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The Effect Of Subject Sophistication On Ratio And Discrimination Scales

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The Effect Of Subject Sophistication On Ratio And Discrimination Scales

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Content THE EFFECT OF SUBJECT SOPHISTICATION ON RATIO AND DISCRIMINATION SCALES by Richard Kenneth Eyman A Dissertation Presented to the FACULTY OF THE GRADUATE SCHOOL UNIVERSITY OF SOUTHERN CALIFORNIA In Partial Fulfillment of the Requirements for the Degree DOCTOR OF PHILOSOPHY (Psychology) January 1966 UNIVERSITY O F SO U T H E R N CALIFORNIA THE GRADUATE SC H O O L UN IV ER SITY PARK LO S A N G ELES, C A L IFO R N IA 9 0 0 0 7 This dissertation, written by R ich ard ..K en n eth .E y m a n ............................ under the direction of hi.s..„.Dissertation Com mittee, and approved by all its members, has been presented to and accepted by the Graduate School, in partial fulfillment of requirements for the degree of D O C T O R OF P H I L O S O P H Y D ean D ate January.,....! 966 DISSERTATION COMMITTEE ■ ~ ) )..A « . ■ c ACKNOWLEDGMENTS The author wishes to express his appreciation to Dr, Norman Cliff and Dr, J, P. Guilford for their significant contributions to the design and execution of this study. Appreciation is also expressed to Dr. Harvey Dingman for his encouragement and support throughout the study. This research was supported in part by. the National Institute of Mental Health Grant No, MH-O8667: Socio- Behavioral Study Center for Mental Retardation, Pacific State Hospital, Pomona, California, Computing assistance was obtained from the Health Sciences Computing Facility, UCIA, sponsored by NIH Grant No. FR-3. Finally, the writer wishes to express his apprecia tion to his wife, Vivian, for her continuous help and encouragement in the preparation and accomplishment of this study. ii TABLE OP CONTENTS Page ACKNOWLEDGMENTS.................................. 11 LIST OF TABLES.................................... v LIST OF FIGURES.................................. vli Chapter I NATURE OF THE PROBLEM................. 1 Need for Study Definitions II REVIEW OF THE LITERATURE............... 8 Evidence Supporting Ratio Scales Evidence Supporting Discrimination and Category Scales Attempts to Reconcile Differences Between Ratio and Magnitude Scales and Discrimination and Category Scales Methodological Problems Possible Psychological Factors Affecting Discrimination and Ratio Scales III METHOD AND PROCEDURE................... 37 Purpose Method Subjects Procedure and Directions in the Administration of Tasks Scaling Methods and Scaling Results ill TABLE OF CONTENTS (continued) Chapter IV BESULTS....................... Page 79 Psychophysical Function of the Pair Comparison Scale Psychophysical Function of the Constant Sum Scale Comparison of the psychophysical Functions for the Pair Comparison and Constant Sum Scales Comparison of the Accuracy of Judg ments of the Patient, Normal, and Sophisticate Groups V DISCUSSION ...... .................. 115 VI SUMMARY AND CONCLUSIONS.................. 126 APPENDIX A ..................................... 131 APPENDIX B ...................................... 150 APPENDIX C ...................................... 157 APPENDIX D .............. ........................ 159 BIBLIOGRAPHY...................................... 16k lv LIST OP TABLES Table Page 1 Comparisons Required for Stimulus Series A, B, and C , ....••••• *+0 2 Comparisons Required for Stimulus Series D ........................... 42 3 Pair Comparison Scale Values and Associated Chi Square Values (goodness of fit) for Length of Lines for Patient, Normal, and Sophisticate Groups............................... 57 4 Pair Comparison Scale Values and Associated Chi Square Values (goodness of fit) for Weight for Patient, Normal, and Sophisticate Groups............... 59 5 Geometric Means, Standard Deviations, and Number of Subjects Included for Each of the 44 Point Assignments Used for the Constant Sum Scale of Length and Weight--Patients ................. 66 6 Geometric Means, Standard Deviations, and Number of Subjects Included for Each of the 44 Point Assignments Used for the Constant Sum Scale^of Length and Weight— Normals.................. 67 7 Geometric Means, Standard Deviations, and Number of Subjects Included for Each of the 44 Point Assignments Used for the Constant Sum Scale of Length and Weight— Sophisticates............. 68 v LIST OP TABLES (continued) Table Page 8 Constant Sum Scale Values for Length of Lines and Weights for Patient, Noxmal, and Sophisticate Groups, ....... 75 9 Regression Coefficients and Standard Error of Estimates for Eighteen Linear Equations Fitted to Psychophysical Functions Based on Pair Comparison Scales • • •• •• ••• .• •. ..• 82 10 Exponents (n), Scale Constants (c), Additive Constants (k). and Standard Error of Estimates (SE) of Best Fitting Power Functions for Length and Weight ^ = c (s + k)n ............... 92 11 Slopes (first derivatives) of the Power Functions in Table 10 for Length and Weight at Specified Values of These Stimuli.......................... 101 12 Analysis of Variance of Individual Accuracy Scores Connected with the Pair Comparison Scales for the Patient, Normal, and Sophisticate Groups.......................... 112 13 Analysis of Variance of Individual Accuracy Scores Connected with the Constant Sum Scales for the Patient, Normal, and Sophisticate Groups. . . . 114 vi LIST OF FIGURES Figure Page 1 Relationship Between Geometric Means (GM) and Standard Deviations (SD) of the Constant Stun Judgments for Length . . ............................. 70 2 Relationship between Geometric Means (GM) and Standard Deviations (SD) of the Constant Stun Judgments for Weight................................. 71 3 Illustration of Stimuli Selected for Scaling, Using Pair Comparison and Constant Stun Methods .••••••••« 7^ k Pair Comparison Scale of Patients, Normals, and Sophisticates for Subjective Lengths ( ) as a Function of Physical Length for Three Ranges of Stimuli.......... 30 5 Pair Comparison Scale of Patients, Normals, and Sophisticates for Subjective Weight ( v ) as a Function of Physical* Weight for Three Ranges of Stimuli.......... • • . 31 6 Constant Sum Scale of Patients for Subjective Length as a Function of Physical Length......................... 35 7 Constant Sum Scale of Normals for Subjective Length as a Function of Physical Length. 86 vii LIST OP FIGURES (Continued) Figure Page 8 Constant Sum Scale of Sophisticates for Subjective Length as a Function of Physical Length . .................. 87 9 Constant Sum Scale of Patients for Subjective Weight as a Function of Physical Weight.......................... 88 10 Constant Sum Scale of Normals for Subjective Weight as a Function of Physical Weight......... 89 11 Constant Sum Scale of Sophisticates for Subjective Weight as a Function of Physical Weight • 90 12 Constant Sum Scale of Patients for Subjective Length as a Function of Physical Length in Log-Log Coordinates • 9^+ 13 Constant Sum Scale of Normals for Subjective Length as a Function of Physical Length in Log-Log Coordinates . 95 14 Constant Sum Scale of Sophisticates for Subjective Length as a Function of Physical Length in Log-Log Coordinates........................ 96 15 Constant Sum Scale of Patients for Subjective Weight as a Function of Physical Weight in Log-Log Coordinates. . . . . . . . . . . . . . . 97 16 Constant Sum Scale of Normals for Subjective Weight as a Function of Physical Weight in Log-Log Coordinates...................... ....... 98 17 Constant Sum Scale of Sophisticates for Subjective Weight as a Function of Physical Weight in Log-Log Coordinates. ........... 99 viii LIST OP FIGUBES (continued) Figure Page 18 Comparison of Psychophysical Functions of Constant Sum (CS) and Pair Com parison (PC) Scales for Length— Patients .................. ...... 104 19 Comparison of Psychophysical Functions of Constant Sum (CS) and Pair Com parison (PC) Scales for Length— Normals. ......................... 105 20 Comparison of Psychophysical Functions of Constant Sum (CS) and Pair Com parison (PC) Scales for Length— Sophisticates.......... 106 21 Comparison of Psychophysical Functions of Constant Sum (CS) and Pair Com parison (PC) Scales for Weight— Patients .......... 107 22 Comparison of Psychophysical Functions of Constant Sum (CS) and Pair Com parison (PC) Scales for Weight— Normals.......... 108 23 Comparison of Psychophysical Functions of Constant Sum (CS) and Pair Com parison (PC) Scales for Weight— Sophisticates. ................... 109 ix CHAPTER I NATURE OP THE PROBLEM Stevens (1960a) notes that modem efforts to quan tify subjective magnitudes have led to three types of scales— all three designed to show how sensation grows as a function of stimulus values. Category and discrimi nation scales have generally been shown to be nonlinearly related to magnitude scales. Guilford asks* "Are we to be left with a purely operational view, accepting different psychophysical laws for different scaling methods? Is there any way to bridge the gaps between the different kinds of scales? In the face of discre pancies like this, it is not so much a question of which is correct, but how to reconcile the differences. Until we do, we have not learned as much about the phenomena as we should know" (Guilford, 1961, p. 115)* Need for Study It is well known that in psychophysical judgments 1 subjects may give "biased" responses as a function of training, maturation and other experimentally mani pula table factors (Ekman & SJdrberg, 1965; Parduccl, 1963; Stevens, 1958). Senders and Sowards (Carso, 1963) found that subjects yield proportions of Judgments in accordance with their expectations based on their knowledge of the experimental situation. No study has systematically investigated the effect of subject’s knowledge of the experimental situation and general sophistication on the relation of either discrimination or magnitude scale values to the stimulus continuum. An investigation is proposed which will attempt to maximize the chances of finding the same psychophysical function between stimulus and response, using a discrimi nation and ratio scale by taking account of the following two factors: 1. Subject’s sophistication and knowledge of the experimental problem. 2. The manner stimuli are presented and subsequently Judged. In addition, by studying the psychophysical function of the two scales over levels of subject sophistication, it is hoped to disoover some of the psychological factors involved concerning the inconsistency between the two scales. Definitions 1. Discrimination— confusion scales are based on the subject*s ability to discriminate between stimuli, Thurstons (1927) Introduced Ingenious methods for pro cessing data on the variability of discriminations, con fusions, or Just noticeable differences (Jnd) In order to erect Interval scales of psychological magnitude. Having determined a measure of the variability Involved, It Is possible to count off distances between stimuli on a psychological scale. In the method of Pair Comparison (Case V), equal measures of variability or discrlminal dispersions are assumed to represent equal distances on a scale of subjective magnitude. Judgments for the con struction of this scale are collected by presenting the subject with a set of stimuli, two at a time where his task is to rate one of the stimuli over the other con cerning magnitude, preference, etc. Prom these Judgments we assume that "equally often noticed differences are psychologically equal." 2. Category— partition scales are based on the subject’s ability to assign numbers, to partition, or to categorize a segment of a stimulus continuum Into equal intervals. In other words, categories are assigned so that Intervals between magnitudes of stimuli correspond to numerical differences between the corresponding assigned categories. 3. Batio scales are based on the subject's ability to assign numbers so that the ratios between perceived magnitudes of stimuli correspond to numerical ratios between the corresponding assigned numbers. It is assumed the scale will have a zero point which Is not arbitrary and on which ratios therefore have a meaning. In other words, the numbers assigned to stimuli are proportional to the subjective magnitudes of the stimuli. Among the available methods, two are of current Interest and are described here— magnitude estimation and ratio estimation. Magnitude estimation refers to a procedure In which the subject makes direct numerical estimations of a series of subjective impressions. As used, the subject is presented with a "standard stimulus" and told that he is to assign a number proportional to the apparent magnitude, as he perceives It. Batio estimation calls for the presentation of two or more stimuli and the observer names the value of the apparent ratio between them. The constant sum method is a special Instance of this procedure. This method re quires the observer to divide 100 points between two stimuli in such a way that the division between the points reflects the apparent ratio between sensations. Batlos between stimuli are averaged over subjects (or occasions for the same subject) and subjective scales are constructed from these averages. 5 Fechner* s Law or logarithmic function is fre quently used to describe the psychophysical relationship found between discrimination or category scales and the corresponding stimulus continuum. Fechner*s Law is usually stated (Guilford, 195^» P* 331)* where S is the stimulus continuum and yJ represents the subjective scale. 5. Power Law describes the psychophysical relation usually found between ratio scales and the corresponding stimulus continuum. The Power Law can be simply stated: "n" is the exponent. An exponent greater than one des cribes a positively accelerated curve and an exponent less than one gives a negatively accelerated ourve. 6. Cross Modality Matching was thought of and used by Stevens (1962, p. 32) for "testing the validity of the power functions, and their exponents, by methods that would eliminate the verbal behavior." The method requires the subject to adjust the stimulus magnitude of one modality as a function of the stimulus magnitude presented to him on a second modality. 7. Metathetic Continue is a name "coined" by Stevens for a class of subjective continue which he refers are defined as in equation (1) above and to as "qualitative" and relates to "what" and "where." Examples are pitch, position, inclination, proportion, etc. Stevens defines these continue operationally through evidence that the subject's sensitivity to differences (discriminal dispersions) is constant over the subjective scale. 8. Prothetlc Continue are a second class of subjec tive continue which Stevens uses to describe "quantita tive," intensitive continue which relate to "how much." This class is operationally characterized as those modalities where the subject's sensitivity to differences is good at the low end and poor at the high end of the scale. Examples are length of lines, weight, reflectance, loudness, etc. 9* For purposes of brevity, generalizability, and to simplify later discussion, ratio estimation scales and magnitude estimation scales are described as essentially being equivalent to one another as are category and dis crimination scales. Although in both Instances the operations for obtaining judgments from the subject are different, there is direct evidence demonstrating that the two ratio methods furnish equivalent results as is true also of the two interval methods. Kflnnapas and WlkstrOm (1963) scaled occupational preferences using ratio estimation and magnitude estimation. The results demon strated that the two methods gave the same scale values. Onley (i960) also found agreement between the methods for brightness scales. Eisler (1962b; 1963b; 1963d) has shown that the category scale constitutes a pure dis crimination scale and Is essentially Identical to a Thurstonlan discrimination scale. Goude (1962) has demonstrated that category scale values are equivalent to discrimination scale values using lifted weights and visual Judgments. He also demonstrated this equivalence between magnitude estimation and constant sum Judgments. It would appear then, that based on evidence largely from the Stockholm Psychological Laboratory, the methods connected with interval scaling and ratio scaling give reasonably consistent results. CHAPTER II REVIEW OF THE LITERATURE Numerous studies have shown that category and dis crimination scales are nonlinearly related to ratio scales of subjective magnitude (BjiJrkman, I960; Ekman, 1962; Ekman & Ktlnnapas, 1962a; 1963a; 1963b; Engen & MeBurney, 196^; Kflnnapas & Wikstrflm, 1963; Perloe, 1963; Schneider & Lane, 1963; Stevens & Mack, 1959; Stevens, 1957; 1958; 1960a; 1962; Stevens, Carton, & Shlckman, 1958; Stevens & Gallanter, 1957; Stevens & Gulrao, 1963; Stevens & Stone, 1959; Torgerson, i960). The most common finding has been that scales based on discrimination data (Jnd scales) or category ratings are approximately logarithmically related to a physical continuum. Ratio and magnitude estimates of stimuli generally yield a power function between the physical continuum and psychological continuum. Finally, a negatively accelerated curve was generally found to describe the relation between the discrimination and 9 ratio scales. Evidence Supporting Batio Scales Stevens (1962) and Stevens and Galanter (1957) argue that It Is the magnitude methods which furnish the correct psychophysical relationship. According to Stevens, the results obtained by either category or confusion methods are caused by the subject confusing discrimlnablllty with psychological distance. Stevens and Galanter (1957) have accumulated considerable evidence to support the empirical generality of the power function as determined by magni tude estimation. They have also relied heavily on cross modality matching as an additional measure of the "correctness** of their scales. Stevens outlines this argument as follows* If two continue are governed by power functions, then neglecting constants: fz ' *Z* where p = psychological scale value S = stimulus Now if the subject equates with p resulting "equal sensation" matching function will have the form The matching relation is a power function with the exponent n/m. Ekman, impressed by Stevens* work, has stated: After a hundred years of almost general acceptance and practically no experimentation, Fechner’s Logarithmic law was replaced by the power law. The amount of experimental work performed in the 1950's on this problem by Stevens and other research workers was enormous, and the outcome was an outstanding success. The power law was verified again and again, in literally hundreds of experiments. As an experimental fact, the power law is established beyond any reasonable doubt, possibly more firmly established than anything else in psychology (Ekman, 1964*, p. 1). The generality of the power law concerning a variety of ratio and magnitude estimation methods has certainly been substantiated by an Impressive number of investiga tions (Baker & Dudek, 1955; Bartoshuk, 1961*; Beloff, 1962; Eisler, 1963c; Engen & McBumey, 196^; Guilford, 195**; Hawkes, I960; Ktbinapas, i960; Onley, I960; Perloe, 1963; Pitz, 1965; Schneider & Lane, 1963; Stevens, 1955; 1956; 1957; 1959a; 1959b; 1959c; 1962; Stevens, Carton & Snickman, 1958; Stevens & Galanter, 1957; Stevens & Gulrao, 1962; 1963; Stevens & Harris, 1962; Stevens & Mack, 1959; Stevens, Mack & Stevens, i960; Stevens & Stevens, I960; Strougert, 1961). 