -m a x im u m sta n d a rd iz e d v a lu e s u su ally w ill be betw een z e ro and one. The lac k of c o n stra in t a sso c ia te d with the s p e c ie s - m e a n sta n d a rd iz a tio n can be c o n tro lle d by a tra n s fo rm a tio n p rio r to the sta n d a rd iz a tio n , and the ra n g e of v alu es w ill becom e m o re p re d ic ta b le . The m a in advantage of th is m ethod when c o m p a re d to the c e ll- s u m m a r y m ethod lie s in the fact that the in fo rm a tio n fo r a ll the s e p a ra te s ite s and sp e c ie s is p re s e rv e d . It is ju st th is in fo rm a tio n w hich n o rm a lly is r e q u ir e d to decide w h e re to d e lim it the gro u p s in th e f ir s t p lace (with th e help of the d e n d ro g ra m s, of c o u rs e ). 4. O rdination two-way ta b le s and w eighted - a v e ra g e s c o r e s T w o-w ay ta b le s also can be v e ry u se fu l in illu s tra tin g the re la tio n s h ip s betw een the s p e c ie s p a tte rn s and the s ite s c o r e s on the v a rio u s o rd in atio n a x es. The seq u en ce of colum ns (site s) in su ch a tab le w ill sim p ly c o rre s p o n d to the ra n k o rd e r of the s ite s c o r e s fo r a cho sen axis. The seq u en ce of row s (sp e cie s) w ill depend upon the ra n k o rd e r of the sp e c ie s acc o rd in g to so m e 180 m e a s u re of the a v erag e axis s c o re of the s ite s at w hich each s p e c ie s o c c u rs . Such a m e a s u re would be as follow s : n Si = , (6-1) w h e re S j^ is the w e ig h te d -a v e ra g e s c o re fo r s p e c ie s i , Aj is the s c o re fo r s ite j on the axis of in te r e s t, Wj^j is so m e m e a s u re of the re la tiv e abundance of sp e c ie s i in site j , and n is the n u m b er of s ite s . The m e a s u r e of abundance (W) could be the raw abundance counts, although a tra n s fo rm a tio n would be advisable u su a lly to avoid d isto rtio n s w hich m ight be c a u se d by a few e x tre m e v a lu e s . F o r each ax is of in te re s t, a s e p a r a te tw o-w ay tab le m u st be c o n stru c te d . The r e s u lts of two or th r e e axes can be su m m a riz e d in one fig u re by plotting the w e ig h te d -a v e ra g e s c o re s fo r each sp e c ie s, w ith the d im en sio n s of the plot re p re s e n tin g the w e ig h te d -a v e ra g e sp e c ie s s c o r e s c o rre sp o n d in g to d ifferen t axes. Such plots c a n be r e la te d d ire c tly to the n o rm a l o rd in atio n space, sin ce the s p e c ie s s c o r e s a re c alcu late d fro m th e site s c o re s . One m ethod of o rd in atio n c a lle d r e c ip r o c a l av eraging (Hill, 1973; M oore, 1974) o r c o rre sp o n d e n c e a n a ly sis (Melguen, 181 1974) involves sim u lta n eo u s c alc u la tio n of s c o r e s for both the s ite s and the sp e c ie s . H ere the sp e c ie s s c o r e s would be c o m p a ra b le to the sp e c ie s w eighted a v e ra g e s (eq. 6-1). The sp e c ie s s c o r e s fro m the r e c ip ro c a l-a v e ra g in g m ethod could be ra n k e d and u se d in the c o n stru c tio n of an o rd in atio n tw o-w ay ta b le in the sa m e m a n n e r as is done w ith the w eighted a v e ra g e s. It should be noted th at th is m ethod of tw o-w ay table c o n stru c tio n could be u se d w ith any o th er technique from w hich site o r sp e c ie s s c o r e s a re g e n e ra te d . F o r exam ple, the site s c o r e s fro m a d isc rim in a n t a n aly sis could be u se d to build su ch tw o -w ay ta b le s . 182 CHAPTER VII ANALYSIS OP THE SPEC IES USING CLASSIFICATION AND W EIGHTED DISCRIMINANT ANALYSIS A. In tro d u ctio n The p u rp o se h e re is to d isc u ss m ethod s of d ire c tly re la tin g the sp e c ie s p a tte rn s to the en v iro n m en t. The a n a ly sis can be c a r r i e d out at two levels : 1) The s p e c ie s groups as defined in the in v e rs e c la ss ific a tio n can be analyzed, and 2) a n u m b er of in d i vid ual sp e c ie s can be analyzed to g e th e r. The sp e c ie s u se d in the la tte r type of a n a ly sis can be cho sen by th e eco lo g ist or, as is d e m o n s tra te d in C h ap ter VIII, the sp e c ie s w ithin each sp e cie s group can be an aly zed to g eth er. Individual sp e c ie s have been re la te d su c c e ss fu lly to e n v iro n m e n ta l p a tte rn s u sin g unm odified m u ltip le d isc rim in a n t an aly sis by G reen (1971, 1972, 1974). M odifications of the m ethod p ro p o sed h e re should im p ro v e the pow er, a cc u ra c y , and app licab ility of the d isc rim in a n t a n a ly sis, e sp e c ia lly w hen sp e c ie s show ing a la rg e am ount of o v e rla p or co m p lete o v e rla p a re included (as is the c a se in the p re s e n t data). In th e in v e rs e c la ssific a tio n , c e r ta in m odification s in the 183 c alcu latio n s of the in te r - s p e c ie s d istan c es a lso a re p ro p o sed . The p u rp o se h e re is to a lle v ia te so m e of the p o ten tial d isto rtio n s c au sed by the sa m p lin g p a tte rn . B. M ethods 1. The c alcu latio n of the in te r - s p e c ie s d ista n c e s fo r in v e rs e c la ss ific a tio n a) D efinition of d istan ce, s im ila r ity and o v erlap The e co lo g ic al d istan ce betw een each p a ir of sp e c ie s w ill be b a se d on a c rite r io n of the re la tiv e am ount of o v e r lap betw een the s p e c ie s . O verlap, in tu rn , is defined as the extent to w hich the sp e c ie s being c o m p a red s h a re a given s e t of h a b itats. The w o rd s "d istan ce, " " s im ila rity , " o r "o v e rla p " w ill be u sed in te rch a n g ea b ly in th e follow ing d isc u ssio n s. S im ila rity and o v e r lap a re c o n sid e re d synonym ous, and a s im ila r ity m e a s u r e can be c o n v erted e a sily to a d ista n c e m e a s u r e by su b tra c tio n from an a p p ro p ria te co n stan t (see C h ap ter II, sectio n on B ra y -C u rtis index). T h e re a re both qualitative and quantitative co m ponents in th e o v e rla p . F o r exam ple, the two p a ir s of sp e c ie s in F ig. 7 - l a , b have an id e n tic a l am ount of o v e rla p in a qualitative s e n se , sin c e in both c a s e s the habitat of one sp e c ie s is wholly contained w ithin the h ab itat of the o th er; th u s, th ey com m only sh a re th e sa m e p e rc e n ta g e of the habitat. The sp e c ie s p a ir in Fig. 7 - la . 184 F ig . 7-1. H ypothetical c u rv e s d e m o n stra tin g the q u alitativ e and quantitative com ponents of in te r - s p e c ie s o v e rla p . 185 0 ) § s < E n v iro n m e n ta l g ra d ie n t — ► b. E n v iro n m e n ta l g ra d ie n t c. E n v iro n m e n ta l g ra d ie n t 186 h o w ev er, show s g r e a te r o v e rla p (than th at in Fig. 7 - lb) in a quanti tativ e s e n se , sin c e g r e a te r p ro p o rtio n s of the individuals of each sp e c ie s o c c u r in th e sa m e h ab itat. An a n aly sis b ased so le ly on the q u alitativ e data (p re s e n c e -a b s e n c e ) w ould of c o u rs e fail to detect th is quantitative com ponent. F o r th is re a so n , an an aly sis w ith su ch data is not c o n sid e re d s e rio u s ly any fu rth e r as a b a s is for c a lc u lating the in te r - s p e c ie s d ista n c e s. F inally, the sp e c ie s p a ir in F ig . 7 - l c , in both a qualitative and quantitative se n se , show s le s s o v e rla p th an th e o th e r two sp e c ie s p a ir s in th e fig u re. b) The c alcu latio n of in te r-sp e c ific overlap In th e lite ra tu r e th e re a re s e v e r a l in d ices using s p e c ie s abundances that have been p ro p o se d to m e a s u re sp e c ie s o v e rla p . E x am p le s a re th o se b a se d on in fo rm a tio n th e o ry (Horn, 1966; C olw ell and F utuy m a, 1971; P ielo u , 1972), p ro b ab ility th e o ry (M o risita, 1959a, b; Lloyd, 1967), or so m e sim p le d istan ce o r s im ila r ity index applied to s p e c ie s -s ta n d a rd iz e d counts (Schoener, 1970; the B r a y -C u rtis index; the C a n b e r r a - m e tr ic index). The c la ss ific a tio n m eth o d u se d h e re r e q u ir e s that th e o v e rla p betw een a p a ir of sp e c ie s is s y m m e tric a l, i. e. , the o v e rla p of sp e c ie s A w ith sp e c ie s B is equal to the o v erlap of sp e c ie s B w ith sp e c ie s A. An exam ple of an a sy m m e tric index 187 is that p ro p o sed by L evins (1968), w hich is only s y m m e tric a l when the "b rea d th " of the two sp e c ie s being c o m p a re d is equal. In the p re s e n t study, each site is c o n sid e re d as a s e p a ra te h ab itat. T h ere a r e so m e p ro b le m s w ith th is appro ach sin c e the sa m e o r v e ry s im ila r h a b itats can be sa m p le d re p e a te d ly w hile o th e rs can be re la tiv e ly u n d e r-s a m p le d . T his w ill lead to d isto rtio n s in th e c alcu latio n s but c o rre c tiv e p ro c e d u re s to be d is c u sse d below can be applied. The m eth ods u se d h e re w ill at le a s t m e a s u re the re la tiv e am ount of in te r - s p e c ie s h ab itat o v erlap in the a r e a sam p led . The c o rre la tio n coefficient often is u se d in conjunc tion w ith c la ss ific a tio n (E beling et al. , 1970; L ittle r and M u rray , 1975), fa c to r a n a ly sis (Lie and Kelley, 1970; A ngel and F a sh a m , 1973; P a r k e r , 1975), or PCA (C a ssie and M ichael, 1968) as a m e a n s of d elim itin g "co m m u n itie s" or groups of sp e c ie s sh a rin g the sa m e h a b ita ts. In th e s e c a s e s , the c o rre la tio n coefficient is u se d as if it w e re a m e a s u r e of o v erlap . It can be d e m o n stra te d th at the c o rre la tio n co efficien t is an u n stab le and in co n siste n t m e a s u r e of o v e rla p . In F ig. 7 - 2a, five id e n tic a l sp e c ie s c u rv e s a re shown w ith four of th e s e c u rv e s having v a ry in g am ounts of o v e rla p w ith th e fir s t c u rv e . T able 7-1 show s the in te r c o rre la tio n s of the four sp e c ie s w ith sp e c ie s A. The c o rre la tio n s 188 F ig . 7-2. D e m o n stra tio n of the in sta b ility of the c o rre la tio n coefficien t as a m e a s u re of in te r - s p e c ie s o v e rla p . See T able 7-1 fo r in te r - s p e c ie s c o rre la tio n s . 189 5 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 a . F iv e id e n t ic a l s p e c i e s c u r v e s w ith v a r y in g d e g r e e s o f o v e r la p . b . F iv e n o n - id e n t ic a l s p e c i e s c u r v e s w ith v a r y in g d e g r e e s o f o v e r la p . 190 S pecies C o rre la tio n (xlQQ) P a i r s 17 Sites 27 Sites D ifference A -B +57 +50 -07 A -C -01 -20 -19 A -D -23 -44 -21 A -B -2 5 -46 -21 F -H -47 -25 +22 F -G +51 +58 +07 I - J -06 -04 -02 T able 7-1. C o rre la tio n s (xlOO) betw een so m e of the sp e c ie s c u rv e s in F ig . 7-2. 191 a re c a lc u la te d w ith 17 to ta l s ite s and again w ith 27 to ta l s ite s . Since the 10 additional s ite s a re devoid of th e se s p e c ie s , the o v e r lap should be unaffected by th e ir addition. The c o rre la tio n , how e v e r, can change quite d ra s tic a lly w ith the additional s ite s included. The am ount and d ire c tio n of the d ifferen ce depends on the am ount of o v e rla p and the c u rv e involved (see F ig. 7 - 2b, w h ere d ifferen t c u rv e s a re included). F u r th e r m o r e , a c o rre la tio n of z e ro does not in d ic ate z e ro o v e rla p . The c o rre la tio n betw een sp e c ie s A and C (with 17 s ite s ) is a p p ro x im a te ly z e ro , and th e s e two sp e c ie s a re s till o verlapping. In fact, th e r e is no se t value of c o rre la tio n w hich c o rre sp o n d s to z e ro o v erlap . S pecies p a irs A and E (Fig. 7-2a)\ F and H , and I and J (Fig. 7 -2b) show no o v e rla p but th e ir in te r c o rre la tio n s v a ry betw een -.0 4 0 and -.4 6 7 (Table 7-1), depending on th e n u m b er of s ite s u sed and the c u rv e s being c o m p a re d . Note also th at sp e c ie s I and J a re c o m p lete ly non overlap p in g , yet th ey show h ig h e r in te r - c o r r e la tio n s than o ther s p e c ie s p a irs w ith so m e o v e rla p . T h ese p ro b le m s a ris e from two s o u rc e s . 1) The c a lc u la tio n s a re influenced by the joint ab se n c e s of the sp e c ie s being c o m p a re d . The r e s u lt of th e ir in c lu sion is an in c r e a s e in the c o rr e la tio n w ithout any in c r e a s e in the a ctu al o v e rla p . 192 2) The c o rre la tio n coefficien t a ssu m e s lin e a r re la tio n s h ip s betw een the e n titie s (sp e c ie s) being c o m p a red . If b e ll-c u rv e d istrib u tio n s (Fig. 7-2) a re p re s e n t in the data, th is a ssu m p tio n is not m e t and the c o rre la tio n s beco m e le s s m eaningful. If a ll sa m p le s a re tak en fro m a n a rro w ran g e of the lo c a l e n v iro n m e n ta l g ra d ie n ts ( i . e . , w ithin the sa m e "co m m unity"), one w ould expect few z e ro s in th e biotic data m a trix , and only the m o re lin e a r, o r at le a s t m onotonie p o rtio n s of m o st of the sp e c ie s c u rv e s w ould be included. In su c h c a s e s , the c o r r e l a tion coefficient w ould be m o re u se fu l as a m e a s u r e of the re la tio n sh ip s betw een th e sp e c ie s (e d ito r's note in Goodall, 1973:146). c) The re la tio n sh ip betw een o v e rla p and raw abundance The re la tio n sh ip betw een abundance and o v erlap is illu s tr a te d in F i g s . 7-3 and 7-4. S pecies A and B a re alm o st c o m p lete ly o verlapping. H ow ever, the abundance counts of th ese two sp e c ie s d iffer g re a tly and the c o m p a ris o n of the raw n u m b ers alone w ill be m isle a d in g . Raw v alu es a re u se d to calcu late the B ra y -C u rtis d istan ce betw een the sp e c ie s and the r e s u lts a re show n in F ig . 7 - 3c. The c a lc u la te d d istan ce betw een sp e c ie s A and C is a ctu ally le s s than th a t betw een A and B . T his is due so le ly to the fact th at the abundances of sp e c ie s A and C a re on a co m p a ra b le s c a le - not due to g r e a te r o v e rla p . When a ll sp e c ie s a re 193 F ig . 7-3. R elatio n sh ip betw een raw abundance counts and calcu late d in te r - s p e c ie s o v e rla p . 194 140 120 ^ 100 80 60 40 20 8 3 5 6 2 7 1 4 E Sites a. T h re e h y p o th etica l s p e c ie s c u rv e s w ith ra w abundance counts S ites 1 2 3 4 5 6 7 8 A 25 55 100 130 100 50 12 0 B 10 20 30 30 25 15 8 0 C 0 0 20 70 120 120 70 20 b. D ata m a tr ix A B A B C X . 59 . 46 X . 65 X c. B r a y - C u rtis d ista n c e m a tr ix 195 F ig . 7-4. R e la tio n sh ip betw een s p e c ie s - to ta l sta n d a rd iz e d data and c a lc u la te d in te r - s p e c ie s o v e rla p . 196 30 .2 5 .20 . 15 . 10 . 05 2 1 3 4 5 7 6 8 E S ites a. T h re e h y p o th e tic a l c u rv e s w ith counts s ta n d a rd iz e d by s p e c ie s to ta l Sites 1 2 3 4 5 6 • 7 8 A . 06 . 13 . 23 . 30 .23 . 12 .03 .0 B . 07 . 14 . 22 .2 2 . 18 . 11 .0 6 .0 C .0 . 0 . 05 . 17 .2 9 . 29 . 17 . 05 D ata m a trix A B C A X . 10 . 46 B X . 44 C X c. B r a y - C u r tis d ista n c e m a tr ix 197 sta n d a rd iz e d to p ro p o rtio n s of th e ir to ta l abundance (s p e c ie s -to ta l sta n d a rd iz a tio n ), the d ista n c e s a re m o re c o n siste n t w ith the o v e rr lapping of th e se sp e c ie s (Fig. 7-4). H ere the effect of the s c a le of the counts for th e d ifferen t sp e c ie s is som ew hat equalized by the sta n d a rd iz a tio n . O ther sp e c ie s s ta n d a rd iz a tio n s would have a s im ila r effect, but the only sp e c ie s-m a x im u m sta n d a rd iz a tio n w ill put each s p e c ie s on a sc a le w ith an id e n tic a l u pper lim it. To m e e t th is sc a lin g p ro b lem , the calcu latio n s fo r the in fo rm a tio n th e o ry in d ices r e f e r r e d to above include a s p e c ie s - to ta l sta n d a rd iz a tio n . The p ro b a b ilistic in d ices also in clude so m e type of b u ilt-in sta n d a rd iz a tio n . The c o rre la tio n co efficien ts include a sta n d a rd iz a tio n by the sp e c ie s s ta n d a rd deviation. The " sim p le " index r e f e r r e d to in C olw ell and F utuym a (1971) and G oodall (1973), and u se d by Schoener (1970) w ill give r e s u lts id e n tic a l to th o se obtained w ith a B r a y -C u rtis index and a p rio r s p e c ie s - to ta l sta n d a rd iz a tio n of the data. When a s im ila r ity or d istan c e index (such as the B ra y -C u rtis o r C a n b e rra m e tric ) is u tilize d , it is up to the u s e r to apply an a p p ro p ria te sp e c ie s sta n d a rd iz a tio n b efo reh an d . R e su lts w ith so m e of the d ifferen t sp e c ie s sta n d a rd iz a tio n s a re c o m p a re d in the next section. 198 d) C o m p a riso n of v a rio u s m e a s u r e s of in te r-s p e c ie s ov erlap H e re d ista n c e m a tr ic e s g e n e ra te d by a v a rie ty of in d ices and sp e c ie s sta n d a rd iz a tio n s a re c o m p a re d . The indices u se d a re show n in T able 7-2. W ith the B ra y -C u rtis index, s e p a ra te a n aly se s a re ru n w ith d ata sta n d a rd iz e d by sp e c ie s total, n o rm , m ax im u m , and m e a n . B in a ry data also a re included. With the C a n b e r r a - m e tr ic index, d ata sta n d a rd iz e d by sp e c ie s m ax im um and to ta l a re included. The s p e c ie s -m e a n sta n d a rd iz a tio n is p r e ceded by a c u b e -ro o t sta n d a rd iz a tio n to c o n tro l som ew hat th e sc a le of the final sta n d a rd iz e d v a lu e s. The o th er sta n d a rd iz a tio n s a re p re c e d e d by a s q u a r e - r o o t tra n s fo rm a tio n . F o r the sa k e of u n i fo rm ity , s q u a re - ro o t tr a n s fo r m e d data a r e u sed w ith the M o risita and in fo rm a tio n in d ic es. The 70 sp e c ie s w hich o c cu r at m o re than five s ite s and a re id en tified to at le a st the g en eric lev el a re u se d h e re . F o r each of the v a rio u s m eth o d s, an in te r - s p e c ie s d ista n c e m a trix is c alcu late d , and each of th e s e d ista n c e m a tr ic e s is s ta n d a rd iz e d by to ta l (as in C h ap ter IV w ith th e in te r- site d ista n c e m a tric e s ). The e le m e n ts of the sta n d a rd iz e d d istan ce m a tr ic e s a re c o n sid e re d th en as a ttrib u te s and the m etho ds of c alcu latio n a re the e n titie s in a PCA. The r e s u lts a re show n in F ig. 7-5. 199 Index In fo rm a tio n th e o ry M o r is ita 's index (p ro b ab ilistic ) Sim ple index B r a y - C u rtis index C a n b e r r a - m e tr ic index R e fe re n c e s C olw ell and F utuym a, 1971 (Horn, 1966 fo r a s im ila r index) M o risita , 1959a, b ( Lloyd, 1967 fo r a s im ila r index) S choener, 1970; C olw ell and F utuym a, 1971; Goodall, 1973 C h ap ter II C h ap ter II A b b rev iatio n U sed in F ig. 7-5 IN FO MGR SIM BC CM T ab le 7-2. I n te r - s p e c ie s o v e rla p in d ic es to be co m p ared . 200 F ig . 7-5. PCA of th e v a rio u s m ethods of c alcu latin g the i n t e r sp e c ie s o v e rla p . The e le m e n ts of the sta n d a rd iz e d i n te r - s p e c ie s d ista n c e m a tric e s a re u se d as a ttrib u te s. The sym bols fo r the in d ices a re show n in T ab le 7-2. All sta n d a rd iz a tio n s in d icated a re sp e c ie s s ta n d a rd iz a tio n s. The sy m b o ls fo r the sta n d a rd iz a tio n s a re : TOT = total; MAX = m ax im u m . 201 A M S n (24% ) M O R • # IN F O > B C -M A X • B C - N O R M B C -T O T • SIM AVT.q T (57% ) • B C - M E A N C M -M A X • • C M -T O T # B C -B I N A R Y 2 0 2 It a p p e a rs th at the m o st im p o rta n t c o n sid e ra tio n in the c alcu latio n of th e in te r-s p e c ific d istan c es is the index u sed. The f ir s t axis (F ig. 7-5) m o stly s e p a ra te s the d ifferen t in d ic es fro m one a n o th er. F o r in stan c e, a ll the C a n b e r r a - m e tr ic and B ra y -C u rtis index r e s u lts (with th e exception of the b in a ry data) a re in re la tiv e ly c lo se p ro x im ity r e g a r d le s s of the sp e c ie s s ta n d a rd iza tio n u se d . The seco n d axis s e p a r a te s the B ra y -C u rtis , b in ary d a ta m ethod, and the C a n b e r r a - m e tr ic index r e s u lts fro m the r e m ain in g m eth o d s. A lso, th e data sta n d a rd iz a tio n s w ithin th e B ray - C u rtis and C a n b e r r a - m e tr ic m ethods a re s p re a d som ew hat out along th is ax is. In su m m a ry , the p r im a r y c o n sid e ra tio n when c alcu latin g th e in te r -s p e c ific d istan c es a p p e a rs to be the type of index u sed . T his in d ic a te s that it is im p o rta n t to m ak e s u r e that the p ro p e rtie s of the index chosen a re co m patible w ith the o b jec tiv e s of the a n a ly sis. Of se c o n d a ry im p o rta n c e is the s ta n d a rd iz a tion u sed w hen th e m eth o d allow s a choice of sta n d a rd iz a tio n . The r e s u lts u sin g the b in a ry data a re different som ew hat from the o th er r e s u lts w ith the sa m e index (B ra y -C u rtis ). T hus, it a p p e a rs that the e lim in atio n of the quan titativ e com ponent of the o v e rla p can a lte r the r e s u lts sig n ifican tly . 203 e) A d ju stm en ts in the c alcu latio n s fo r the sh o rtc o m in g s of the sa m p lin g p a tte rn C olw ell and F u tu y m a (1971) point out som e im p o rta n t c o n sid e ra tio n s w hen w ork ing w ith o v erlap in d ic e s. Con s id e r the h y p o th etical tr a n s e c t in F ig. 7 - 6a. It is a ssu m e d that the b io ta change continuously w ith th e w a te r depth. The f ir s t se v en o r so tr a n s e c t sa m p le s a re tak e n at a p p ro x im ate ly the sa m e depth (F ig. 7 -6b). If th e sp e c ie s in th e a re a change w ith depth in the m a n n e r show n in F ig . 7 - 6c, it w ill ap p ea r th at the re la tiv e ly shallow sp e c ie s A and B a r e m o re o v erlap p in g th an a re th e s p e c ie s in d e e p e r w a te r. T his w ill be due to the m o re in te n se sa m p lin g in the sh a llo w -w a te r h ab itat, and not due to g r e a te r o v e r lap, sin c e the o v e rla p is co n stan t fo r a ll s p e c ie s . A s im ila r p ro b lem could even a r is e if the s ite s w e re sp a ce d r e g u la rly along th e e n v iro n m e n ta l continuum . T his would happen if the e n v iro n m en ta l change did not affect the g e n e ra l biotic change in a continuous, lin e a r m a n n e r. To o v erco m e th e s e p ro b le m s, C olw ell and F u tu y m a (1971) p ro p o se d th at the o v e rla p calcu latio n s include a w eighting (for each site) w hich is d ire c tly p ro p o rtio n a l to the en v iro n m e n ta l u n iq u en ess of the site . If th is is done, th e m o re e n v iro n m en ta lly "red u n d an t" s ite s ( e . g . , s ite s 1 - 7 in Fig. 7- 6a, b) w ill be d e e m p h a siz e d and the m o re unique s ite s w ill be given 204 F ig . 