11 Elsler (1963c) has demonstrated the validity of the power law for the whole dynamic range of the stimulus. In this study, subjective force exerted by pushing a pedal with the foot was scaled by the method of magnitude estimation. The power law accounted for subjective force as a function of physical force up to the strongest forces the subjects were capable of exerting. In addition to demonstrating the consistency of a power function using either direct reporting or cross modality matching, some of these studies (Stevens & Guirao, 1962; 1963; Stevens & Harris, 1962; Stevens & Stevens, I960; Torgerson, i960) have also demonstrated that ratio estimation data show a reciprocality between two inverse aspects of the stimulus— longness, shortness, etc. In the scaling of tactual roughness and smoothness, Stevens states: The power function has asserted itself not only on continue that involve well known stimulus variables, but also on a continuum, tactual roughness, for which we had at first thought there would be no metric stimulus correlate (Stevens & Harris, 1962). Our first guess proved delightfully in error, for we found that apparent roughness grows as the 1.5 power of the diameter of the abrasive particles on standard emery cloths. When the observers Judged the apparent smoothness of the same emery cloths, the exponent turned out to be nearly equal in magnitude, but opposite in sign. We thereby demon strated the observers remarkable ability to Judge a continuum in terms of its reciprocal function. ...Furthermore functions were produced rather exactly in two cross modality experiments. ...The testimony of the reciprocal Judgments adds another 12 dimension to the network of evidence supporting the power law (Stevens, 1962, PP. 38-39). Finally, Ekman and SJflbferg (1965) noted that whereas it is generally recognized that a category scale is largely affected by the selection of stimuli, and their spacing in particular, corresponding effects on scales obtained by direct magnitude and ratio estimation pro cedures are less known and to some extent neglected. However, Stevens (1960a; 1960b) has stated that ratio scales are relatively invariant and that this character istic is a strong support for the validity of the ratio scales. Evidence concerning this point will be reported later in this chapter. Evidence Supporting Discrimination and Category Scales Torgerson (i960) has stated "I have been rather hoping that a champion would come along for the category scales, if only to even things out. It seems to me that there is much that can be said in their favor” (p. 2*0. A number of investigators (BJflrkman, 1958; 1959a; 1959b; Eisler, 1963a; Ekman & Ktbmapas, 1963c; Ekman & Llndman, 1962; in press; Gamer, 1958; Goude, 1962; Guilford, 1961; Kttnnapas, 1961; Luce & Edwards, 1958; Parducci, 1963; Bambo, 1962; 19634 SJflberg, 1963; 1965a; 1965b; 1965c; Torgerson, I960; Warren < S t Poulton, 1962; Warren & Warren, 1963) have advanced arguments in defense of discrimination and category scales and have demonstrated their value in specific scaling experiments. Luce and Edwards (1958) have also pointed out that numerous psychophysicists feel that discrimination scaling is the better method. For example, Gamer (1958) has used his own work on loudness judgments to show that fractionation and ratio estimation scales are frequently unreliable variables, and peculiarly subject to context effects unless extensively trained observers are used. Discrimination scales were obtained from untrained subjects and animals. Context effects appeared to be eliminated or greatly reduced by discriminability scaling. Parducci (1963) found that the function between category Judgments of numerousness and stimulus magnitude is linear rather than concave downward when the Judgments are rescaled to equate for range. This finding was also colaborated by Goude (1962) on lifted weights for a limited range of weights. BJdrkman (1958; 1959b)» using the discrimination scaling approach has developed scales for verbal rote learning which have been found to be extremely stable in spite of varying experimental conditions. Ktinnapas (1961), Ekman and Llndman (1962; in press) have also applied discrimination scaling successfully to new situations of perception and learning. SJdberg (1962; 1963; 1964; 1965a; 1965b; 1965c) has done a considerable amount of theoretical and empiri cal work with Thurstonian Scaling. He has developed a scaling method (the correlational scaling method) where the usual assumptions concerning variances and covariances of the responses are not necessary. The method does presuppose the application of the tetrachorlc correlation coefficient to pair comparison data. His methodological studies CSiflberg, 1963; 1965b; 1965c) have demonstrated the applicability and usefullness of this scaling approach concerning both preference data and learning data in situations where discriminal dispersions were clearly unequal. Since a physical continuum did not exist the psychophysical functions could not be checked. Finally, Luce and Galanter (1963) have shown that category and discrimination scales also account for the results of cross modality matching which Stevens cites as supporting ratio scaling methods and the power law. The argument proceeds as follows: Using j and 7 2 * S2 as representing a psycho logical scale and stimuli respectively for two modalities, we have s (P - a-, log and 7 1 bj. then S- 15 and si ” cs2V ax Khepe A \a2^al ° = bl | b2j Thus both the power function of the ratio scale and the logarithmic hypothesis of the discrimination and category scales predict a power relation for the matching data. It Is apparent from the evidence supporting discri mination and category scales that these scales are currently finding widespread use and that their validity, general applicability, and invariance are supported by comparable evidence as has been advanced in connection with the magnitude estimation methods. Consequently, Fechner*s Law, generally confirmed through the use of discrimination scales, still must be given serious con sideration along with the Impressive evidence amassed for the power law In connection with magnitude estimation. In other words, the "correct" scale and psychophysical law Is very much an Issue and a "resolution of" or "reason for" the different results furnished by discrimi nation and ratio scales Is a current problem. Attempts to Reconcile Differences Between Batio and Magnitude scales and Hfllscrimlnatlon and dategory Scales Guilford (1961) has pointed out that it is not so 16 much a question of which type of soale Is correct, but rather Is It possible to reconcile the differences, Torgerson (i960) suggests that discrimination soales and ratio estimation soales refleot two standard ways subjects have of regarding number or quantity. Ratio scales would be most useful concerning quantitative rela tions among objects (proportional change) and discrimina tion or category soales would be most useful concerning absolute differences between objects (linear change). Fagot (1963) derived a general formulation of the power law which admits negative exponents and still preserves the necessary monotone property of the law. Furthermore, he shows that the power law approaches the log law as a limit as the exponent approaches zero. Luce (1959b) and Eisler (1964) have obtained a ratio scale from pair comparison Judgments through the applica tion of set theory. Both Luce and Eisler base their respective scales on Luce's choice axiom, which defines either scale in terms of proportions of preferred choices among stimulus pairs. Luce (1959b) has demonstrated that his V scale is a power function of physical magnitude, in agreement with what has been found empirically for magnitude scales of prothetlc continue. These choice models handle the choices or preferences as ratios between pairs of subjective magnitudes. Unfortunately, however, the exponents of a power function calculated for the 17 scales are more than one order of magnitude away from that obtained by direct (ratio) scaling experiments. It would appear possible from these choice models that the scaling approach Is the principal factor involved. However* as Eisler (19<&) has stated, the discrepancy between discri mination and ratio soales Is not as yet resolved by choice models. Luce (1959a) has attempted to Implement a better understanding of the psychological notions underlying the discrimination and ratio scales and the relations between them by proposing some explicit rules for the application of theories. Using Luce's theoretical principles, the number of permissible functional relations between Interval and ratio scales Is very limited. For example, If the subjective (dependent) scale Is measured on an Interval scale and the physical (indepen dent) variable Is measured on a ratio scale, the only possible laws are power or logarithmic functions; and If both are measured on a ratio scale, then the only possible law is a power function. However, Bozeboom (1962) has taken Issue with Luce's argument and shows that Luce's whole line of reasoning depends on the part of his principle which demands strict Invariance of the substantive theory under admlssable transformations. Bozeboom argues that there Is no reason to make such a demand which can lead to absurb conse 18 quences. Ekman and SJflrberg (1965) note that although It may be unreasonable and unnecessary to demand strict Invariance of substantive theories, some kind of Invar iance seems highly desirable. It will be seen that Luce has pointed out a very real problem and that even though Bozeboom*s criticisms are valid, some Interesting studies have resulted from Luce's paper. In an effort to amplify Luce's (1958) treatment and to explore the consequences for the relations between discrimination and ratio scales, Helm, Messlck and Tucker (1961) developed a rationale for a logarithmic transforma tion to rectify the observed nonllnearlty between the two scales. The rationale Is based on the fact that the discrimination models assume the proportion of times one stimulus Is Judged greater than another Is a function of differences In subjective scale values. If, however, such Judgments reflect ratios of subjective scale values instead of differences, the discrimination model would introduce an Implicit logarithmic transformation. This would imply the necessity for an antilog transformation to return the discrimination scale to an appropriate scale representing the subjects true responses. However, Helm et al. (1961) consider three other possibilities from their foregoing rationale, 1. The ratio model may be biased wherein analysis of responses by the usual ratio assumption would require 19 a log transformation to return to the correct "difference" scale representing the subjects true responses, 2. Subjects may actually be judging differences between stimuli on a logarithmic scale which would make judging ratios indistinguishable from a claim of differ ence judgments on the logarithmic scale. 3. The two types of scaling procedures generate different kinds of scales* tapping different sensory processes (Luce & Edwards, 1958). Galanter and Messlck (1961), employing the arguments advanced by Helm et al. (1961) presented evidence from loudness judgments supporting the validity of the ratio assumptions and the consequent power law. Both ratio and category scales were derived from the loudness judgments, A successive intervals analysis of the category judg ments was undertaken to account for unequal dlscrlmlnal dispersions and category widths. This correction only served to accentuate the curvilinearity. An antilog transformation was applied to the "corrected" oategory values which produced a linear relation between the ratio and category scales. Theoretical considerations developed by Luce (1959*) in conjunction with Galanter and Messlck's (1961) experimental results lead these author's to conclude that the appropriate form of the psycho physical scale is GS * a (4 + b)n 20 where "GS" is a ratio scale, 'V is the stimulus and "b" is an additive constant. Eisler (1962a) has argued that, given oertain assumptions, the category scale must be a logarithmic function of the magnitude scale. He has advanced evidence supporting the contention that the category scale is a real Fechnerian discrimination and that its deviation from the logarithmic function of the ratio scale can be explained by deviations of the Weber function for sub jective continue from Weber’s law. Eisler (1963a) defines a Weber function as any function that relates a measure of uncertainty of a psychophysical variable to its central values. Weber's law is a special case of a Weber function. Eisler goes on to state that the category's scale deviation from the ratio scale is a function of changes in discrimi nation with changes in magnitude. He concludes that in the category rating situation, the subject constructs a scale whose units consist of measures of his uncertainty, not units of his magnitude scale. In a subsequent report, Eisler (1963d) argues that the usual goodness of fit for pair comparisons and categorical judgments is insensitive to the form of the latent distribution. Supposedly, an infinite number of distribution forms to which the goodness of fit test is not sensitive can exist. As already noted, Eisler is interested in the lognormal distributions of discxlmina- tion scales with standard deviations proportional to their means. A case VI model of comparative judgment is suggested which cannot be distinguished from case V on the basis of fit. The case VI model produces a "logarith mic Interval scale" which is linear with ratio scales. Superficially, this is identical to the logarithmic relationship developed by Helm et al. (1961) although the rationale for this relationship is quite different. Eisler reasons that the difference between scales is the result of the difference between intra- and lnter- indlvldual standard deviations yielding a different Weber function. Intraindividual standard deviations furnish a linear generalization of the Weber law which gives a scale consistent with ratio estimation while lnterlndividual standard deviations (as used in pair comparison) furnish a scale assuming a pure Weber law and thus logarithmically related to ratio estimation. Ekman and Sjdberg (1965) also note that even though there is general agreement as to the logarithmic relation between discrimination and ratio scales, the theoretical interpretation of the relation is still very much a matter of controversy and uncertainty. Numerous other comparative investigations of the two main subclasses of scaling methods have been undertaken (Bjdrkman, 1959a; I960; Eisler, 1963b; Ekman, 1962; Ekman, Goude & Waem, 1961; Ekman & Kttanapas, 1962a; 1962b; 1963a* 1963b; 19630; Engen & Me Burney, 196*4-; Gamer, 1956; Kflnnapas & WlkstrOm, 1963; Parduccl, 1963; Perloe, 1963; Schneider & Lane, 1963; Stevens et al., 1958; Stevens & Galanter, 1957; Stevens & Guirao, 1963; Stevens & Harris, 1962; Stevens & Stone, 1959; Torgerson, I960; Warren A Poulton, 1962). All but four of these studies (BjSrkman, 1959a; Ekman & Kdnnapas, 19630; Parduocl, 1963; Warren & Poulton, 1962) confirmed the logarithmic relationship between category and discrimination scales and ratio and magnitude scales, Ekman and Sjflrberg (1965) point out that it is in line with the logarithmic relation that the two scales defined in opposite terns (loudness and softness) have a recipro cal relation when constructed by ratio methods, and a complementary relation when constructed by category methods, Torgerson (i960) first demonstrated this result with respect to judgments of reflectance. Subsequent studies (Eisler, 1962b; Schneider & Lane, 1963; Stevens & Guirao, 1963; Stevens & Harris, 1962) have confirmed this finding with length of lines, tactual, and loudness judgments. Special note should be taken of studies conducted at the Psychological Laboratory, University of Stockholm, Several of these investigations (Bjflrkman, 1959a; Eisler, 1962b; 1963a; 1963b; Ekman, 1962; Ekman et al., 1961; Ekman & Kttnnapas, 1962a; 1962b; 1963a; 1963b; 1963c; 23 Goude, 1962; Ktlnnapas & Wlkstrdm, 196 3) have compared scales constructed by the method of pair comparisons (Case V) with scales constructed using the constant sum method (ratio estimation). The results, with three exceptions (BJdrkman, 1959®; Ekman & Ktlnnapas, 1963c; Goude, 1962) were Identical with the findings regarding category scales, e.g., logarithmic function between the discrimination and ratio scales. The unique aspect of these studies Is that -Thurstonlan and ratio scales are usually applied to different kinds of continue. All scales except one (Goude, 1962) were based on subjective continue— aesthetic value, moral judgment, political Influence, occupational preference, etc.