7-6. The p ro b lem of sa m p le sp acin g in re la tio n to the m e a s u re m e n t of in te r - s p e c ie s o v erlap . 205 W a te r • • • • • • • • • • 1 2 3 4 5 6 7 8 9 10 L. H y p o t h e t i c a l b e n t h i c t r a n s e c t w i t h s i t e n u m b e r s . 500 i 400 -o 300 i 200 1 100 a 1 2 3 4 5 6 7 89 10 S ite # b . P lo t o f d e p th on t r a n s e c t f 2 Z f S i 7 D ep th 100 ' I I I 2 0 0 3 0 0 4 0 0 500 C h a n g e in v a r ia b l e s a s s o c ia t e d w ith d e p th ----- c . P o s it io n s o f s i t e s and s p e c i e s o n d e p th c o n tin u u m . E a c h b e ll c u r v e r e p r e s e n t s a s p e c i e s . 206 g r e a te r w eight, the fin al re s u lt of w hich a re b e tte r e s tim a te s of o v erlap . The e n v iro n m e n ta l u n iq u en ess o r w eight of e ach site m u st be c a lc u la te d fro m the sp e c ie s data on the assu m p tio n that the biota w ill change in re s p o n se to changes in th e m o re im p o rta n t en v iro n m e n ta l fa c to rs . Dahl et al. (1967) show ed a high c o r r e l a tion betw een biotic change (as m e a s u re d by i n te r - s i te ecological d ista n c e s) and e n v iro n m e n ta l change (as m e a s u r e d by d istance in d is c rim in a n t sp ace). If e n v iro n m en ta l m e a s u re m e n ts w e re to be u se d to c a lc u la te the w eig hts, the re la tiv e im p o rta n c e of the dif fe re n t v a ria b le s w ould have to be d e te rm in e d b efo reh an d to avoid usin g ir r e le v a n t o r re la tiv e ly u n im p o rtan t v a ria b le s in d e te rm in in g the w eights. Since d e te rm in a tio n of im p o rta n t e n v iro n m en ta l v a ria b le s e n ta ils r e f e r e n c e to the biota, it w ill m ak e se n se u su ally to u se the biotic d ata d ire c tly in the w eight c a lc u la tio n s, even if en v iro n m en ta l m e a s u re m e n ts a re av ailab le. F u rth e rm o r e , it is p o ssib le th at v a ria b le s r e la te d to som e of the m o re im p o rta n t e n v iro n m en ta l fa c to rs have not even been m e a s u re d ; in this c a se the re s u ltin g w eights w ould be inadequate. C olw ell and F u tuy m a (1971) re c o m m e n d e d that the w eights be c alc u la te d fro m the sp e c ie s d ata u sin g a m e a s u re re la te d to in fo rm a tio n th e o ry (calculation fo rm u la e shown in C h ap ter IX). The w eights can be applied to one of two p ro p o se d in d ices of 207 o v e rla p . Both in d ices u tiliz e s p e c ie s - to ta l sta n d a rd iz e d abun d ances . In a p r e lim in a r y a n a ly sis w ith the p re s e n t data, the m ethods p ro p o sed by C olw ell and F u tu y m a (1971) w e re te s te d and ev alu ated . The re s u ltin g site w eights did not a g re e w e ll w ith the u n iq u en ess of the s ite s and, fo r th is re a s o n , an a lte rn a tiv e m eth o d of c alcu latin g th e site w eights is p ro p o sed . A second objection to the m ethod a r is e s fro m th e u se of s p e c ie s -to ta l s ta n d a rd iz e d data in the calcu latio n s of o v erlap . T his is d isc u sse d below . Since each site w ill be w eighted, it is im p o rta n t th at the s ta n d a rd iz e d v alu es at each s ite co n trib u te to the o v e rla p c a lc u la tio n s in an eco lo g ically m eaningful p a tte rn . C o n sid er, fo r exam p le. F ig . 7 - 7a. The two sp e c ie s depicted have id e n tic a lly shaped c u rv e s w hich peak at th e sa m e site , but differ in c u rv e w idth. In F ig . 7 - 7b the s p e c ie s - to ta l sta n d a rd iz e d v alu es a re plotted. Note that th e s ite at w hich the sta n d a rd iz e d values fo r the two s p e c ie s differ the m o st is the peak s ite (site #5). T his m ean s th a t the v a lu e s at this site would c o n trib u te the m o st to a c a lc u la te d d istan c e (or la c k of o v erlap ) b etw een th e s e two sp e c ie s. If such a site w e re h e av ily w eighted due to its r e la tiv e en v iro n m en tal u n iq u en e ss, th en th is d ifferen ce in sta n d a rd iz e d values would be 208 F ig . 7-7. C o m p ariso n of sta n d a rd iz e d c u rv e s u sin g s p e c ie s -to ta l and s p e c ie s -m a x im u m sta n d a rd iz a tio n s . F o r th e raw data, the c u rv e sh a p es a re constant, but the w idths v a ry . S traig ht lin es a re u sed to a s s u re that the sh ap es of the o rig in a l c u rv e s a re id en tical. 209 15 10 5 Raw d a ta . 4 . 3 . 2 . 1 9 7 8 3 5 4 6 1 2 b. S pecies to ta l s ta n d a rd iz a tio n 1 . 5 3 2 4 5 6 7 8 A c. S p e c ie s-m a x im u m sta n d a rd iz a tio n 210 m ag n ified and the sp e c ie s w ould show m isle a d in g ly low o v erlap v a lu e s. In the exam ple, both sp e c ie s o c c u r in h ig h est abundance in the sa m e h ab itat and the c alcu latio n s involving this site should lead to a re la tiv e in c re a s e in m e a s u re d o v e rla p , not a d e c re a s e . F u rth e r m o r e , the positions of the points of id e n tic a l sta n d a rd iz e d values (on e ith e r sid e of th e peak in Fig. 7 - 7b) seem to be of no eco lo g ical sig n ific an c e. The r e a l d ifferen c e betw een th e se two s p e c ie s o c c u rs m o re to w a rd the o u ter p a rts of th e c u rv e s w h ere sp e c ie s D o c c u rs and sp e c ie s E does not. The sp e c ie s-m a x im u m sta n d a rd iz e d v a lu e s show n in F ig. 7 - 7c show a m o re m eaningful p a tte rn of v a lu e s in th is situ ation , sin c e the d ifferen ce is m in im a l at the peak and g ra d u a lly in c r e a s e s to w ard the o u te r p o rtio n s (away fro m the peak) of the sta n d a rd iz e d c u rv e s . In F ig. 7-8, th re e sp e c ie s c u rv e s a re show n in w hich the w idth is held constant, but the shape is v a rie d . F o r both the sp e c ie s to ta l and m axim um sta n d a rd iz e d v alu es (Fig. 7-8b, c), the c u rv e s for sp e c ie s B and C a re s im ila r . Both th e se c u rv e s a re re la tiv e ly "flat" or convex on one side (when view ed from above). F o r th e s p e c ie s - to ta l sta n d a rd iz e d d ata (Fig. 7 -8b), the c u rv e w ith the concave s id e s (sp e cie s A ) show s th e g re a te s t dif fe re n c e from the o th e r c u rv e s at the peak. The shape is a c h a r a c te r is tic of the e n tire c u rv e and th e re is no lo g ic al or ecological 211 F ig . 7-8. C o m p ariso n of sta n d a rd iz e d c u rv e s u sin g th e s p e c ie s - to ta l and s p e c ie s -m a x im u m sta n d a rd iz a tio n s. The o rig in a l c u rv e s v a ry in shape but a re co n stan t in w idth. 212 140 120 ( D g 100 n j T S 80 c 5 60 < 40 20 Raw d ata .3 . 2 . 1 9 8 5 6 7 3 4 2 1 b. S p e c ie s -to ta l s ta n d a rd iz a tio n . 0 . 8 . 6 B / . 4 3 5 6 9 4 7 8 1 2 c. S p e c ie s-m a x im u m sta n d a rd iz a tio n 213 r e a s o n why so m uch of the sta n d a rd iz e d d iffe re n c e s should be con c e n tra te d at the peak of the c u rv e s . As b e fo re , th is situ a tio n could becom e a p ro b lem when the s ite s a re w eighted. In F ig. 7 - 8c, it can be se en th at no su ch p ro b lem e x ists w ith the sp e c ie s -m a x im u m sta n d a rd iz e d data, w h e re the peak values a ll a re id e n tic a l, and the d iffe re n c e s a re m o re evenly d istrib u te d over th e w hole c u rv e s. E ven w hen the s ite s a re not w eighted, the sp e c ie s- m ax im u m sta n d a rd iz a tio n m ay s till be the m o st a p p ro p ria te sta n d a rd iz a tio n . W ith a ll the o th er sp e c ie s sta n d a rd iz a tio n s , d ifferen t sp e c ie s w ill tend to end up on d ifferent s c a le s . W ith the s p e c ie s -to ta l and no rm sta n d a rd iz a tio n s , the m o re w id e sp re a d s p e c ie s w ill ten d to be on a low er sc a le th an the r a r e r s p e c ie s , and w ith the s p e c ie s - m e a n sta n d a rd iz a tio n , the sp e c ie s w ith the m o re lep to k u rtic c u rv e s (see F ig. 4-14) w ill be on a h ig h e r s c a le . V a ria tio n in the s c a le s of the v a rio u s sp e c ie s a lre a d y h a s been show n to lead to u n d e s ira b le r e s u lts as fa r as m e a s u re m e n t of o v e rla p is c o n ce rn e d (see F ig s . 7-3 and 7-4). f) P ro p o s e d m eth o d s fo r c alcu latin g the site w eights and the in te r - s p e c ie s d ista n c e s To d e te rm in e the w eighting of a s ite , th e a v erag e o r to ta l d istan c e betw een th at site and all o th er s ite s is calcu late d . The d ista n c e s em ployed a re th e sa m e in te r - s ite d ista n c e s as u se d in th e n o rm a l a n a ly sis. T his w ill be a m e a s u re of the u n iq u en ess 214 of a site sin ce a re la tiv e ly d istin c tiv e site w ill be at a g r e a te r a v e ra g e d istan c e from m an y of the o th er s ite s . The m o re " r e dundant" s ite s w ill be c lo s e r to s e v e r a l o th er s ite s and th e a v e ra g e d ista n c e w ill be le s s . T h e re is a p ro b lem w ith the sc a le of the w eig h ts. F o r exam ple, the a v e ra g e w eights for the s ite s in th is study v a ry betw een . 59 and .7 8 . If th e s e n u m b e rs a re u se d d ire c tly as w eights, the m o st unique site w ill re c e iv e only 1.32 (i. e . , . 7 8 /. 59) tim e s the w eight of the le a s t unique s ite . T his s e e m s too r e s tr ic te d . The a v erag e d ista n c e s can be c o n v e rte d to any s c a le d e s ire d as follow s: ~ ------- X (S - 1) + 1 , (7-1) m ax m in w h e re is the sc a le d w eight of site i , T^ is the a v erag e o r to ta l d ista n c e betw een site i and a ll o th e r s ite s , is the s m a lle s t c a lc u la te d T, is the la r g e s t c a lc u la te d T, and S is th e u p p er lim it of th e d e s ire d sc a le . The low er lim it of the s c a le w ill be one. In a p re lim in a r y a n a ly sis w ith the p re s e n t data (using 70 sp e c ie s, s e e C h ap ter VIII), w eight s c a le s of 2, 3, 4, 6, 8, 10, and 12 w e re u se d in in v e rs e a n a ly se s (using m eth ods ex plained below ). The re s u ltin g groups w e re su b jectiv ely ev alu ated by o b se rv in g tw o-w ay coincidence ta b le s and by c o n sid e rin g the 215 p a tte rn s of the s p e c ie s abundances (with th e aid of m ap s of the sam p lin g a r e a in w hich the abundances fo r each sp e c ie s w e re d is played). A s c a le of 10 w as ch o sen for the p re s e n t data. The w eights re s u ltin g fro m th is m ethod fit the sa m p lin g a re a b e tte r than did the w eights c a lc u la te d fro m the m ethod of C olw ell and F utuy m a (1971). R easo n s for th is, along w ith a m o re o b jectiv e m ethod of d e te rm in in g the w eight s c a le , a re e x p lo red in C h ap ter IX. Once th e w eights a re calcu late d , the in te r - s p e c ie s d ista n c e s a re c a lc u la te d u sin g a B r a y -C u rtis index and s p e c ie s - m ax im u m sta n d a rd iz e d data. The m ethod of applying the w eights is d is c u s s e d below . The advantages of the s p e c ie s -m a x im u m s ta n d a rd iza tio n have been d is c u s s e d a lre a d y . The B r a y -C u rtis index is u se d b e c a u se it is s e n s itiv e to the m agnitu de of th e data. C o n sid er the th re e sp e c ie s c u rv e s in F ig . 7-9. At points 1, 2, and 3, w h ere the c u rv e s in te r s e c t, the values a re id e n tic a l. W ith the B ra y - C u rtis index, th e la r g e r n u m b ers w ill r e c e iv e h e a v ie r w eight and, thus, th e id en tity at the peak (point 1) w ill re c e iv e g r e a te r w eight th an th e o th e r id e n titie s at points 2 and 3. It is of g r e a te r eco lo g ic a l in te r e s t th at two sp e c ie s find the s a m e habitat o p tim al than th at th e y both find th e sa m e h ab itat sub o p tim al. W hen c alcu latin g th e d istan c e betw een two sp e c ie s, the w eights can be applied by m ultiplying the d ifferen ce in the 216 F ig . 7-9. T h ree h y p oth etical s p e c ie s c u rv e s u sed to d e m o n s tra te re a s o n s fo r u sin g the B r a y -C u rtis index in the in v e rs e a n a ly sis. 217 Sp e cie s --m axim um S ta n d a rd iz e d A bundance 1 5 E . 5 -- 218 sta n d a rd iz e d v a lu e s (of the two sp e c ie s) at a site by the w eight of the s ite . T his is equivalent to re p e a tin g the d ifferen ce W tim e s in the c a lc u la tio n s, w ith W being equal to the site w eight. T his is illu s tr a te d in F ig. 7-10. The e a s ie s t w ay to im p lem en t th is p ro c e s s is to sim p ly m u ltip ly each colum n (re p re s e n tin g a site ) in the sta n d a rd iz e d data m a trix by the c o rre sp o n d in g w eight, and c a lc u la tin g the d ista n c e s from the re s u ltin g m a trix . The sa m e p ro g ra m th a t c a lc u la te s the B ra y -C u rtis d ista n c e s can be u se d w ithout a lte ra tio n to include w eights b e ca u se the w eights w ill be in c o rp o ra te d in the data m a trix . It could be a rg u e d th a t th is w eighting sc h e m e is an e x e r c is e in c ir c u la r re a so n in g , sin c e th e w eights a re c alc u la te d fro m the sa m e sp e c ie s th at a re being c o m p a re d for o v erlap . C olw ell and F u tu y m a (1971) re c o m m e n d e d th at for each c o m p a ris o n the w eights be re c a lc u la te d w ithout including the data for the s p e c ie s being c o m p a re d . At le a s t w ith the p re s e n t data, it should m ake little d ifferen ce w h eth er o r not the w eights a re re c a lc u la te d fo r each c o m p a riso n . In Fig. 7-11, the effect on the in te r- site d ista n c e s (from w hich the w eights a re calcu lated ) of rem o v in g s p e c ie s ra n d o m ly fro m the d ata m a tr ix is shown. By o b se rv in g the s te e p e s t slope on the in fo rm a tio n -lo s s c u rv e (when g r e a te r th an 10 p er cent of the sp e c ie s a re c o n sid e re d ), it can be se e n that, at 219 F ig . 7-10. The applicatio n of site w eights to th e B ra y -C u rtis index c alcu latio n s in th e in v e rs e a n a ly sis. Note th at the s im p le s t m e a n s of applying the w eights is to m u ltip ly the colum ns of th e data m a trix by the re s p e c tiv e site w eights, and then u sin g th is m a trix in the u su a l m a n n e r of c alcu latio n (as in c and d). 220 a. S ta n d ard ize d data m a tr ix and s ite w eig h ts. Site S pecies A S pecies B 1 . 2 . 5 1 2 W eights b. W eighted c alcu latio n s u sin g th e B ra y -C u rtis index, D , = 4 (1 - .2) + 2 (1 - .5 ) 4 (1 + .2) + 2 (1+.5) _ (4- . 8 ) + (2- 1) _ 4. 2 ^ (4 + . 8) + (2+1) 7. 8 c. S ta n d ard ize d d ata m a tr ix w ith colum ns m u ltip lie d by the re s p e c tiv e w eights. Site 1 2 S pecies A S pecies B d. C alcu latio n s u sin g m a trix in c. D AB (4 - .8) + (2 - 1) (4 + . 8) + (2 + 1) = . 54 221 F ig . 7-11. In fo rm a tio n -lo ss p ro file from ran d o m e lim in a tio n of sp e c ie s. S p e c ie s-m e a n sta n d a rd iz e d d a ta w ith a p rio r s q u a r e - ro o t tra n s fo rm a tio n a re u sed (as in the n o rm a l an aly sis). See C h ap ter V for an explanation of in fo rm a tion lo s s . 222 100 90 80 70 to O î 60 c o " S I 50 o 3 30 20 10 10 20 30 40 50 60 70 80 90 100 % 22 44 67 89 111 133 155 178 2 0 0 S p e c ie s in c lu d e d in c a lc u la t io n o f th e i n t e r - s i t e d is ta n c e m a t r ix 22 2 # S p e c ie s 223 m o st, the in fo rm a tio n lo ss w ill in c r e a s e by about one p e r cent when two sp e c ie s a re e lim in a te d . F ro m th is it can be a ssu m e d that the w eights would change v e ry little if th ey w e re re c a lc u la te d each tim e w ithout two of the sp e c ie s . T his is im p o rta n t sin c e, w hen s e v e r a l sp e c ie s a re involved, th e se re c a lc u la tio n s w ill lead to a la rg e in c re a s e in com putation tim e. 2. The re la tio n sh ip s betw een the sp e c ie s o r sp e c ie s gro u p s and the m e a s u re d e n v iro n m en ta l v a ria b le s A group of site s can be defined acc o rd in g to the p re s e n c e of a sp e c ie s o r a sp e c ie s group. The re la tio n sh ip s b e tw een two or m o re such groups and the m e a s u re d e n v iro n m en ta l p a ra m e te r s then can be analyzed by m ultiple d is c rim in a n t a n aly sis (G reen, 1971, 1972, 1974). F ig u re 7-12 illu s tr a te s the fo rm a tio n of groups c o rre sp o n d in g to each sp e c ie s in a sa m p le data m a trix . The ex am ple is show n s p e c ific a lly fo r use w ith ind iv id u al sp e c ie s , but a s im ila r p ro c e d u re can be applied to sp e c ie s g ro u p s, as w ill be d is c u s s e d la te r. T his m eth od can he im p ro v e d upon. C o n sid e r the d istrib u tio n s of th e two sp e c ie s dep icted in Fig. 7-13. U sing the m eth o d of group fo rm a tio n d e s c rib e d above, all but two of the site s c o rre sp o n d in g to th e s e s p e c ie s w ould be the sa m e s ite s . The e n tire d is c rim in a n t an aly sis (as fa r as th e s e sp e c ie s a re co n cern ed ) would 224 F ig . 7-12. F o rm a tio n of gro u p s of s ite s fo r studying the en v iro n m e n ta l re la tio n s h ip s of the sp e c ie s w ith m u ltip le d is c r im in a n t a n a ly sis. 225 a. D ata m a trix . Sites 1 2 3 4 5 A 3 1 0 0 2 S pecies B 2 0 0 2 1 C 0 0 3 2 0 b. Sites in groups re p re s e n tin g each s p e c ie s , i . e . , the s ite s in w hich each sp e c ie s o c c u rs. A Species B C Sites in group 1, 2, 5 1, 4, 5 3, 4 226 F ig . 7-13. Two h y p o th etical sp e c ie s c u rv e s u se d to d e m o n stra te the u se fu ln e ss of w eighted c alcu latio n s in m u ltip le d is c rim in a n t a n aly sis. 227 10 9 8 7 E 228 depend on the e n v iro n m e n ta l m e a s u re m e n ts at th e s e two site s (2 and 9) w h e re th e sp e c ie s do not o v e rla p . Two p ro b le m s can a r is e in such a situ atio n . a) Sites like 2 and 9 m ay not have b een sam p led . In th is c a se the s ite s c o rre sp o n d in g to th e two sp e c ie s a re a ll id e n tic a l and the in fo rm a tio n c o n ce rn in g the e n v iro n m e n ta lly re la te d d iffe re n c e s betw een th e s e s p e c ie s w ill be lost as fa r as the d is c rim in a n t a n a ly sis is co n ce rn e d . b) The s p e c ie s abundances fo r th e s e two sp e c ie s a re re la tiv e ly low in s ite s 2 and 9. T his in c r e a s e s th e ch an ces that the o c c u rre n c e s of th e se sp e c ie s w ill be m is s e d even if s ite s 2 and 9 a re sa m p le d . As above, the groups th en w ill be id e n tic a l and in fo rm a tio n is lost. Thes'è p ro b le m s could be o v e rc o m e if th e calcu latio n s in the d isc rim in a n t a n a ly sis a re w eighted by th e r e la tiv e abundance of th e c o rre sp o n d in g sp e c ie s in each site . In th is way, the e n v iro n m e n ta l in fo rm a tio n at a s ite in w hich a sp e c ie s is re la tiv e ly abun dant is c o n sid e re d m o re ty p ic a l of th at sp e c ie s th an is th e e n v iro n m e n ta l in fo rm a tio n at a s ite in w hich the s p e c ie s is le s s abundant. T hus, co m p letely o v erlap p in g sp e c ie s (in a qualitative se n se ) can be d istin g u ish ed as long as th e ir re la tiv e abundances a re not id e n tic a l in all s ite s . In o th er w o rd s, the w eighted v e rs io n would 229 u se both the q ualitative and q u an titativ e com ponents of th e biotic data, w hile the unw eighted m etho d w ould u tilize only the q u alitativ e com ponent. The additional in fo rm a tio n u se d in a w eighted m etho d would lead to a m o re pow erful and ro b u st a n aly sis if quan titativ e s p e c ie s data w e re available. A m ethod of w eighting the d isc rim in a n t a n a ly sis c alcu latio n s is p roposed. See Appendix A for d e ta ils of the m ethod and som e fu rth e r d is c u s s io n . The sp e c ie s groups r a t h e r th an individual sp e c ie s also can be analyzed for e n v iro n m en ta l re la tio n sh ip s in a w eighted m u ltip le -d is c rim in a n t a n a ly sis. The sa m e p ro c e d u re u se d fo r individual sp e c ie s is u tilized, except th at at each site a sta n d a rd iz e d re la tiv e abundance m e a s u re is u se d for each group r a th e r than for e ach s p e c ie s . The abundance m e a s u r e u se d h e re fo r each s p e c ie s group at each site is the s p e c ie s -g ro u p m ax im u m sta n d a rd iz e d sum of the sp e c ie s-m a x im u m s ta n d a rd iz e d v a lu e s for each sp e c ie s in the group in question. S pecifically, th is is Si 4 =-------------------------------------------------------------------- - (7-2) J m ax Si w h e re Sij is the value re p re s e n tin g th e re la tiv e am ount of sp e c ie s group i in site j , n is the n u m b er of sp e c ie s in group i , 230 is the s p e c ie s -m a x im u m sta n d a rd iz e d value for sp e c ie s k in site j , and is the m ax im u m value of Z x o v er a ll the s ite s fo r sp e c ie s group i. The sp e c ie s-m a x im u m sta n d a rd iz e d v a lu e s, r a th e r th an the stra ig h t abundance d a ta fo r each sp e c ie s, a re u se d in th e n u m e ra to r sin ce th e r e is no a p rio r i r e a s o n to a ssu m e th at a m o re abundant sp e c ie s is m o re r e p re s e n ta tiv e of a s p e c ie s group than is a le s s abundant sp e c ie s . An efficient s tr a te g y fo r studying the re la tio n s h ip s betw een the sp e c ie s and th e e n v iro n m en t would be as follow s. a) A nalyze the la r g e - s c a le p a tte rn of th e sp e c ie s by p e rfo rm in g the w eighted d isc rim in a n t a n aly sis on the m ain sp e c ie s g ro ups defined in the in v e rs e c la s s ific a tio n a n a ly sis. b) A nalyze the s m a ll- s c a le p a tte rn s of the sp e c ie s by p e rfo rm in g a s e p a r a te w eighted d isc rim in a n t a n a ly sis fo r the s p e c ie s w ithin each sp e c ie s group. The re la tio n sh ip s betw een the s m a ll- s c a le p a tte rn s and the e n v iro n m en t m ay be o b s c u re d by the la r g e - s c a le p a tte rn s if a ll the sp e c ie s w e re to be analyzed to g e th e r. If so m e of the m ain sp e c ie s g ro u p s ch o sen a re la rg e and h e te r o geneous, it m ay be m o re efficient so m e tim e s to f ir s t analyze su b group s of the m ain group and th en analyze the individual sp e c ie s in the su b g ro u p s s e p a ra te ly . 