— where no physical continuum is available. Using the pair com parison method and the constant sum method was an attempt to keep the operations of the two scaling classes as similar as possible. The logarithmic relation between the Thurstonlan and ratio scales would seem to indicate that ldentiflcal factors are operating with respect to both category— Thurstonlan scales vs. ratio scales irrespective of the method of presentation of the stimuli. As already indicated, five studies (BjCrkman, 1959a; Ekman & Ktlnnapas, 1963c; Goude, 1962; Farducci, 1963; Warren & Poulton, 1962) did not find a curvilinear relationship between discrimination and ratio scales. Ekman and Ktlnnapas (1963c) reported a linear relation 24 between a Thurstonlan scale and constant sum scale on a continuum of conservatism. Bjflrkman (1959a), investigating fruit preference, found that for six out of eight stimuli, a linear relationship was appropriate for relating a Thurstonlan and ratio scale. In both instances, the linear relationship was rationalized to be a function of metathetic continue, where the case V is applicable and provides an undistorted scale. Goude (1962) compared the methods of pair comparisons and magnitude estimation using lifted weights. A very narrow range of stimuli was used— 307.5 to 363.5 grams. Employing an apparatus suggested by Baker and Dudek (1955) Goude found a linear relation between the two scaling procedures. Scale values based on both methods were found to be somewhat irratic in terms of not progressing in magnitude as the weights became heavier. Serious limitations were noted with this study. Only three subjects were used and the range of stimulus magnitude judged was too narrow to be a fair test of the relation between the two scales. Parducci (1963) found a linear relationship between ratio and category scales of numerousness in spite of various context effects. Parducci explains his results on the basis of a range-frequency compromise although as Parducci admits, this finding was secondary to the principal purpose of the study involving adaptation level. Consequently, the results were far from conclusive. Warren and Poulton (1962) made a special effort to main tain comparable experimental conditions In comparing ratio and a partition Judgment for lightness. The results, based on differences between medians and means, did not support a distinction between the two types of scales. Warren and Poulton consider the curvilinear relation between ratio and category soales to be a result of artifacts Introduced by Incidental differences In experimental procedure. Warren and Warren (1963) reason that both scales reflect stimulus correlates rather than sensory magni tudes. Warren postulates a "physical correlate theory" In which subjects have learned from their prior exper ience certain relations between their sensory experience and physical stimulus. Warren cites experimental data which he argues support his position. Ekman and Sjflrberg (1965) note that basically, Warren does not accept the psychological scales as psychological in the usual sense. In any event, Ekman and SjOrberg conclude that the relation between the two classes of scales still awaits its final interpretation. They believe it Is consistant with the log normal distribution stated by Eisler (1963d), but are not convinced that ratio estima tion (constant sum) scales are sufficient, or even necessary criteria. Attempts to reoonclle differences between discrimina tion and ratio scales have resulted in some further insights into the problems Involved but the basic issue still remains. We have seen that Luce (1959a; 1959b), Eisler (196*0, and Fagot (1963) have attempted to deal with the problem in terms of alternative axiomatic schemes or specified properties of the two laws. This has helped in clarifying some of the Issues involved but certainly has not resolved the problem. Perhaps the most useful finding to emerge from all of these attempts is the reasonably reliable finding of an approximate logarith mic relation between discrimination and ratio scales. The question is, which of the two scales is biased and needs a transformation to bring it in line with a "correct" estimate of a specified sensory process. To date, this question is still an issue (Ekman & SjCrberg, 1965). It would appear that either Torgerson's (i960) line of reasoning is a likely solution, wherein the two scales reflect two ways Individuals have of regarding quantity, or that Eisler's (1963a) argument will prove correct, e.g., in a discrimination situation, the subject con structs a scale whose units consist of measures of his uncertainty, not units of his magnitude scale. The latter alternative suggests an antilog transformation to return the discrimination scale to an appropriate scale repre senting the subject's true estimates in sensing stimulus 27 magnitude. The former alternative provides a "psycholo gical explanation" for the facts and suggests the Investi gator select the scale most suitable to his particular application. Hopefully* the present Investigation will explore this possibility further. Methodological Problems Galanter (1962) listed four kinds of criteria commonly used to justify the validity of a result— In this Instance category and ratio scales. They are: 1. The consistant repeatability of the result. 2. The explanation of the result In terms of some basic theory. 3. The prediction of new findings based upon the result. 4. Qhe Invariance of the result in the face of manipulation of ostensibly nonessential characteristics of the experiments. Galanter shows that the first three criteria are met by both of the methods. He then cites evidence to show that the category scale is seriously distorted when the particular stimuli that are used for constructing the scale are changed. The magnitude scale, on the other hand, Is invariant with changes in the stimulus ensemble. However, this stated Invariance is by no means a general characteristic of ratio or magnitude scales. 28 Ekman and SJdrberg (1965) note that It has been known for a long time that a magnitude scale may vary to some extent with the choice of standard stimulus. Engen and Levy (1955) found that the subjective range of magnitude scales of brightness and lifted weights to vary as a function of the value of the selected standard. Other Investigations (Dlnnerstein, 1965; Ekman, 196lb; Erickaen & Hake, 1957; Parducci, 1963; Parducci & Marshall, 1961) have demonstrated the effects of anchor stimuli In both ratio and discrimination Judgments. As Ekman and SJdrberg suggest, "common practice now Is to use either a standard somewhere In the middle of the subjective range, or no fixed standard at all" (1965* p. ^62). The effects of range and grouping of stimuli have been given particular emphasis concerning ratio scales. Garner (195^) and Engen and Tulunay (1957) showed that the group of stimuli selected had a strong effect on the scale values using factlonatlon techniques. In general ratio methods of scaling which have supported the power function have been found to be effected by the range of stimuli used. The exponent of a power function decreases as a function of a larger range of stimuli (Beck & Shaw, 1961; BjSrkman & Strangert, i960; Kflnnapas, I960; Strangert, 1961; Wong, 1963). Ekman and SJflrberg (1965) rationalize that this finding could be interpreted in at 29 least two ways: a) It may reflect a response bias, a kind of error In the subject’s handling of the numbers. In this Instance, the numbers given by the subject are distorted, b) The finding may reflect a genuine psychological effect. The effect here is the result of adaptation. Other studies (Bevan, Maler & Helson, 1963; Bevan & Turner, 1965; diLollo, 1964; Miller & Engen, I960; Over, 1963; Pierrel, 1963; Poulton & Simmonds, 1963; Boss, 1964; Weinfraub & Hake, 1962; Weiss, 1963; Weiss & Hodgson, 1963) have also shown that any contextual arrangement of the experimental situation is likely to have significant effect on both discrimination and ratio scales. Engen and Levy (1958), however, found the primary factor Influencing constant-sum Judgments was the magnitude of the stimuli themselves. Still, the majority of studies suggest quite conclusively that context has an adaptation effect consistant with the findings of Parducci (1963) and Helson (1959) concerning both classes of scales. Apart from the effects of the experimental situation, there has been a recent emphasis on individual differences in psychophysical scaling (Jones & Marcus, 1961; McGill, I960; Pradham & Hoffman, 1963; Schiffman & Messlck, 1963). Schiffman and Messlck (1963) refer to two major approaches in analyzing scaling data. One approach has 30 been to compute group averages and generalize the finding to a population. A second procedure has been to analyze each person*s responses separately. The first method has come under criticism due to the possible masking of Individual differences. The second method of treating each Individual separately also presents difficulties» particularly in describing the results for groups of Individuals and In comparing the results for several individuals and groups. Jones and Marcus (1961) modified individual psycho metric functions to include a constant characteristic of the individual. If is the appropriate constant for an Individual, these authors suggest a psychophysical function of^ = aSbc*, where c^ is an individual constant. Jones and Marcus conclude that the data now may be averaged if either a logarithmic transform is computed prior to averaging or the geometric mean is used. Both should logically furnish unbiased results. McGill (i960) has shown that certain loudness funotions are "personal equations." He verified that they reveal a good deal about the process of estimation in the individual but not much about loudness per se. These individual functions were brought Into line with Steven’s loudness function wherein subjects were found to agree on a reference point but disagree about the length of the scale. McGill concluded that the appropriate treatment of 31 Individual data can "bring meaningful results, e.g., similar to results found for groups. Pradham and Hoffman (1963) take issue with this view point and appear to demonstrate with lifted weights that individual psychophysical functions do not follow the power law although averaging over subjects does yield a power function. The conclusion is reached that the power law is an artifact of grouping. However, a serious problem exists with this study. The authors tested linearity of regression using the equation Log ^ « Log a + N log S This solution does not consider an additive constant which has been found to be usually necessary for a reasonable fit (Ekman, 1961a; Fagot, 1963; Galanter & Messlck, 1961; Lewis, i960). Fagot (1963) concludes that in fitting a power relation using a log-log plot, an assumption is made that the function passes through the origin and the assumption that the threshold is zero introduces an unknown error into the estimate. If an additive constant is introduced into the power law, then a simple graphicalsolution is no longer feasible. In this case, it is necessary to estimate the additive constant from a graph of a psychophysical function before a log-log plot can be made, and the estimation of this constant is very unreliable (Lewis, i960). Dudek (1964) has demonstrated that scale values based on pooled judgments of subjects' rating of adverbs closely approximate the median value where scale values were determined for each subject separately. In a study of lifted weights (Baker & Dudek, 1955), the distributions of the direct estimates of ratios were similar to the scale value distributions for individual subjects. It would appear then that grouping Individuals in the calculation of scale values does not seriously distort any measure of central tendency taken for Individual results. However, the foregoing evidence suggests that serious methodological problems exist In fitting psychophysical functions from either individual data or group data. It Is also evident that scaling based on either of the discrimination or ratio estimation techniques is sensitive to range of stimuli, grouping, and the context of the experimental situation. The idea of varying subject sophistication and studying its effect relative to contextual effects, etc., should provide useful information on the general applicability of discrimination and ratio scales. Possible Psychological Factors Affecting fAscrlmlnatlon and Batlo Scales It would have been quite logical to inolude a dis cussion of these factors under methodological problems connected with the contextual arrangements of the experimental situation. However, there are two psycholo- 33 gical factors, which have not heen extensively studied hut which deserve special consideration due to their possible theoretical Impact on an evaluation of ratio and discrimination scales. These are motivation and training. Galanter (1962) notes from articles by Swets (1961) and Shipley (1960) two kinds of errors that can be made in connection with detecting a stimulus. These are failures to detect a stimulus when it is there and reports of a signal when it is not there. Galanter discusses these errors in terms of the subject’s motiva tion or "outcome structure" of the experiment. This is known as a payoff function. He concludes from the work of Swets (1961) and Shipley (i960) that varying signal strengths give rise to the same detection probability if we but vary the motives and expectations of the subjects appropriately. The solution for the subject is some compromise between his values, his expectations, and the sensory limits of his eyes and ears. Kinchla and Atkinson (1964) confirmed this compromise by showing that false information feedback increases both the probability of a hit and of a false alarm, and at the same time Increases the number of incorrect responses. Other studies (Hblowinsky, 1964; Swets & Sewall, 1963) have found motivation to have little effect on psychophysical judgments concerning length discrimination 3^ or the detection of tone hurst. In both Instances motivation in the form of monetary rewards and instruc tions of consequences had little effect on performance. Hblowinsky (1964-) did find, however, that ability to discriminate length varied significantly with IQ. This was found to hold for subjects with IQ»s below 45. Swets and Sewall (1963) conclude that their negative findings concerning the effect of motivation or detection of tone bursts are well supported by other recent evidence (Blackwell, 1963; Lukaszewski & Elliott, 1961; Swets, 1961; Zwlslockl, Maire, Feldman, & Bubin, 1958). Unlike the status of motivation whose effect is inconclusive, training, practice, or sophistication is almost uniformly found to effect psychophysical Judgments (Blackwell, 1963; Carso, 1963; Gundy, 1961; Luce & Edwards, 1958; Lukaszewski & Elliott, 1961; Parducci, 1963; Poulton & Simmonds, 1963; Sprague, 1961; Swets & Sewall, 1963; Verplanck & Biaugh, 1958; Zwislocki et al., 1958). Negative evidence (Loeb & Dickson, 1961) has nevertheless been found in connection with tone bursts. The preponderance of findings still support a practice effect— at least insofar as is possible regarding adapta tion (Parducci, 1963). The effect and its extent is by no means clear in all instances. As Swets and Sewall (1963) conclude, "For audition the effect of practice is limited to the first session and is no more than 2db 35 for other than low frequencies'* (p. 126). Goude (1962) used extensive practice sessions to minimize its effect on weight lifting. Thus, practice, although generally- agreed to have an effect on Judgment, is still difficult to assess and is confounded with context effects of the experimental situation, motivation, and sophistication. Ekman and SJdrberg (1965) have concluded "that subjects in psychophysical experiments are usually remarkably cooperative and may prefer to give a biased estimate rather than no response at all when the experimental arrangement does not provide an adequate response alternative" (p. *f62). This implies that knowledge of the experimental situation is quite important and that practice is but one aspect of the total problem in making psychophysical Judgments. To date, the sophistication of the subject has not been systematically Investigated or controlled with the exception of a study by Poulton and Simmonds (1963) which raises doubts about the generality of direct measurements of sensory magnitude using unsophisticated subjects. The evidence concerning possible psychological factors affecting discrimination and ratio scales would appear to confirm the Importance of subject sophistica tion in making psychophysical Judgments. Sophistication of the subject is certainly confounded with training and practice. It would seem, that, by studying psychophysical scales relative to the subjects* knowledge of the situation and his intelligence, a good estimate could be made concerning what the overall effect of training and prac tice might be. In other words, the design of this study is such that if the effect of subject sophistication is found to be substantial regarding the obtained scales, at least three factors would be known to predominate* (1) intelligence of the subject, (2) knowledge of the subject reflected in his past training, and (3) the effect of practice given (1) and (2). CHAPTER III METHOD AND PROCEDURE Purpose Considerable evidence has been cited showing that subjects tend to bias their responses in psychophysical experiments as a function of training and other experi mentally manlpulable factors. The aims of this study are: 1. To compare the psychophysical functions of discrimination and ratio scales and evaluate their applicability over three levels of education and intelli gence. a. Compare psychophysical functions based on scales using the method of pair comparison (discrimination scale) and the constant-sum method (ratio scale) for subjects with knowledge of these methods, subjects of normal intelligence but who have no knowledge of scaling methods, and subjects Institutionalized as mentally retarded. b. Evaluate discrimination and ratio scales in terms of their suitability for retarded as well as for sophisticated subjects in judging length of lines and heaviness of weights. 2. To investigate possible differences in the observers ability to Judge the magnitude of stimuli as a function of his intelligence and sophistication. a. Compare accuracy of judgments using ratio and discrimination responses over three levels of education and intelligence. b. Study the effect of experimental conditions on accuracy of judgments over the three levels of sophistication. Method Judgments of length of lines and heaviness of weights constituted the two tasks used in this study. Both modalities are considered "prothetic" by Stevens. Judgment of Length. Strips of white felt (1 cm. wide) were laid against a black felt wall. The strips were viewed two at a time and were next to each other in a horizontal direction. Pour series of line lengths were judged by all subjects. The four ranges of strip lengths used are presented Subject*s View 39 Strips of white felt ® rH 11 below as series A, B, C, and D. Series A, B, and C were used In connection with pair comparison Judgments. Series D, representing the total range sampled, was used Stimulus No. Series A Series B Series C Seri i 1 10.0(1°30*) 30.0(4°30*) 90.0(13°28* ) 4.0 2 10.1 30.2 90.4 10.0 3 10.2 30.4 90.8 10.4 4 10.3 30.6 91.2 10.8 5 10.4 30.8 91.6 30.0 6 10.5 31.0 92.0 30.8 7 10.6 31.2 92.4 31.6 8 10.7 31.4 92.8 90.0 9 10 11 10.8(1°38*) 31.6(4°44*) 93.2(13°56* )91.6 93.2 140 with constant sum Judgments. Visual angles for some of the stimuli are given in parentheses. In Series A, B, and C, the subject Judged which of the two strips of felt presented was longer. A pretest ( 40 demonstrated that the length of lines had to he very olose to one another in order for the subject to have a dis crimination problem as discriminal dispersions were generally small. Thus* because of the large number of comparisons needed to include even a limited range of the stimuli, the following design was used which entailed Judging only those pairs of stimuli close together in magnitude. Each of the three portions of the total range (Series A, B, C) of length were presented according to the experimental method shown in Table 1, Table 1 Comparisons Bequired for Stimulus Series A, B, and C Stimuli K 6 1 2 8 3 5 9 ? Each MXW in Table 1 represents a comparison between two stimuli. Note that Xs below the diagonal involve the same comparison as their counterparts above the diagonal. A total of 15 comparisons were needed for each of Series A, B, and C, using this procedure. Porty-five comparisons (3 x 15) were needed to evaluate the scales for length over the total range. The stimuli were presented in random order with the exception that no stimulus was presented on successive comparisons. In Series D, the subject used the constant sum method dividing 100 points between the two lines in accordance with his best Judgment of the absolute ratio of the greater to the lesser. In presenting the stimuli, the following design was used: Series D was presented according to the experimental method shown in Table 2. Each ”X" in Table 2 represents a comparison Judg ment between two stimuli. The Xs below the diagonal Involve the identical comparisons as their counterparts above the diagonal. Twenty-eight Judgments were re quired for Series D using this procedure. The design presented was selected for scaling in order to economize on the number of Judgments necessary, i.e., 28 instead of the 55 which would be required if all possible comparisons were made. Four separate ranges of the length of lines had to be 42 Table 2 Comparisons Required for Stimulus Series D 1 2 3 4 5 6 7 8 9 10 11 1 - X X X X 2 X - X X X X X X X 3 X - X 4 X X - X X X X 5 X X X - X X X X X 6 X - X 7 X X X X - X X 8 X X X X X - X X X 9 X - X 10 X X X X X X - 11 X X X X - used to compare the two scaling methods. The constant sum method is suitable for a large range of stimuli required to investigate the questions under study. The method of pair comparisons is only applicable to relatively small ranges of stimuli due to small dlscrimlnal dispersions for this type of task. Thus, for pair comparison Judgments for both length and weight, three ranges of stimulus magnitude were selected within the total range sampled by the constant sum method. The three limited ranges of stimulus magnitude selected for use with the 43 pair comparison method sampled the low, middle, and upper range of the stimuli used in connection with the constant sum method. Consequently, a total of four ranges of stimuli were needed for each modality. The testing room measured 14 feet 3 inches in length, 11 feet wide, and 9 feet high. The subject was placed 12 feet 6 inches from the wall where the stimuli were presented. The lengths for Series A, B, and C were determined by a pretest using 10 undergraduate psychology and sociology students. The total range and spacing between ranges A, B, and C used for Series D was deter mined by previous studies (Guilford & Dingman, 1954) with the exception that the first and last stimuli of the pair comparison ranges were included in Series D. Judgment of Weight. Pour series of weights were also judged by all subjects. As with length, the weights were presented two at a time using an apparatus suggested by Baker and Dudek (1955). This consisted of a box-like table, enclosed on all sides except the experimenter's side. Two holes were drilled in the top of the table and a string was extended through each hole. At the top end of each string was a knob to which the weights were attached. The subject saw two knobs, labeled A and B and lifted the knobs one at a time with the same hand after appropriate weights were attached by the experi menter. Visual cues were thus eliminated. 44 Like the lengths, the four ranges of weights were labeled A, B, C, and D, where A, B, and C were used for the pair comparison judgments and D was used with the constant sum method. The total range of weight sampled was proportionately identical to the length of lines. The weights in grams by series were: Stimulus No. Series A Series B Series C Series 1 100 300 900 40 2 103 305 910 100 3 106 310 920 112 4 109 315 930 124 5 112 320 940 300 6 115 325 950 320 7 118 330 960 340 8 121 335 970 900 9 124 340 98O 940 10 980 11 1400 As with judgments of length, in Series A, B, and C, the weights were presented two at a time and the subject judged which was heavier. The Identical design was used for weight, as shown in Table 1, page 40, as the discri- mlnal dispersions for weight were also small. Thus, 15 comparisons were needed for each of Series A, B, and C. Series D for weights followed the same procedure described 45 for length in Table 2, page 42. Twenty-eight Judgments were required using this economical approaoh. Hie stimuli were presented in a random order with the exception that no stimulus was presented on successive comparisons. The weights for Series Af B, and C were determined by a pretest using the same subjects as were used for the length of lines. Again the total range and spacing was determined by Guilford and Dingman (1954) except that the first and last stimuli of ranges A, B, and C, were Included in series D. This allowed for a more direct comparison between the two methods of scaling. Subjects Three groups of subjects were used. Group I. Twenty mildly retarded subjects were selected who were resident at Pacific State Hospital (an institution for the mentally retarded). All patients had IQ's between 45 - 70, a diagnosis of undifferentiated or familial, and were over 21 years of age. They also had less than a 5th grade education. There were 11 males and 9 females. Their mean age was 30 wlbh a standard deviation of four years. These subjects were administered two scales of the WAIS"*" (Wechsler, 1955)---arithmetic and vocabulary— to ^■Wechsler Adult Intelligence Scale determine their potential ability for reporting the judgments required. A minimum total scale score of 3*5 was used for the selection of patients. This was a very liberal minimum and was by no means a significant crlter- ian in ultimately determining who was capable of making consistant judgments. These measures were useful, how ever, as a control for ability in the evaluation of psychophysical Judgments. The patients had a mean arithmetic scaled score of 3.5 and a standard deviation of 1.16. Their mean vocabulary score was 4.95 with a standard deviation of 1.81. The mean total score was 8.70 with a standard deviation of 2.58. The general norms for the WAIS scales were based on a reference group of 500 cases between the ages 20 - 34. For each of the two scales the average or mean score was 10 with a standard deviation of 3. The average total score (arith metic and vocabulary) was 20 and a standard deviation was not specified for the combination of these scales. Group II. Twenty normal subjects were recruited from the psychiatric technicians employed by Pacific State Hospital to care for the patients. These individuals were relatively homogeneous with respect to education and general health. All had a high school education, good health (no handicaps) and had no acquaintance with scaling methods. Each subject was carefully interviewed so that no one who had any college experience participated 47 in the study. There were three males and seventeen females. Their mean age was 33 with a standard deviation of 6. These subjects also were administered the vocabulary and arithmetic scales of the WAIS in order to control for these abilities. The mean arithmetic scale score was 10.70 with a standard deviation of 2.25; the mean vocabulary score was 12.20 with a standard deviation of 2.31; and the total scale score was 22.9 with a standard deviation of 4.17. Group III. Fifteen sophisticated subjects volun teered for the study. Eight subjects had Ph.D.s and seven subjects had MAs. All degrees were either in psychology, sociology or mathematics. All subjects were familiar with scaling and had at least one course on the subject. This group had 11 males and 4 females. Their mean age was 32 and standard deviation was 5. These subjects were not given the WAIS due to their familiarity with the test materials and the homogeneity of their education which was well above that considered adequate for making the judgments used. Procedure and Directions in the Administration of flasks Introductory instructions given to subjects were somewhat different for each group. Patients. "We are going to do an experiment where ^8 you lift weights and tell which one is heavier and look at lines and tell which one is longer. Have you ever done anything like this before? We Just want to find out how people tell the difference between weights, etc. It’s not a test and it's not going to go on your record. It’s not to see how smart you are. Both technicians and patients are taking the experiment, and I wanted you to be in it because I thought you’d like it and would do a good Job. Before we begin the experiment I want to ask you some questions." The arithmetic and vocabulary scales of the WAIS were then administered. Psychiatric Technicians. "This is an experiment in which we’re trying to determine people’s ability to tell differences in heaviness of weights and length of lines. There are six parts to the experiment, involving three series of weights and three series of strips, or lines. There are two groups— patients and technicians. We’re mainly interested in whether mental retardation affects the ability to make these distinctions, and we’re using the technicians as a sort of control, or normal group, against which to compare the retardates. There aren’t any gimmicks in the experiment. We’re not going to give you anything using optical illusions, and it’s really true that no two weights weigh the same amount and that no two lines are the same length. Before we begin the experiment there are some questions to answer concerning ^9 your vocabulary or arithmetic knowledge. This Information gives us an Idea of how these factors affect your Judg ments, All results are confidential," Sophisticates. No special Instructions were given since all of these subjects were acquainted with the writer and dissertation topic. All subjects were given five practice attempts for length and weights preparatory to making the pair com parison Judgments, Ten practice trials were given for the constant sum method. For this method broad limits of the correct answer were set up to determine whether or not the subject was capable of making the Judgments. For example, two stimuli with a ratio of magnitudes of 55*^5 would be considered correct within 50*50 to 75*25 limits. If an individual could not succeed in getting ten correct practice trials within these limits he was eliminated from the experiment. Thirty patients were tested before twenty were selected capable of doing ratio Judgments, Normal and sophisticate subjects had no difficulty at all. No test stimuli were Included in the practice series. In addition, all subjects were given careful Instructions on lifting weights and Judging lines lengths. For example, regarding weights* "You may pick up the knobs as many times as you wish, but always begin by lifting the left one and then the right one. Always 50 use the same hand and use the hand you use most for other things. Do not rest your wrist or elbow on the edge of the table, but keep your arm raised like this." (Efemon- dtratlon given). In the instance of pair comparison judgments subjects were told merely to select the longer line or heavier weight. For constant sum Judgments subjects were instructed to: "Try and decide how heavy the weights are in comparison to each other. Give me a number for each, Indicating their relative weights. The two numbers must add to 100. Always give the most points to the heavier weight and in proportion to how much heavier...O.K., how would you assign points to these?" One examiner was used for all subjects throughout the study. This individual was well acquainted with both the hospital staff and patients, being employed on various research projects at the hospital to make anthropological observations. She was an undergraduate student in anthropology. The presentation of the stimuli was carefully counter balanced for all groups. Three conditions were controlled in this manner: 1. Order of task— length presented first or weight presented first. 2. Order of Series A, B, or C pair comparison Judgments presented first or Series D constant sum Judg- 51 ments presented first. 3. Order of stimuli— the larger stimulus presented first or the smaller stimulus presented first. However, the order of presenting the pairs of stimuli within each series was not varied over the three groups. Possible adaptation effects were possible although such effects were not considered during the design of the study. Finally, all subjects were screened concerning 20/20 vision and other possible handicaps. This was particularly important concerning the patients. Medication records were also checked for the patients. In general, all subjects were in good health with no handicaps. Even though the presentation of the stimuli were counterbalanced, their possible effects were nevertheless investigated where feasible. Order of task, order of stimulus series, order of stimuli within the series, and the total WAIS score (applicable to only patient and normal groups) were included for study. All four variables were dichotomized + 1 or - 1 for analysis. The various orders of presentations were dichotomized accord ing to their description given on pages 50 and 51. Subjects were split into high (+ l) and low (- 1) sub groups at the median WAIS scores. The approach used to study the effect of these variables on the subjects* Judgments was a stepwise multiple regression analysis. The independent variables were the four factors listed above. The possible effect of the interaction between any combination of these factors was handled by entering them in the regression analysis as additional independent variables. For example, the Interaction between order of stimuli and task (S x T) could be determined by coding (S i T) + 1 or - 1, If both S and T are + 1 or - 1, then the interaction term is coded + 1. If either are - 1 while the other is + 1, the interaction Is coded - 1. In this manner interaction between any of the four experimental conditions could be investigated by considering each of them separately as an independent variable. With four experimental conditions there are eleven possible interaction terms. In the case of the sophisticate group there were only three conditions studied giving four possible Interaction terms. The dependent variable used was an actual response on a specified trial. For the pair comparison data, Judgments were coded according to which stimulus of a pair was selected, e.g., in the comparison "2 x 3," if ”2” was selected over w3»" the response was coded M0;M if W3W was selected, the response was coded "1." For the constant sum data, the actual number of points assigned to the larger of the two stimuli on a given trial was used for the dependent response. A total of thirty-six responses were selected for 53 analysis from the pair comparisons among Series A, B, and C. Specifically, two responses were sampled from each of the three series for length and weight in each group. To summarize, there were 3 series of stimuli x 2 tasks x 2 responses x 3 groups which equals 36 separate analyses. Twenty-four responses were selected for analyses from Series D concerning the constant sum responses. In this case four responses were sampled from Series D for each task in each group. More specifically, there were 2 tasks x 2 responses x 3 groups (xl series of stimuli) which equals 2b separate analyses. All selections of responses within Series A, B, C, and D were random. Although a stepwise multiple regression approach was used in the analysis of the data, this approach was employed as a convenient method for doing an analysis of variance. Brownlee (i960) discusses In detail the multiple regression approach to analysis of variance. Briefly, two sources of Tariation are evaluated* sums of squares due to regression and the remainder sums of squares (deviation about the regression). If the P ratio based on mean squares calculated from these sources of variation is significant, the sums of squares due to regression can be further partitioned according to the effects of each of the independent variables and evaluated separately. In this study, the independent