231 C H A PT ER VIII A D ETA ILED ANALYSIS OF THE DATA A. In tro d u ctio n The m eth o d s to be u se d h e re have been d isc u sse d and ra tio n a liz e d in C h a p te rs IV th ro u g h VII. The data a re a p p ro ach ed fro m m an y d ire c tio n s and m u ch m o re d e ta il is given than would in te r e s t m o st r e a d e r s . The p u rp o se h e re is to d e m o n stra te the m eth o d s and show w hat kind of in fo rm a tio n c an be e x tra c te d fro m the d a ta if so d e s ire d . B. M ethods 1. N o rm a l a n aly sis a) Biotic data The s p e c ie s -m e a n sta n d a rd iz e d data w ith a p r io r s q u a r e - r o o t tra n s fo rm a tio n a re u se d . T his is the sam e data m a tr ix u se d in C hapter IV (95 sp e c ie s). The r e s u lts of th e d ata re d u c tio n study show th at the elim in a tio n of a ll but the 95 m o st fre q u e n tly o c c u rrin g sp e c ie s is a s s o c ia te d w ith only about th re e p e r cent in fo rm a tio n lo ss (F ig. 5-1 and T able 4-3). The s q u a r e - r o o t tr a n s fo rm a tio n is applied to c o n tro l som ew hat the ra n g e of the s ta n d a rd iz e d values (see 232 C h a p te r II). A few s p e c ie s w ith la rg e flu ctu atio n s in th e ir counts have so m e re la tiv e ly la rg e sta n d a rd iz e d v a lu e s. W ith the sq u a re ro o t, th e se sta n d a rd iz e d v alu es a re s till la r g e r but m o re co m p a ra b le to the o th e r sta n d a rd iz e d v a lu e s. W ithout the tr a n s f o r m a tion, the h ig h est and seco n d h ig h est sta n d a rd iz e d v alu es in the data m a trix a re 15. 73 and 9. 66, re s p e c tiv e ly . W ith the sq u a re ro o t, the la rg e s t s ta n d a rd iz e d value is 6. 71 and th e second la r g e s t is 4. 80. 2. In v e rs e a n a ly sis - c la ss ific a tio n a) B iotic data The sp e c ie s w hich o c cu r at m o re than five s ite s and a re id en tified at le a s t to the g e n eric le v e l a re used. T h ese a re th e sa m e 70 s p e c ie s u se d in C h ap ter VI. D ata re d u c tio n w ith the in v e rs e a n aly sis r e q u ir e s d ifferen t c o n sid e ra tio n s than that in the n o rm a l a n a ly sis. In th e n o rm a l a n a ly sis, the sp e c ie s w hich a re re m o v e d a re a ttrib u te s of the e n titie s (site s), but in the in v e rs e a n aly sis th e s p e c ie s a re th e e n titie s th e m s e lv e s . R em oval of a sp e c ie s h e re is equivalent to e x p re s s in g a re la tiv e lack of in te r e s t in studying th at sp e c ie s in the a n a ly sis. S p ecies m ay be e lim in a te d fo r two re a s o n s : 1) The data for m an y of the le s s freq u en tly o c c u rrin g s p e c ie s u su a lly a re in su fficien t fo r finding m eaningful e n v iro n m en ta l c o rr e la tio n s . 233 2) It is convenient to keep the n u m b er of sp e c ie s to a re a so n a b le lev e l sin ce com putation, sto ra g e , d ata handling, and d isp lay a re fa c ilita te d . T he d ata a re tr a n s fo rm e d f ir s t by th e sq u a re ro o t (se e C h ap ter 11) and th en sta n d a rd iz e d by the s p e c ie s -m a x im u m . B e fo re c alcu latio n of the B r a y -C u rtis d ista n c e s , the sta n d a rd iz e d v a lu e s at each site a re m u ltip lie d by the w eight of the s ite . As d is c u s s e d in C hapter Vll, the w eight is a m e a s u r e of the re la tiv e u n iq u en e ss of the site and the w eights a re se t on a sc a le fro m 1 to 10. The e lem en ts of the m a trix u se d to c alcu late the in te r s p e c ie s d ista n c e s also a re u se d as the w eights in th e w eighted d is c rim in a n t a n aly sis (see Appendix A). 3. T ra n sfo rm a tio n of abiotic data V a ria b les show ing a skew ed d istrib u tio n a re logg tr a n s f o r m e d (B a rn e s, 1952; Dahl et al. , 1967). T his in clu d es depth, sulfide, Hg, and DDT. Such a tra n s fo rm a tio n can help m ake the d istrib u tio n s of th e v a ria b le v a lu e s c lo s e r to the d istrib u tio n s a ssu m e d in the m u ltip le - r e g r e s s io n and d is c rim in a n t-a n a ly s is m e th o d s. A lso, the d isto rtin g effects of a few e x tre m e v a lu e s a re d e c r e a s e d by the tra n s fo rm a tio n . F o r re a s o n s d is c u s s e d in C a ssie and M ichael (1968), the m e a s u re m e n ts of p e rc e n ta g e sand, silt, clay, and n itro g e n a re tr a n s fo r m e d by the a rc sin s q u a re ro o t. 234 U nless sta te d o th e rw ise , the tra n s fo rm e d v a lu e s o r p a ra m e te r s d e riv e d from tr a n s fo r m e d values a re r e p o rte d in the su b seq u en t ta b le s and fig u re s w hich d e al w ith abiotic data. 4. O th er c o n sid e ra tio n s The m ethod of c a lc u la tin g th e re la tiv e am ount of each sp e c ie s group at each site is d e s c rib e d in C hapter Vll, eq. (7-2). T his in fo rm a tio n is u se d in the w e ig h te d -d is c rim in a n t an aly sis of the sp e c ie s groups and also on the m ap s show ing the p a tte rn s of the sp e c ie s groups in the sa m p lin g a r e a (F ig. 8-5). C . R e su lts 1. C la ssific a tio n a) N o rm al an aly sis The d e n d ro g ra m is show n in Fig. 8-1. E leven groups of in te r e s t a re d e lim ited and the po sitio n s of the groups on the sa m p lin g g rid a re show n in F ig. 8 -2 . E ach group is signified by a le tte r of the alphabet. The groups w e re cho sen w ith the aid of both th e d e n d ro g ra m and th e tw o -w ay coincidence tab le (Fig. 8-7). The d istrib u tio n of th e groups in Fig. 8-2 in d i c a te s th at fa c to rs r e la te d to depth and the sew age effluents no doubt a re influencing th e d istrib u tio n of the biota. A ll but two groups a re at a sin g le depth. T h re e groups a re only in the im m ed iate v icin ity 235 C la s sific a tio n of the 40 s ite s . Fig. 236 16 24 26 23 29 22 38 34 33 20 32 39 CD LU 30 40 49 44 43 42 41 46 45 47 45 CD 36 35 27 26 25 CD o o oo o o ' o ' ' oj o o (OOl x ) 30NVISia o C D (V 237 F ig . 8 -2 . P o sitio n s of the s ite s in the 11 site gro u p s defined by the n o rm a l c la s s ific a tio n a n a ly sis. 238 u . _ J LL. « N LL U. 239 of the outfall, and at the 200-, 500-, and 1000-ft depths, the group s a re on both sid e s of the outfall s ite s . T he p a tte rn of the groups show s an ex ten sio n of the outfall s ite s to w a rd the n o rth w est, in d icatin g th at som e m a te r ia ls from the o utfall (which affect the biota) a re tr a n s p o r te d in th at d irec tio n m o s t p ro b ab ly by the lo cal c u rr e n ts (see a lso F ig. 3-3). b) In v e rs e a n aly sis 1) Site w eights T he site w eights a re show n in Fig. 8-3. It a p p e a rs th at the m o r e unique s ite s a re the 100-ft s ite s distan t from the outfall, s e v e r a l of the 1000-ft sites, and the s ite s in the a re a of the pipe situ a te d m o re to w a rd the n o rth w est. 2) D en d ro g ram The d e n d ro g ra m is shown in Fig. 8-4. Seven m a jo r g ro u p s a re d e lim ited . As w ith the n o rm a l a n a ly sis, the gro ups w e re chosen w ith the aid of the tw o-w ay table (Fig. 8-7). The re la tiv e in te n sitie s of each sp e c ie s group in each site a re shown in F ig. 8 -5 . E a c h sp e c ie s group show s a d istin ctiv e p attern ; at f ir s t glance, depth and o u tfa ll-re la te d fa c to rs a p p ear to be r e lated to th e s e p a tte rn s . 240 D istrib u tio n of the site w eightings. •<o + g 242 C la ssific a tio n of the sp e c ie s. Fig. 243 CD ID H . H C O CM — lT C Z CMLOEIA PiNNRTfi LUMBRINERIS INDEX LUCINOMA ANNULRTA LUMBRINERIS LACUNAE HETEROMASTUS FILOBRANCHUS NRSSRRIUS INSCULPTUS ANAITIDES •MUCOSA” PRIONOSPIO CIRRIFERA NEPHTTS CORNUTA FRANCISCANA CTCLOCRRDIR VENTRICOSR GLTCINDE PDLTGNRTMA RGLAJA SP. LAONICE CIRRATA PARRPRIONOSPIO PINNRTA CEREBRATULUS SP. SPIOPMANES FIMBRIATA GLTCERA BRANCHIOPODA MACOMA CARLOTTENSIS CYLICHNA DIEGENSIS DECAMASTUS GRACILIS PECTINARIR CALIFORNIENSIS AXINOPS I DR SP. NOTOMASTUS TENUIS TELLINA CARPENTER I PARVlLUCINfi TENUISCULPTA HRRMOTMOE "LUNULATR” LISTRIOLOBUS PELODES DORVILLJDAE S P .1 SOLEMTA PANAMENSIS ARMRNDIA BIOCULRTR CAPITELLA CAPITATA ANAITIDES MULTISERIATA GTPTIS ARENICOLA GLABRA GLTCERA AMERICANA NASSRRIUS MEND I eus NEREIS SP. 1 EUMIDR •SANGUINEA” SHISTOMERINGOS LONGICORNIS PROTOTHACA LACINIATA OLIVELLR BAETICR THTASIRA SP. THARYX •M U L T IF IL IS * LUMBRINERIS •LR TR E ILLI" SPIOPMANES 7MISSI0NENSIS LEPTOCHELIA S P .1 MELINNA HETERODONTA COSSURR CANDIDA PHERUSA PAPILLATA DIOPATRA ORNATR COMPSOMYAX SUBDIAPHANA KURTZIELLA BETA LUMBRINERIS JAPONICA RICTAXIS PUNCTOCAELATUS LISTRIELLR S P .I PERIPLOMA PLANUISCULUM CAPITITA AMBISETR NEPHTYS FERRUGINEA TELLINA MODESTA SPIOCHAETOPTERUS COSTARUM PRIONOSPIO MALMGRENI MITRELLA TUBEROSA OPHIODROMUS PUGETTENSIS CAULLERIELLA ALATA THARYX "PARVUS" COOPERELLA SUBDIAPHANA AMPHARETE LABROPS PHOTIS SP. CHONE ECAUDATA CONUS CRLIFORNICUS CREPIDULA ONYX (O O l X ) 3 0 N V 1 S I0 244 F ig . 8 -5 . S p e c ie s-g ro u p p a tte rn s in th e sa m p lin g a re a . The s p e c ie s -g ro u p lev e l at a site is c a lc u la te d by eq. 7-2. 245 9 - Kl a w C L tD ' D. & D CL U ) 246 c) F u r th e r su b division of sp e c ie s group 1 Since group 1 is la rg e and a p p e a rs h e te ro geneous, it w as subdivided b e fo re fu rth e r a n a ly sis (involving the in d iv id u al sp e c ie s)w a s p e rfo rm e d . The su b d iv isio n s a re in d icated in F ig . 8- 7, and th ey c o rre sp o n d to d istin c t gro u p s on the d e n d ro g ra m (F ig. 8-4). The d istrib u tio n s of th e s e su bgrou ps in the sa m p lin g a r e a a re show n in Fig. 8-6. d) T w o-w ay tab le The tw o -w ay ta b le is show n in F ig. 8-7. L ines a re d raw n to d e lim it the six m ain s ite g roups and the seven m ain s p e c ie s g ro u p s. F o r the sa m e groups, tw o-w ay tab le s u m m a rie s a re show n in Fig. 8-8. 2. D isc rim in a n t a n aly sis of the n o rm a l c la ss ific a tio n r e s u lts a) P r e lim in a r y a n aly sis To stu d y the p a tte rn of the d is c rim in a n t-a n a ly s is r e s u lts at s e v e r a l lev e ls of the d e n d ro g ra m , a s e p a ra te a n aly sis w as p e rfo rm e d at the 2- through 1 1 -group le v e ls . The p a tte rn of the co efficien ts of s e p a ra te d e te rm in a tio n on the f ir s t two axes w as fa irly c o n siste n t from the 6- to the 1 1 -group lev e ls on the d e n d ro g ra m . In g e n e ra l, th e sa m e s ite s w e re s e p a ra te d in th ese two d im e n sio n s r e g a r d le s s of the n u m b er of grou ps into w hich th ey a re 247 P a tte r n s of th e su bgroups of sp e c ie s group 1. Q C j O . D. b c 249 F ig . 8-7. T w o-w ay co incidence table. See F ig. 8-1 for the n o rm a l a n aly sis and F ig. 8 -4 for the in v e rs e a n a ly sis. Sym bols c o rre s p o n d to sp e c ie s-m e a n sta n d a rd iz e d v a lu e s w ith a p rio r c u b e -ro o t tra n s fo rm a tio n . Sym bols : for v alu es > 2 j "+" fo r v alu es >1 and <2; " for v a lu e s >. 5 and <1; " for v alu es > 0 and < . 5. 2 50 S I T E G R O U P S S P E C I E S G R O U P S A B ^ C D + E F + G H 4 1 J K -* S U B G R O U P S C R E P I D U L A O N Y X C O N U S C A L I P O R N ! e u s C H O N E E C A U D A T A P H O T I S S P . ____ A M P H A R E T E L A B R O P S _ 1 C O O P E R E L L A S U B D I A P H A N A T H A R Y X " P A R V U S " C A U L L E R I E L L A A L A T A - O P H I O D R O M U S P U G E T T E N S I S . M I T R E L L A T U B E R O S A - - 4.+ P R I O N O S P I O M A L M G R E N I » -» + 4. 4. 4. + 4. . 4. S P I O C H A E T O P T E R U S C O S T A R U M » » » » » » w w 4. T E L L I N A M O D E S T A » » w 4.4. 4. 4. 4. 4. . * N E P H T Y S F E R R U G I N E A » * » ^ 4, 4. 4. . 4. » - 2 C A P I T I T A A M B I S E T A * 4 -4 -. -4 -4 - * 1 P E R I P L O M A P L A N U I S C U L U M L I S T R I E L L A S P . l R I C T A X I S P U N C T O C A E L A T U S L U M B R I N E R I S J A P O N I C A K U R T Z 1 E L L A B E T A C O M P S O M Y A X S U B D I A P H A N A - 3 D I O P A T R A O R N A T A P H E R U S A P A P I L L A T A C O S S U RA C A N D I D A M E L I N N A H E T E R O D O N T A L E P T O C H E L I A S P . l S P I O P M A N E S 7 M I S S I 0 N E N S I S L U M B R I N E R I S " L A T R E I L L I " L T H A R Y X " M U L T I F I L L S " T H Y A S I R A S P . - + - 2 O L I V E L L A B A E T I C A P R O T O T H A C A L A C I N I A T A S H I S T O M E R I N G O S L O N G I C O R N I S . E U M I D A " S A N G U I N E A " N E R E I S S P . l N A S S A R I U S M E N D 1 e u s ' G L Y C E R A A M E R I C A N A 3 G Y P T I S A R E N I C O L A G L A B R A A N A I T I D E S M U L T I S E R I A T A C A P I T E L L A C A P I T A T A A R M A N D 1 A B I O C U L A T A ; S O L E M Y A P A N A M E N S I S D O R V I L L I D A E S P . l - - A L I S T R I O L O B U S P E L O D E S - . - H A R M O T H O E " L U N U L A T A " - - - . - z P A R V I L U C I N A T E N U I S C U L P T A T E L L I N A C A R P E N T E R 1 - + 4 N O T O M A S T U S T E N U I S - + * 4 . - - - - - - « 4. 4. 4. A X I N O P S I DA S P . * ^ _ + .4 . 4 4. 4 + + P E C T I N A R I A C A L I F O R N I E N S I S » ^ ^ ^ * * - 4 — « « - - 4. - 4. D E C A M A S T U S G R A C I L I S 5 C Y L I C H N A D I E G E N S I S 4 . - - 4. - M A C O M A C A R L O T T E N S I S V V * 4» 4-+ -4 4- 4 . G L Y C E R A B R A N C H I O P O D A — » — » » 4-4 4 - 4 4 4 - 4 S P I O P M A N E S F I M B R I A T A » — » - - - 4 - 4 4 4 4 4 4-f -f C E R E B R A T U L U S S P . P A R A P R I O N O S P I O P I N N A T A L A O N I C E C I R R A T A A G L A J A S P . - - - - — - G L Y C I N D E P O L Y G N A T H A - • 4 C Y C L O C A R D I A V E N T R I C O S A N E P H T Y S C O R N U T A F R A N C I S C A N A * _ 4 - - 6 P R I O N O S P I O C I R R I F Ç R A A N A I T I D E S " M U C O S A " N A S S A R I U S I N S C U L P T U S H E T E R O M A S T U S F I L O B R A N C H U S L U M B R I N E R I S L A G U N A E - -------- L U C I N O M A A N N U L A T A 7 L U M B R I N E R I S I N D E X C H L O E I A P I N N A T A . — — j — » 251 F ig . 8-8. S u m m a rie s of the c e lls of the tw o -w ay tab le (Fig. 8-7). E n trie s a re in p e rc e n ta g e of row or colum n to ta l. 252 Site grou ps 1 2 3 4 5 6 1 2 2 3 11 22 61 2 0 0 0 0 33 67 3 47 9 2 8 13 22 S pecies g roups 4 8 : 9 )6 ■ .5 53 20 5 7 11 17 31 19 15 6 1 12 53 15 8 11 7 0 41 10 37 10 2 a. E m p h a sis is on the d istrib u tio n of each sp e c ie s group throughout the s ite g ro u p s. E ach row to ta l equals 100%. Site g ro u p s 1 2 3 4 5 6 1 5 7 8 18 36 60 2 0 0 0 0 2 2 3 S p ecies groups 4 79 1 27 2 3 2 10 1 16 8 15 2 5 14 43 54 61 34 17 6 1 8 32 5 3 3 7 0 12 2 5 2 0 b. E m p h a sis is on the sp e c ie s group m ake group. E ach colum n to ta l equals 100%. -up in each site 253 divided. The su b seq u en t ax es each s e p a ra te d a p a ir of gro u p s w hich w e re not w e ll d istin g u ish ed by th e f ir s t two ax es. The gro u p s s e p a r a te d in each c a se w e re ad jacen t on the d e n d ro g ra m . T h is p a tte rn of r e s u lts su g g e sts th at the m o st efficien t a n aly sis would c o n sist of two ste p s. 1) A nalyze only the gro u p s of s ite s w hich a re d istin g u ish ed w ith two axes at the 11-g ro u p level. F ig u re 8-9 show s the plot of the d is c rim in a n t s c o r e s for the s ite s in the f ir s t two d im e n sio n s using th e 11 g ro u p s. The g e n e ra l groupings w hich se e m to be b e st s e p a ra te d a re outlined. T h ese sev en g ro u p s c o r re s p o n d to the 7 -group lev e l on the d e n d ro g ra m and a re u se d in a s e p a r a te d isc rim in a n t a n a ly sis. T h is w ill c o n ce n tra te the full pow er of the a n aly sis on d e te rm in in g th e e n v iro n m en ta l c o rre la tio n s w ith th e " la r g e - s c a le " biotic p a tte rn s . 2) E ach of the su b seq u en t axes (III, IV, and V at the 1 1 -group level) s e p a r a te one p a ir of groups not a lre a d y w e ll s e p a ra te d by th e f i r s t two a x es. The position of th e se axes is r e s tr ic te d som ew hat by th e p o sition of the e a r lie r axes, sin c e all axes m u st be at rig h t angles to one a n o th er, and the s c o r e s a re u n c o rre la te d w ithin and betw een g ro u p s. T hese r e s tr ic tio n s can be re m o v e d e a sily by p e rfo rm in g a s e p a ra te d isc rim in a n t a n a ly sis fo r each p a ir of g ro u p s in qu estio n . H ere the full pow er of the 2 54 F ig . 8-9. The se v en m a in gro u p s of s ite s w hich a re s e p a ra te d by the f ir s t two axes of th e 1 1 -group d isc rim in a n t a n a ly sis. The le tte r s c o rre s p o n d to the gro u p s as show n in Fig. 8-2. 255 A x is II A x is I 256 d isc rim in a n t a n a ly sis can be focused on th e " s m a ll- s c a le " biotic p a tte rn w ithout in te r f e r e n c e from th e " la r g e - s c a le " p a tte rn , b) A n a ly sis at th e 7 -group lev e l T he p o sitio n s of the se v en groups a re d elim ited on a m ap in F ig. 8-10. The co efficien ts of s e p a r a te d e te rm in a tio n for the fir s t th r e e axes a re shown in T ab le 8-1, and the plots of the d isc rim in a n t s c o r e s a re in F ig s. 8-11 and 8-12. In F ig . 8-11 it can be se e n th at the sev en groups a re fa irly w e ll s e p a r a te d by th e v a ria b le s c o rre sp o n d in g to the fir s t two axes. The f ir s t axis is re la te d m ain ly to depth and se c o n d a rily to se d im e n t c o a r s e n e s s . The second axis is r e la te d to sed im en t c o a r s e n e s s , clay, DDT, and eH. Sedim ent c o a rs e n e s s in c re a s e s to w ard the low er end of A xis I and to w ard the u p p e r end of A xis II. T h is re le g a te s the groups w ith c o a r s e r se d im e n ts to the upper left- hand c o rn e r of the sp a c e and th e groups w ith fin er se d im e n ts to the low er rig h t-h a n d c o rn e r . T his oblique re la tio n sh ip of the sed im en t c o a r s e n e s s to th e f ir s t two axes show s that the effect of depth, clay, DDT, and eH cannot be c o n sid e re d w ithout r e fe r e n c e to the s e d i m en t c o a r s e n e s s . If the lev el of sed im en t c o a rs e n e s s o r depth is held con stan t, it can be se e n in Fig. 8-11 th at c la y and DDT in c r e a s e and eH d e c r e a s e s c o n siste n tly tow ard the bottom of Axis II. In F ig . 8-12, Axis III is shown to fu rth e r se p a ra te grou ps A, B, and C (outfall site s ) from the o th er s ite s at the sa m e 257 F ig . 8-10. P o sitio n s of the site group s at the 7-g ro u p lev el. The lin es d e lim it th e g ro u p s. The le tte r s c o rre s p o n d to the groups at the 1 1 -group level. 258 li. 259 Abiotic Axis v a ria b le s I II III % G ra v el 0 .0 0. 4 0. 7 % Sand 2. 3 6. 7 0 .0 % C lay 0. 7 21. 8 2 .3 Std Dev (phi) 0 .3 0 .6 0. 5 Sed c 2 3 .4 3 4 .8 12. 5 Depth 68. 6 2. 7 11. 6 O rganic n itro g e n 1. 4 6 .8 4. 2 Sulfide fa c to r 2 .0 1.8 2 9 .0 M e rc u ry 0. 6 0 .9 0. 9 DDT 0. 5 12. 8 1. 2 eH 0. 2 10. 5 37. 1 T ab le 8 -1 . C oefficients of s e p a ra te d e te rm in a tio n fro m the d isc rim in a n t a n a ly sis of sev en site g ro u p s. 260 F ig . 8-11. D isc rim in a n t a n aly sis of s ite s at the 7 - group level. B esid e each group, the m e a n v a lu e s of the v a ria b le s im p o rta n t on th e f ir s t two axes a re included. Sym bols: D = depth, Sed c o r SC = se d im e n t c o a r s e n e s s , C = clay . The u n tra n s fo rm e d v alu es fo r depth a re u se d h e re . The f ir s t two axes a re show n. The co efficien ts of s e p a ra te d e te rm in a tio n for the v a ria b le s a re in d ic ate d in p a re n th e s e s . 261 A xis II D = 100 C = .3 3 D D T = 3 . 2 \ eH = -96 D = 1000 SC = 2. 8 C = . 48 D D T = 2 . 4 eH = 109 4 62. 5 1000 C = . 59 D D T = 3 . 7 eH = 26 C = . 46 DDT = 3 . 4 eH = 24 500 9 ffi SC D = 175 SC = 1. 1 C = .6 2 D D T = 4 . 3 eH = - 3 7 54 D DT eH D = 200 SC = 1 D D T eH 214 A x is D ep th <69) Sed c (23) 262 F ig . 8-12. D isc rim in a n t a n a ly sis of s ite s at th e 7-group lev el. The f ir s t and th ird axes a re shown. 263 A xis I 02 Q . C M CO 4 A x is in eH (37) S u lfid e (29) 264 depth. T his axis is r e la te d m ain ly to the eH and sulfide le v e ls, w hich a re m o re e x tre m e n e a r the outfall. c) A n aly sis of the group p a irs not s e p a ra te d at the 7 - group level The c o efficien ts of s e p a ra te d e te rm in a tio n fo r all the c o m p a riso n s a re in T able 8-2. A n o n -p a r a m e tric , u n iv a ria te sig n ific an c e te s t (M ann-W hitney U -te s t, Siegel, 1956) of group d iffe re n c e s also is in clu d ed fo r each v a ria b le in each a n a ly sis (T ab les 8-S; 8-4, 8-5). 1) G roups F and G T able 8-3 show s the m ean , m ax im u m , and m in im u m fo r the v a ria b le s w hich c o rre s p o n d to the s e p a ra tio n of th e s e two g ro u p s. A lthough th e coefficien t of s e p a ra te d e te rm in a tio n fo r DDT is not as high as so m e of th e excluded v a ria b le s , it is in clu d ed sin c e th e r e is no o v e rla p betw een th e group s in th e v alu es of th is v a ria b le . In fact, th e re is no o v e rla p b etw een any of the four v a ria b le s show n. G roup G, w hich is c lo s e r to th e outfall, has le s s sand and m o re n itro g e n , Hg, and DDT th an does group F. 2) G roups H and I See T able 8-4 fo r a su m m a ry of the m o re im p o rta n t v a ria b le s s e p a ra tin g th e s e g ro u p s. On a u n iv a ria te level, sulfide and eH sig n ific an tly differ in the two g ro u p s, but on 265 C o m p ariso n F - G H - I J - K G ra v e l 00 03 01 Sand 12 08 01 C lay 06 14 02 SD (phi) 01 1 1 03 Sed c 05 02 01 D epth 00 07 - N itro g en 11 02 02 Sulfide 00 26 28 Hg 16 02 02 DDT 05 1 1 43 eH 02 l i 17 T able 8-2. C o efficien ts of s e p a ra te d e te rm in a tio n fo r the s e p a ra te a n a ly se s of group p a irs not w ell s e p a ra te d by the f i r s t two axes in the 1 1 -group d isc rim in a n t a n a ly s is . 266 G roups V a ria b le s F G Significance p (U -te st) m in . 56 . 03 . 018 SAND m ax . 77 . 17 m ean . 67 . 38 m in . 42 . 70 . 018 N m ax . 59 .8 5 m ea n . 52 . 76 m in .3 6 1. 26 . 018 Hg m ax 1. 05 1. 74 m ean . 70 1. 47 m in 1. 86 4. 07 . 018 DDT m ax 4. 01 5. 01 m e a n 3. 16 4. 55 T ab le 8 -3. C o m p ariso n of the m o re im p o rta n t v a ria b le s s e p a ra tin g groups F and G. 267 G roups V a ria b le s H I Significance p (U -test) SU LFID E m in m ax m e a n 3 .53 4 .3 9 3. 87 4.32 5.88 5. 06 . 036 eH m in m ax m ea n -15 8 -1 -106 -3 -59 .036 CLAY m in m ax m ea n . 56 . 85 . 74 . 25 . 82 . 55 . 196 SD m in m ax m e a n 2. 20 2. 70 2. 43 1. 50 2. 50 2. 00 . 161 DDT m in m ax m ea n 3. 99 5. 08 4. 56 2. 77 5. 22 4. 17 .286 T able 8 -4 . C o m p a riso n of the m o re im p o rta n t v a ria b le s s e p a ra tin g groups H and I . 268 th e m u ltiv a ria te lev el, su lfide is show n to be the m o re im p o rta n t v a ria b le (T able 8-3). 