variables were manipulated as far as possible by counter-balancing to produce orthogonal relationships between the independent effects investigated. The counter-balanced design was utilized to remove the effect of possible biased responses from one group in comparison to another. It does provide a convenient device, however, for the interpretation of any significant effects. To the extent orthogonality between the independent variables exist, it is easier v to interpret the effect of any one of these variables. The actual analysis of the effects of the experi mental conditions on the sample of 60 (36+24-) responses selected for study was accomplished by evaluating 60 separate F ratios. Of the total of 60 analyses, only one showed a significant effect to be present. The normal subjects* Judgment of the heavier of two weights in Series B (300 gm. weight vs. 310 gm. weight) was apparently significantly affected by the order of the task, the order of the presentation of stimuli, level of ability, and the interaction between these conditions. In general, if lifted weights were given first and/or the heavier stimulus presented first, the subjects with the least aptitude Judged the lighter stimulus as "heavier." The interaction between these factors was also signifi cant which further complicates this explanation. However, because only one out of sixty independent tests for the effect of experimental conditions emerged significant, 55 there is little point in giving detailed attention to this single significant finding. A table of P values for the sixty analyses undertaken is presented in Appendix C. Scaling Methods and Scaling Results Pair Comparison. The method of pair comparison was used to scale each task in terms of discrimination judgments. The method of pair comparison is discussed in detail by Guilford (195^) and Torgerson (1958). In its complete form, the method involves pairing each stimulus (weight or line strip) with every other stimulus. The subject "Judges” one of the two stimuli as "heavier,M "longer," etc. Based on the proportion of times one stimulus is Judged over others, it is possible to esti mate its relative psychological distance from the other stimuli, given specified assumptions (Guilford, 195^; Torgerson, 1958). As Torgerson and Guilford point out, it is not necessary to compare every stimulus with every other stimulus to erect a scale based on the method. The number of observations can be reduced by selecting a limited number of stimuli as standards or given the stimuli in rank order, Judgments can be obtained only for those pairs that are close together in scale value. The latter approach was used in this study. Gulliksen's (1956) least-squares solution for incomplete data was used to scale the pair comparison data. This solution assumes Thurstone's Case V but still represents the most satisfactory analytical approach available for an incomplete data matrix of the type used in this study. Being a least-square solution, it will solve for scale values of a set of stimuli for which the sum of squares of the discrepancies between the observed unit normal deviates corresponding to the number of times stimulus K is Judged greater than stimulus J and estimates of these normal deviates are a minimum. Within the limitations of the assumptions involved in using the unit normal deviates as Thurstone (Guilford, 19540 pro posed in his law of comparative Judgment, the solution will provide a "best fit" for the observed data. Goodness of fit tests discussed by Guilford (19540 and Torgerson (1958) were applied to check the internal consistency of the solution. However, as Eisler (1963d) has shown, these tests are insensitive to the form of a possible latent distribution and measure only to what extent the distributions can all be normalized with some arbitrary transformation of the scale axis. Tables 3 and 4 present the pair comparison scale values and associated chi square values showing goodness of fit for length and weight of the three groups studied. Two findings are very apparent. The scale values are somewhat unstable in a few instances, particularly in Series C for lengths. In other words, the scale values Table 3 Fair Comparison Scale Values and Associated Chi Square Values (goodness of fit) for Length of Lines for Patient, Normal, and Sophisticate Groups cm. Patients Normals Sophisticates Series A 1 10.0 .000 .000 .000 2 10.1 .140 .143 .524 3 10.2 .245 .236 .670 A 10.3 .447 .592 1.011 5 10.4 .514 1.016 1.223 6 10.5 .616 1.428 1.796 7 10.6 1.298 2.277 2.222 8 10.7 1.190 2.555 2.755 9 10.8 1.644 2.805 3.036 x2=3,397 x2=0.631 x2=5.599 df=13 df»13 df=13 Series B 1 30.0 .000 .000 .000 2 30.2 .122 .162 .180 3 30.4 .137 .616 .696 4 30.6 .369 .844 1.346 5 30.8 .583 1.018 1.629 6 31.0 .872 1.467 2.003 7 31.2 1.085 1.821 2.467 8 31.4 1.181 2.183 2.180 9 31.6 1.199 2.327 2.155 x2=4.570 x2=2.450 x2=14.185 df = 13 df = 13 df = 13 (Continued on following page) 58 Table 3— continued cm. Patients Normals Sophisticates Series C 1 90.0 .000 .000 .000 2 90.4 .541 .163 .168 3 90.8 .554 .487 .516 4 91.2 .614 .785 .790 5 91.6 .296 1.318 .750 6 92.0 .293 1.462 1.185 7 92.4 .235 1.297 1.050 8 92.8 .280 1.397 1.104 9 93.2 .01? 2.105 1.700 x2=8.807 x2=0.209 x2=1.020 df=13 df=13 df=13 do not always Increase in value with an Increase In 2 stimulus magnitude. Secondly, the x values are all Insignificant Indicating that by the traditional criteria, Thurstone’s Case V was appropriate and the "goodness of fit" of the model was adequate. Similar findings were apparent In Goude's (1962) attempt to use the pair com parison method with lifted weights. The matrices of joint proportions Indicating the percentage of times one stimulus was chosen greater than another stimulus for Series A, B, and C for length and weight are given in Appendix A. These proportions substantiate the scale values In revealing that discrimi nation at times appeared to decrease as the stimuli got further apart. This causes the scale values to decrease Table 4 Fair Comparison Scale Values and Associated Chi Square Values (goodness of fit) for Weight for Patient, Normal, and Sophisticate Groups gms. Patients Normal Sophisticates Series A 1 100 ,000 .000 .000 2 103 .643 .570 .214 3 10 6 .706 .720 .301 4 109 1.136 1.014 .667 5 112 1.426 1.373 1.754 .493 6 1X5 1.439 .735 7 118 1.623 2.292 1.462 8 121 1.836 2.337 1.204 9 124 1.922 2.568 1.860 x2*3.409 x2=4.810 x2=3.273 df=13 df=13 df=13 Series 5 1 300 .000 .000 .000 2 305 .249 .485 .548 3 310 .276 .443 .725 4 315 .200 .757 1.349 5 320 .400 .930 1.605 6 325 .534 1.172 1.570 7 330 .967 1.471 1.503 8 335 1.209 1.531 1.678 9 34 0 1.473 1.438 2.021 x 2 = 9 .054 x2=5.671 x2=5.450 df=13 df=13 df=13 (Continued on following page) 60 Table 4— continued gms. Patients Normals Sophisticates Series C 1 900 .000 .000 .000 2 910 .2 .391 .106 3 920 .555 .380 .231 k 930 .180 .7 87 • 37^ 5 9^0 .86^ 1.058 .781 6 950 1.000 1.191 .917 7 960 .9^ 1.285 1.232 8 970 .716 1.382 1.599 9 980 i;JW4 1.671 2.051 x2=3.103 df=13 x2= s2.955 df=13 x2=4.450 df«13 as a function of stimulus magnitude and results from poor discrimination on the part of the subjects. Some of this difficulty could have been avoided if the stimuli were spread out more although the results of the pretest did not indicate such a possibility. The relationship between length and weight stimuli and the associated pair comparison scale values were investigated by considering two factors: 1. The type of function describing the relationship between each series of lengths and weights and their respective scale values for each of the three groups was determined by attempting to fit a number of well known functions and accepting the one giving the smallest standard error of estimate. The following functions 61 were fitted to each of the 18 resulting psychophysical curves (3 groups x 2 modalities x 3 series of stimuli): a. y>= a S + K b. (ft** a S2 + b S + K c. i / J = a e1 *3 + K d. a Log S + K e. a (S + K)n where ^ = Pair comparison scale values S = Stimuli a, b, K, and n = constants determined by best fitting equation All of the above functions were fitted analytically using a computer program which essentially applies an appropriate transformation for the function to become linear, and then determines a least-square solution. Standard errors of estimate are given for each of the best fitting equations representing each of the five functions used. 2. Since the ranges of stimuli covered within each series by the method of pair comparison were very restric ted, the psychophysical function would need to be investi gated by evaluating the slopes at each of the three points of stimulus range studied in Series A, B, and C. As was noted in Goude*s (1962) study, a linear relation between a discrimination scale and stimulus magnitude within a 62 narrow stimulus range is not a true test of the psycho physical function over a larger sample of the stimulus range. The values of the slopes of the psychophysical function at the three points over a large range of stimuli would be Indicative of the type of function that exists. If the function is curvilinear then the first derivative at any stimulus value in Series A, B, or C would give an evaluation of the slope. Specifically, if the overall psychophysical function for the discrimination scale is linear, then the slope at stimulus Series A would equal the slope at Series B and Series C, Two other possibilities exist. If "a" is the slope and A, B, and C represent the low, middle, and high sections of the total range, then aA > aB > aC would describe a negatively accelerated curve indicative of a power function f = a (s + b)n where the exponent n Is less than one or a logarithmic function y*= Log S + K. However, if a aA< aB< aC then overall function would have to be positively accelerated and therefore would likely be a power function as above where the exponent n is greater than one or an ; \ 63 exponential function y = aebs + K The linear function already described would be aA = aB = aC which would satisfy the power function where the exponent n is equal to exactly one. The quadratic function due its flexibility could satisfy any of the previously mentioned possibilities. However, the quadratic function has little theoretical value. In this manner, it is also possible to evaluate the similarity of the discrimination scale based on three samples of the total range and the ratio scale covering the total range. Meaningful comparison among the three ranges of stimuli for each task was achieved by use of "standard ized" measures which are results of a linear transforma tion involving translation (expected value) and rotation (variance), following Gullickson*s (1956) approach. Constant Sum Method. The constant sum method as presented by Torgerson (1958) was used to scale each task in terms of ratio judgments. This method is discussed in detail by Guilford (195*0 and Torgerson (1958). Basically, it entails requesting the subject to divide 100 points between stimuli presented in pairs in accor dance with the absolute ratio of the greater to the lesser. Every stimulus is compared with every other stimulus. 64 The number of points assigned to stimulus J when compared to stimulus K is then averaged over individuals. Scale values are derived by taking an average of the ratios of the mean point assignments for each pair of adjacent stimuli. Two modifications of the constant sum method as proposed by Comrey (Guilford, 195*0 were employed. The first involves using the geometric mean of the point assignments instead of the arithmetic mean customarily used. This procedure has been suggested by both Stevens (1962) and Coombs (1964). The second modification was suggested by Torgerson (1958) and involves substituting the geometric mean for the arithmetic mean of the derived ratios. In both instances it has been shown that the geometric mean provides more stable, internally con sistent averages for ratio numbers than does the arith metic mean (Coombs, 1964). Because of the problem considered by Pradham and Hoffman (1963) and Schiffman and Messick (1963) concerning averaging individual responses, an additional measure was taken to insure the "representiveness" of the geo metric mean of the individual point assignments. It was desirable in this Instance to exclude "blunders” or "gross errors" of a subject on a specified trial from the geometric average of the point assignment for that trial. In this manner the "efficiency" of this average 65 and the degree to which it was representative of the individual point assignments would he improved. Dixon and Massey (1957) suggest a criterion for eliminating these extreme blunders on any given trial from the sample. For samples of 1^ to 30 observations the following formula is given* r = "3 - *’ 22 Xk-2 “ X1 where r22 *s a statistic referred to a probability table (Dixon and Massey, 1957) for determining the level of significance regarding the chances out of 100 that the extreme value belongs to the same population of values; (the criterion used to exclude values in this study was type I error, Of £ .01); x^ = a specified observation or judgment where the subscripts on the observations indi cate that k observations have been arranged in order of size from the smallest to the largest and numbered 1 to k, xx < x2 ... < Each value can then be tested. In actual practice, no more than three Judgments were excluded from the average point assignment on any trial. Approximately 35 per cent of the trials had one to three extreme Judgments removed based on all subjects tested. Tables 5 through 7 present the geometric means, standard deviations, and number of subjects included for 66 Table 5 Geometric Means, Standard Deviations, and Number of Subjects Included for Each of the 44 Point Assignments Used for the Constant Sum Scale of Length and Weight Patients Length Weight GM SD N GM SD N 1 (1x2) 82.17 18.849 20 72.67 16.749 20 2 0x5) 90.53 19.257 20 87.62 17.356 20 3 (1x8) 95.12 15.323 20 96.77 10.463 19 A (lxll) 97.21 7.636 18 97.55 4.645 18 5 (2x4) 69.20 15.^98 19 47.41 13.002 19 6 (2x5) 86.91 16.951 20 82.04 17.811 20 7 (2x7) 86.76 16.802 20 72.28 15.030 20 8 (2x8) 93.80 9.422 19 91.17 14.172 20 9 (2x10) 94.44 4.204 17 94.36 11.063 20 10 (2x11) 96.72 6.980 19 97.76 2.692 18 11 (4x5) 83.OO 15.3*0 20 77.59 15.753 20 12 (4x7) 83.60 13.580 20 85.75 16.931 20 13 (4x8) 91.24 9.148 20 93.77 13.262 20 14 (4x10) 94.64 11.948 20 94.61 14.639 19 15 (5x7) 69.OO 17.796 20 59.26 18.957 20 16 (5x8) 84.15 IO.965 19 87.88 15.17k 20 17 (5x10) 86.65 13.421 20 90,84 12.692 20 18 (5x11) 92.92 12.388 20 93.98 4.772 18 19 (7x8) 81.65 12.405 20 87.39 13.019 20 20 (7x10) 84.85 10.956 20 90.04 12.040 19 21 (8x10) 62.08 23.417 20 71.97 21.055 20 22 (8x11) 83.87 14,442 20 77.86 14.54? 20 67 Table 6 Geometric Means, Standard Deviations, and Number of Subjects Included for Each of the 44 Point Assignments Used for the Constant Sum Scale of Length and Weight Normals Length Weight GM SD N GM SD N 1 (1x2) 80.25 10.171 20 79.37 17.554 20 2 (1x5) 92.53 4.983 19 89.05 11.888 20 3 (1x8) 96.94 1.736 18 97.12 3.007 20 4 (lxll) 97.76 1.830 19 97.16 3.562 20 (2x4) 52.55 2.179 17 53.73 2.909 17 6 (2x5) 73.00 9.644 20 76.63 12.987 20 7 (2x7) 7 2,96 11.036 20 74.11 11.989 20 8 (2x8) 92.60 5.912 19 92.88 5.305 19 9 (2x10) 92.00 5.126 18 94.82 3.663 17 10 (2x11) 94.75 3.685 18 97.96 3.247 18 11 (4x5) 72.60 6.077 19 72.52 7.625 20 12 (4x7) 79.20 10.333 20 83.99 10.547 20 13 (4x8) 87.99 7.758 20 89.55 7.381 20 14 (4x10) 93.58 5.666 20 94.39 6.471 19 15 (5x7) 53.82 2.940 18 54.62 5.599 19 16 (5x8) 73.02 7.397 20 79.27 10.867 20 17 (5x10) 72.55 4.661 20 82.65 11.378 20 18 (5x11) 83.29 10.549 20 94.21 5.248 18 19 (7x8) 74.79 4.890 19 79.48 9.085 19 20 (7x10) 78.66 7.907 20 85.94 7.422 20 21 (8x10) 52.30 2.948 19 54.65 4.630 20 22 (8x11) 67.76 7.720 20 63.24 10.633 20 68 Table 7 Geometric Means, Standard Deviations, and Number of Subjects Included for Each of the 44 Point Assignments Used for the Constant Sum Scale of Length and Weight Sophisticates Length Weight GM SD N GM SD N 1 (1x2) 71.72 4.111 15 68.75 9.749 15 2 (1x5) 87.35 3.977 15 82.37 11.135 15 3 (1x8) 94.88 1.840 14 96.90 2.503 14 A (lxll) 96.47 0.942 15 97.32 2.788 14 5 (2 x4) 53.67 1.630 14 58.59 3.249 13 6 (2x5) 74.55 2.752 15 72.5^ 9.285 15 7 (2x7) 73.84 4.063 15 76.52 9.741 15 8 (2x8) 88.06 2.984 15 93.53 4.028 13 9 (2x10) 88.83 2.916 15 92.83 4.614 15 10 (2x11) 92.44 1.869 15 95.33 2.985 15 11 (4x5) 70.99 4.205 15 68.99 7.183 15 12 (4x7) 71.15 4.209 15 77.34 9.076 15 13 (4x8) 85.03 1.000 12 90.98 4.926 15 14 (4x10) 87.48 3.179 15 94.12 2.907 14 (5x7) 53.55 2.095 14 53.51 2.061 13 16 (5x8) 71.53 3.130 14 79.41 9.544 15 17 (5x10) 73.53 3.320 15 82.89 4.213 13 18 (5x11) 83.15 2.972 14 91.06 6.323 15 19 (7x8) 69.98 3.249 15 79.83 7.189 15 20 (7x10) 71.54 2.823 15 77.10 5.906 15 21 (8x10) 53.34 2.644 15 53.26 3.134 13 22 (8x11) 70.00 0.000 13 59.58 3,826 15 each of the forty-four comparison judgments used for the constant sum scales of length and weight. The rela tionship between these geometric means and the vari ability of the point assignments associated with these means is presented in Figures 1 and 2 for length and weight. It can be noted from Tables 5» 6, and 7 that the number of subjects varies from trial to trial. It Is also evident that the variability of judgments was greatest for the patients and least for the sophisti cates. It can be observed from Figures 1 and 2 that for both length and weight the relationship between the geometric means and standard deviations of the constant sum judgments differs somewhat between the groups studied. The patients clearly show the highest standard deviations although for them this variability is inversely related to the value of the geometric mean of the judgments. The normal subjects exhibited their greatest variability of judgments in the middle of the range of point assignments. Finally, the sophisticates show somewhat the same relationship between the geometric means and standard deviations as the normal subjects with the exception that in the case of length judgments the variability was extreme throughout the whole range of I 1 f s 72 point assignments. For all groups variability in Judgment was least where the point assignment given to one stimulus relative to a second was large, e.g., 90 points or greater. Dudek and Baker (1956) found identical kind of results concerning constant sum Judgments of length and weight. However, these authors found larger standard deviations for the closer Judgments than seen in Figures 1 and 2 concerning geometric means of 70 or less for the normal and sophisticated groups. Scaling was done on five subsets of the 11 stimuli used for each task. These subsets of stimuli were selected so that all 11 stimuli could be scaled even though an incomplete pair comparison design was used (see Table 2, page 42) in the interest of economizing on the number of Judgments required. Using weight as an example for both tasks, the following sets of stimuli were scaled* 1. Weights 40 gm., 100 gm., 300 gm., 900 gm., and 1400 gm. 2. Weights 100 gm., 124 gm., 300 gm., 340 gm., 900 gm., and 980 gm. 3. Weights 100 gm., 112 gm., and 124 gm. 4. Weights 300 gm., 320 gm., and 340 gm., 5. Weights 900 gm., 940 gm., and 980 gm. Subsets 3» and 5 were a poor choice, being too few stimuli to scale and resulting in extremely irratic scale values in comparison with subsets 1 and 2. Con sequently, subsets 3» and 5 were discarded. Subset 1 included two anchor stimuli at the extremes of the stimulus range employed and three weights in the middle range. Subset 2 stimuli all represent the middle range and were selected to be identical to the two end stimuli used in Series A, B, and C for the pair comparison approach. Figure 3 illustrates the stimuli sampled for scaling over the total stimulus continuum using the pair comparison and constant sum methods for weight. Table 8 presents constant sum scale values for length and weight for each group. Scale values are presented as computed from the two submatrices of point assignments (see Figure 3), hence two scale values are given for the same stimulus in three instances. As can be noted in Table 8, generally all scale values are consistent even though calculated separately from two subsets of overlapping stimuli. The rationale for combining the scale values in this manner was based upon the added stability of a psychophysical function provided by including scale values associated with the two extreme stimuli ( * J - and 1A0 cm. lines; *1-0 and 1*1-00 gram weights). This was especially important since, as seen in Table 8, scale values associated with the stimuli identical to those used for the pair comparison scale, tended to Figure 3 Illustration of Stimuli Selected for Scaling, Using Pair Comparison and Constant Sum Methods 40 Total Range Grams 100 124 300 340 } 900 980 3 PC Series A PC Series B PC Series C 1400_L PC = Pair Comparison CS = Constant Sum = Series D— Total range X-L...X^ Dl = Series D— Limited range Y^.