3) G roups J and K The v a ria b le s c o rre sp o n d in g to group s e p a r ation a re s u m m a riz e d in T able 8-5. G roup J , w hich is c lo s e r to the outfall, is h ig h er in DDT and su lfid e, and low er in eH. T h e re is no o v e rla p of the DDT m e a s u re m e n ts in the two grou ps. 4) G roups D and E Since th e re is only a sin g le site in group E, the d isc rim in a n t a n a ly sis and U -te s t a re in ap p lica b le . Site E , how ever, is s e p a r a te d fro m the s ite s in group D along Axis II in F ig . 8-11 (the 7 -g ro u p d is c rim in a n t a n a ly sis). The v a ria b le s im p o rta n t on th is axis a re c o m p a re d fo r th e two grou ps in T able 8 -6 . A ll th e DDT m e a s u r e m e n ts a re h ig h er in group D than in site E. The o th er v a ria b le s show c o n sid e ra b le o v e rla p . The r e m ain in g se v en v a ria b le s w e re checked also fo r group d ifferen c e s and nothing outstand in g w as found. T hus, of a ll the v a ria b le s , only DDT s e e m s to show a p a tte rn c o n siste n t w ith th e group se p a ra tio n . T his tre n d should be in te r p re te d w ith e x tre m e caution since th e re is only one site in one of the gro u p s. d) S u m m a ry of the re s u lts - The la r g e - s c a le p a tte rn of the group s is c o rr e la te d m ain ly w ith the p a tte rn s of depth and se d im e n t c o a r s e n e s s , but at 269 G roups Signifie ance V a ria b le s J K p (U -test) m in 3.42 1. 62 . 014 DDT m ax 4. 99 2. 66 m e a n 4. 15 2 .2 5 m in 4. 73 3 .5 8 .029 SU LFIDE m ax 5. 42 4 .8 8 m e a n 5. 14 4. 40 m in -165 -79 . 029 eH m ax - 57 -47 m e a n -133 -58 T able 8-5. C o m p a riso n of the m o re im p o rta n t v a ria b le s s e p a ra tin g groups J and K. 270 V a ria b le s G roups D E m in DDT m ax m ea n 1 .7 4 1.26 3 .7 6 1.26 2 .6 2 1.26 m in eH m ax m e a n 1 ’ 134 178 ' 134 104 134 m in SED C m ax m e a n 2 3 3 3 28 3 m in CLAY m ax m ea n .2 6 .4 4 .6 6 .4 4 .4 9 .4 4 T able 8-6. C o m p a riso n of the p o ten tially im p o rta n t v a ria b le s s e p a ra tin g groups D and E. T h e re is only one site in group E . 271 any given depth and se d im e n t c o a r s e n e s s , the p re s u m e d outfall- re la te d v a ria b le s such as clay, DDT , and eH a re c o rr e la te d w ith group se p a ra tio n . The s m a lle r - s c a le p a tte rn is re la te d to the s e p a r ation of grou ps in w hich the s ite s involved a re m o stly at the sa m e depth but v a ry in th e d istan ce fro m the outfall. The im p o rta n t v a ria b le s a re not the sa m e at all depths. At the sh allo w est (100 ft) and the d e ep e st (1000 ft) depths, DDT is c o n siste n tly h igh er in the group n e a r e r the outfall. DDT also is se c o n d a rily re la te d to se p a ra tio n of both m id -d e p th g ro u p s. Sulfide s e e m s to be the m o st im p o rta n t at the 2 0 0 -ft le v e l and of se c o n d a ry im p o rta n c e at the 100-ft level, w ith g e n e ra lly h ig h er sulfide le v e ls at th e s ite s c lo s e r to the outfall. The eH m e a s u re m e n ts show a p a tte rn s im ila r to that of sulfide in the r e s u lts , but in the d is c rim in a n t a n aly sis, the sulfide is given h e a v ie r w eight. At th e 500 -ft lev el, stro n g g ra d ie n ts of sulfide and eH a re lacking beyond the im m e d ia te a re a of the o u tfall (Fig. 3-3). H ere the group c lo s e r to the outfall is hig h er c o n siste n tly in th e am ounts of n itro g en , H g , and DDT , and low er in the am ount of san d in the se d im e n t. Although th e re also a re g ra d ie n ts of n itro g e n and Hg at the 2 0 0 -ft lev el (Fig. 3-3), the low -oxygen and toxic conditions a sso c ia te d w ith the p re s e n c e of sulfide m ay be m o re im p o rta n t h e re , and no re s p o n s e re la te d to th e s e v a ria b le s (N and Hg) could be d etected . 272 3. W eighted d isc rim in a n t a n aly sis of th e sp e c ie s and sp e c ie s groups a) U sing th e sev en m a in sp e c ie s groups F ig u re 8- 13 show s the s p e c ie s groups in the f ir s t two d im en sio n s of d is c rim in a n t sp a c e . Axis III (not shown) m ain ly s e p a ra te s groups 1 and 2 , and is m o stly re la te d to the le v e l of DDT (which is h ig h er in group 2). It s e e m s th at the m a in groups of s p e c ie s p r im a r ily a re s e p a ra te d by th e ir re la tio n sh ip s to depth- r e la te d f a c to rs , and se c o n d a rily to o u tfa ll-re la te d fa c to rs (as in d icated by the e H , N , and Hg m e a s u re m e n ts ). T hese con c lu sio n s a re c o n siste n t w ith the p a tte rn s show n in F ig s. 3-3 and 8-5. b) U sing th e four su b group s of sp e c ie s group 1 In F ig. 8-14, the su b g ro u p s a re shown in d is c rim in a n t sp a c e . T h e re is a fa ir am ount of o v erlap h e re ; th is w ould be ex p ected sin c e th e se groups m ak e up a single c lu s te r at a h ig h e r lev e l of the d e n d ro g ra m . Along Axis I , subgroups 1 and 2 a re s e p a ra te d fro m su bgrou ps 3 and 4 in th at the fo rm e r two extend fu rth e r into a re a s w h e re the sulfide and n itro g e n a re h ig h er and the eH low er. The second axis s e p a r a te s su bgrou ps 2 and 3 fro m 1 and 4, w ith the la tte r two su bgrou ps tending to be found 273 F ig . 8-13. The se v en m a in sp e c ie s grou ps in th e w eighted d is c rim in a n t sp a c e . The lin es d e lim itin g the groups in d icate th e ap p ro x im ate lo catio n s of th e top 50% of e ach group in te r m s of s p e c ie s -m a x im u m sta n d a rd iz e d v a lu e s. The w eighted group m e a n s fo r the im p o rta n t v a ria b le s a re in d icated . The co efficien ts of s e p a ra te d e te rm in a tio n a re shown in p a re n th e s e s . 274 D eH N Hg 4. 61 -109 . 49 . 68 D = 4. 87 eH = -57 N = . 51 Hg = . 71 D eH N Hg 5. 85 -5 . 57 . 83 D = 6 .4 0 eH = 70 N = . 52 Hg = . 68 eH = 18 Hg = . 87 ^ D = 5. 3 eH = -4 0 eH = - 1 1 0 N = . 67 Hg = 1. 18 eH , D epth (21) (62) 275 F ig . 8 -14. The four su b g ro u p s of sp e c ie s group 1 in the w eighted d isc rim in a n t sp a c e . The c o rre sp o n d in g co efficien ts of s e p a ra te d e te rm in a tio n a re shown in p a re n th e s e s . The w eighted m e a n s of the im p o rtan t v a ria b le s also a re in d icated . 276 4. 19 T O C \] N = . 49 Sul = 4. 66 eH = - 66 Sed c = 5 .6 Sand = . 95 N = .4 9 Sul =4.37 eH = -49 Sed c = 4. 4 Sand = .8 6 N = . 55 Sul = 4. 58 eH = - 63 Sed c = 4. 2 Sand = .4 5 N , (21 ) Sulfide (17) 277 re la tiv e ly m o re often in c o a r s e r s e d im e n ts . T h ese p a tte rn s a re c o n siste n t w ith th o se shown in F ig s. 3-3 and 8-6. c) U sing th e indiv idu al sp e c ie s in the v a rio u s groups At th is sta g e of the a n a ly sis, a ll the sp e c ie s in a sin g le an aly sis w ill be highly overlapping, w hich is the r e a s o n th at th ey c lu s te re d into groups in the f ir s t place. The d iffe re n c e s show n in th e s e r e s u lts w ill be due m o s tly to (1) sp e c ie s having p e ak abundances in d iffe re n t h a b ita ts , a n d /o r (2) sp e c ie s extending into d ifferen t h a b itats in th e a r e a s of n o n -o v e rla p . The d istrib u tio n of each ind ividual s p e c ie s o v e r th e sa m p lin g a r e a is show n in Fig. 8-1 5 . R e fe re n c e to th is fig u re and F ig . 3-3 should be helpful in u n d e rsta n d in g and in te rp re tin g the d is c rim in a n t-a n a ly s is r e s u lts to be show n below . 1) S pecies group 1 T h is group c o n s is ts of sp e c ie s w hich o ccu r m o s tly in shallow w a te r and g e n e ra lly avoid the site s in the ou tfall a r e a w ith the high sulfide m e a s u re m e n ts (Fig. 8 - 5a). The plots of th e m e a n s c o r e s fo r each of the sp e c ie s in each of the four su b g ro u p s a r e show n in F ig . 8-16. The m e a n values fo r the im p o rta n t abiotic v a ria b le s a re shown in T able 8-7. T his tab le can be u se d to c h ec k the tre n d s of th e v a ria b le m e a s u re m e n ts in the d is c rim in a n t sp a c e . In c o n tra s t to the l a r g e r - s c a l e p a tte rn (Fig. 8-13), 278 F ig . 8-15. P a tte r n s of th e sp e c ie s abundances in the sam p lin g a re a . The sp e c ie s a re shown in the sa m e o rd e r as th ey o c c u r on the d e n d ro g ra m in F ig. 8-4. The c o r resp o n d in g sp e c ie s group is in d ic a te d follow ing the sp e c ie s n a m e . The n u m b er follow ing the hyphen in d i c a te s the subgroup. The s q u a r e - r o o t abundance is u se d to m ak e th e p a tte rn s m o re lin e a r and th e s e values a re m u ltip lie d by 10 to elim in ate d e cim a l points. 279 T 3 €0, 281 M l 1 CO 282 Q ., X2i y - / 283 JO 3" tft C O 285 to § 286 â 0 3 £ l CO CD . 4 288 F ig . 8-16. W eighted d isc rim in a n t a n aly sis of the sp e c ie s in the four su bg roup s of sp e c ie s group 1 . W eighted av erag e s c o r e s fo r each sp e c ie s plotted. The c o rre sp o n d in g co efficien ts of s e p a ra te d e te rm in a tio n a re shown in p a re n th e s e s . See T able 8-7 fo r the w eighted m ean s of the in d ic ate d v a ria b le s . 289 a. Subgroup 1 b. Subgroup 2 A xis II A xis II C. ecaudata a M. tu b erosa C. alata CL CO T. "parvus A. labrops P h otis sp. # C. subdiaphana C. onyx O. pugettensis C. ca lifo rn icu s • A xis I Hg, N , Clay Sed c.' L istr ie lla sp. (36)(17) (11) (14) S. co st arum • • p . m alm gren i C. am b iseta # T .m o d e s ta N. ferruginea • P . planuisculum - N (13) Sand ■ (44) -Axis I A xis n A xis II R. punctocaelatus M. heterodonta L-eptoct^lia sp. 1 C. C an d id a (» S. m issio n e n sis D. ornata L. " ia treilli P . papillata L. japonic a T. m u ltifilis C. su b d ia p h a n a C Ü K. beta • T h yasira sp. A xis I A xis I Sand Clay, Hg N , Depth, Clay (52) (15) c. Subgroup 3 (13) (10) d. Subgroup 4 (24) 290 1. » U U t S p e c ie s 1 21 Q œ c u S u b g ro u p 1 C r e p id u la on y x . 32 6. 1 . 46 . 57 4. 61 - 7 8 C o n u s c a lif o r n ic u s .32 6. 3 . 45 . 54 4 . 61 - 7 6 C h o n e e c a u d a ta . 35 5. 7 . 46 . 56 5. 02 - 3 5 P h o t is s p . .33 6. 4 . 47 . 55 4. 61 - 7 5 A m p h a r e te la b r o p s . 31 6. 7 . 45 . 56 4. 61 - 81 C o o p e r e lla su b d ia p h a n a . 28 6. 2 . 44 . 57 4. 61 - 8 2 T h a r y x " p a r v u s" . 42 4 . 9 . 51 . 71 4. 85 - 4 9 C a u lle r i e ll a a la ta . 47 4 . 9 . 57 .8 1 4. 95 - 6 2 O p h io d r o m u s p u g e t t e n s is . 54 4 . 1 . 60 .9 5 4 . 93 - 5 2 M itr e 11a t u b e r o s a . 44 4 . 7 . 54 . 83 4 . 94 - 6 0 0 ) T3 m 3. S u b g ro u p 2 P r io n o s p io m a lm g r e n i .8 2 S p io c h a e t o p te r u s c o s t a r u m . 82 T e llin a m o d e s t a . 93 N e p h ty s f e r r u g in e a . 92 C a p itita a m b is e t a . 73 P e r ip lo m a p la n u is c u lu m . 67 L i s t r i e l l a s p . 1 . 72 S u b g ro u p 3 R ic t a x is p u n c to c a e la tu s . 40 L um b r in e r i s ja p o n ic a .3 8 K u r t z ie l la b e ta . 49 C o m p s o m y a x s u b d ia p h a n a .3 6 D io p a tr a o r n a ta . 27 P h e r u s a p a p illa t a . 33 C o s s u r a C an d id a . 51 54 55 51 51 58 61 56 48 49 57 50 43 45 56 63 67 89 69 55 62 80 4. 98 4. 91 4. 93 4. 89 4. 98 5. 21 4 . 85 4 . S u b g ro u p 4 M e lin n a h e te r o d o n ta .3 0 1. 03 L e p t o c h e lia s p . 1 .2 6 1 .1 0 S p io p h a n e s ? m is s i o n e n s is .4 0 .8 5 L u m b r in e r is " I a tr e illi" .3 1 1 .0 1 T h a r y x m u l t i f i l i s .2 6 1 .1 1 T h y a s ir a s p . .3 2 .9 3 45 42 48 44 45 49 4 . 74 4. 61 5. 28 4 . 99 4 . 64 4. 94 4.22 4 . 4 5 4. 13 4 . 15 4 . 29 3 . 75 T a b le 8 - 7 . W e ig h te d m e a n s o f th e m o r e im p o r ta n t v a r ia b l e s fo r th e s p e c i e s in s p e c i e s g r o u p 1. 291 the se d im e n t v a ria b le s se e m to be m o re im p o rta n t in the se p a ra tio n of th e sp e c ie s in th is group. The fir s t axis fo r each subgroup is re la te d to one o r m o r e se d im e n t v a ria b le . 2) Species group 2 T h ese sp e c ie s a re unique am ong the s hallow - w a te r sp e c ie s in th at th ey o c cu r a lm o st ex clu siv ely at 100-ft site s w hich a re c lo se to the o utfall (Fig. 8-5). Due to th e s m a ll s iz e of the group, the sp e c ie s a re not analyzed s e p a ra te ly . 3) S pecies group 3 T he sp e c ie s in th is group tend to o ccu r p r e dom inantly in the outfall a re a (Fig. 8 - 5c). F ro m F ig . 8 - 17a and T able 8-8, it can be s e e n that th e s e sp e c ie s a re d iffe re n tia te d m ain ly by v a ria b le s a s s o c ia te d w ith depth. T he sp e c ie s on the left- hand end of Axis I ten d to occu r re la tiv e ly le s s freq u e n tly at the 100-ft s ite s w h e re th e N , Hg , and sulfide m e a s u re m e n ts also a re g e n e ra lly lo w er and the se d im e n t c o a r s e r (co m p a red to the im m e d ia te outfall s ite s ). The sp e c ie s a s s o c ia te d w ith the rig h t- hand end of A xis I tend to o ccu r m o stly at shallow s ite s . If sulfide is c o n sid e re d s e p a ra te ly , the sp e c ie s o c c u rrin g to w a rd th e upper rig h t-h a n d c o rn e r of the d isc rim in a n t sp ace ten d to be found at s ite s in w hich the sulfide is re la tiv e ly low c o m p a re d to the outfall s ite s . 292 F ig . 8-17. W eighted d isc rim in a n t an aly sis of the sp e c ie s in four of the sp e c ie s g ro u p s. The coefficients of s e p a ra te d e te rm in a tio n a re in d ic ate d in p a re n th e s e s . 293 a. S p ecies group 3 A xis II A. m u ltiseria ta G. a. glabra ^ I b. S p ecies group 5 A xis II ■ O m N ereis sp. 1 S. pan em en sis # • • S. langicornus A. bioculata N. m endicus # # • C. capitata # i D orviU id sp. 1 G. am erican a E . " s^ g u in e a " i ------------------------------------------------------ — A xis Depth, N , H g, Sulfide Sed c.—* • j P . ca lifo rn ien sis M. ca rlo tten sis » C erebratulus sp. T. carp en teri P. pinnata A jdnopsida sp. • D. g r a c ilis • S. fim briata P . ten uiscu lpt a N. tenuis Sulfide G. branchiopoda C. d ie^ en sis (33) (12)(10) (10) ( 10 ) ■Depth, Sulfide, Hg, N (37) (13) (12) (12) Sand- ( 12 ) .A xis I c. S p ecies group 6 A xis II Q C 5 a. m ü cosa g. polygnatha N. c. fran ciscan a O » H. filobranchus N. inscuiptus # L. lagunae P . c ir r ife r a d. S p ecies group 7 A xis II — Depth (37) N , Sulfide ■ (15) (14) A xis I L. annulata C. pinnata • L. index A xis I DDT, Sed c . , Clay Depth (2 8 ) (2 1 ) ( 16 ) (14) 294 I C /D 8 S p e c ie s g r o u p 3 S c h is t o m e r in g u s lo n g ic o r n u s E u m id a " sa n g u in e a " N e r e is s p . 1 N a s s a r i u s m e n d ic u s G ly c e r a a m e r ic a n a G y p tis a r e n ic o la g la b r a A n a itid e s m u l t is e r ia t a C a p it e lla c a p it a t a A r m a n d la b io c u la ta S o le m y a p a n a m e n s is D o r v iU id s p . 1 S p e c ie s g r o u p 5 P a r v ilu c in a t e n u is c u lp t a T e llin a c a r p e n t e r i N o to m a s t u s te n u is A x in o p s id a s p . P e c t i n a r ia c a l i f o r n i e n s i s D e c am a s t u s g r a c i l i s C y lic h n a d ie g e n s is M acorn a c a r l o t t e n s i s G ly c e r a b r a n c h io p o d a S p io p h a n e s f im b r ia t a C e r e b r a t u lu s s p , P a r a p r io n o s p io p in n a ta S p e c ie s g r o u p 6 S p e c ie s g r o u p 7 L u c in o m a a n n u la ta L u m b r in e r is in d e x C h lo e ia p in n a ta 63 68 71 65 78 64 60 60 52 57 68 72 G ly c in d e p o ly g n a th a . 74 2 . 15 C y c lo c a r d ia v e n t r i c o s a . 66 2 . 27 N e p h ty s c o r n u t a f r a n c is c a n a . 70 2 . 45 P r io n o s p io c i r r i f e r a . 63 2 .3 3 A n a itid e s " m u c o sa " . 70 2 . 08 N a s s a r i u s in s c u ip t u s . 55 2 . 29 H e t e r o m a s t u s filo b r a n c h u s . 60 2 . 49 L u m b r in e r is la g u n a e . 56 2 . 46 U 1 3 ( U CO 2 . 8 3 . 4 3 . 4 4. 7 3 . 4 3. 0 3 . 4 1. 4 1. 8 1 . 1 1. 0 2 . 8 3 . 0 3 . 4 2. 9 3 . 9 2. 9 2 . 6 2 . 2 1. 8 2. 5 3. 1 3. 2 2. 24 1. 52 1. 19 0 ) " O B Q % ë 5 . 23 . 70 5. 17 5. 02 . 67 5. 42 5. 16 . 64 5. 03 4 . 77 . 57 5. 07 5. 00 . 62 5. 05 5 .3 9 . 66 4. 84 5. 32 . 59 4. 56 5 . 59 . 76 5 .4 8 5. 30 . 76 5. 45 5 . 61 . 75 5. 33 5. 4 4 . 87 5. 77 5. 62 . 61 4. 23 5 .8 3 . 56 4 . 11 5. 52 . 54 3 . 91 6 . 17 . 54 3. 53 5. 89 .4 7 3 . 65 6 . 04 . 55 3 . 59 5 . 68 . 57 3. 90 6 . 27 . 60 4. 20 5 . 94 . 63 4 . 19 5 . 97 .6 1 4. 06 5. 95 . 56 3. 99 5 . 66 . 56 4 . 35 6 . 47 . 46 3. 21 6 .8 2 . 48 2. 92 5. 90 . 56 4. 32 5. 60 . 58 4. 4 5 6 . 07 . 56 3 . 98 6 . 84 . 53 3. 18 6 . 86 . 49 3 . 60 6. 67 . 52 3. 66 5. 99 4. 09 6 .4 1 3 . 32 6.66 4. 57 U 1.3 1.3 1 . 0 .9 1 . 1 1 . 1 . 9 1. 5 1.4 1. 4 1.4 . 96 .81 . 74 . 72 .60 . 73 .83 .87 1.03 . 92 . 76 .83 3. 8 3 . 4 3. 3 3.2 2. 7 3. 1 3. 5 3. 6 3. 9 3. 6 3.3 3. 5 4.19 .56 3.82 .51 3.4 0 .43 T a b le 8-8. W e ig h te d m e a n s o f th e im p o r ta n t v a r ia b l e s fo r th e s p e c i e s in th e in d ic a t e d s p e c i e s g r o u p s . T h e s e v a lu e s c a n b e r e la t e d to th e d is c r im in a n t a n a ly s is r e s u l t s in F ig . 8-17. 295 Som e of the sp e c ie s in th is group so m e tim e s a re c a lle d " in d ic a to rs " of d istu rb e d o r "polluted" conditions (R eish, 1959, 1973; S tirn, 1971) or as "o p p o rtu n istic " sp e c ie s (G ra ssle and G ra s s le , 1974). Since th e se sp e c ie s a re able to exist in re la tiv e ly h a rs h e n v iro n m e n ta l conditions th at m an y of th e ir p o ten tial c o m p e tito rs and p re d a to rs cannot to le ra te , they can becom e quite abundant in su ch s t r e s s e d a re a s . The d e g re e to w hich the sp e c ie s in th is group a re confined to the highly s t r e s s e d a re a (as in d icated m ain ly by the sulfide m e a s u re m e n ts ) v a r ie s , w ith the sp e c ie s m o st r e s t r i c te d occupying the low er left-h an d c o rn e r of the d isc rim in a n t sp a c e (Fig. 8 - 17a). 4) S pecies group 4 T h ese two sp e c ie s a re found in shallow depths w ith se d im e n ts containing re la tiv e ly high n itro g e n m e a s u r e m e n ts (Fig. 8-13). T hey a re not found g e n e ra lly in the a re a s w ith the v e ry h ig h est n itro g e n m e a s u re m e n ts sin c e they ap p aren tly cannot to le ra te the s t r e s s f u l conditions in the im m e d ia te outfall area. T he two sp e c ie s show unu su ally s im ila r p a tte rn s of o c c u rre n c e (Fig. 8-15). H arm othoe often is known to fo rm c o m m e n sa l re la tio n sh ip s w ith o th er in v e rte b ra te s (D avenport, 1953), and such a re la tio n sh ip w ith L istrio lo b u s would explain the o b se rv e d coinciden ce of p a tte rn s . The group is not analyzed fu rth er. 296 5) Species group 5 The sp e c ie s in th is group tend to be found at m id -d e p th s and avoid th e outfall a r e a (F ig. 8-5). The a n aly sis h e re is ru n w ithout L aonice c i r r a t a and Agla.ja s p . sin ce th e s e s p e c ie s a re re la te d only lo o se ly to th e re m a in in g sp e c ie s in the group (F ig s. 8 -4 and 8-15). F ro m F ig . 8 - 17b and T able 8-8, it can be se en th a t the sp e c ie s differ in (1) th e ir d e g re e of avoidance of the s ite s w ith high sulfide m e a s u r e m e n ts , w ith the sp e c ie s to w ard the rig h t end of Axis I avoiding th e h ig h er sulfide; (2) the sp e c ie s in the u p p e r left-h an d c o rn e r of th e d isc rim in a n t sp a c e tend to be found on th e a v erag e in d eep er w a te r; and (3) the sp e c ie s to w a rd the low er left-h an d c o rn e r of the d isc rim in a n t sp a ce ten d to be found in fin e r se d im e n ts w ith hig h er le v e ls of N , H g , and DDT. 6) Species group 6 T h ese sp e c ie s a re found p red o m in an tly in deep w a te r and away from the o utfall (F ig. 8 - 5f). The sp e c ie s on the rig h t end of Axis I (F ig. 8 - 17c) a lso a re found in sh a llo w -w a te r s ite s w h e re the N and sulfide a re re la tiv e ly hig h er c o m p a re d to the deep s ite s . The sp e c ie s plotted to w a rd th e bottom of Axis II tend to be found on the a v e ra g e in fin e r se d im e n ts. If m o re exact in fo rm a tio n w e re d e sire d on the sp e c ie s p re s e n t a lm o st ex clu siv ely at the d e e p -w a te r site s , th e a n a ly sis should be re p e a te d w ith the 297 sp e c ie s found in sig n ifican t n u m b e rs at any s hallow - w ate r s ite s elim in a te d . 7) Species group 7 T h ese sp e c ie s a re found m o stly in deep w a te r but o ccu r c lo s e r to the outfall th an the sp e c ie s in group 6 (F ig s. 8-5 and 8-13). As the d e n d ro g ra m (Fig. 8-4), tw o-w ay tab le (F ig. 8-7), and F ig . 8-15 in d ic ate , th is is a h e tero g en eo u s group w h ere th e a v erag e am ount of o v e rla p is not v e ry g re a t. L ucinom a annulata and L y m b rin e ris index both o ccu r at 500-ft site s adjacent to the o utfall a r e a although th e f o rm e r is m o re w id e sp re a d . C hloeia pinnata o c c u rs m o stly at the 1000-ft s ite s d ire c tly opposite the outfall, and in th is it is unique am ong a ll the sp e c ie s analyzed. In F ig. 8-17 the depth tre n d on the f ir s t axis is c o n siste n t w ith the above d isc u ssio n . The o th e r v a ria b le s a s s o c ia te d w ith th is axis a re re la te d to the fact th at the 1000-ft a r e a w h e re C. pinnata i s m o st abundant has fine se d im e n t w ith re la tiv e ly low c la y and DDT content for a site n e a r the outfall. The second axis re fle c ts the fact th at index is absent from s ite s w ith v e ry high sulfide, w h e re a s the o th er two sp e c ie s a re found at one o r two su ch s ite s . 298 d) M o re d etailed in fo rm a tio n on individ ual sp e c ie s ■The re la tio n sh ip s betw een the s e p a ra te sp e c ie s have been d isp lay ed in th e d isc rim in a n t sp a c e in F ig s . 8-16 and 8 -1 7 u sin g the m e a n d isc rim in a n t s c o r e s . No in d icatio n of the d istrib u tio n of the s c o r e s in the sp a c e is given. T his in fo rm a tio n can be v e ry u se fu l and is d e m o n s tra te d h e re . E ach sp e c ie s can be show n on a s e p a ra te plot, and each site in the d isc rim in a n t space can be re p r e s e n te d by a sy m b o l w hich is r e la te d to th e abundance of the sp e c ie s in q u e stio n at the site . As an exam ple, the plots for the sp e c ie s in sp e c ie s group 7 a re shown in F ig . 8-18. The s y m bols in the fig u re w e re g e n e ra te d by (1) dividing up the ra n g e of the sp e c ie s-m a x im u m sta n d a rd iz e d abundances fo r the sp e c ie s in question into 10 equal in te rv a ls , and (2) a ssig n in g a sy m b o l of 0 - 9 to each s ite depending on the c o rre sp o n d in g sta n d a rd iz e d abundance value at th e s ite . The hig hest abundance fo r a sp e c ie s is given the sy m b o l 9 and the low est the sy m b o l 0 . It is in te re s tin g to note th a t th e b e ll-c u rv e m odel of the re la tio n sh ip betw een sp e c ie s fitn e ss (abundance) and the en v iro n m e n t w ould p re d ic t a p a tte rn like that show n in F ig. 8- 18a, c (and to a l e s s e r d e g re e F ig . 