^Yg CS Series dT_jcj + X2 • x3 X, -^5 Sei 4 75 Table 8 Constant Sum Scale Values for Length of Lines and Weights for Patient, Normal, and Sophisticate Groups Length Patients (^) Normals ((p) Sophisticates (ip ) 4 .127 .108 .162 10 .2 74 .3 67 .395 10 .185 .308 .348 10.8 .281 .318 .411 30 .982 1.200 1.103 3° .866 .975 .981 31.6 1.200 1.015 1.054 90 3.131 3.480 2.964 90 3.626 2.959 2.447 93.2 140 5.112 3.485 2.761 9.315 6.072 4.772 Weight 40. .135 .123 .154 100. .264 .285 .309 100. .285 .270 .276 124. .234 .289 .324 300. .778 .844 .793 300. .730 .856 .806 340. .845 .956 .993 900. 3.639 3.963 3.334 4.117 900. 3.806 3.536 980. 6.383 4.702 3.959 1400. 9.906 8.561 6.446 increase lrratioally in some instances over a small range of stimulus magnitude. For example, scale values for weights 900 and 980 grams increased from 3.806 to 6.383 in the patient group. This finding is in contrast to a scale value of only 9*906 for 1^00 grams in the patient group. Good internal consistency was demonstrated for the constant sum scales by plots of empirically derived ratios as a function of expected (derived) ratios determined from scale values. In all Instances these plots showed a linear relation between the two sets of ratios and generally little variation was found around the regression between them. Figures illustrating the Internal consistency of the constant sum scales for all groups concerning length and weight can be found in Appendix B. The relationship between length and weight stimuli and the associated constant sum scale values was investi gated in the same manner as was previously discussed for pair comparison scales.2 One exception should be noted regarding the constant sum scale. After some initial attempts at fitting these curves, it was necessary to employ a non-linear least squares estimation program for some of the functions 2 See page 61. (Hartley, 1961). The Increments in stimuli and scale values were such that a satisfactory fit could not always be obtained using a least squares linear solution on specified transformations of the data as was employed for the discrimination situation. A more detailed dis cussion of this problem is presented in Appendix D. The program employed for the exponential, log, and power functions concerning the constant sum data was an Iterative procedure that yields a sequence of parameter values which have a maximum likelihood of converging to the true values of the parameters of the function examined. This routine finds a "best fit" in the direc tion of the steepest descent. The linear, quadratic, exponential, logarithmic, and power functions were fitted to combined scale values (Dj and Dj^, Figure 3) from the two submatrices used. Although combining scale values calculated separately from two submatrices of point assignments can be a questionable approach, there was sufficient overlap in the stimuli scaled to produce very consistent scale values. This was checked empirically before the scale values determined from the subsets of stimuli were combined. As with the pair comparison scales, the criterion used for selecting the appropriate function was the standard error of estimate. Similarly, the power and logarithmic functions represented the preferred equations 78 other factors being equal for theoretical reasons. Investigating the slope of the best fitting function at specified points of the stimuli (ranges covered in stimulus Series A, B, and C using the pair comparison method) offered a means of oomparing the Thurstonian discrimination scale with the constant sum scale. If the psychophysical function is linear, the slope is directly apparent. The slope of a curvilinear function can be evaluated simply by differentiating the scale (^) with respect to a specified stimulus value (S). Appropriate comparison of the relationship between the pair comparison and constant sum scale values was investigated by this procedure. The internal consistency of the constant sum method was checked by plotting observed derived ratios as a function of expected derived ratios determined from the scale values. This was done for the three groups. CHAPTER IV RESULTS The first question Investigated was the psychophysi cal function characterizing the pair comparison and constant sum scales over the three levels of subject sophistication for the stimuli under study. Psychophysical Function of the Pair Comparison Scale Figures A and 5 illustrate the pair comparison scales of the three groups for subjective length and weight as a function of their physical magnitude for the three ranges of stimuli used. Table 9 complements these figures in showing the regression coefficients and standard error of estimates for the eighteen best fitting linear equations illustrated in Figures 4 and 5. An attempt to fit the linear, quadratic, exponential, logarithmic, and power functions to these data revealed that in all instances the linear equation provided the best fit or was at least as good as alternative best 79 SERIES A SERIES B SERIES C S oph. Normal 3.0 2.8 2.6 2.4 2.2 2.0 P ts V o o o o o o o o o Soph P ts // 0 > o Ch> o C O Co CO CO o CO Noma I Soph P ts tO o to o to o to to to IS5 to rs> — roco^cncn'sjoD L e n g t h f n c e n tim e te r s FIGURE 4. PAIR COMPARISON SCALE OF PATIENTS, NORMALS, AND SOPHISTICATES FOR SUBJECTIVE LENGTH ( y ) AS A FUNCTION OF PHYSICAL LENGTH FOR THREE RANGES OF S T IM U LI. C O O SERIES A S ERIES B SERIES C 3 .0 Normal 2.8 2.6 2 .4 P ts .2 .0 o Soph .8 .6 .4 .2 .0 .4 .2 □ D O O — — O C O O) (D N) (fl ro ro t o t o t o f o t o t o t o c n t o cn o x ^ < 0 O o 1 0 < 0 ro < 0 to ( O Cn O < 0 o ( O o W e ig h t in g r a m s FIGURE 5. PAIR COMPARISON SCALE OF PATIENTS, NORMALS, AND SOPHISTICATES FOR SUBJECTIVE WEIGHT ( V ) AS A FUNCTION OF PHYSICAL WEIGHT FOR THREE RANGES OF S T IM U LI. CD H 82 Table 9 Regression Coefficients and Standard Error of Estimates for Eighteen Linear Equations Fitted to Psychophysical Functions Based on Pair Comparison Scales m Length Weight 4> t e CO Patients Normals Sophis ticates Patients Normals Sophis ticates Regression Coefficients A 2.000 3.896 3.788 .074 .108 .071 B 1.858 1.521 1.556 .035 .038 .044 C -.077 .609 .468 .015 .020 .025 Standard Error of Estimates A .158 .224 .130 .164 .120 .205 B .091 .081 .312 .168 .135 .244 C .212 .202 .155 .229 .105 .144 fitting functions. Since the scale values were erratic in several instances and difficult to fit using any function, the linear equation was the preferred equation, other factors being equal. The standard error of estimate was used as the criterion for evaluating the best fitting function. Figures 4 and 5, and Table 9* show clearly that a negatively accelerated curve describes the psychophysical function over the full range of stimuli studied. This 83 finding holds for both length and weight for the patient, normal, and sophisticated groups. In other words, were it possible to tie together the scale values for the three ranges studied (Series A, B, and C), it would be evident that as stimulus magnitude Increased, the slope of the curve would decrease. The flatter slopes shown for stimulus Series B and C in comparison to Series A in Figures 4 and 5 illustrate this point. In general, the steeper the slope, the better the discrimination shown by the subjects involved. It can be noted that generally the slopes of the lines in Figures ^ and 5 are steeper for the sophisticate and normal groups than for the patient group. This is particularly apparent in Figure k concerning length. Consequently, the discrimination shown by the sophisti cate and normal groups was usually superior to that shown by the patients. This point can be further verified by noting the values of the slopes (regression coefficients) given in Table 9. It can also be seen in Table 9 that the standard error of estimates are reasonably small. An analysis of variance technique suggested by Dixon and Massey (1957) was used to test for linearity of regression. Only one function did not satisfy the requirements of linearity. This function was unique in having an essentially zero regression coefficient (length, patients, 8b Series C). In this Instance, the sum of squares due to regression was approximately zero. Hence, the Indication of nonllnearlty resulted from the pair comparison scale breaking down due to a complete lack of discrimination between stimuli. Still, the standard error of estimate for this function was far below those for the four competing functions tried. The finding Is nevertheless conslstant with Weber's Law, the classical assertion that discrimination becomes poorer with greater magni tudes of stimuli. Psychophysical Function of the Constant Sum Scale Figures 6 through 11 Illustrate the relationship between the constant sum scales for length and weight and the magnitude of these stimuli for the three groups. The straight line on these figures represents the true function in that it reflects what would be expected if the subjects gave correct judgments to all stimuli, resulting In a one to one correspondence between stimulus magnitude and scale values. It can be seen in Figures 6, 7, and 8 for length that the patients grossly over estimated the longer strips whereas the normal and sophisticate groups gave much more realistic judgments. Furthermore, it Is Interesting that the normal group slightly overestimated the longer strips while the sophisticate group slightly underestimated the longer HHI € 3 ! Qrr U r i U W W I > I I 91 strips. Concerning weight* it can he observed that the heavier weights were overestimated in all three groups. However* an Interesting result is apparent. The patients gave the most gross overestimate of the heavier weights. The normal subjects also grossly overestimated the heavier weights* although less so than the patients. Finally* the sophisticate group gave by far the most accurate judgments and only slightly overestimated the heavier weights. Also, it can be seen in Figures 7 and 8 that the relationship between length and the constant sum scale for the normal and sophisticate groups is very close to linear. However, the relationship between weight and the constant sum scale is concave upward, distinctly not linear. Five functions (linear* quadratic* exponential* logarithmic, and power) were tried in fitting the con stant sum scales for length and weight to stimulus magnitude. The power function in all instances was the best fitting function, although in two occasions (patients) there was not much to choose between this function and an exponential or quadratic function. Still, as pre viously stated, other factors being equal the power function was the preferred function due to its wide spread use and theoretical value. The standard error 92 of estimate was used to evaluate the goodness of fit for all functions. Table 10 presents the exponent (n), scale constant (c), additive constant (k) and the standard error of estimate (SE) of the best fitting power functions for these data. For both length and weight a very evident decrease in the value of the exponent is apparent, going from the patients to the sophisticate groups. These findings are consistent with the overestimation of weights and lengths previously pointed out from Figures 6 through 11. The standard error of estimates show satisfactory Table 10 Exponents (n), Scale Constants (c), Additive Constants (k), and Standard Error of Estimates (SE) of Best Fitting Power Functions for Length and Weight = c (s + k)n Length Weight n c k SE n c k SE Patients 2.?4 .0000050 51.40 .469 1.91 .0000085 73.04 .505 Normals 1.52 .0028214 15.03 .160 1.69 .0000394 6.71 .266 Sophis ticates 1.37 .0044656 17.37 .158 1.30 .0005097 11.28 .160 93 fits for all groups, although the power function for the patients were far less satisfactory than for the normal and sophisticate groups. It is also apparent from Table 10 that all the exponents (n) are greater than one and hence describe a positively accelerated curve. Figures 12 through 1? illustrate the relationship between the constant sum scales for length and weight and the magnitude of these stimuli for the three groups in log-log coordinates. This is the traditional way of showing a power function where in log-log coordinates the relationship should be linear and the slope repre senting the regression of the constant sum scale on the stimulus magnitude is equivalent to the exponent (n) in this function. These figures Illustrate the findings presented in Table 10. The straight line on these figures represents the best fitting power function in log-log coordinates. Comparison of the Psycho-physical Functions for the Pair Comparison and Constant Sum Scales The relationship between the pair comparison scales and the constant sum scales was investigated by comparing their respective psychophysical functions. This was done as previously discussed by comparing the slopes of the psychophysical functions at particular values of the stimuli— length and weight. Specifically, the regression 6o rf Q rf we H 40 r 4 ( 100 coefficients presented in Table 9 for the pair comparison scales for three parts of the stimulus range were com pared with slopes of the associated power functions within these same stimulus ranges. Slopes for the power functions were calculated by differentiating the constant sum scale with respect to a specific value of the stimulus HS" (length or weight). Thus: = c (S + k)n d i/VdS = nc (S + k)*1"1 For example, from Table 5 (normal group— length): = .0028214 (S + 15.03J1*52 At a length of 10 cm. d /dS = 1.52 (.0028214) ^T10 + 15.03J7*52 = .017 Table 11 gives the slopes of the power functions (constant sum scales) listed in Table 10 for length and weight within stimulus ranges A, B, and C. These slopes are listed adjacent to the slopes determined from the pair comparison scales for these ranges of stimuli. The values of two slopes can be seen at the two ends of each range of stimuli listed in connection with the constant sum scale. Since the power functions for the constant sum scales change with Increments in stimulus magnitude, two estimates of the slopes were presented within each of the ranges listed. 