8 - 18b) w hen two im p o rta n t independent e n v iro n m en ta l fa c to rs a re r e p r e s e n te d by the axes of the plot 299 F ig , 8-18. The d istrib u tio n of the sp e c ie s abundances in the d is c rim in a n t sp a ce fo r the sp e c ie s in sp e c ie s group 7. The sy m b o ls a re d ire c tly p ro p o rtio n a l to the sp e c ie s abundance at the c o rre sp o n d in g s ite . A pproxim ate iso p le th s a re draw n in the sp ace to e m p h a size the d istrib u tio n of abundances in the sp a c e . See Fig. 8-17 for the e n v iro n m e n ta l v a ria b le s w hich a re re la te d to the ax es. 300 A x isn AXIS I AXIS I a . L u c in o m a a n n u la ta b, C h lo e ia p in n a ta AXIS I O c . L u m b r in e r is in d e x 301 (P ian k a, 1974; 191). T his kind of a p a tte rn also is c alle d a "G a u ssia n re s p o n s e s u rfa c e " (W hittaker, 1956; A lder dice, 1972). 3. O rdination - n o rm a l a n aly sis P lo ts of the o rd in atio n s c o r e s fo r each site on the f ir s t th re e axes a re shown in F ig . 8-19. The site groups fro m the c la s s ific a tio n a n aly sis a re outlined for c o m p a riso n of r e s u lts . Note th at th e s ite s in a group also a re in re la tiv e ly c lo se p ro x im ity in the o rd in a tio n sp a ce of the f ir s t two axes. F o r each axis, th e o rd in atio n s c o re s a re plotted on a m ap of the sam p lin g a r e a (F ig s. 8-20, 8-21, 8-22). F ig u re 8-20 show s that the s c o re s fo r th e f ir s t axis a re re la te d c le a rly to depth, as m o st of th e iso p leth s a re e n tire ly p a ra lle l to the sh o re . The s c o r e s for s ite s 18 and 19 (clo sest to L os A ngeles H a rb o r at 100 ft) a re m o re s im ila r to the adjacent s ite s at 200 ft than th ey a re to th e o th er s ite s at 100 ft. Unlike the o th er 100-ft s ite s , fine se d im e n t is found h e re , in d ic atin g that the axis m ay be r e la te d also to se d im e n t siz e . The iso p leth s fo r A xis II (Fig. 8-21) show a d istin c t p a tte rn , w ith the s ite s c o rre sp o n d in g to th e m o st stro n g ly negative s c o r e s n e a r the outfall. The s c o re iso p leth s a re re m a rk a b ly s im ila r to the iso p le th s for th e sulfide m e a s u re m e n ts as show n in F ig . 3-3. 302 F ig . 8-19. P lo ts of the o rd in atio n s c o r e s on the f ir s t th r e e axes. 303 a. AXIS n ( 1 8 % ) + .3 0 3 42 30 20 32, [49 39 40 44 29 21 22 43 38 12/14 33 34 .487 45 17 28 23 46 47 24 37 35 36 27 26 2 5 ,487 304 b. A X I S n + .303 42 30 20 32 49 39 40 44 29 21 22 43 34 - . 2491- 48 45 17 28 23 46 47 24 37 36 27 26 25 .487 305 The p a tte rn of site s c o re s on o rd in atio n Axis I. LU ec i s /•} 'O # ' o < o / 307 The p a tte rn of site s c o r e s on o rdin ation A xis II. 309 8 - 22 . The p a tte rn of site s c o r e s on o rd in atio n Axis III. .- o • 2 < c O O •l o o 311 The p a tte rn of s c o r e s on Axis III (Fig. 8-22) is not so e a sy to in te r p r e t. The negative s c o r e s m o s tly o ccu r in r e l a tiv ely fine se d im e n ts (Fig. 3-3) and at the 200- and 500-ft le v e ls. F ig u re 8-23 co ntains the iso p leth s for both A xis III and sulfide. It can be se e n th at the m o s t stro n g ly negative s c o r e s on Axis III lie ju st beyond the a re a of high sulfide le v e ls . It is p o ssib le that the p a tte rn defined by th is axis is c a u se d by sp e c ie s w hich a re en hanced in abundance by the o rg an ic m a te r ia ls from the outfall, but at the sa m e tim e a re re la tiv e ly in to le ra n t of the fa c to rs a sso c ia te d w ith high sulfide le v e ls (low oxygen, re d u c in g c h e m ic a l p o ten tials, etc. ). One would expect to find peak n u m b e rs of such sp e cie s clo se to the outfall (the so u rc e of the o rg a n ic s), but not c lo se enough to be affected a d v e rse ly by the s u lfid e -re la te d fa c to rs . P o sitiv e s c o re s m ay in d ic a te low er lev els of such sp e c ie s and negative s c o r e s could be a s s o c ia te d w ith re la tiv e ly h ig h er abundances. The sp e c ie s involved in su ch a p a tte rn can be d e te rm in e d from F ig. 8-15 and from the o rd in atio n tw o-w ay tab le (Fig. 8-26). 4. M ultiple r e g r e s s io n of the o rd in a tion s c o r e s and abiotic v a ria b le s The sta n d a rd iz e d m u ltip le r e g r e s s io n co efficients fo r each axis a re show n in T able 8-9. T hese r e s u lts a re c o n sisten t w ith w hat has been d e te rm in e d su b je ctiv e ly from the p a tte rn s on the m a p s. On Axis I , the coefficient for depth by far is th e la rg e s t 312 F ig . 8-23. C o m p ariso n of the sulfide p a tte rn w ith the p a tte rn of s c o re s on o rd in atio n Axis III. 313 m / / / LU 314 AXIS I AXIS II AXIS III g ra v e l -2 8 -18 san d 12 -9 -17 c lay 4 2 -34 SD -7 10 17 se d c -19 (.02) -5 41 (. 1) depth 101 ( « .0 0 1 ) -46 (. 01) 47 (. 1) N -2 -28 -16 sulfide 6 -49 (. 001) 53 (.02) Hg 8 -25 4 DDT -6 1 -38 eH -9 42 (. 01) -11 T able 8-9. S ta n d ard ize d m ultiple r e g r e s s io n co efficien ts (xlOO). The dependent v a ria b le s a re the o rd in a tio n axes and the independent v a ria b le s a re th e abiotic v a ria b le s . The sig n ifican ce lev els of the co efficien ts a re indicated in p a re n th e s e s if they a re le s s than o r equal to .1 . 315 and m o st sig nifican t, and th at of se d im e n t c o a rs e n e s s is of se c o n d a ry but sig n ifican t (p < .0 2 ) im p o rta n c e . The hig h est co efficien ts on Axis II a re sulfide, depth, and eH. E viden tly the sulfide and eH lev els m u st be c o n sid e re d in conjunction w ith the depth level. On Axis III, sulfide is a sso c ia te d w ith a positive coefficient, in d icatin g that sp e c ie s o c c u rrin g at s ite s w ith negative s c o r e s ten d to avoid the high sulfide le v e ls. The po sitive coefficient fo r depth in d ic a te s th at su ch sp e c ie s avoid the d e ep e st s ite s , and the p o sitiv e coefficient for sed im en t c o a r s e n e s s is c o n sisten t w ith the fact th at th e s e sp e c ie s also avoid s ite s w ith c o a r s e sed im en t. T hus, the sp e c ie s involved in the "en h an cem en t-av o id a n ce " situ atio n d isc u sse d in re la tio n to th is axis ten d to avoid sulfide, c o a rs e se d i m e n ts, and th e 1000-ft depths, sin c e they a re c o n c e n tra te d at site s w ith negative s c o r e s (F ig s. 8-22, 8-23). The fact th at N , clay, and DDT have negative co efficien ts is c o n siste n t w ith the h ypothesis that th e se sp e c ie s a re "enhanced" by so m e m a te r ia ls fro m the out fall. T h ese th r e e v a ria b le s often a re a sso c ia te d w ith h ig h er lev e ls of o rg an ic m a tte r. 5. R e la tio n sh ip s betw een the o rd in a tion r e s u lts and the sp e c ie s a) O rd ination tw o-w ay ta b le s The m ethod u se d h e re is d isc u sse d in C hap ter VI. F ig u re s 8-24 to 8-26 contain the tw o-w ay ta b le s fo r the f ir s t th re e 316 O rdination tw o -w ay tab le for Axis I. SI TES SCORES + SCORES - 3 3 5 2 2 1 2 2 2 1 1 1 1 3 7 5 2 1 5 8 1 * 0 1 5 0 8 1 8 2 0 6 4 3 41 * 1 * 4 4 3 3 3 3 3 2 2 2 2 2 1 1 1 1 1 8 6 4 0 9 2 7 6 9 3 6 4 7 3 9 9 7 5 1 2 1 . NASSARIUS INSCULPTUS - - - - + - + 2. c h l o e ia p in n a t a --♦++- — - 3 . HETEROMASTUS FILOBRANCHUS - - - ++- 4 . CYCLOCARDIA VENTRICOSA + - - 5 . LUMBRINERIS LACUNAE - - ♦ +- - 6. LUMBRINERIS INDEX -+ - + + ♦ 7 . GLYCINDE POLYGNATHA ♦ - 8. MACOMA CARLOTTENSIS + + - - - - - - - - - 9 . LUC I NOMA ANNULATA + + + - + — ♦ - +- +- - - 10 . AX INOPSI DA SP. — “ +— — ++★♦ +— ♦++■+★ + — — — — — — — — — — — 11. D EC AM AS TU S GRACILIS — *•* + + +*+— + - — — - — - — 12 . GLYCERA BRANCHIOPODA — +— +- — + +++— —+ + -- — +++ —+— — 13 . ANAl TIDES "MUCOSA" +- - 14 . CEREBRATULUS SP. — +++ + +— ++— — — — — +— — 15 . CAPI TELLA CAPITATA — +— — * + — — — — — — 16 . SPIOPHANES FIMBRIATA — — — — — +— ++— 4- - ♦ + 4-+— — - + -4--+ + + 4- +- - — 17 . SOLEMYA PANAMENSIS - - - - - 4 - *44 4-4 -4 4 18 . TE L LINA CAR PE NTERI — — — — — 4 — 444 — — 4 — —4 * —* — — — 4— — — 4 — 4 — — — — 44— — 44 19 . PECTINAP.IA CALI FORNt ENSIS ~ - - 4 4 4 - 4 - * ★ - - 4 - — — 4 - - - - 4 20 . DORVILLI DAE S P .l - - « - - - * 21 . PARAPR IONOSPI 0 PINNATA 4— — — - — 4444- — 4 4 —4-4 — - 4 — — — 44 — 4— — 22 . NEPHTYS CORNUT A FRANC ISCANA - - - • - - - — -4 4 4 - - 4 23 . CYLICHNA DIEGENSiS • - + - 4 - 4 4 - - -4 - - - 24 . PARVILUCINA TENUISCULPTA — — — — — —, — — 44 — — 4 — 44444— — — 4— — — *4 — — 4— 4 — — 44 — 4 25 . AGLAJA SP, - - - - - - - - 4 26 . LAONICE CIRRATA - 4 4 - 4 4- 444 - - - 27 . NOTOMASTUS TENUIS 4— — — — 4 — — * — — 4 — 4—44 — 44— — — 4 — 44 28 . GYPTIS AR EN I CO LA GLABRA — — 4444 4 4- 44 — 44 — 4— — 4— 4 29 . ARMAND IA BIOCULATA - 4- - - - 4 4 - * 4 - - 4- 4 - - - - 30 . ANAlTIDES MULTISERIATA 4 - - -4 - 4 - - - - 4 31 . HARMOTHOE " LUNULATA" - - 4 - 4 44 4 - - 4 32 . SHISTOMERINGOS LONGICORNIS — ~ - — — 4 - 4 — —4—* * — - 4 — — - —4— - 4 —44— 444 33 . PRIONOSPIO CIRRI FERA - + - 4- - 4 4 - 4 34 . NEREIS SP. 1 — - — 4- - - — 4 — 4 - - 4 — — 4-44 444 35 . LISTRIOLOBUS PELODES - - - - - - - + - * - 4 4 - 4 - - - * 36 . CQSSURA CANDIDA - - + - 4 4 37 . PERIPLOMA PLANUISCULUM — - - — — 4 - - 4 — — 4 4 -4 4 4 - — — — — 4—* 38 . SPIOPHANES ?MI SSIONENSIS - — - - - 4— 4 - 4 39 . PRIONOSPIO MALMGRENI - - 4- 4- 444444444-4 AO. GLYCERA AMERICANA - 4- - 4— -44 A i. EUMIDA "SANGUINEA" 4 - - - - 4 - - 1 *2 . CAPITITA AM BI S ET A - - - » - - - — 4 + - * * * A3 . THYASIRA SP. - - - + + - AA. OPHIODROMUS PUGETTENSIS - - - 4- A5 . RICTAXIS PUNCTOCAELATUS 44 - - - - - 4 4-44 A6 . NEPHTYS FERRUGINEA +- - 4 44-4444 - A7 . SPIOCHAETOPTERUS COSTARUM - --'f - - 444 - 4444*4 AE. TELLINA MOD EST A — — - — —- — 4 - - — 444444-4 A9 . LISTRIELLA S P .l - - - — 4- 44 - - 4 -4 5 0 . KURTZ IELLA BETA ♦- - -+ - - - 4 5 1 . CAULLERIELLA ALATA - - - - - 4 - — - - 4 4 - 5 2 . LUMBRINERIS "LATREI LLI" - - + 4 4- 5 3 . THARYX "PARVUS" — — — 4 — * 4 4 - - — — 4— —* - # ♦ 5A. COMPSOMYAX SUBDIAPHANA' - - - * - +- + -- 4 5 5 . PHERUSA PAP ILLATA - - + + + + 5 6 , PROTOTHACA LAC I NI ATA + + - -+ + 5 7 , CHON E ECAUDATA - - - - -+ —4 4 *- 5 8 . MITRELLA TUBEROSA - - +++ 5 9 , NASSARI US MEND ICUS - - - - - - - 4-44 444 6 0 . LUMBRINERIS JAPONI CA - - ' - - - - 4-44 5 1 . MELINNA HETERODONTA - - + + + - 6 2 . DIOPATRA ORNATA - - - +++ 6 3 . COOPERELLA SUED IAPHANA + - - - 4 - 4 - 4 6 4 . OLIVELLA BAETICA - - -+ + - 5 5 . THARYX "MULTI FI LIS" - - + - + - 6 6 . CREPIDULA ONYX - 4- 4- - 4+ 5 7 . PHARETE LABROPS - - - 4- 44- 5 8 . PHOT IS SP. - - - - 4 - 6 9 . CONUS CALIFORNICUS +------- ++ 70 . LEPTOCHELIA S P .l +- + + -- 318 F ig . 8-25. O rdination tw o-w ay table for A xis II. 319 SITES SCORES + SCORES - 4 2 3 1 U 4 2 2 1 1 2 1 U 1 2 4 2 3 2 2 1 0 1 0 0 4 9 2 2 4 4 9 5 7 3 5 4 5 7 5 4 3 3 4 3 1 2 1 4 3 3 1 4 1 2 1 4 3 3 2 20 2 9 9 1 1 3 3 8 3 5 8 6 8 8 7 7 6 6 CYCLOCARDIA VENTRICOSA - + - + + NASSARiUS INSCÜLRTUS - - + + GLYCINDE POLYGNATHA - - + - - % - HETEROMASTUS FILOBRANCHUS + - + - LUMBRINERIS "LATREILLI" - - + + - - + LUMBRINERIS LAGUNAE + + - - - AX I NOPS I DA SP. 4 * 4 4 + ■ * 4 — 4— — 4 — — — — — — 4—4 4 — — — — — — THYASIRA SP. - 4 - — 4 DECAMASTUS G RAC I LIS - 4 - 4 4 - - * - - - - - 4 — 4 - - PECTINARIA CALIFORNIENSIS + - * - + * + + - + - - 4 - -4 - - - - - - - RICTAXIS PUNCTOCAELATUS + - - + + + — +- 4 - MELINNA HETERODONTA + + - - + COSSURA CANDIDA -+ - -+ + SPIOPHANES ?M) SSIONENSiS — -4 4 - - 4 — - — — - THARYX "MULTI FI LIS" + + - DIOPATRA ORNATA - + + + LEPTOCHELIA S P .l + + NOTOMASTUS TENUIS — — — —44— — * 444444 4— — — — — 4 — — — — 4 CYLICHNA DIEGENSiS - + ++ - - - - 4 — - - — - 4 LUMBRINERIS JAPONICA 4 - 4- 4 - PHERUSA PAPILLATA - 4 - 4 4 4 THARYX "PARVUS" -+ - - + - * 4 4 *- * --------------------------- - CONUS CALIFORNtCUS + - + 4 - CHONE ECAUDAT A - - 4 4 — — » 4 — — - COMPSOMYAX SUBDIAPHANA - 4 4 - 4 - - - - * NEPHTYS CORNUTA FRANC ISCANA * - - - + + — — — - 4 - 4 - - - — CREPIDULA ONYX + - + 4 - - 4- NEPHTYS FERRliG I NEA - - - + + - + + 4 44 4 CEREBRATULUS SP. — 44 — 4—44— — 44 — 4 + —4—44 — 4 — — — — — — — — — KURTZ I ELLA BETA “ ~ -44 - - - 4 TELLINA CARPENTERI —4 * 4444— 44 — — 4—4— 44 — — — 4 — — —4— 4 — — — ———— —— — — LUMBRINERIS INDEX + - - 4 4 -4 CAPITITA AMBISETA — — — — — — — + *4*4 4— — 4— * — — 44 — — — — — PARVI LUCINA TENUISCULPTA - 4 4 4 4 4 - 4 4 .4- * 444- - 4444- - - - - - - - - - - - - - - - - - TELLINA MODEST A — —4 — 4 4 —4 — — 4— 4 — 44 — — — ANA I TIDES "MUCOSA" 4 - - PRIONOSPIO CIRRIFERA -+ - - + 4 4 4- PHOTIS SP. SPIOPHANES F IMBRI ATA 44+ 4 — — 4— —44 — 4— 4 — 444 — — — 4 — 4— 4 — — — — — AMPHARETE LABROpS - - - 4 4 - +- PRIONOSPIO MALMGRENI — — — — 4— 4 4 — 44 — — 4 — — 44 444—4 — — — — LISTRIELLA S P .l - + 4 — 44 - — 4 — - - - COOPERELLA SUBDIAPHANA - + 4 - 4 - - - 4 GLYCERA BRANCHIOPODA 4444— 4 — — 44 4— 4 4— —4+ — 4 — — — — 4 — — — — — LUC1 NOMA ANNULATA +- - - 4- - 4 44 4 - 4 - PARAPRIONOSPI0 PINNATA —4444 — — — — — — — — — 44 4 — 4 — 4— 4 4— —44 — — SPIOCHAETOPTERUS COSTARUM - - - - - 4 -44 4 -4 — — 44 4 4 -4 - - - - OPHIODROMUS PUGETTENSIS 4 - - 4 - -------- AG LAJ A SP. — — +— — — — — — MITRELLA TUBEROSA + 4 4 - - — - - MACOMA CARLOTTENSIS — 44 — 4*4 — 4 — — 4— — — — 4—4— 44 — — 44 — — — — — 44 — 444 — — — LAONICE CIRRATA - — 4 - - 4 4 -4 - 4 — 4 4 PERIPLOMA PLANUI SCULUM — — — — — 4 — *4 — — 44 — — 4 — — 44 — — — 4 — ANAITIDES MULTISERIATA - - - + - - - 4 - - 44 CAULLERI ELLA ALATA - - - - * 4 - — - 4 - - LISTRIOLOBUS PELODES — — — — — 4*4 — — — — — 4* — 4 — — OLIVELLA BAETICA - 4 - 4 - - PROTOTHACA LAC I NI ATA + -+ +- 4 HARMOTHOE "LUNULATA" - +++ - - - 44 - - 4 - - NASSARIUSMENDICUS - 4 4 4 4 4 4 - - - - - - - CHLOEIA PINNATA - ' - - - + 44- GLYCERA AMERICANA — — — 4 4 — — — 4—4 — — — 4 NEREIS SP. 1 — — — — 4 4 4 4 —4— 44 — — — —4 — 44 — — GYPTIS AR EN I CD LA GLABRA — 4—44 - 44-4 — - 4 -4 4+444- SHISTOMERlNGOS LONG I CORN IS — — — — — — — — — — 4— +44444— — 444 — — — —— — ♦♦ * EUMIDA "SANGUINEA" - - - - 4 - - - +4 + ARMAND IA BIOCUL AT A — — — — — — — — — —44 — 4 — —4*44 SOLEMYA PANAMENSIS — — — 4 44 — — — 4— — 4 + 44 CA PI TE LLA CAPITATA — " — — — — — — — — — — — — — 44 — 4 — — + * * * DORVILLIDAE S P .l - - - - 320 - 2 6 . O rdination tw o-w ay tab le fo r Axis III. s I TES SCORES + SCORES - 1 * U A 1 1 1 1 1 * 2 2 1 1 3 2 2 2 3 3 3 2 0 5 2 U 0 3 5 9 5 7 8 9 7 3 1 8 0 U 82 U 1 1 U 1 * 1 U 1 2 L 1 * 2 3 3 2 3 2 3 3 3 3 2 6 1 U 1 7 7 5 8 5 0 5 6 L 1 9 2 3 9 PHOTIS SP. AMPHARETE LABROPS + + + ------- - - CHONE ECAUDATA * » - - - + ++- - CONUS CAL!FORNICUS + - +-+ - CREPIDULA ONYX + + - +-+ - - LEPTOCHELIA S P .l - + ++- - THARYX "MULTIFILIS" + - + - - - COOPERELLA SUBDlAPHANA + _ +_ + - OLIVELLA BAETICA + + - - HETEROMASTUS FILOBRANCHUS + MITRELLA TUBEROSA + + - - - - LUMBRINERIS "LATREI L U " + - ++- DIOPATRA ORNATA + + -+ NASSARlus MEND ICUS + + + - + ++ - - - - - - - EUMIDA "SANGUINEA" _ _ + + CYCLOCARDIA VENTRICOSA + . + + - - - CHLOEIA PINNATA - + + - - + - NASSARI US INSCULPTUS - - - + + NEPHTYS CORNUTA FRANC I SCANA - - - * - + - - + - - GLYCINDE POLYGNATHA - + SPIOPHANES ?MISSIONENSIS - - - - +++ - - - LUMBRINERIS JAPONICA + - -++ - - TELLINA MODESTA - + -+ +++ ++ - - - PROTOTHACA LACINIATA + + - - + + MELINNA HETERODONTA + ++- LUMBRINERIS LAGUNAE - - ++ - - - AG LAJA SP. - - - - - GLYCERA AMER ICANA + + - — + - +- - - - - - - + CAULLERIELLA ALATA * - + - - - - + - - » THARYX **PARVUS" * — — + ** — — — — — — — — — — — — PHERUSA PAPILLATA + + + + - - SPIOCHAETOPTERUS COSTARUM + + - + -++ ++ - - - + +- --------- ---------- NEREIS SP. 1 4 4 4 - 4—44— — — 4 — — — — 4— 44 — — — NEPHTYS FERRUGINEA + + ++ +- - ++ PECTINARIA CA LlFORNIENSt S -44 - 4— - - 4 - - - 4 - —- - 4* - * 4 - COMPSOMYAX SUBDlAPHANA - - +++ - * RICTAXIS PUNCTOCAELATUS + - +++ - - - - + +- CEREBRATULUS SP, + — — - — 4 -4 4 — 4— - — 4 — 4 - — — - 4— — — 44444- DORVILLIDAE S P .l — - * - - - - - ANAITIDES MULTISERIATA + - - 4 4 4 -- — - CAP I TE LLA CAPITATA —4- — — - * * - * — 4 — — 4 4 PRIONOSPIO CIRRIFERA - + -+ + - - -+ 4 SHISTOMERINGOS LONG I COR N I S + + + — — + 4 4 - * * — 4— — — 444— — LISTRIELLA S P .l - -44 - - + - —-4 - 4 ANAITIDES "MUCOSA" - - - - -+ PARAPRIONOSP10 PINNATA — — — - 4 —4 - — — 4- 4— — 44 —44- - — ++ — 4+ 4- GYPTIS ARENICOLA GLABRA + - — - 4 - 4 —4 4— 4 444- 4 4 44 - CAPITITA AMBISETA 4 4 - —4* *4 - — — — — — — — 4— — 4 - — — 4— — 4 PRIONOSPIO MALMGRENI - 4 4 444 44 - 4—4- — —- 4 —4— - — - - - — 4 ARMAND I A BIOCULATA - - - - + 44 * + - - 4 - 4 - —- OPHIODROMUS PUGETTENSIS + - - - - 4 — - KURTZI ELLA BETA - - -+ — 4 - 4 PERIPLOMA PLANUf SCULUM — - - 4 — — * — — — — — 4 — 44 — 4 4— —4—— 4 SOLEMYA PANAMENSIS - - - 44 4 - + --4 4 4 4- - MACOMA CARLOTTENSIS 4——4— —4—+———4— —4— —4—4— ——444— 4444+4— TELLINA CARPENTERI 44-4— — — — — — ——— — — 4*4 — * + - —++- NOTOMASTUS TENUI S + — — —4—44 — — — — 4— 4 — 4444—4— — — *4 THYASIRA SP. +- - - - + SPIOPHANES FIMBRIATA — — 44 — — — 4— — — — — — 4 — — — 44 — 444444 + 44 HARMOTHOE "LUNULATA" - - — +- - - ♦+ ++ - + COSSURA CANDIDA + ++ - - AX I NO PS IDA SP. — — — — 4 * — 4— — — 4 - 4 — —— —* — — 4444 + 4- LlS TR l0LOBUS PELODES — — * —— — — — — — * 4 — 44 — — — 4 LAO NI CE ClRRATA - - - - - ++ —+ ++- ++ PARVILUCINA TENU ISCU L PTA ~ . —-- — 4 4 - 4 4 4 - - 4 —- — — — — * 4 —4+444444 LUMBRINERIS INDEX - - + + ++- GLYCERA BRANCHIOPODA — — - — — — — — — 4—+ -4 —— — — 4444444—4444 CYLICHNA DIEGENSIS - - - — - - — + + -++ - -4 DECAMASTUS GRACILIS - - - — - - — - —4— —444 —+ * LUC I NOMA ANNULATA - 4 4 — - 4- — 4444— - 322 o rd in atio n a x e s. T h ese ta b le s give a c le a r p ic tu re of the re la tio n sh ip s betw een the sp e c ie s and the axis s c o r e s . In F ig . 8-24 it can be se en th at the sp e c ie s to w a rd the bottom of the tab le a re a sso c ia te d w ith sh a llo w -w a te r s ite s and negative o rd in atio n s c o re s ; the sp e c ie s a sso c ia te d w ith d e e p e r-w a te r s ite s a re to w a rd the top of the tab le. In F ig . 8-2 5 th e sp e c ie s th at can to le ra te fa c to rs a sso c ia te d with sulfide (which is h ig h est at s ite s w ith the m o re n eg ativ e s c o re s ) a re to w a rd the bottom of th e ta b le . In F ig. 8-26 th e sp e c ie s to w ard the bottom of the ta b le a re a s s o c ia te d w ith s ite s w ith negative s c o re s and, as such, a re found m o s tly at m id -d e p th s and could be involved in th e "e n h a n c em en t-av o id a n ce " situ atio n d isc u sse d in the o rd in atio n r e s u lts sectio n (see F ig . 8-23). 6. P lot of the sp e c ie s w e ig h te d -av e r age s c o re s in the o rd in atio n sp a ce The sa m e w e ig h te d -a v e ra g e s c o r e s u se d to d e te rm in e the seq u en ce of sp e c ie s in the ord in atio n tw o-w ay ta b le s a re plotted in F ig . 8-27. T his sp a c e is d ire c tly c o m p a rab le to th at shown in F ig . 8-19a,w hich is a plot of the site s c o re s . F o r exam ple, the s p e c ie s tow ard th e neg ative end of Axis 1 in F ig. 8-2 7 te n d to be found in th e sh a llo w -w a te r s ite s w hich a re to w a rd the negative end of A xis 1 in F ig. 8 - 19a. W hen c o m p a re d w ith the o rd in atio n tw o -w ay ta b le s, th is type of d isp lay h a s th e advantage that p a tte rn s in m o re than one 323 Fig. 8-27. Plot of the w e ig h te d -a v e ra g e s c o r e s for each sp e c ie s in the o rd in atio n sp a ce of the f ir s t two axes. The p ositions of the sp e c ie s in the sp ace a re d ire c tly co m p a ra b le to the site p o sitio n s in the sp ace in Fig. 8-19. The n u m b e rs c o rre sp o n d to the sp e c ie s n u m b ers shown in F ig. 8-24. 324 AXIS H +.187 52 36 62 27 23 70 “ II 48 49 “ 56---47 44 22 67 66 39 63 4 + .253 26 25 37 AXIS I 39 64 56 59 40 28 32 29 20 -.361 325 d im en sio n can be view ed in one plot. The disadvantage is th at the s p re a d of sp e c ie s o c c u rre n c e s and abundances in the sp ace is lo st sin ce e ach sp e c ie s is r e p re s e n te d by only a single point. T his in fo rm a tio n is of c o u rs e availab le in the o rd in atio n tw o-w ay ta b le s . U sed to g e th e r, th ese two tech n iq u es co n stitu te an effective m ea n s of in te r p re tin g the biotic p a tte rn s along the o rd in atio n axes. 326 CHAPTER IX FU R TH ER CONSIDERATIONS O F THE SITE-W EIGHTING PR O B L EM A. In tro d u ctio n T h re e p ro b le m s re m a in from the p rev io u s c h a p te rs . They a re as follow s. 1. In C h ap ter VII, the w eights c a lc u la te d by the m ethod of C olw ell and F u tuy m a (1971)(from h e re on, r e f e r r e d to as in fo rm a tion w eights) w e re re je c te d b e ca u se they did not m e e t the a u th o r's im p re s s io n of the s ite s ' u n iq u en ess. The w eights b a sed on the d is tan ce m a trix (to be c a lle d d istan ce w eights from h e re on) seem to be s u p e rio r. The p o ssib le re a s o n fo r th is situ atio n is one of the topics of th is c h a p te r. 2. A nother p ro b lem re m a in s c o n ce rn in g the d istance w eights. A m o re objective m ea n s of d e te rm in in g the sc ale of the w eights w ould be d e s ira b le . Such a m ethod is p ro p o se d and applied to the data. 3. In C hapter IV, the s p e c ie s -m e a n sta n d a rd iz a tio n is re c o m m e n d e d in th e n o rm a l a n aly sis b e c a u se it gives g r e a te r w eight to the m o re re lia b le sp e c ie s counts when u se d w ith the B ra y -C u rtis 327 index. H ow ever, th is sta n d a rd iz a tio n is se n sitiv e to uneven h abitat sam p lin g . The s p e c ie s - to ta l and s p e c ie s -n o r m sta n d a rd iz a tio n s a lso a re affected by the problem , but this is not p u rsu e d fu rth e r b e c a u se th e s e sta n d a rd iz a tio n s have been r e je c te d for o th er re a s o n s (C hapter IV). F ig u re 9 - l a show s a sp e c ie s c u rv e along an e n v iro n m e n ta l g ra d ie n t. If all 33 s ite s in d ic ate d a re sam p led , the r e su ltin g sta n d a rd iz e d sp e c ie s c u rv e is the one lab eled "evenly sa m p le d " in F ig . 