101 Table 11 Slopes (first derivatives) of the Power Functions in Table 10 for Length and Weight at Specified Values of These Stimuli Patients Normals Sophisticates d^/dS(cs) Begr. Coeff. (PC) d^/dStcs) Begr. Coeff. (PC) d^/dS(os) Begr. Coeff. (PC) LENGTH 10 cm 10.8 cm .017 .018 2.000 .023 .023 3.896 .021 .022 3.788 30 cm 31.6 cm .028 .029 .858 .031 .032 1.521 .026 .026 1.556 90 cm 93.2 cm .074 .077 -.077 .048 .049 .609 .035 .036 .468 WEIGHT 100 gm 124 gm .0018 .0020 .074 .0017 .0019 .108 .0028 .0029 .071 300 gm 340 gm .0036 .0040 .035 .0035 .0038 .038 .0038 .0039 .044 900 gm 980 gm .0087 .0094 .015 .0073 .0077 .020 .0053 .0054 .025 The slopes (regression coefficients) from the pair comparison scales (Table 9) are also presented for com parison purposes. CS = Constant Sum Scale PC = Pair Comparison Scale 102 It should also be pointed out that a direct comparison of the slopes connected with the constant sum scale and pair comparison scale Is Impossible due to a different scaling factor for each of the scales. In other words, the absolute values of the slopes presented are contin gent on the range of stimuli and associated scale values. The range of scale values for the pair comparison scale Is more "stretched out" than Is true for the constant sum scale. Each pair comparison scale was determined separately for each of the three ranges of stimuli selected. The constant sum scale was determined from the total range of stimuli studied and specified sections of this scale (ranges A, B, and C), well within this total range, were then compared with the pair comparison scale. Hence, the scaling factors are necessarily different for the two scaling methods In this study. Comparisons between the slopes from the two scaling techniques must be based upon their relative change in value over the three ranges of length and weight inves tigated. For all groups in both length and weight it can be seen from Table 11 that the values of the slopes Increase for the constant sum scale with Increases in stimulus magnitude. In contrast to this finding, the values of the slopes decrease for the pair comparison scale with Increases In stimulus magnitude. This was discussed 103 previously In connection with the conclusion that the psychophysical function for the pair comparison scale is negatively accelerated for all groups in both length and weight. The fact that the slopes of the power functions increase with Increases in stimulus magnitude is obvious from the exponents of these functions des- crlbed in Table 10. In all Instances the exponents are greater than one and clearly describe positively accele rated curves. It is further apparent that the psycho physical functions for the pair comparison scale Is clearly different from the psychophysical functions connected with the constant sum scale. The former is negatively accelerated and the latter is positively accelerated. This finding is illustrated in Figures 18 through 23 showing a comparison of psychophysical functions of constant sum and pair comparison scales for length and weight. The scale values plotted in these figures consist of the "predicted" values based on the best fitting linear and power functions. Thus, the two scaling methods could be directly compared over ranges A, B, and C for length and weight. In addition, random variations in the actual scale values are absent and values based on a smooth function substituted. Finally, a crude attempt was made visually to adjust for the different scaling factors discussed before. PC cs 4 .0 6 4 .0 4 4 .0 2 4 .0 0 3 .9 8 3 .9 6 3 .9 4 3 .9 2 3 .9 0 3 .8 8 3 .0 3 .8 6 2.0 3 .8 4 3 .8 2 3 .0 2.0 A PC 2.0 A PC u CO u < £ > CD O O - M U C c t n a N O I o CD o 0 > o CT> 0 0 FIGURE 18. COMPARISON OF PSYCHOPHYSICAL FUNCTIONS OF CONSTANT SUM (CS)ANO PAIR COMPARISON (PC) SCALES FOR LENGTH - PATIENTS *701 3 .0 2.0 1.0 .0 - . 5 3 .0 2.0 1. 0 1 PC CS 3 .5 6 3 .5 4 3 .5 2 . 3 .5 0 3 .4 8 3 .4 6 3 .4 4 3 .4 2 3 .4 0 3 .3 8 3 .3 6 3 .3 4 - 3 .3 2 !£3-3 1 .9 6 ■ .9 4 - .9 2 .0 - .5 | 3 . 2.0 1.0 ■90 l *4T .4 0 - .0 -.5 1 .361 CS PC X X X .C S XPC X X «• X PC . .cs o oooooooo • • ••■•••• o — ro c j ^ c/i < J > ^ co tr o o "tr o "tr o CO o C O o CO o CO ro C O U . “o FIGURE 19. COMPARISON OF PSYCHOPHYSICAL FUNCTIONS OF CONSTANT SUM (CS) AND PAIR COMPARISON (PC) SCALES FOR LENGTH - NORMALS PC cs 4.02 4 .0 0 3 .9 8 3.96 3 .9 4 3 .9 2 3 .9 0 CS 2.88 2.86 2 .84 3 .0 2 .82 2.0 2 .8 0 2.78 - t e . 76 - . 5 3 .0 9 CS 2.0 3.<r “ 2.0 44 <0 CO CO CO < 0 CO CO ( 0 80 CO CD o o > GM(O>CnO)sJ0D O B ro o > ro FIGURE 20. COMPARISON OF PSYCHOPHYSICAL FUNCTIONS OF CONSTANT SUM (CS) AND PAIR COMPARISON (PC) SCALES FOR LENGTH - SOPHISTICATES cs •cs PC 5 .0 0 4 .9 5 4 .9 0 4 .8 5 4 .8 0 4 .7 5 4 .7 0 4 .6 5 3 .0 . 4 .6 0 2 .0 4 .5 5 1 .0 4 .5 0 .0 ■ 4 .4 5 • fc * 3 .0 - .8 5 2 .0 - .8 0 1 .0 • .7 5 .0 • .7 0 - . 5 1 3 .0 .3 0 2 .0 • .2 5 1 .0 .2 0 .0 .1 5 PC • CS PC It?:":: A PC , c s o o o o — — — ro ro 0 ( 0 9 > I O M ( A Q D ^ » 6 ) U O U U G » ( a U U Oo -* r o r o c o 6 j # ( n o c n e c n o o i o ■vw < D o < 0 ro o < 0 CO o < 0 * o 10 cn o ( 0 ■ a * o to * s l o FIGURE 2 1 . COMPARISON OF PSYCHOPHYSICAL FUNCTIONS OF CONSTANT SUM (C S ) AND PAIR COMPARISON (P C ) SCALES FOR WEIGHT - PATIENTS 107 I 9 8 0 PC cs 4 .5 5 4 .5 0 4 .4 5 4 .4 0 4 .3 5 4 .3 0 4 .2 5 4 .2 0 4 .1 0 3 .0 4 .0 5 2.0 4 .0 0 0 3 .9 5 0 3 .9 0 • CS 2.0 * PC 0 2.0 0 . *C S O J 04 o 64 <0 o < o 64 < 0 C4 < 0 (0 ( 0 (0 C4 C4 cn O o ro o FIGURE 22. COMPARISON OF PSYCHOPHYSICAL FUNCTIONS OF CONSTANT SUM (CS) AND PAIR COMPARISON (PC) SCALES FOR WEIGHT - NORMALS PC cs cs 4 .0 5 4 .0 0 3 .9 5 3 .9 0 3 .8 5 3 .0 3 .8 0 PC 3 .7 5 2.0 3 .7 0 3 .6 5 •CS 3 .0 1.0 5 2.0 1.00 3 .0 2 .0 ©PC CO CO < 0 < 0 CD o> CD CD o o o Crt o C n o o VQ FIGURE 2 3 . COMPARISON OF PSYCHOPHYSICAL FUNCTIONS OF CONSTANT SUM (C S ) AND PAIR COMPARISON (P C ) SCALES FOR WEIGHT - SOPHISTICATES 110 It can be observed, from Figures 18 through 23 that for range A, the slope of the pair comparison scale Is generally steep In comparison with the constant sum scale; for range B, the two slopes are almost the same demonstrating that this part of the stimulus range would appear to give consistant results using the two scales; and for range C the slope of the pair comparison scale is essentially flat whereas the slope of the constant sum scale becomes steep in comparison. These results illustrate the findings shown in Tables 9, 10, and 11. For example, it can also be noted that the psychophysical function for the sophisticates is the most linear with the least increase in the magnitude of the slope in stimulus range C. The findings clearly show that the psychophysical functions for the constant sum scales are different from those for the pair comparison scales. The constant sum scales were related to length and weight by curves concave upward (positively- accelerated). The pair comparison scales were related to length and weight by curves concave downward (negatively accelerated). The degree of upward curvature in the psychophysical functions for the constant sum scales was inversely related to subject sophistication. The derived nega tively accelerated curves describing the relation between the pair comparison scales and the stimulus continue Ill were also associated with subject sophistication. In this Instance, the steepness of the slopes describing the extent of the downward concavity of these psycho physical functions was generally less for the patients. Comparison of the Accuracy of Judgments of the Patient. Normal, and Sophisticated Groups A second goal of the study was to Investigate possible differences in the observers* ability to Judge the magnitude of stimuli as a function of his intelli gence and sophistication. In other words, an important aspect of the evaluation of the psychophysical functions previously reported is a comparison of the Judgment accuracy of patient, normal, and sophisticated subjects. This was accomplished by a separate Investigation of scores based on the number of correct choices from the pair comparison data and scores based on the accuracy of point assignments given in connection with the constant sum scale. Table 12 presents the results from an analysis of variance of the total number of correct choices given by a subject with respect to the pair comparison situation. Thus, each individual was assigned a score according to the number of correct choices he made from the forty-five comparisons used in Series A, B, and C for the pair comparison scale. The results of the analysis of variance of the pair comparison data show a significant difference at the .05 112 Table 12 Analysis of Variance of Individual Accuracy Scores Connected with the Pair Comparison Scales for the Patient, Normal, and Sophisticate Groups Length Weight N Mean SD Mean SD Patients 20 26 4.8 26 3.8 Normals 20 30 4.1 28 3.9 Sophisticates 15 29 3.6 28 2.8 F=5.8 (Slg. at level) .05 F=l.63 (Non-signi- fleant) A Subject's Score = <£T w*iere C = stimulus k>J (a high score signifies good discrimination). level between group means for length. No significance was found between groups concerning weight. A glance at the group mean scores shows the patients with the lowest average of correct Judgments for both tasks. In length this difference was significant whereas with weight it was not. The normal and sophisticated groups were within one point of having the same average scores for the two tasks. Although the overall significance of these results is questionable, they do support the 113 psychophysical functions of the pair comparison scales. The accuracy of the point assignments from the constant sum Judgments were studied, by constructing a difference score for each individual. This score was simply the total difference in points over the twenty- eight comparisons administered between the correct number of points describing the ratio between any two stimuli and the number the subject gave. The higher the individual’s score the poorer were his judgments of the twenty-eight pairs of stimuli presented. Table 13 gives an analysis of variance of the mean difference scores for the three groups. Significance was discovered between the average accuracy of judg ments between groups for both length and weight. Prom the mean scores for each group it can be easily assessed that for both tasks the patients were by far the least accurate in their estimation of the stimulus ratios and the sophisticates were by far the most accurate in their estimations. It is evident that these findings are consistent with the psychophysical functions of the constant sum scales for these groups previously dis cussed. The fact that the findings are significant though, gives added information. Table 13 Analysis of Variance of Individual Accuracy Scores Connected with the Constant Sum Scales for the Patient, Normal, and Sophisticate Groups Length Weight N Mean SD Mean SD Patients 20 379 75 384 94 Normals 20 157 126 218 106 Sophisticates 15 83 30 159 50 F=53.1 (Sig. at level) .001 F=31.0 (Sig. at Level) .001 l Subject*s Score = ^ (V - A ) where V.k = number of points out of 100 assigned to stimulus k when compared to stimulus j. Ajk = actual number of points out of 100 specifying the ratio of stimulus k to stimulus 3 (a low score signifies accurate point assignments). CHAPTER V DISCUSSION It may be recalled that the major purpose of this study was to compare psychophysical functions based on scales using discrimination judgments (method of pair comparison) and ratio judgments (constant sum method) for well educated subjects with a knowledge of these methods, subjects of normal intelligence but just high school educations, and subjects Institutionalized as mentally retarded. Judgments of relative length and weight were selected as two well studied tasks easily administered. The results of this study have clearly indicated that the psychophysical function of the dis crimination scale is negatively accelerated over length and weight regardless of the sophistication and intelli gence of the subjects. The negative acceleration of the psychophysical function of the pair comparison scale was demonstrated by an evaluation of the slopes of best fitting linear 115 equations over three ranges of stimulus magnitude for length and weight. The relationships between scale values and stimuli used was first established as linear, then the slopes of these functions were found to decrease in value over the low, middle, and high sections of the total stimulus range sampled. This, of course, describes a curve concave downward over stimulus magnitude and was found to hold for both tasks and all subjects. This type of a psychophysical function characterizes the customary logarithmic relation between the psychological continuum and stimulus magnitude reported in the litera ture, It was evident, however, that the steepness of the slopes describing the extent of the downward con cavity of this psychophysical function was related to the subjects* sophistication. In general, the steeper slopes characterized the psychophysical functions of the sophisticated and normal subjects, indicating superior discrimination between stimuli compared to the patients. In contrast to the pair comparison scale, a positively accelerated curve described the psychophysical function of the constant sum scale for both length and weight for all subjects, A power function provided the best fitting equation out of five attempted. Based on arguments advanced by Galanter and Messick (1961), Stevens (1962), Fagot (1963)» and Ekman and Sjflrberg (1965),^ — (S + K)n, represented the form of the power 117 function used. In these data, the additive constant was very necessary for a good fit and a number of iterations were needed for determining the optimum values of "n" and "k" in terms of a minimum standard error of estimate. Although all of the psychophysical functions observed for the constant sum scale were positively accelerated over stimulus magnitude, two alternative functions could also be argued as “appropriate.” These are quadratic and exponential functions. In this study they did not provide as good as fit as the power function, although for the patients estimation of weight, there was little difference In the standard error of estimates. Of course, the crucial factor is the explanatory value of the power law: "Equal stimulus ratios correspond to equal sensation ratios.” It was particularly interesting to note that the value of the exponents "n" in the fitted power functions varied systematically with subject sophistication for both length and weight. The power functions for the patients were characterized by an exponent in excess of 1.90, whereas ”n =s I.69 to 1.60" was found for the normal group and the exponents were less than 1.40 for the sophisticate group. Actually, these exponents describe the extent of overestimation applied to the relative "longness" and "heaviness" of length and weight by the patients 118 relative to the normals and sophisticates, etc. It would definitely appear from this finding that the education and intelligence of the subject plays an Important role In making ratio Judgments. Not only was overestimation of stimulus magnitude a function of subject sophistication, but variability of Judgments also showed this relationship. The patients, by far, were the most "unsure” and variable in their Judgments. The normal group were less variable but still far short of the precision of the sophisticated subjects. This is as would be expected. Certainly education and intelligence play an important role in estimating the value of stimulus ratios. It is now possible to discuss the relationship between the pair comparison and constant sum scales. Slopes of the power functions of specified stimulus magnitudes regarding the psychophysical relationships of the constant sum scales were determined for the same magnitude of stimuli as was used for the pair comparison scales (d y/dS = nc (S + K) . Thus, the change in the slope of the psychophysical function over three separate ranges of length and weight was evaluated for both classes of scales. Table 11 and Figures 18 through 23 illustrate clearly the negative acceleration of the relation between the subjective continuum and stimulus magnitude for the discrimination scale, and the positive 119 acceleration of the relation between the subjective continuum and stimulus magnitude for the ratio scale. The former psychophysical function is concave downward over stimulus magnitude and the latter is concave upward over stimulus magnitude. Interpretations given by Eisler (1963a) and Torgerson (i960) would appear to apply to these results. Torgerson (i960) argues the two scales reflect two ways we have of Judging relative differences between stimuli. Batio scales are most useful concerning quantitative relations among objects— relative gains, etc. Numbers are used to specify the ratio of one stimulus to another in such a way that errors are proportional to the diff erences between the stimuli Judged. These errors increase exponentially as our "round number tendency" is such that the numbers are less precise and get rounder as we go up the scale. Prom the results given here regarding length and weight, it was noted that overestimating increases with larger stimuli, and does so as a function of the subjects* ability to use numbers— quite in line with Torgerson*s argument. The pair comparison and category scales are believed to be pure discrimination scales by both Eisler (1963a) and Torgerson (i960). Eisler states that the category’s scales deviation from the ratio scale is a function of changes in discrimination with changes in magnitude. The 120 subjects construct units of their "uncertainty” not units reflecting ratios on a magnitude scale. Torgerson (i960) also notes that this "uncertainty" is based on "differ ences," not "ratios." We specify Just noticeable gains in stimuli and report on precision to plus or minus so many units. In contrast to a scale based on ratio estimation, Judgments of extreme stimuli reflecting the top of the scale are no rounder than those at the bottom of the scale. Hence, the use of numbers to specify a ratio is not an issue. This interpretation was supported by the evidence presented here in which for all subjects discrimination became poorer with increases in length or weight. The normal and sophisticated subjects appeared a little better at noticing differences than the patients, but still, all groups manifested the typical logarithmic type of psychophysical function characteristic of reporting differences instead of ratios. Finally, it can also be suggested that keeping the experimental operations as close as possible in teims of the manner stimuli are presented and Judged for both discrimination and ratio estimation scales did not make these scales any more similar. In other words, presenting stimuli two at a time, for Judgments of one relative to the other in both the pair comparison and constant sum situations did appear to Influence the nature of these scales relative to their physical continue. As already 121 pointed out, the results were the same as would be expected concerning the majority of studies noted in the literature review regarding category and magnitude esti mation approaches. Having discussed the relationship between the pair comparison and constant sum scales, relative to subject sophistication, it is now possible to consider the suitability of these scales for the three groups of subjects used in connection with the judgment of length and weight. As was pointed out in the "Results," the pair comparison scale demonstrated considerable insta bility for all subjects. The scale values did not Increase sequentially with increments in length and weight. This would imply that the magnitude of the stimuli were selected too close together. Still, by noting the joint proportion