9 -lb . This is the id ea l c u rv e w hich m ay o r m ay not be a p p ro x im ate d in r e a lity depending on w hich of th e 33 s ite s a ctu ally a re sa m p le d . If the s ite s to w a rd the e x tre m itie s of the g rad ien t a re sa m p le d m o re in te n siv e ly th an th e s ite s to w ard the m id d le, th e sp e c ie s m e a n w ill be d e p re s s e d . The re s u lt w ill be d isp ro p o rtio n a te ly high sta n d a rd iz e d v a lu e s fo r the s ite s n e a r the m id d le. T his i s d e m o n s tra te d w ith th e c u rv e lab eled "tails o v e r sa m p le d . " Ju st the opposite happens w hen the s ite s to w ard the m iddle a re o v e r-s a m p le d , as is also d e m o n s tra te d in Fig. 9 - lb. T h is p ro b lem could be so lved by u sin g a w eighted s p e c ie s -m e a n sta n d a rd iz a tio n , w ith the w eights p ro p o rtio n a l to the site u n iqueness as b e fo re . W ith th is ad ju stm en t, th e u n d e rs a m p le d h ab itats can be m o re h eav ily w eigh ted and the sp e c ie s m e a n w ill not be dom inated by the o v e r-s a m p le d h a b ita ts. 328 F ig . 9-1. D e m o n stra tio n of th e effect of uneven h ab itat sam p lin g on the s p e c ie s - m e a n sta n d ard iza tio n . 329 50. E n v ir o n m e n t a l g r a d ie n t a. A b u n d a n c e o f h y p o t h e t ic a l s p e c i e s a lo n g an e n v ir o n m e n t a l g r a d ie n t . T ! 0 ) N T > is I » £ si 6 T a il s o v e r - s a m p le d 5 4 3 E v e n ly s a m p le d P e a k o v e r - s a m p le d 2 1 E n v ir o n m e n t a l g r a d ie n t b . T h r e e s p e c i e s - m e a n s t a n d a r d iz e d c u r v e s fr o m s a m p lin g d if f e r e n t s i t e s a lo n g t h e e n v ir o n m e n t a l g r a d ie n t . T h e p o in t s on th e c u r v e s in d ic a t e th e p o s it io n s of th e s i t e s s a m p le d to fo r m t h e r e s p e c t i v e c u r v e s . 330 The m ain p ro b lem w ith th is ap p ro ach is the fact that the b e st w eights a re g e n e ra te d fro m the i n te r - s ite d ista n c e s, w hich th e m se lv e s a re c a lc u la te d from the s p e c ie s -m e a n sta n d a rd iz e d v a lu e s. A solution to th is p ro b lem is p ro p o se d below . B. M ethods 1. O bjective d e te rm in a tio n of the sc a le of the d istan c e w eights As m en tio n ed in C hapter VII, an u n se a le d s ite w eight is sim p ly the a v e ra g e o r to ta l of the d ista n c e s betw een the c o r r e sponding site and all o th er site s . T his is illu s tra te d in F ig . 9 - 2a. The colum n (or row ) su m s r e p r e s e n t th e to ta l d istan c e fo r the site c o rre sp o n d in g to th e colum n. If th e se su m s a re to be co n v erted to a s c a le of, say , 2 (using eq. 7-1), th e site w eights would be 1.38, 1 , 1. 012 , and 2 , re s p e c tiv e ly , fo r site s A - D. W hen site D is w eighted by a value of 2 , th is m ean s th at the data fro m th is site is u se d tw ice in the c alcu latio n s applying the w eig h ts. T his is equivalent to adding one additional site id e n tic a l to site D to th e d a ta m a trix . J u s t c o n sid e rin g the w eight of site D , the d istan c e m a tr ix now can be r e p re s e n te d as in F ig . 9 - 2b. The d ista n c e s betw een the new h y p o th etical site (D') and th e o th e r s ite s a re th e sam e as for D . Notice th at th e addition of th is site changes the row and colum n to ta ls fo r the o th er site s sin c e an additional row and colum n have been added. 331 F ig . 9-2. The effect of w eighting on the colum n to ta ls of the d ista n c e m a trix . 332 Site A B C D A 0 20 33 69 B 20 0 14 57 Q ) 4 - 3 m C 33 14 0 47 D 69 57 47 0 122 91 94 173 O rig in al d ista n c e m a tr ix w ith colum n A B C D D' A 0 20 33 69 69 B 20 0 14 57 57 C 33 14 0 47 47 D 69 69 47 0 0 D' 69 57 47 0 0 191 148 147 173 173 b. Scime d istan c e m a trix w ith the addition of the hyp oth etical site D' , w hich is id e n tic a l to site D. 333 In m a th e m a tic a l te r m s , the ad ju stm en t of each colum n to ta l from the addition of h y p o th etical s ite s (or fra c tio n s of a site) is T - /i = (W. - 1 )D .. , (9-1) w h ere is the am ount added to the to ta l of colum n j ( c o r r e sponding to site j ) fro m the addition of - 1 s ite s id e n tic a l to site i , is the w eight of site i , and is the distance betw een s ite s i and j . The value of 1 m u s t be su b tra c te d from the w eight b e ca u se one su ch site a lre a d y is p r e s e n t and only the d ifferen ce actu ally is added. Since fro m eq. 7 -1 , W. = ^ i - ^ m in (S - 1) + 1 , th en from eq. 9-1 , T - T ' m ax ^m in T ./. = — i----(S-1) , (9-2) J /1 T - T • - iJ m ax ^ m in w h e re is the o rig in a l colum n to ta l for colum n i, and T m ax the s m a lle s t and la rg e s t of all the o rig in a l colum n to ta ls, and S is th e u p p e r lim it of the ch osen sc a le . T his c a lc u la tion is re p e a te d fo r each site and the final colum n to ta l is the sum 334 of th e o rig in a l colum n to ta l plus the additions from a ll the hypotheti c a l s ite s , i. e . , = Tj Tj/, . (9-3) 1-1 w h e re T j is the final colum n to ta l for site j , and n is the n u m b er of o rig in a l s ite s . The o rd e r in w hich the site to ta ls a re ad ju sted is not im p o rta n t b e ca u se the am ount added each tim e is dependent on the v alu es of the v a ria b le s Tj^ and (eq. 9-2), n e ith e r of w hich is affected by p rev io u s calcu latio n s. If a ll h a b itats (each r e p r e s e n te d by a site) a re sa m p le d w ith equal in te n sity , the colum n to ta ls would ap p ro ach equality. Since the p u rp o se of the w eighting sc h em e is to equalize the r e p r e se n ta tio n of th e v a rio u s h a b ita ts, a good w eighting would ten d to c au se the colum n to ta ls to co n v erg e when applied in the m a n n e r d e s c rib e d above. S e v e ra l w eighting s c a le s can be te s te d in th is m a n n e r and the sc a le re s u ltin g in the m o s t equal colum n to ta ls could be c h o sen fo r u se in subsequent a n a ly se s. A sim p le m e a n s of evaluating th e con verg ence of the colum n to ta ls is to c alcu late the sta n d a rd deviation (SD) of the m e a n d ista n c e s in the colum ns (after th e addition of the hypothetical s ite s ) as show n on th e follow ing page. 335 f n , 2 / n i 2 SD . /n (9-4) n - 1 w h ere t / = 'Dj' . (9-5) M M is the to ta l n u m b er of s ite s (o rig in al plus hypothetical), o r in m a th e m a tic a l te r m s , n M = n - l + y * (W. - 1). (9-6) i = l The o rig in a l n u m b er of s ite s co n trib u te n - 1 in ste a d of n to the to ta l b e ca u se one of the d ista n c e s (that betw een each site and itself) w ill alw ays be z e ro . The a v e ra g e d istan ce (Tj ) is u se d in eq. 9-4 in ste a d of the to ta l d istan c e b e ca u se the colum n to ta ls w ill in c re a s e as la r g e r s c a le s a re te s te d . T his could in c r e a s e the SD and m ake r e s u lts from the d ifferen t s c a le s in c o m p a ra b le . The g e n e ra l lev els of th e m e a n d ista n c e s w ill not fluctuate g re a tly betw een applications of th e d ifferen t s c a le s . F ig u re 9-3 show s sa m p le c alcu latio n s fo r s c a le s of 2 and 3 w ith th e d a ta of F ig . 9-2. A s im ila r p ro c e s s is applied for sc a le s of 4 , 5 , and 10 (calculations not shown), and the re s u ltin g SD 's fo r each s c a le a re plotted in Fig. 9-4. It can be se e n that a 336 F ig . 9 -3. Sam ple c alcu latio n s fo r the application of two d ifferen t s c a le s to the d istan ce m a trix in Fig. 9-2. 337 I o % x ! CD T J (D m CTi 0 2 oTrt T 5 C olum n D istan ces <D Ü in II 0 2 (D 4 k A Site B C D C om m ents 0 122 91 94 173 In itia l colum n to ta ls (Tf) 2 A' . 38 0 7. 6 12. 5 26.2 A dditional colum n B' 0 0 0 0 0 d ista n c e s fro m h y p o th etical s ite s C ’ . 012 1. 2 V 5 0 1.7 ( T j /i) D » 1 69 57 47 0 192. 2 156. 1 153. 5 200. 9 F in a l colum n to ta ls ( T j* ) 4 3.8 3 5 .5 3 5.0 4 5.7 A v erag e colum n d ista n c e s (Tj ) SD ^t = 5.6 122 91 94 173 In itia l colum n to ta ls ( T j ) 3 A' . 756 0 15. 1 2 4 .9 52.1 A dditional colum n B' 0 0 0 0 0 d ista n c e s from h y p o th etical s ite s c . 074 2. 4 1. 0 0 3.5 ( q / i ) D' 2 138 114 94 0 262. 4 221. 1 212.9 . 228.6 F in a l colu m n to ta ls (Tj') A v e rag e colum n d ista n c e s ( T j ) SDpjr' - 3. 76 338 4 5 .0 37. 9 36.4 39.2 F ig . 9-4. S tand ard deviations of the m ean d ista n c e s (SD;^ ) of the s ite s v s. the sc a le of the w eights. Note th at a sc a le of 3 r e s u lts in the m o st hom ogeneous se t of d ista n c e s and as such would be the p r e f e r r e d sc a le . 339 14 - 12 10 1 2 3 4 5 6 7 9 SCALE 340 s c a le of 3 is a s s o c ia te d w ith th e m in im a l SD and would be chosen for fu rth e r u s e . A ctually the b e st sc a le i s so m e w h e re betw een s c a le s of 2 and 4. S cales of, say, 2.25 , 2 . 5 , 3 . 0 , and 3.50 could be te s te d and the m inim um SD could be pinpointed with g r e a te r a c c u ra c y if d e s ire d . 2. C alculation of the in fo rm a tio n w eights ' The m eth o d of C olw ell and F u tu y m a (1971) for c a lc u lating site w eights is Xj (log Xj - log Z) - ^ N j_j log (N ij/Y i) Wj = ___ i z l _________________ , (9-7) n Xj log Xj - Z log Z 3 = 1 w h e re Wj is the u n iq u en ess o r w eight of site j , Xj is the sum of the sp e c ie s counts in site j , Z is the to ta l of all counts in th e d ata m a trix , N^j is the count fo r sp e c ie s i in site j , is the to ta l of all counts for sp e c ie s i in all s ite s , s is the n u m b er of sp e c ie s, and n is the n u m b er of site s . The w eights a re sta n d a rd iz e d to fra c tio n s of the to ta l of all w eights : > Wi W4 = 3 . ( 9 - 8 ) ^ n 3 = 1 341 The m o re com m on sp e c ie s w ill dom inate the above c a lc u la tio n s. If th is is not d e s ira b le , each sp e c ie s can be w eighted: ! ^ i ^ i i Nij = (9-9) w h e re Rj_ is the w eight d e s ir e d fo r sp e c ie s i : N^j in ste a d of Nij th en is u se d in th e c a lc u la tio n s of W. In a p re lim in a ry a n a ly sis, th is m ethod w as te s te d w ith th e 40- site data. The w eights w e re c alcu lated u sin g raw n u m b e rs , s q u a r e -r o o t tra n s fo r m e d n u m b e rs, sp e c ie s w eighted by th e log of the m axim um value, and equal w eights fo r each s p e c ie s . The p a tte rn of the w eights in the sa m p lin g a re a did not v a ry m uch, but the sc a le of the w eights did fluctuate som ew hat. The r e s u lts in w hich the sp e c ie s a re w eighted by th e log of th e ir m axim um value w e re fa irly ty p ic a l and a re u se d fo r c o m p a riso n w ith the d istan c e w eights la te r in the c h a p te r. 3. W eighting of the sp e c ie s m ea n s fo r the s p e c ie s -m e a n sta n d a rd iz a tio n To help o v e rc o m e p o ten tial p ro b lem s from uneven h a b ita t sam pling, a w eigh ted s p e c ie s m e a n can be defined: _ Xi = 1=1------------ , (9-10) 3=1 342 w h e re X^j is the count fo r sp e c ie s i in site j , n is the n um ber of s ite s w h ere X^j is g r e a te r than z e ro , and Wj is the site w eight fo r the jth site . The sta n d a rd iz e d value for sp e c ie s i in site j is X ij/X i . 4. C alculation of in te r - s ite d istan ces for the purpo se of site w eighting In the c a se of uneven habitat sam p lin g , th e i n te r - s i te d ista n c e s u sed fo r w eighting can be c alcu lated as follow s. a) S p e c ie s-m ax im u m sta n d a rd iz e d d ata could be u sed in s te a d of sp e c ie s m ea n . In C hapter IV th is sta n d a rd iz a tio n w as r e je c te d b e ca u se its s c a le is in se n sitiv e to the k u rto sis of the s p e c ie s cu rv e . If the h a b ita ts a re sa m p le d unevenly, how ever, this c an becom e a positive fe a tu re b e ca u se the a p p aren t shape of the c u rv e can be m isle ad in g , as d e m o n stra te d in F ig. 9-1. In fact, if one is re lu c ta n t to u se w eighting sc h em e s and the h a b ita ts in the sa m p lin g a re a a re known to be sam p led unevenly, the s p e c ie s - m ax im u m sta n d a rd iz e d data should be c o n sid e re d as a re a so n a b le a lte rn a tiv e in the c la s s ific a tio n an aly sis of the s ite s . b) U nw eighted s p e c ie s -m e a n or sp e c ie s-m a x im u m sta n d a rd iz e d v alu es could be u se d to gain an in itia l e stim a te of the i n te r - s ite d ista n c e s. T h ese d istan c es could be u se d to c alcu late site w eights, w hich in tu rn could be u se d to c alcu late new w eighted s p e c ie s -m e a n sta n d a rd iz e d v a lu e s. This p ro c e s s could be re p e a te d 343 (with the w eighted s p e c ie s -m e a n data) u n til su c c e ss iv e w eight e s ti m atio n s co n v erg e. T e s ts w ith the 4 0 -site data show that the in itia l d ista n c e s a re not changed m uch by th is ite ra tiv e p ro c e d u re when applied to s p e c ie s -m e a n sta n d a rd iz e d data w ith a p r io r s q u a re -ro o t tra n s fo rm a tio n . The p a tte rn of site w eights b a sed on unw eighted sta n d a rd iz e d v a lu e s is alm o st id en tica l to that b a se d on five i t e r a tio n s, as d e m o n s tra te d by a c o m p a riso n of F ig s . 8-3 and 9-6. T his, how ever, m a y not alw ays be the c a se and the ite ra tio n s should be a ttem p ted as a p rec au tio n . A lso, it m a k e s no differen ce if the in itia l w eights a re c alcu late d from sp e c ie s-m a x im u m sta n d a rd iz e d valu es o r unw eighted s p e c ie s -m e a n s ta n d a rd iz e d data. The final r e s u lts w ill be id e n tic a l in e ith e r c ase . C. R e su lts and EtiLscussion 1. Scale of d istan ce w eights for the 4 0 -site data A ty p ic a l plot of the SD?p v s. the sc a le of the w eights is shown in Fig. 9-5. The c u rv e has not yet re a c h e d a m inim um at a s c a le of 40 . H ow ever, the c u rv e s t a r ts to flatten out a fte r a sc a le of 5. The w eight sc a le su b jectiv ely ch o sen in the calcu latio n s of the in te r - s p e c ie s d ista n c e s (C hapter VII) s e e m s to be rea so n a b le ; a s c a le of around 10 is beyond the a r e a of ra p id change in the cu rv e , and a m uch h ig h er sc a le would not im p ro v e the r e s u lts ap p rec iab ly , sin c e the c u rv e a lre a d y has flatten ed out in th is ran g e . 344 F ig . 9-5. P lo t of v s. the sc a le of the w eights u sin g th e 40- site data. The d istan c es u sed in the c alcu latio n s h e re a re d e te rm in e d w ith s p e c ie s -m e a n sta n d a rd iz e d data. A cu t-o ff slope of 3 .4 x 1 0 is u se d in the subsequent ite ra tio n s to d e te rm in e the final w eights. 345 o o ''oo lO C M O CM OO O 0 3 lO oo o co oo s u c/] CM (g.oTx) ^ a s 346 E v id en tly^ w ith la r g e r and m o re com plex m a tric e s , the SD;jr w ill only slow ly r e a c h a m in im u m if it e v e r does so (the sc a le w as not te s te d beyond 9 7). In light of th e s e r e s u lts , an ade quate sc a le of th e w eights could be d e te rm in e d by se le c tin g an adequately s m a ll slope of the c u rv e as a cu t-o ff point. W hen the c u rv e re a c h e s the se le c te d slo pe, the left-h an d point of the two points u se d to d e te rm in e th e slope w ill be c o n sid e re d the final sc a le . It would se em to be advisab le to keep the s c a le below ex tr e m e v alu es; o th erw ise the in fo rm a tio n at th e s ite s w ith low w eights would ten d to be e lim in a te d co m p letely . F u r th e rm o re , the u pper lim it of the s c a le should not be g r e a te r than the n u m b er of site s sin c e , w hen c o m p a re d to the m o st u n d e r-s a m p le d habitat, the m o st h eavily sa m p le d h ab itat cannot be o v e r-s a m p le d by a g re a te r n u m b e r of s ite s th an a re actu ally p re s e n t in the su rv e y . In the c alc u la tio n s of th e w eight s c a le s w ith the 40- site data, a c u t-o ff slope of 3 .4 x 1 0 " ^ is u se d . T his is in the v icin ity of a sc a le of 10 , as in d icated in Fig. 9-5. The final w eights (on a sc a le of 9) .a re shown in Fig. 9-6b. F ive ite ra tio n s a re p e rfo rm e d w ith w eighted s p e c ie s - m e a n sta n d a rd iz e d data (with a p rio r s q u a re - ro o t tra n s fo rm a tio n ) to obtain the fin al w eights. 347 2, C o m p ariso n of in fo rm a tio n and d istan ce w eig hts fo r the 4 0 -site data and sim u la te d te s t data F ig u re 9-6 show s the p a tte rn s of the w eights as c a lc u lated by the two d ifferen t m eth o d s and applied to th e 4 0 -site data. T h e re a re s e v e ra l d is c re p a n c ie s in the r e s u lts . The m o st se rio u s is the re la tiv e ly high w eight given to s e v e ra l of the 1000-ft s ite s by th e d istan c e m ethod and the g e n e ra lly low w eights for th e s e site s w ith the in fo rm atio n m ethod. The hig h er w eights given by the in fo rm a tio n m ethod seem to be at s ite s w h e re a n u m b er of an im als becom e m a x im a lly abundant (see Fig. 8-15). If th e w eight c a lc u lations a re influenced som ehow by the lo cal re la tiv e abundance le v e ls independently of the actu al site u n iq u en ess, th is would ex plain the low w eights fo r the 1000-ft s ite s , at w hich few er sp e c ie s have peak abundances (see F ig s. 8-7, 8-15). To te s t th is id e a and to c o m p a re the two m ethods in a situ atio n w h ere th e site u n iq u en ess is known, w eights a re calcu late d fro m th e sim u la te d data shown in F ig. 9 - 7a. The to ta l and re la tiv e abundances a re se t d e lib e ra te ly to v a ry along th e e n v iro n m e n ta l g rad ie n t as is the c a se in the 4 0 -site data. The s a m p le s a re sp aced evenly along th e g ra d ie n t fo r even sam p lin g of the h a b itats, except fo r the s ite s to w a rd the e x tre m itie s . The end s ite s a re re la tiv e ly u n d e r-s a m p le d b e c a u se th ey have few er ad jacen t (and th e re fo re 348 F ig . 9-6. The p a tte rn s of site w eights for the 4 0 -s ite data, c a lc u lated by th e two a lte rn a tiv e m ethods. 349 a. In fo rm atio n m ethod (w eights x 10 ) VO •XI to SI 77 b. D istan ce m ethod (w eights x 10; sc a le = 9 ) 350 F ig . 9-7. The p a tte rn s of site w eights c alcu late d by the two a lte rn a tiv e m ethods when u sing sim u la te d data. 351 40 o u 1 É 20 . < 9 19 7 13 15 17 1 3 11 21 5 S it e s o n e n v ir o n x n e n ta i g r a d ie n t a. A s s o r t e d s p e c ie s - a b u n d a n c e c u r v e s a lo n g a h y p o th e t ic a l e n v ir o n m e n ta l g r a d ie n t . " o 0) c C Q s o m o o o D is t a n c e I n fo r m a tio n o 3 9 5 7 13 17 19 21 11 15 1 S it e s b. S ite w e ig h t s c a lc u la t e d b y th e in f o r m a tio n and d is ta n c e m e t h o d s . D is t a n c e s c a lc u la t e d fr o m th e s o e c i e s - m e a n s ta n d a r d iz e d v a lu e s . 352 sim ila r) sa m p le d h a b ita ts. F o r in sta n c e , site 1 has only one neighboring habitat sa m p le d (site 2 ), but any o th er site (except site 22 } would have two neighboring h ab itats sa m p le d . Thus, the id eal w eights for th e s e hypothetical data would be a bit h ig h er tow ard the e x tre m itie s and ap p ro x im ately even and low er tow ard the m iddle of the g rad ie n t. F ig u re 9 - 7b show s a plot of the site w eightings from the two m eth o d s. The d ista n c e m ethod p ro d u ces w eights which g e n e ra lly a re c o n siste n t w ith expectation s. The in fo rm atio n - m ethod w eights a re e r r a tic . W eights fo r s ite s 1, 5, 6, 7, and 8 a re d e p re s s e d ab n o rm ally . As a ru le , th e s e a re s ite s which contain no sp e c ie s w hich a re at o r v e ry n e a r th e ir peak abundance counts. T hus, it s e e m s that the in fo rm atio n m ethod is se n sitiv e to the quantitative d istrib u tio n of the sp e c ie s counts independent of the actu al u n iq u en ess of the s ite s . T his m ethod, thus, is c le a rly u n su itab le for u se w ith eco log ical data. 3. R e ca lc u latio n of the n o rm a l a n aly sis u sin g the w eighted s p e c ie s -m e a n sta n d a rd iz a tio n and the 4 0 -site data The d istan ce w eights shown in F ig. 9-6 a re b ased on a w eighted s p e c ie s -m e a n sta n d a rd iz a tio n and s q u a r e - r o o t tra n s fo rm e d data. T h ere a re m in o r changes only in the d e n d ro g ra m when c o m p a re d w ith the unw eighted s p e c ie s -m e a n a n a ly sis. Thus, with 353 th e s e data, it s e e m s th at the unw eighted sta n d a rd iz e d values a re not m uch affected by uneven habitat sam p lin g . H ow ever, the a u th o r's e x p erien c e w ith o th er se ts of d a ta has shown th at th is is not alw ays th e c a se . A w eighted s p e c ie s -m e a n sta n d a rd iz a tio n a lso is applied to the ra w data. W hen c o m p a re d to the c la ss ific a tio n w ith the p r io r s q u a r e - r o o t tra n s fo rm a tio n , the r e s u lts ap p ear to be m o re affected by the application of the w eights. A s q u a re - ro o t t r a n s f o r m atio n b e fo re the sta n d a rd iz a tio n would c au se so m e con v erg en ce in the shape of the sp e c ie s c u rv e s and, as such, p ro b ab ly would act as a "b u ffer" to the effects of uneven h abitat sam pling. T his would explain the g r e a te r effect of the w eighting on the s p e c ie s -m e a n sta n d a rd iz a tio n of raw data. 3 54 C H A PTER X DISCUSSION A. What Is Being M e a su re d I. The d elim itin g of co m m u n ities B efore one can study th e s tr u c tu r e of a com m unity, it is n e c e s s a r y f ir s t that th e com m unity c an be id en tified . Depending on the e c o lo g is t's definition of a "com m unity, " co m m u n ities could be r e p re s e n te d e ith e r by the site groups fro m the n o rm a l an aly sis o r the sp e c ie s groups fro m the in v e rs e a n a ly sis. If one c o n sid e rs a com m unity as an "entity defined in the neighborhood of a point location in the la n d sc a p e " (W hittaker et al. , 1973:328) or "a lim ite d a r e a (of vegetation) which se e m s hom ogeneous" (W hittaker, 1967: 211), then th e s ite groups would define the sp ace w h ere the com m unity i s lo cated . The sp e c ie s in su ch a com m u nity would, of c o u rs e , be the sp e c ie s p re s e n t at th e se s ite s . T his in fo rm a tio n is available e a sily in tw o-w ay coincidence ta b le s ( e . g .. Fig. 8-7). A m o re com plex and le s s exact m ethod of obtaining m u ch of the sa m e sp e c ie s in fo rm a tio n i s to u se the site grou ps in a m u ltip le d is c rim in a n t a n a ly sis w ith the sp e c ie s abundances as the v a ria b le s 355 re p re s e n tin g th e s ite s (N o rris and B ark h am , 1970; G ringal and G oldstein, 1971; C a ssie , 1972; P o lg a r, 1974; G ringal and Ohm ann, 1975). H ere the tre n d s of the sp e c ie s abundances in d isc rim in a n t sp a ce can be re la te d to the positions of the v a rio u s groups in the sp ace. On th e o th er hand, if one defines a com m unity as "a group of sp e c ie s w hich a re often found living to g e th e r" (P ag er, 1963: 418), th en the sp e c ie s groups would define the c o m m u n ities. The two ap p ro ach es to the co m m u nity concept often w ill give quite different r e s u lts . T his is w ell illu s tra te d w ith the p re s e n t d ata by exam inatio n of th e tw o -w ay ta b le in Fig. 8-7. If the site g roups c o rre sp o n d to the c o m m u n ities, then a single c o m m unity can c o n s is t of sp e c ie s in s e v e r a l d ifferen t sp e c ie s gro u p s. F o r exam p le, the sh a llo w -w a te r co m m u nity (groups J + K ) would include m e m b e rs of a lm o st all the sp e c ie s g ro u p s. If the sp e c ie s groups a re the c o m m u n ities, th en the c o m m u n ities, as a ru le , a re d istrib u te d o v e r m o re than one s ite group. T hus, the fo rm e r definition lead s to a com m un ity re la tiv e ly r e s t r i c te d in sp ace but le s s r e s t r i c te d as to sp e c ie s com position, and the la tte r definition leads to ju st the opposite. The u se of sp e c ie s groups m a y have som e p ra c tic a l advantages as fa r as fu rth e r a n aly sis of the data w ithin each 356 co m m u n ity is c o n ce rn e d . The sp e c ie s groups u su a lly co v er a la r g e r n u m b er of sites, and thus m o re in fo rm a tio n about the sp e c ie s in su c h groups would be available w hen c o m p a re d to the site -g ro u p defined c o m m u n ities. A lso, the s m a lle r n u m b er of sp e c ie s in a defined s p e c ie s -g ro u p com m unity w ould m ak e an an aly sis le s s c u m b e rso m e . On the o th er hand, th e sp e c ie s in the different sp e c ie s groups do o v e rla p in sp ace and can in te ra c t w ith each o th er. The in tra -c o m m u n ity a n aly sis m ay be in co m p lete without taking th is into c o n sid e ra tio n . In th is c a se , th e sp e c ie s re p re s e n ta tiv e of a site group m u st be chosen and analyzed to g eth er, usin g only the s ite s in the c o rre sp o n d in g site group. If m o re site s a re re q u ire d to m ak e th e a n aly sis m eaningful, fu rth e r sam p lin g could be c a r r ie d out in the a re a s defined by the site g roupings, or la r g e r site groups could be d elim ited in the c la ss ific a tio n a n aly sis (choose groups fro m a h ig h er lev e l on th e d en d ro g ram ). 2. H abitat and niche d im en sio n s W h ittak er et al. (1973) provide a convenient and thoughtful fra m e w o rk in w hich to view the r e s u lts of the types of an aly sis u se d in p re s e n t study. H abitat d im en sio n s (m aking up a h ab itat h y p e rsp a c e ) a re defined as c o rre sp o n d in g to th e e n v iro n m e n ta l v a ria b le s w hich se p a ra te the d ifferen t co m m u n ities. Niche d im en sio n s (defining a niche h y p e rsp ac e) w ould c o rre sp o n d to the 357 v a ria b le s w hich s e p a ra te th e sp e c ie s w ithin the com m unity. T his is the h y p e rsp a c e definition of th e niche fo rm u la te d by H utchinson (1958). F in ally , the ecotope hyp er sp ace is defined as niche and h ab itat h y p e rs p a c e s c o n sid e re d jointly. The v a ria b le s which s e p a ra te sp e c ie s betw een c o m m u n ities a re not n e c e s s a r ily the sa m e as th o se th at s e p a ra te sp e c ie s w ithin each com m unity, and s e p a ra te a n a ly se s should be c a r r ie d out at each lev el. The d isc rim in a n t a n aly sis of site groups o r sp e c ie s groups would u su a lly be an a n aly sis of habitat d im en sio n s. The o rd in atio n a n a ly sis likew ise is a study of habitat dim en sio n s, D isc rim in an t a n aly sis of the sp e c ie s w ithin a defined c o m m unity would be an a n a ly sis of the niche. The d ata could be studied at the ecotope le v e l by u sin g a ll sp e c ie s in a single d isc rim in a n t an a ly sis. The f ir s t s e v e r a l axes would p ro b ab ly define v a ria b le s a sso c ia te d w ith h ab itat d im en sio n s and th e la te r axes would c o r r e spond to so m eth in g c lo s e r to niche d im en sio n s. Such an an aly sis would be c u m b e rso m e and difficult to in te r p r e t if m an y sp e c ie s and s e v e r a l d ifferen t c o m m u n ities a re involved. It w ill be difficult so m e tim e s to define the habitat or niche d im en sio n s in n o n-b iotic t e r m s only sin ce the d im en sio n s can be re la te d to biotic fa c to rs w hich a re u n re la te d to the p h y sical n a tu re of the en v iro n m en t. F o r exam ple, F e n c h el (1975) could not 358 explain co m p letely the d istrib u tio n of in te rtid a l m ud sn a ils by con s id e rin g th e p h y sical en viro nm ent alone. M ig ratio n s, colonization, in te rs p e c ific com petition, and lo c a l extinctions w e re hyp oth esized as im p o rta n t in the sp e c ie s d istrib u tio n s. On the o th er hand, M ak arew icz and Liken (1975) w e re able to s e p a ra te the niches of zooplankton sp e c ie s by c o n sid e rin g only fa c to rs re la te d to food siz e, depth, and tim e . T h e re is , of c o u rs e , no r e a s o n why biological p a ra m e te r s cannot be included as e n v iro n m en tal v a ria b le s in the d isc rim in a n t a n aly sis. Quite often niche d im ensions s e p a ra tin g sp e c ie s w ithin a com m unity w ill be obvious w ithout com plex m a th e m a tic a l t r e a t m en t. F o r in stan c e, filte r-fe e d in g and d ep o sit-feed in g an im als can be a ssu m e d to be s e p a ra te d by dim ensions re la te d to the type of food taken in. M akarew icz and L iken (1975) s e p a ra te d c a rn iv o re s fro m h e rb iv o re s , and w ithin th e s e c a te g o rie s la rg e sp e c ie s w e re s e p a ra te d from s m a ll sp e c ie s, on the assu m p tio n that e ach s e p a r a tio n is a sso c ia te d w ith a differen t type o r siz e of food in g ested . Once the sp e c ie s in a co m m u n ity a re b ro k en down into the obvious su b g ro u p s, fu rth e r se p a ra tio n m ay be obtained by m a th e m a tic a l tre a tm e n t (such as w eighted d isc rim in a n t an aly sis) if re le v a n t v a ria b le s have been m e a s u re d . The r e s u lts can give clu es con c e rn in g the n a tu re of the niche dim ensions and lead to m o re d etailed 3 59 stu d ies such as in F e n c h el et al. (1975). As pointed out by G reen (1974), the s ta tis tic a l d e s c r ip tion of m u ltip le -d is c rim in a n t axes is id e n tic a l to H utchinso n's (1957) definition of niche d im en sio n s, i . e . , "the n u m b er of in d e pendent fa c to rs th at s e p a ra te groups (sp e c ie s)" (G reen, 1974: 77). T his being th e c a se , the r e s u lts m ay lead to a quantification of niche d im en sio n s which would be u navailab le o th erw ise. The d is c rim in a n t sp a ce could th en be u se d to d e te rm in e niche p a ra m e te rs (breadth, o v erlap , sh ap e, e tc .) and to te s t h ypotheses re la te d to com m unity s tr u c tu r e (M aguire, 1967; G reen, 1974). S im ila rly , when studying the re la tio n sh ip s betw een co m m u n ities, the d isc rim in a n t sp a ce w ill be id e n tic a l to the habitat sp a ce as defined by W h ittak er et al. (1973). The study of niche d im en sio n s s o m e tim e s w ill re q u ire th at the e n v iro n m en ta l and biotic v a ria b le s be m e a s u re d on a m uch fin er sc a le th an is the u s u a l p ra c tic e . S m a ll-s c a le en v iro n m en tal h e te ro g e n e ity w ill be re la te d so m e tim e s to th e c o ex iste n c e of dif fere n t sp e c ie s w hich u tiliz e or occupy different p a rts of the m ic r o en v iro n m en t. In th e p re s e n t study, the re p lic a te biotic sa m p le s at a site a re pooled and the e n v iro n m en ta l v a ria b le s e ith e r a re m e a s u re d only on one re p lic a te o r a re from pooled re p lic a te s . T his p ro c e d u re u su a lly w ill be sufficient to study the h ab itat dim en sio n s. 360 but som e of the m o re su b tle fa c to rs re la te d to the niche d im en sio n s m ay be m is se d . B. P re d ic tio n The r e s u lts of the a n aly se s can be used in m o d els w hich a re designed to p re d ic t the com m unity com position, given th at the re le v a n t e n v iro n m en ta l conditions can be e stim a te d or m e a s u re d fo r the eco sy stem involved in the p re d ic tio n s. T his could s e rv e two u se fu l functions. 1. If the r e s u lts lead to re a so n a b ly a c c u ra te p re d ic tio n s, then th e ecologist can be m o re confident th at (a) the r e s u lts of the o rig in a l a n aly se s a re due to so m e m eaningful re la tio n sh ip s and not ju st to so m e sp u rio u s c o rre la tio n s , and (b) the p ro p o sed m o d el is re a s o n a b le in light of th e a ctu al situ atio n in n a tu re . 2. E co lo g ists a re in c re a s in g ly being asked to p re d ic t the im p a c t of e n v iro n m en ta l changes due to p ro p o se d hum an activity. The developm ent of p re d ic tiv e m odels w ill be v e ry helpful in th is endeavor. The d isc rim in a n t an aly sis m o d el is su ited v e ry w ell fo r p r e dictive situ atio n s. D isc rim in a n t coefficients (see eq. A - 11) can be c a lc u la te d from a re p re s e n ta tiv e se t of data, and th e r e a f te r any future sa m p le can be given a position in the d isc rim in a n t sp ace if the sa m e type of e n v iro n m e n ta l m e a s u re m e n ts o r e s tim a te s of such 361 (as in the in itia l a n aly sis) a re available for th at sam p le (Cooley and L ohnes, 1971:262-286). If the d isc rim in a n t an aly sis is con c e rn e d w ith site g ro u p s, th is in fo rm a tio n can lead to calcu late d p ro b a b ilitie s that the sam p le w ill be in each site group. The sam p le could be p re d ic te d to be biotic ally s im ila r to the m o st probable site group. If the d isc rim in a n t an aly sis involves sp e c ie s or sp e c ie s g roups, the p ro c e s s would be a bit m o re com plex since th e re u su a lly w ill be o v e rla p betw een the c o rre sp o n d in g groups of s ite s . H ere the p ro b a b ilitie s of belonging in each group could be tra n s la te d d ire c tly in to a level of o c c u rre n c e or abundance of the sp e c ie s o r sp e c ie s grou ps in question. M ost p re d ic tio n s w ill involve p e rio d s of tim e w hich a re different from the tim e p erio d s in w hich the o rig in a l data a re g a th e red . B ecau se of th is, the d isc rim in a n t co efficients should be b a sed on d ata w hich include te m p o ra l v a ria tio n and som e te m p o ra lly v a ry in g v a ria b le s (such as te m p e ra tu re , etc. ). A nalysis over tim e is d isc u sse d fu rth e r below. C. A nalysis of D ata G athered Over T im e T his se ctio n c o n c e rn s the an aly sis of the data when a su rv e y is re p e a te d at d ifferen t tim e s. W ith the ty p es of an aly se s u se d in the p re s e n t study, th e re a re four w ays to ap p ro ach th is situatio n. 362 1. A s e p a ra te a n a ly sis can be p e rfo rm e d fo r each su rv e y and the d ifferent r e s u lts th en can be c o m p a red su b jectiv ely . The la r g e - s c a le shifts in biotic o r en v iro n m en tal p a tte rn s should be evident. 2. A ll the data can be put to g eth er in one la rg e data m a trix and the a n aly sis then is p e rfo rm e d as u su al. T his w ill r e s u lt in (# sites) X (#tim es) e n titie s, w hich m a y becom e e x c e ssiv e as m o re s u rv e y s a re taken. E x p e rie n c e has shown that the m ixing of tim e and sp ace in the n o rm a l a n aly sis often leads to r e s u lts w hich a re difficult to in te r p re t. W eighted d isc rim in a n t an aly sis of the sp e c ie s or sp e c ie s groups w ith d ata taken o v er tim e should be a valuable tech n iq u e. G reen (1974) p ro p o sed and d e m o n stra te d th e u se of m u ltiv a ria te an aly sis of c o v arian c e as a m ea n s of se p a ra tin g the te m p o r a l and sp a tia l com ponents of the data p rio r to the m u ltip le d is c rim in a n t a n a ly sis. P re su m a b ly , the sam e m ethod could be u sed w ith w eighted d isc rim in a n t a n aly sis, and te s ts along th e s e lines c u rre n tly a re in p r o g r e s s . 3. E ach tim e p e rio d can be c o n sid e re d as a s e p a ra te entity. To a cc o m p lish th is , the data for each sp e c ie s can be su m m ed o r a v erag e d o v e r a ll s ite s at each tim e (W illiam s and Stephenson, 1973). The an aly sis can p ro c e e d as u su a l w ith the re s u ltin g data m a trix . If the sam p lin g a r e a is re la tiv e ly h etero g en eo u s, it m ay be 363 advisable to avoid su m m atin g over a ll s ite s , b e ca u se different te m p o ra lly -r e la te d p a tte rn s m ay be p re s e n t in d ifferen t p a rts of the sam p lin g a re a . In su ch c a s e s , a s e p a ra te a n aly sis would be re q u ire d for each d istin c t su b a re a . 4. The s ite s can be c o n sid e re d as s e p a ra te e n titie s. H ere the data fo r each sp e c ie s a re su m m ated o r a v erag e d o v e r tim e for each s ite (W illiam s and Stephenson, 1973). D. The P ro b le m of Uneven H abitat Sam pling As d e m o n s tra te d in C h ap ters VII and IX, th e re a re poten tia lly a d v e rse effects fro m uneven sa m p lin g of th e d ifferent habitats in the study a re a . The adjustm en ts p ro p o sed to o v e rc o m e th e se effects a re b a sed on ro u g h approxim ations w hich r a r e ly , if ev er, w ill be as good as sa m p lin g the h ab itats evenly in th e f ir s t place. Thus it is im p o rta n t to attem p t to plan an a p p ro p ria te sam p lin g p a tte rn . Random sa m p lin g of th e study a re a u su a lly w ill not lead to even habitat sam p lin g . A flexible sy ste m a tic sy ste m m ay be the m o st a p p ro p ria te . H e re th e sa m p le s a re sp aced evenly w ithin sub- a re a s , but the sam p lin g is m o re in ten se in the s u b a re a s of g re a te r e n v iro n m en ta l and biotic h e te ro g e n e ity (L am b ert, 1972). The h e te ro g e n e ity of an a r e a would have to be e stim a te d in th e field or be b a sed on p r io r su rv e y s. 364 If tim e p e rio d s a re being u se d as e n titie s (see C. ), s im ila r p ro b le m s could a ris e if the d ifferen t p e rtin e n t tim e p e rio d s a re unevenly sa m p le d . E. Site v s. Species S tandardization of the D ata in the N o rm al A nalysis In C h ap ter IV a sp e c ie s sta n d a rd iz a tio n of the data is r e c o m m en d ed in ste a d of e ith e r a site sta n d a rd iz a tio n o r no sta n d a rd iz a tio n . T h e re a re so m e situ atio n s, how ever, w h e re the site sta n d a rd iz a tio n m ay be p re fe ra b le . 1. A site sta n d a rd iz a tio n can help o v erco m e som e of the p ro b le m s c a u se d by the assu m p tio n s a sso c ia te d w ith m ethods b a sed on a E u c lid e a n -d ista n c e m o d el (C h ap ters II and IV; Or loci, 1967a). W hen such m ethods a re used, the a n aly sis is b u rd en ed w ith a site sta n d a rd iz a tio n , not b e ca u se it is the m o st ecologically m eaningful p ro c e d u re , but only to m ak e up fo r the inadequacy of the b a sic m o d el ch o sen fo r the a n a ly sis. If the E u c lid e a n -d ista n c e m o d el is avoided, g r e a te r flexibility in re la tio n to data sta n d a rd iz a tio n can be re a liz e d . 2. As m entioned in C h ap ter II, a site sta n d a rd iz a tio n can be u se fu l when the sa m p le siz e is v a ria b le am ong th e s ite s . In so m e c a s e s , th e effectiv e sa m p le siz e w ill not be equal fo r each site even w hen th e sam p lin g effo rt is equal. D istu rb a n ce s o r o b stru c tio n s w ithin an indiv idu al sa m p le a r e a can e lim in a te th e p o ssib ility of 365 finding c e rta in o rg a n is m s in the affected p a rt of the a re a . T his w ould be a g re a te r p roblem in e c o sy ste m s w h e re th e re c o v e ry tim e and grow th of the lo c a l sp e c ie s a re re la tiv e ly slow . Such prob ably would be the c a se in som e fo re s t e c o s y s te m s . 366 C H A PTER XI SUMMARY AND CONCLUSIONS Som e of the bio logical a sp e c ts involved in the n u m e ric a l an aly sis of eco lo g ical su rv e y data a re c o n sid e re d . The em p h asis is on finding the im p o rta n t c o rre la tio n s betw een the biotic and en v iro n m e n ta l p a tte rn s . B ecau se of th e ir g r e a te r app licability and pow er, m u ltiv a ria te m ethods a re used . A. P a tte r n s in th e B iological Data P a tte r n s in the biolog ical d ata (sp e c ie s abundance counts, b io m a ss , etc. , at se le c te d sam p lin g s ite s ) can be elucidated w ith c la ss ific a tio n and o rd in atio n tec h n iq u e s. C la ssific a tio n involves delim itin g (a) groups of biotic ally s im ila r sam p lin g s ite s (n o rm al a n aly sis), and (b) groups of sp e c ie s w hich ten d to be found to g e th e r in the sa m e h a b itats (in v erse a n aly sis). O rdination is u sed to place th e s ite s along continua w hich m ay be re la te d to the m o re im p o rta n t e n v iro n m e n ta l g rad ie n ts in th e sam p lin g a re a . Both m eth o d s re q u ire an input of a d istan ce m a trix , which conveys in fo rm a tio n on the m e a s u re d biotic d ifferen c e s betw een each p a ir of e n titie s being c o n sid e re d . In th e n o rm a l a n aly sis, the 367 e n titie s a re the site s ; in the in v e rs e a n aly sis, th e e n titie s a re the s p e c ie s . At th is point in th e an aly sis, the eco lo g ist m u st choose an a p p ro p ria te d istan c e index, and then d e te rm in e which, if any, d a ta tra n s fo rm a tio n s and sta n d a rd iz a tio n s a re w a rra n te d . The b io lo g ical c o n sid e ra tio n s involved in th e se ch o ices a re exam ined in d etail. The d isc u ssio n and reco m m e n d atio n s a re b a se d on a m o d el in which sp e c ie s abundance p a tte rn s follow b e ll-sh a p e d c u rv e s along th e im p o rta n t en v iro n m en tal g ra d ie n ts. As fa r as the calcu latio n s of the i n te r - s i te d istan ces (n o rm al an aly sis) a re co n cern ed , it is d e m o n stra te d that the s p e c ie s -m e a n sta n d a rd iz a tio n , u se d in conjunction w ith the B ray- C u rtis index, w ill give g r e a te r weight to the m o re re lia b le sp e cie s counts (C hapter IV). T his w ill r e s u lt in d ista n c e s w hich a re m o re c o m m e n s u ra te w ith the e n v iro n m en ta l d ifferen c e s betw een the re s p e c tiv e s ite s . When c o m p a re d with o th er a lte rn a tiv e s , the im p ro v e m e n t re a liz e d fro m th is technique w ill v a ry w ith the sc ale of the sam plin g p a tte rn . If the different habitat types in the sam p lin g a re a a re sa m p le d unevenly, the s p e c ie s - m e a n sta n d a rd iz a tion m ay be su b ject to so m e d isto rtio n s, and a c o rre c tiv e p ro c e d u re is su g g ested (C hapter IX). In the in v e rs e a n a ly sis, the use of the B ra y -C u rtis index w ith sp e c ie s-m a x im u m sta n d a rd iz e d data is shown to be co n sisten t 368 w ith b iologically m ean in g fu l a ssu m p tio n s. The c alcu latio n s h e re a lso can be affected by uneven habitat sam pling, and m eth o d s of ad ju stm en t a re d isc u sse d and evaluated (C h ap ters VII and IX). B. R elatio n sh ip s betw een the Biotic P a tte rn s and the E n v iro n m en t The biological p a tte rn s shown in the c la ss ific a tio n an aly sis can be re la te d to the en v iro n m en ta l p a tte rn s w ith m u ltip le d is c rim in a n t an aly sis (C hapter II). In the n o rm a l an aly sis, th is m eth o d is shown to be e sp e c ia lly u sefu l when u se d in conjunction w ith an a g g lo m e ra tiv e -h ie ra rc h ic a l c la ssific a tio n technique (C h ap ters IV and VIII). The re la tio n sh ip s betw een the sp e c ie s g roups (in v erse a n aly sis) o r the sp e c ie s and the environm ent a re b e st studied w ith a m od ified form of m ultiple d isc rim in a n t a n aly sis. The m odifications include site w eightings w hich allow the u tiliza tio n of both the qualitative and quantitative com ponents of the biological d ata (C hapter VII and Appendix A). The c alcu latio n s h e re also can be affected by uneven habitat sam pling, and a c o rr e c tiv e p ro c e d u re is p ro p o sed (Appendix A). The biotic p a tte rn s shown by the o rdination r e s u lts can be re la te d to the e n v iro n m en ta l m e a s u re m e n ts w ith m ultiple r e g r e s s io n a n a ly sis (C hapters II and VIII). 369 c. D ata R eduction In the n o rm a l a n aly sis, the data fo r so m e sp e c ie s can be e lim in a te d u su a lly w ithout sig n ifican tly affecting the r e s u lts . M ethods of finding w hich sp e c ie s c o n trib u te the le a st am ount of in fo rm a tio n to th e calcu latio n s a re d is c u s s e d and d e m o n stra te d (C hapter V). The effects of d ifferent lev e ls of sam p le re p lic a tio n on the n o rm a l and in v e rs e an aly ses also a re d em o n stra te d . D. E co n o m ical D isplay of the R e latio n sh ip s betw een the Site and Species P a tte rn s T w o-w ay coin cid ence ta b le s can be u se d to show the re la tio n sh ip s betw een the site and sp e c ie s gro u p s fro m the c la ssific a tio n a n a ly sis (C hapter VI). Two m o d ificatio n s of the u su a l p ro c e d u re a re su g g ested . The fir s t involves the u se of sym bols in place of n u m b e rs in the body of the tab le. The seco nd c o n c e rn s a m ethod of su m m a riz in g th e c e lls of the tab le. T he site o rd in atio n r e s u lts also can be r e la te d to the sp e c ie s p a tte rn s with tw o-w ay ta b le s. The d e ta ils of c o n stru c tin g such ta b le s a re given in C hapter VI. In addition, the p a tte rn s of the sp e c ie s in the site o rd in atio n sp a ce can be plotted using a re la te d technique. E. The C o rre c tiv e P ro c e d u re for Uneven H abitat Sam pling As noted above, m any of the calcu latio n s involved in th e se m ethods a re affected by uneven h abitat sam pling. All the c o rre c tiv e 370 p ro c e d u re s su g g ested involve the calcu latio n of site w eights w hich a re d ire c tly p ro p o rtio n a l to th e biotic and en v iro n m en ta l uniqueness of the site in question. A m ethod of calcu latin g such w eights w as su g g e ste d by C olw ell and F u tu y m a (1971). T his m ethod is evaluated and re je c te d b e ca u se it is dependent on fa c to rs which a re u n re la te d to site u n iq u en ess. An a lte rn a tiv e m ethod b a se d on i n te r - s i te d ista n c e s is p ro p o sed and fav o rab ly evaluated. F . U sefulness of the M ethods The r e s u lts from th e se m ethods can be u se fu l in 1. The d elim itin g of co m m u n ities and th en re la tin g th ese co m m u n ities to th e ir h a b ita ts. 2. Studying the h a b itat and niche re la tio n sh ip s of s e ts of individual sp e c ie s. 3. The c o n stru c tio n of h e u ris tic and p red ic tiv e m o d e ls. 371 A ld e rd ic e, D. F ., 1972. " F a c to r com bination s. R e sp o n se s of m a rin e p o ik ilo th erm s to en v iro n m en tal fa c to rs acting in c o n c e rt. " In: M arin e E cology, Vol. 1. E n v iro n m en ta l F a c to r s , P a r t 3, O. Kinne (ed): 1659-1722. New York: John W iley and Sons. A nderson, A. J. B. , 1971. "O rdination m ethods in ecology, " J. 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IV: A m ethod for the elucidation of s m a ll- s c a le fo re s t p a tte rn s ," J. E col. , 57: 635-654. W illia m s, W. T. and W. Stephenson, 1973. "The a n aly sis of th re e -d im e n s io n a l d ata (sites x sp e c ie s x tim e s) in m a rin e ecology, " J. Exp. M ar. Biol. & E col. , 1 1 :2 0 7 -2 2 7 . 383 APPEN D IX A M U L T IPL E DISCRIMINANT ANALYSIS - CALCULATIONS AND IN T ER PR E TA TIO N OF RESULTS A. In itial C alculations The m u ltip le -d is c rim in a n t a n aly sis m ethod re q u ire s an in itia l input of th re e su m s of sq u a re s and c ro s s -p r o d u c ts m a tr ic e s . Both the unw eighted and w eighted (see C hapter VII) v e rsio n s of the calcu latio n s a re show n below. 1. The value in the jth row and the m th colum n of the to ta l su m s of s q u a re s and c ro s s - p ro d u c ts m a trix (T) is X ^ r / - - 1 T jm = j Ç [ <^ijk - Xj) (Xinik - j (A-1) w h e re j = l , 2 , 3 , ................v , m = l , 2 , 3 V, Tjm = '^mj ' and Tjm is the unw eighted sum of the c ro s s p ro d u cts of env iro n m e n ta l v a ria b le s j and m (when j=m , it w ill be a sum of sq u ares), g is the n u m b er of gro u p s, N^ is the n u m b er of s i t e s in group k , X ijk is th e value of v a ria b le j in the ith site in group k , v is 384 th e n u m b er of en v iro n m en tal v a ria b le s m e a s u re d at each site, and and Xj a re the grand m ea n s of v a ria b le s j and m , r e sp ectiv ely , over all s ite s in a ll g ro ups, w hich for v a ria b le j is X- = k=l i= l ijk (A -2) In th e w eighted v e rsio n , each site in each group can be w eighted by the c o rre sp o n d in g species-abun dance m e a s u re s : g % r jm (Xijk - Xj ) (Xini^ - X jn)W ik (A -3) Nk / g % \IJ-^ \ ^ XijkXimkWik - g^Xiji, Wikj|Ê X ^imk W ikj X è Wik k=l i= l w h e re W j_]^ is som e m e a s u re of the re la tiv e abundance of sp e c ie s k in the ith site in group k , and g % X X ^ i j k ^ i k k=l i- 1______ ,__ g ^ (A -4) 385 2. The e le m e n ts of the unw eighted w ith in -g ro u p sum s of sq u a re s and c ro s s -p ro d u c ts m a trix (D) a re g % r _ 1 X X (Xijk - Xjk) (Ximk - ^ m k ) > (A- k= l i= l ^ J D jm 5) w h e re is the m ea n of v a ria b le j in group k , defined as Nk Xjk - X Xijk 1 = 1 N- k (A- 6) The w eighted v e rsio n of th is fo rm u la is D g % '"■ Ç iS (Xijk - Xjk) (Xin,k - Xmk) (A- 7) ] k ^ k ^ ^ ^ i- jk ^ ik 3 ^ ^ i m k ^ i k ^ ijk ^ im k W ik " — ---- ^ Ni W ik w here N i ^ j k r ^ijkW ik 1=1 (A -8) 386 3. The b e tw ee n -g ro u p s su m s of sq u a re s and c ro s s - p ro d u c ts m a trix is (in m a trix notation) sim p ly B - T - D . (A-9) T his app lies to both the w eighted and unw eighted v e rs io n s . B. W eighting C o n sid e ratio n s The choice of the re la tiv e abundance m e a s u re to be u sed as w eightings fo r each site in each group is now c o n sid ere d . If s p e c ie s - to ta l sta n d a rd iz e d data w e re utilized, each group (sp ecies) would re c e iv e equal to ta l w eight. T his would be at the expense of the individual o c c u rre n c e s of the m o re abundant and w id e sp re a d sp e c ie s . W ith the sp e c ie s-m a x im u m sta n d a rd iz e d data, each sp e c ie s would re c e iv e equal w eight at the site(s) in w hich the peak abundance counts a re found. H ere the m o re w id e sp rea d sp e c ie s would tend to end up w ith g r e a te r to ta l w eight in the a n a ly sis. The unm odified d isc rim in a n t a n a ly sis m ethod also gives g r e a te r to ta l w eight to the la r g e r gro u p s. The sp e c ie s-m a x im u m sta n d a rd iz e d data s e e m s to have the b e s t p ro p e rtie s for the type of w eighting d e sire d , and a re u se d in th is study. In addition to w eighting the calcu latio n s for sp e c ie s abun dan ces, each site also can be w eighted by the w eights w hich a re b a sed on the site u n iq u en ess (C h ap ters VII and IX). T his would be 387 d e sira b le sin ce the re p e a te d data from re d u n d a n tly -sa m p le d h ab itats m isle a d in g ly can influence the c alc u la tio n s. The m eans (eqs. A -4 and A - 8) w ill tend to be draw n to w ard the m e a s u re d lev e ls of the v a ria b le s in the o v e r-s a m p le d h a b ita ts. The su m s of sq u a re s and c ro s s -p r o d u c ts m a tric e s (eqs. A-3 and A - 7) also w ill tend to d isp ro p o rtio n a te ly re fle c t v a ria b le m e a s u re m e n ts in the o v e r-s a m p le d h a b ita ts. T hus, the w eight for each site in each group is W ik = Wik , (A -10) w h ere W^k is the w eight u se d for site i in the calcu latio n s for group k , is the site w eight re fle c tin g the u n iq u en ess of the site , and is the m e a s u re of re la tiv e abundance of group k (sp e cie s k ) in site i . As m entioned, sp e c ie s-m a x im u m sta n d ard iz ed data w ill be u se d as the m e a s u re of re la tiv e abundance. It tu rn s out th at the fin al w eights to be u sed in the d isc rim in a n t an aly sis a re the e le m e n ts of the w eighted d ata m a trix from w hich the in te r- s p e c ie s d istan c es a re calculated . It should be noted th at the calcu latio n s fo r the unm odified d isc rim in a n t a n aly sis (when using the sp e c ie s o r sp e c ie s groups to define the groups of site s ) also a re affected by uneven habitat sam pling. H ow ever, th e re is no w ay (b a rrin g elim in atio n of 388 s e le c te d site s ) to c o rr e c t fo r th is p ro b lem , sin ce w eights cannot be applied. C. C alculatio n of the D isc rim in a n t S co res The s c o re s a re c alcu late d as follow s: V Sj^jk ^ ^1m ^ im k ' (A -11) m = l w h e re is the sc o re of the ith site in group k on Axis j , V is the n u m b er of en v iro n m en ta l v a ria b le s m e a s u re d at each site , X im k the value of the v a ria b le m at the ith site of group k , and Cjm is th e coefficient c o rre sp o n d in g to v a ria b le m on Axis j . The co efficien ts sa tisfy the m a trix equation (D’ l B - Xj I) Cj = O , (A -12) w h e re Cj is the v e c to r of coefficients for Axis j (one coefficient fo r each v a ria b le ), I is an id en tity m a trix , and Xj s a tis fie s the d e te rm in a n ta l equation | d " ^ B - A j l j = o (A -13) The s c o r e s a re sta n d a rd iz e d to se t the w ith in -g ro u p v a ria n c e of the s c o re s fo r a ll groups equal to one by dividing the coefficients by the w ith in -g ro u p v a ria n c e along th e c o rre sp o n d in g axis. Seal (1968), 389 Hope (1969), and Cooley and Lohnes (1971) should be co nsulted for fu rth e r details of the m ethod. D. R elationsh ips betw een V ariab les and the D isc rim in an t Space The coefficients can be u se d to d e te rm in e w hich v a ria b le s co n trib u te the m o st to th e s c o re s on a p a rtic u la r ax is. T his is s im ila r to the u se of m u ltip le -r e g re s s io n coefficients to d e te rm in e the im p o rtan c e of th e independent v a ria b le s in p red ic tin g the value of the dependent v a ria b le . Since the v a ria b le s u su a lly a re m e a s u re d on d ifferent sc a le s, the coefficients m u st be c o n v e rte d to sta n d a rd - deviation units b efore th ey a re d ire c tly c o m p a rab le. T his is ac co m p lish ed by dividing each coefficient by the to ta l sta n d a rd devia tio n of the c o rre sp o n d in g v a ria b le (Hope, 1969: 127). The v a ria b le s w ith the h ig her sta n d a rd iz e d coefficients would be c o n sid e re d m o re im p o rta n t in the se p a ra tio n of the groups along the c o rre sp o n d in g axis. T his m ethod w as u se d by G reen (1971). The sign of a coefficient so m e tim e s can in d icate the tre n d of the c o rre sp o n d in g v a ria b le along the axis in question. F o r in sta n c e , a v a ria b le w ith a re la tiv e ly la rg e negative coefficient on an axis would add a la rg e negative num ber to the sum (eq. A - 11) in the calcu latio n s of th e site s c o re for a site w ith a high m e a s u re d value for the v a ria b le . T hus, site s w ith high values fo r such a v a ria b le would ten d to be found to w ard the negative end of the axis. 390 E x p erien c e h as shown, how ever, th at th is is not alw ays the c a se . C o n sid er F ig. 2-5 (C hapter II). Both high and low values of v a ria b le B w ill be found at s ite s to w ard e ith e r end of the d is c rim in a n t axis (called Axis I ), and th e sig n of the coefficients a sso c ia te d w ith th is v a ria b le w ill not be in fo rm a tiv e as fa r as in te rp re ta tio n is co n ce rn e d . In fact, the m e a n value of v a ria b le B is about th e sa m e at both the negative end of the axis (where group 1 is found) and at th e positive end (w here group 2 is found). V ariable B w ill be a sso c ia te d w ith a re la tiv e ly high sta n d a rd iz e d coefficient sin ce its in clu sio n in the calcu latio n s would in c re a s e g re a tly the p ro je c te d se p a ra tio n of the gro u p s. Although such a v a ria b le has about the sa m e m ea n value in both g ro u p s, it s till has in te rp re tiv e value sin ce the effect of the o th er v a ria b le (v ariab le A) depends on the lev e l of th is v a ria b le . Put another way, the axis s c o re s would be in te rp re te d to be re la te d to the lev e l of v a ria b le A w ith the level of v a ria b le B held co n stan t. The b e st way to in te r p r e t the re la tio n sh ip s betw een the d is c rim in a n t sp ace and the tre n d s of the im p o rtan t v a ria b le s is to ob se rv e the m ean v alu es fo r each im p o rta n t v a ria b le in each group and r e la te th e s e to the p o sitio n s of the v a rio u s groups in the d is c rim in a n t sp a ce . An exam ple of th is p ro c e d u re is shown in Fig. 8-9. When c o n sid erin g m o re th an a single axis at a tim e , the tre n d 391 of th e m ean values fo r a v a ria b le can be on an angle betw een the axes r a th e r th an along any single axis. O bserving the group m ean s in the space can show th e angle and d irec tio n of th is tre n d . T his sa m e in fo rm a tio n is not alw ays obvious from the sig n s of the coef ficien ts alone. F u rth e rm o re , a situation like th at in Fig. 2-5 with v a ria b le B (see above d iscu ssio n ) w ill be d etected e a sily by th is m ethod. An a ltern a tiv e to u sin g the sta n d ard ize d co efficien ts in the in te rp re ta tio n i s the use of the coefficients of s e p a ra te d e te rm in a tion (Hope, 1969: 159), w hich a re C j = C j T C j 1 ( A - 1 4 ) w h e re Cj is the v e cto r of coefficients of s e p a ra te d e te rm in a tio n for Axis j , Cj is a m a trix w h ere the diagonal e lem en ts a re the co efficien ts of Axis j and the off-diagonal e le m e n ts a re z e ro s , T is as b e fo re (eq. A- 1), and 1 is a colum n v e c to r of u n itie s. The r e s u lts a re e x p re s s e d as p e rc e n ta g e s as follow s: I c ' * 1 P j-j - - — A t — x 1 0 0 , ( A - 1 5 ) V 392 w h ere P ij is the coefficient of s e p a ra te d e te rm in a tio n for v a ria b le i on Axis j e x p re ss e d as a p ercen tag e, is the coefficient of s e p a ra te d e te rm in a tio n for v a ria b le i on A xis j , and v is the n u m b er of v a ria b le s . Sets of v a ria b le s m ay be re la te d to a com m on facto r which is im p o rtan t in the a n a ly sis. H ere the tru e w eight of the com m on c h a r a c te r is tic (as in d icated by the m agnitudes of the coefficients of s e p a ra te d eterm in atio n ) m ay be divided up am ong the v a ria b le s in question (Hope, 1969; 84, 161). F o r th is re a s o n , it is im p o rtan t to c o n sid e r the in te r c o r re la tio n s betw een the v a ria b le s when i n t e r p retin g r e s u lts . E x p erien c e w ith both sta n d a rd iz e d coefficients and coeffi c ie n ts of se p a ra te d e te rm in a tio n has show n th at the coefficients of se p a ra te d e te rm in a tio n u su ally lead to e a s ie r and m o re c le a r-c u t in te r p re ta b ih ty of the r e s u lts . A ccordingly, only the coefficients of se p a ra te d e te rm in a tio n a re u sed in th is study. E. A ssu m ptions of the M ethod G reen (1971) contains a good su m m a ry of the assum ptions involved in the m u ltip le -d isc rim in a n t m ethod. Many of the a ssu m p tions becom e le s s c ritic a l when no s ta tis tic a l te s ts in re la tio n to group se p a ra tio n a re applied, as is the c a se in th is study. U sually the plots of the d isc rim in a n t s c o re s w ill be sufficient to d e te rm in e 393 if the groups a re w ell s e p a ra te d and s ta tis tic a l te s ts would be r e dundant o r, at w o rst, would be a nuisance if e x tre m e m e a s u r e s have to be taken to m e e t a ll the assu m p tio n s. F . E ffect of O utlier G roups To get the m ax im um am ount of in fo rm atio n from an a n a ly sis, often it is n e c e s s a r y to c o n sid er the effect of one or a few groups w hich a re e x tre m e in som e v a ria b le m e a s u re m e n ts when c o m p a re d w ith the o th er groups in th e an aly sis. The se p a ra tio n of such groups from the o th er groups w ill dom inate the r e s u lts of the f ir s t o r f ir s t few ax es. On such axes, v e ry little in fo rm a tio n would be available co n cern in g th e se p a ra tio n of the re m a in in g g roups. T h ese w ill be s e p a ra te d on la te r axes, but it should be re m e m b e re d th at each axis m u st be at rig h t angles to a ll p reced in g axes and the axis s c o r e s m ust be u n c o rre la te d am ong and betw een grou ps. T his p lac es m o re r e s tric tio n s on the la te r axes and the pow er and a c c u ra c y of the a n aly sis w ill su ffe r. It would be m o re efficient if the le s s r e s tr ic te d axes (the f ir s t few) w ere used to s e p a ra te the bulk of the data. The best p ro c e d u re is to ru n the a n aly sis w ith all groups and c h ec k to se e if the plots of the m e a n d isc rim in a n t s c o r e s for each group indicate th at the fir s t axis is dom inated by the se p a ra tio n of a single or re la tiv e ly few groups re p re s e n tin g only a s m a ll 394 p ro p o rtio n of the data. If this is the c a se , th en the an aly sis should be re r u n w ith the o u tlier group(s) elim in ated . The o rig in a l an aly sis, how ever, is not w asted since it contains u sefu l in fo rm a tion co n cern in g the se p a ra tio n of the o u tlie r groups from th e r e m aining g ro u p s. If the second (or subsequent) a n aly sis produces anoth er o u tlie r group, the sa m e p ro ce d u re could be rep e ate d , but the am ount of effort involved in too m any re p e a te d an aly ses m ay becom e e x c e ssiv e when c o m p a red to the ben efits of the p ro c e d u re . S om etim es the o u tlie r groups are of little in te re s t, or the v a ria b le s a sso c ia te d w ith th e ir se p a ra tio n from o th er groups a re obvious w ithout any a n aly sis at all. In th e se c a s e s , the o u th e rs sim p ly could be e lim in a te d b efo re the analysis is s ta r te d if th e ir e x tre m e condition a lre a d y is evident. G. L im its on the N um ber of V ariab les U sed When m o re than two groups a re analyzed, the n um ber of v a ria b le s u se d cannot exceed the nu m b er of to ta l site s in all gro ups. In fact, if th e r e is m uch redu ndancy am ong the v a ria b le s, the r e s u lts a re d isto rte d even before th is lim it is re a c h e d . The n u m b er of v a ria b le s can be red u ced as follow s. 1. C alculate the c o rre la tio n s betw een a ll v a ria b le s and elim inate a ll but one of the se ts of highly in te r c o r r e la te d v a ria b le s (positive and negative). 395 2. A m ethod rec o m m e n d ed by N o rris (1971) involves tra n s la tin g the v a ria b le values to PCA axis s c o re s before u se in the m u ltip le -d isc rim in a n t calcu latio n s. E ach axis is u se d as a s e p a ra te v a ria b le and the axes w ill be few er in n u m b er th an the o rig in a l v a ria b le s if only the axes w ith v a ria n c e am ong the s c o re s a re u sed (i. e. , those w ith eigenvalues >0). 3. V ariab les w hich do not v a ry w ithin all the groups should be re m o v e d in any c a se . A good indication that too m any v a ria b le s a re being u se d is the n e a r identity of the s c o re s for all s ite s in each group. The s c o re plots w ill show a ll points in a group on top of one another. 396 APPEN D IX B LIST OF THE 94 MOST FREQ U EN TLY OCCURRING SPECIES 1. AGLAJA SP. 2. AM PH A RETE ACUTIFRONS (Grube, 1860) 3. AM PH A RETE LABROPS H artm an , 1961 4. AM PHICTEIS SCAPHOBRANCHIATA M oore, 1906 5. AMPHISAMYTHA BIOCULATA (M oore, 1906) 6. ANAITIDES'"M UCOSA" 7. ANAITIDES M ULTISERIATA R ioja, 1941 8. ARMANDIA BIOCULATA H artm an, 1938 9. AXINOPSIDA SP. 10. C A PIT E L L A CA PITA TA (F a b ric iu s, 1780) 11. CA PITITA AM BISETA H artm an , 1947 12. C A U LLER IELLA A LATA (Southern, 1914) 13. CEREBRATULUS SP. 14. CHLOEIA PINNA TA M oore, 1911 15. CHONE ECAUDATA (M oore, 1923) 16. COMPSOMYAX SUBDIAPHANA (C arp en ter, 1864) 17. CONUS CA LIFO RNICUS R eeve. 1844, ex Hinds MS 397 18. C O O PE R E L L A SUBDIAPHANA (C a rp e n te r, 1864) 19. CO;?SURA CANDIDA H artm an, 1955 20. CREPID U LA ONYX Sower by, 1824 21. CRYPTOM YA CALIFORNICA (C onrad, 1837) 22. CYCLOCARDIA VENTRICOSA (Gould, 1850) 23. CYLICHNA DIEGENSIS (Dali, 1919) 24. DECAMASTUS GRACILIS H artm an, 1967 25. DIA STY U S P E ELUCIDA H art,, 1930 26. DIOPATRA OR NAT A M oore, 1911 27. DORVILLIDAE SP. 1 28. EUMIDA "SANGUINEA" 29. EXOGONE LOUREI B erk eley and B erk eley , 1938 30. GLYCERA AMERICANA Leidy, 1855 31. GLYCERA BRANCHIOPODA M oore, 1911 32. GLYCINDE POLYGNATHA H artm an , 1950 33. GOLFINGIA 7H ESPER A 34. GYPTIS ARENICOLA GLABRA (H artm an, 1961) 35. HA PLOSCOLOPLOS ELONGATES (Johnson, 1901) 36. HARMOTHOE "LUNULATA" 37. HETEROMASTUS FILOBRANCHUS B e rk e le y and B erk eley , 1932 38. K U R TZ IEL L A B E TA (Dall, 1919) 39. LAONICE CIRRATA (S ars, 1851) 398 40. L E P T O C H E U A SP. 1 41. LISTR IELLA SP. 1 42. LISTRIOLOBUS PE LODES F is h e r , 1946 43. LUCINISCA NUTTALI (C onrad, 1837) 44. LUCINOMA ANNULA TA (Reeve, 1850) 45. LUMBRINERIS "L A T R E IL L I" 46. LUMBRINERIS CA U FO RN IEN SIS H artm an, 1944 47. LUMBRINERIS CRUZENSIS H artm an, 1944 48. LUMBRINERIS INDEX M oore, 1911 49. LUMBRINERIS JAPONICA (M a re n z e lle r, 1879) 50. LUMBRINERIS LACUNAE (F auchald, 1970) 51. LUMBRINERIS PALLIDA H artm an , 1944 52. MACOMA CAR LOTTE NSIS (White aves, 1880) 53. MAGELONA PACIFICA M onro, 1933 54. M ELINNA HETERODONTA M oore, 1923 55. M IT R E L L A TUBEROSA (C a rp e n ter, 1864) 56. NASSARIUS INSCULPTUS (C a rp e n te r, 1864) 57. NASSARIUS MENDICUS (Gould, 1849) 58. NASSARIUS PERPINGUIS (Hinds, 1844) 59. NEPHTYS CORNU TA FRANCISCANA C la rk and Jo n es, 1955 60. NEPHTYS FERRUGINEA H artm an , 1940 61. NEREIS SP. 1 399 62. NOTHRIA IRIDENSCENS (Johnson, 1901) 63. NOTOMASTUS TENUIS M oore, 1909 64. O U V E L L A BAETICA C a rp e n te r, 1864 65. OPHIODROMUS PUGETTENSIS (Johnson, 1901) 66. OSTRACODA, SM ALL BROWN 67. PARAPHOXUS OBTUSIDENS (A lderm an, 1936) 68. PARAPRIONOSPIO PINNATA (E h le rs, 1901) 69. PARVILUCINA TEN UISCU LPTA (C a rp e n te r, 1865) 70. PECTIN A RIA CAU FO RN IEN SIS H artm an , 1941 71. PEN N A TU LA CEA SP. 1 72. PE R IPL O M A PLANUISCULUM Sow erby, 1834 73. PHERUSA PA PIL L A T A (Johnson, 1901) 74. PHOTIS C A U FO R N IC A Stout, 1913 75. PHOTIS SP. 76. PH Y LLO CH A ETO PTERU S U M ICO LU S H artm an , 1960 77. POLYDORA NUCHA U S Woodwick, 1953 78. PRIONOSPIO C IR R IFER A W iren, 1883 79. PRIONOSPIO MALMGRENI C lap ared e, 1870 80. PRIONOSPIO SP. 1 81. PROTOTHACA LACINIATA (C a rp e n te r, 1864) 82. RICTAXIS PUNCTOCAELATUS (C a rp e n te r, 1864) 83. SHISTOMERINGOS LONGICORNIS (E h le rs, 1901) 400 84. SOLEMYA PANAMENSIS DaU, 1908 85. SPIOCH AETOPTERU S COST ARUM (C lap arede, 1970) 86. SPIOPHANES ? MISSIONENSIS 87. SPIOPHANES FIM BRIATA M oore, 1923 88. STYLATULA ELONGATA V e rrill, 1864 89. TELLIN A C A R PE N TE R I Dall, 1900 90. T E L IIN A MODESTA (C a rp e n ter, 1864) 91. T E L L IN A SP. 2 92. THARYX "M U L T IF IU S " 93. THARYX "PARVUS" 94. THYASIRA SP. 401 UMI Number: DP23625 All rights reserved INFORMATION TO ALL USERS The quality of this reproduction is dependent upon the quality of the copy submitted. In the unlikely event that the author did not send a complete manuscript and there are missing pages, these will be noted. Also, if material had to be removed, a note will indicate the deletion. Dlssertaïion Publishing UMI DP23625 Published by ProQuest LLC (2014). 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Asset Metadata

Creator Smith, Robert William (author)

Core Title Numerical analysis of ecological survey data

School Graduate School

Degree Doctor of Philosophy

Degree Program Biology

Degree Conferral Date 1976-01

Publisher University of Southern California (original), University of Southern California. Libraries (digital)

Tag biological sciences,OAI-PMH Harvest

Language English

Contributor Digitized by ProQuest (provenance)

Permanent Link (DOI) https://doi.org/10.25549/usctheses-c30-213468

Unique identifier UC11227249

Identifier DP23625.pdf (filename),usctheses-c30-213468 (legacy record id)

Legacy Identifier DP23625.pdf

Dmrecord 213468

Document Type Dissertation

Rights Smith, Robert William

Type texts

Source University of Southern California (contributing entity), University of Southern California Dissertations and Theses (collection)

Access Conditions The author retains rights to his/her dissertation, thesis or other graduate work according to U.S. copyright law. Electronic access is being provided by the USC Libraries in agreement with the au...

Repository Name University of Southern California Digital Library

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