matrices used to scale these stimuli (Appendix A), it is evident that in many instances stimuli farther apart in magnitude would have led to proportions of 1.00. The erratic nature of these Judg ments was also found by Goude (1962) and it casts serious doubt on the feasibility of employing a discrimination scale from Judgments of length and weight. It was apparent, however, that a scale of this nature is as feasible for retardates as for sophisticated subjects, although retardates made more errors. The constant sum scale was not characterized by 122 instability as was the pair comparison scale in this situation, but nevertheless did demonstrate serious shortcomings. First of all, scale values did exhibit some irregularity when stimuli close together in magnitude were judged. In general, the slightly longer strip or heavier weight was overestimated disproportionately in comparison to stimuli farther apart in magnitude. Secondly, a clear difference in the value of the para meters of the psychophysical function was evident as a function of subject sophistication for constant sum judgments. Patients appeared to overestimate grossly the longer or heavier stimuli despite a careful effort to ensure a proper understanding of the method. Normal subjects, although superior to the patients in this regard, did not do as well in their judgments as the sophisticate group. The fact that ratio estimations are influenced by education is perhaps a more serious problem than the finding that the retarded had trouble with this task. Turning to the analysis of the accuracy of the judgments will help clarify some of the factors operating on the two classes of scales. The results of an analysis of variance of the pair comparison data showed a sig nificant difference at the 5 per cent level between group means of numbers of correct judgments for length. No significance was found for differences between groups 123 with respect to weight. The actual mean scores reflecting the average number of correct Judgments for each group showed the patients had the lowest scores for both tasks. The normal and sophisticate groups were within one point of having the same average scores for the two tasks. The patients, then, were slightly poorer in their dis crimination than the other subjects. This is consistent with Holowinsky's (1964) finding that severely retarded subjects did significantly worse In discriminating lengths of lines than normal subjects, A difference score between the true ratio and specified ratio of points was used to measure accuracy of the constant sum Judgments. An analysis of variance of these scores for the three groups showed dramatic differences. Differences between group means were significant at the .1 per cent level for both length and weight. It was clearly evident that the most accu rate Judgments were made by the sophisticate group, and least accurate Judgments were given by the patient group. The difference between the scores for the groups was extremely large as can be observed in Table 13. This evidence suggests that for practical purposes, differences between groups in discriminating between stimuli could not account for the large discrepancies between these groups in the accuracy of their ratio estimations. In other words, although discrimination 124 ability was a factor In the pair comparison scale, it was not sufficiently -variable to account for the differ ences in the constant sum scales for each group. From these findings, it appears safe to conclude that the accuracy of ratio Judgments relative to subject sophisti cation was a function of the subjects ability to use numbers as proportions of a standard stimulus rather than a function of the subjects ability to experience these differences in "heaviness" or "longness." It can be recalled that this is precisely what Torgerson (I960) has argued. One additional factor was considered for possible effects on the subjects* Judgments. A number of studies (Bevan, Maier, & Helson, 1963; Beven & Turner, 1965; diLollo, 1964; Over, 1963; Pierrel, 1963; Poulton & Simmons, 1963; Welnfraub & Hake, 1962; Weiss, 1963; Weiss & Hodgson, 1963) have shown that the experimental situation can significantly influence ratio and dis crimination Judgments. Four factors were studied for their effect on the Judgments used in this Investigation: 1) order of the task; 2) order of the ratio-discrimina- tion Judgments; 3) order of the stimuli; and 4) ability within the normal and patient groups. Using a multiple regression approach to an analysis of variance, these effects and their possible interactions were investigated on sixty separate pair comparison and constant sum 125 judgments over the three groups. Thhs sixty P ratios were evaluated for their significance in testing the effect of these factors on a judgment for a specified trial by one of the groups. Only one pair comparison Judgment was found to he significantly affected by these factors. This Involved a response to lifted weights by the normal subjects. Since only one out of sixty Independent tests emerged significant, it was concluded that in general the effects as studied were negligible. It should be pointed out that these factors were counterbalanced among the groups so as to remove the effect of possible biased responses from one group in comparison to another. In addition, the design of the study made any analysis of these effects questionable due to the appropriateness of applying a multivariate technique to a situation where the degrees of freedom to work with were quite small and assumptions difficult to verify. CHAPTER VI SUMMARY AND CONCLUSIONS An investigation was undertaken to study the rela tionship between psychophysical functions based on a pair comparison scale and constant stun scale for length and weight over three levels of subject sophistication and ability. The following results were found; 1. The psychophysical function of the pair compari son scale was negatively accelerated (concave downward) over length and weight for the patient, normal, and sophisticate groups. 2. The psychophysical function of the constant stun scale was positively accelerated (concave upward) over length and weight for the three groups. 3. A power function, (S + k)n, described the relationship of the constant stun scale to the magnitude of length and weight for all groups. The exponent "nM for both length and weight varied systematically with subject sophistication, e.g., the exponent was largest 126 127 for the patients, n > 1.90, next largest for normals, I.69P n > 1.60, and least for the sophisticate group, n<1.40. The magnitude of the exponents describe the degree to which these subjects overestimated the rela tive value of the larger stimuli. 4. A comparison of the value of the slope of the psychophysical function for the pair comparison scale and constant sum soale for three specified subranges of length and weight illustrated the following relation ships: a. For the pair comparison scale, a steep slope characterized the regression of p (scale) on S (stimulus) at the low end of the stimulus continuum and a relatively flat slope was found at the high end of the continuum. b. For the constant sum scale, a relatively flat slope characterized the regression of on S at the low end of stimulus con tinuum and steep slope was found at the high end of the continuum. c. In the middle range of the stimulus continuum, the slopes for the regression of i f ? on S for both the pair comparison and constant sum scales were moderate and similar to one another. 128 These findings were apparent In all groups studied. 5. The accuracy of Judgments for the pair compari son data and constant sum estimates were investigated separately. In general, the patients exhibited poorer discrimination (particularly with respect to length) than the other subjects, although the difference between the three groups was small in comparison to wide discrepancies found for the constant sum estimates. In the latter estimates, the most accurate Judgments were made by the sophisticate group and least accurate re sponses were given by the patient group. 6. Order of task, order of ratio-discrimination Judgments, order of stimuli, and ability within the noimal and patient groups were studied for their possible effect on specified pair comparison or constant sum Judgments. The findings suggested their influence was negligible. In conclusion, a rationale provided by Eisler (1963a) and Torgerson (i960) was used to explain the results of this study. Basically, it would appear that the pair comparison Judgment and ratio estimation reflect two ways we have of Judging relative differences between stimuli. Batio scales are most useful regarding quanti tative relations among the stimuli. The pair comparison scale is based on measures of uncertainty or differences between stimuli. 129 It was further argued that In ratio judgment, the numbers given Increase exponentially as we use them to describe greater differences in the magnitude of two stimuli. In contrast to a scale derived from ratio estimates, discrimination responses are not based on the use of numbers. Discrimination merely becomes poorer with stimulus magnitude. From the results regarding length and weight, it was noted that overestimating increases with larger stimuli and does so as a function of the subjects* ability to use numbers. Also, the evidence shown here indicates that although the sophisticated and normal subjects showed some superiority over the patients in their ability to discriminate between stimuli, it was not enough to account for the variability between groups in making ratio estimations. Thus it would appear that accuracy of ratio judgments relative to subject sophis tication was a function of the subjects* ability to use numbers as proportions of a standard stimulus rather than a function of the subjects* ability to experience these differences in weight and length. Thus, the power law applies to ratio estimation and a negatively acceleration function (logarithmic) typifies the psychophysical relation of the pair comparison scale regardless of the subjects* education and intelligence. Finally, it would also appear that the pair comparison scale has limited application to the Judgment of length and weight because of unstable scale values and that the constant sum scale is definitely affected by the subjects' knowledge and experience in manipulating numbers. APPENDIX A Proportion matrices for lengths of lines and lifted weights over three ranges of stimulus magnitude (series A, B, and C) that were Judged by pair comparisons. Three groups of subjects Judged only those pairs of stimuli where proportions are entered in each of the nine by nine matrices. Each proportion listed in these matrices is equal to the proportion of times stimulus k was Judged greater than stimulus J. 131 Stimulus 132 10.0 10.1 10.2 10.3 10. 4 10.5 10.6 10.7 10.8 Table 1 Length (centimeters) Series A Patients Stimulus It 10.0 10.1 10.2 10.3 10.4 10.5 10.6 10.7 10.8 .500 .650 .500 — .400 .700 .350 .600 — .500 .600 .300 .500 — .550 .550 .400 .450 — .600 .750 .450 .400 — .800 .700 .250 .200 — .350 .750 .300 .650 — .550 .250 .450 — Stimulus 133 30.0 30.2 30.* * " 30.6 30.8 31.0 31.2 31.** 31.6 Table 2 Length (c entimeters) Series B Patients Stimulus k 30.0 30.2 30.** 30.6 30.8 31.0 31.2 31.4 31.6 .650 .**50 .350 — .500 .700 .550 .500 — .**50 .700 .300 *-550 — .600 .650 .300 .**00 — .700 .650 .350 .300 — .600 .650 .350 .*+00 — .650 .**00 .350 .350 — .65) .600 .350 — Stimulus 90.0 90.4 90.8 ■ r , 91.2 91.6 92.0 92.4 92.8 93.2 134 Uable 3 Length (centimeters) Series C Patients Stimulus k 90.0 90.4 90.8 91.2 91.6 92.0 92.4 92.8 93.2 .800 .600 .200 — .500 .650 .400 .500 — .450 .350 .350 .550 — .450 .350 .650 .550 -- .500 .500 .650 .500 ~ .600 .350 .500 .400 — .800 .250 .650 .200 — .550 .750 .450 — Stimulus 135 Table 4 Length (centimeters) Series A Normals Stimulus k 10.0 10.1 10.2 10.3 10.A 10.5 10.6 10.7 10.8 10.0 10.1 10.2 ’10.3 10.4 10.5 10.6 10.7 10.8 .550 .600 .450 — .500 .700 ,4oo .500 — .650 .750 .300 .350 — .250 .300 .700 .800 .650 .900 .200 .350 — .750 .900 .100 .250 — .550 .700 .100 .450 — .600 .300 .400 — 136 Table 5 Length (centimeters) Series B Normals Stimulus k 30.0 30.2 30.4 30.6 30.8 31.0 31.2 31.4 31.6 30.0 30.2 30.4 30.6 ■ r ? « 30.8 | 31.0 •H w 31.2 31.4 31.6 .600 .700 .400 — .800 .650 .300 .200 — .600 .750 .350 .400 — .500 .700 .250 .500 — .750 .750 .300 .250 — .700 .250 .300 — .250 .350 .300 .75 0 .650 .700 .550 .450 — \ 137 Table 6 Length (centimeters) Series C Normals Stimulus k 90.0 90.** 90.8 91.2 91.6 92.0 92.** 92.8 93.2 90.0 -- .550 .700 90.^ .450 — .650 .700 90.8 .300 .350 — .650 .800 91.2 T - > to 91.6 f3 .300 .350 .200 0 0 1 0^ 1 * .700 .750 .550 .500 | 92.° £ 92. .250 .**■50 .500 .550 .**50 .*+50 .550 .800 92.8 .550 .**■50 — .750 93.2 .200 .250 — Stimulus 138 Table 7 Length (centimeters) Series A Sophisticates Stimulus k 10.0 10.1 10.2 10.3 10.4 10.5 10.6 10.7 10.8 10.0 10.1 10.2 10.3 10.4 10.5 10.6 10.7 10.8 .533 .867 .467 -- .333 .733 .133 .667 — .467 .800 .267 .533 — .533 .733 .200 .467 — .733 .867 .267 .267 — .667 .800 .133 .333 — .733 .800 .200 .267 — .600 .200 .400 — Stimulus 139 Table 8 Length (centimeters) Series B Sophisticates Stimulus k 30.0 30.2 30.4 30.6 30.8 31.0 31.2 31.4 31.6 30.0 30.2 30.4 30.6 30.8 31.0 31.2 31.4 31.6 .733 .600 .267 — .933 .733 .400 .067 — .733 .933 .2 67 .2 67 — .400 .733 .067 .600 — .733 .733 .267 .267 — .867 .400 .267 .133 — .533 .400 .600 .467 ~ .467 .600 .533 — 140 Table 9 Length (centimeters) Series C Sophistlcates Stimulus k 90.0 90.4 90.8 91.2 91.6 92.0 92.4 92.8 93.2 90.0 — .600 .66? 90.4 .400 — .667 .733 90.8 .333 .333 — .600 .600 ^91.2 1 £ 92.0 CO 92.4 .26? .400 .400 .467 .533 .600 .733 .600 .400 .267 .400 .600 .400 .533 .467 .733 92.8 .467 .533 — .733 93.2 .267 .267 — 141 Table 10 Weight (grams) Series A Patients Stimulus k 100 103 106 109 112 115 118 121 124 100 — .750 .750 103 .250 — .400 .800 106 1-3 .250 .600 — .550 .750 3 109 .200 .450 — .600 .650 1 112 .250 .400 — .450 .600 w 115 .350 .550 — .500 .700 118 .400 .500 «N* W .650 .500 121 .300 .350 — .650 124 .500 .350 — 142 Table 11 Weight (grams) Series B Patients Stimulus k 300 305 310 315 320 325 330 335 340 300 mm mm .500 .700 305 .500 — 0 0 cn . .600 310 .300 .700 mm mm .500 .400 ^315 ra 3320 .400 .500 .600 0 'A 1 ^ 1 • .450 .850 .400 .600 1 32s ra 330 .150 .600 .400 .350 1 • 1 ON 0 0 0 i n 0 c o in • • .650 335 .150 0 0 in . — .650 340 .350 • 350 — Table 12 Weight (grams) Series C Patients 900 910 920 930 Stimulus 9^0 950 lc 960 970 980 900 ------------ .550 .750 910 .^50 — .550 .500 920 .250 .^50 - - .350 .600 930 T J .500 .650 — .700 .850 0 ) 9^0 .*K>0 .300 — .500 .500 1 95° .150 .500 — .350 .550 •H £ 960 .500 .650 — .300 .650 970 A 50 .700 — .800 980 .350 .200 — Stimulus 144 Table 13 Weight (grams) Series A Normals Stimulus k 100 103 106 109 112 115 118 121 124 100^ — .850 .600 103 .150 — .650 .750 106 .400 .350 — .450 .800 109 .250 .550 — .650 .700 112 .200 .350 — .750 .800 115 .300 .250 -- .750 .700 118 .200 .250 — .550 .600 121 .300 .450 — .600 124 .400 .400 — 145 Table 14 Weight (grams) Series B Normals Stimulus k 300 305 310 315 320 325 330 335 340 300 — .600 .750 305 .400 — .550 .450 310 .250 .450 — .750 .700 ^315 .550 .250 — .450 .750 “ 320 i —1 .300 .550 — .500 .700 I 3 2 5 .250 .500 — .750 .500 •p CO 330 .300 .250 — .650 .500 335 .500 .350 — .450 340 .500 .55 0 — 146 Table 1$ Weight (grams) Series C Normals Stimulus k 900 910 920 930 940 950 960 970 980 900 — .650 .650 910 .350 — .550 .600 920 .350 .450 — .650 .800 T-3 § 930 .400 .350 — .650 .550 H g 940 .200 .350 — .600 .650 •H •P C O 950 .450 .400 — .450 .600 960 .350 .550 — 1 .400 .750 970 .400 .600 — .500 980 .250 .500 — 1*4-7 Table 16 Weight (grams) Series A Sophisticates 100 103 106 109 Stimulus 112 115 k 118 121 12*+ 100 — .66? .533 103 .333 — .*+67 .800 ^106 t o .*+67 .533 — .600 .*+67 | 109 .200 .*4-00 — .600 .*+67 s £ 112 .533 .*400 — .600 .867 C O 115 .533 .*+00 mm ^m .733 .667 118 .133 .2 67 — .*+00 .667 121 .333 .600 — .733 12*+ .333 .267 — Table 17 Weight (grams) Series 5 Sophisticates Stimulus k 300 305 310 315 320 325 330 335 340 300 — .800 .667 305 .200 .667 .800 310 .33 .33 — .733 .800 315 .200 .267 -- .667 .533 W 320 .200 .333 — .600 .400 1 325 ■H m 330 .467 .400 .600 .667 .333 .733 .400 .667 335 .267 .600 — .667 340 .333 .333 — Table 18 Weight (grains) Series C Sophls tlcates Stimulus k APPENDIX B Illustration of the internal consistency of the constant sum scale for length and weight for each group. Each figure consists of a plot of the observed ratios given by the subjects as a function of the expected ratio determined by the scale values. 150 1* CO ■ :* iH CO J* * rH A O CO rH pH so so i!:. \ . .r .... i . j. T-r - J : — . .. : ! ' i : I o ■ ¥ OH co i < i > —r* ■ 4 " " w C*> h . ; rjr t o w n -I--- ~3T . 1 ]W te :: :t r=T |V D 1 ::« OJ o m io cu oj i i ^4- # 1 n T" ** tB I r t ; 1..., ;; 1 i H4 i..; I' £ X or i APPENDIX C Table 1 Analysis of the Effect of Experimental Conditions on the Pair Comparison and Constant Sum Judgments® Patients^ 3 1. Normals Sophisticates0 LENGTH Series A 2 X 3 F=1.935 F=2.213 P=4.67 7 X 9 ^=1.036 P= .686 Pel.779 Series B 3 X 4 F= .951 F=1.177 P=2.435 1 X 3 E=1.454 P=1.268 P=1.467 Series C 7 X 8 F=1.440 F=1.497 F=2.232 5 X 7 F=3.429 P=1.427 P= .587 Series D 2 X 4 F=1.055 F=l.232 P=1.685 2 X 8 F=1.344 F= .797 P=3.122 5 X 6 P= .573 P=1.100 P=1.597 7 X 8 F= .404 ^=1.505 P=1.493 157 158 Table l--Continued Patients13 Normals13 c Sophisticates WEIGHT Series A 1 X 2 F=l.812 pv=2.569 F= .818 4 x 6 Fa .691 F=l.886 F=2.200 Series B 3 x 9 P- .95^ 3 F= .434 F=s2. 500 F- .350 1 x F=24l.32* F= .760 Series C 7 3 T 8 F=l,278 F=2.700 F=l, 857 5 3 C 7 F=1.470 F=3.920 5^3.480 Series D 1 X 8 F=1.744 F= .633 F=1.049 2 x 6 F=2.701 F=1.382 P= .738 5 x 6 F=1.663 F= .921 F= .589 6 x 7 F= .912 F=1.180 F= .528 aF ratios listed were based on a multiple regression approach to an analysis of variance. *>df for regression 15 cdf for remainder for regression 4 7 for remainder = 7 ♦significant at the .05 level or better APPENDIX D Determining a best fitting function between stimulus magnitude (S) and the constant sum scale values ( ^ ) produced some special problems. Initial attempts at finding the proper function were based on the technique used for the pair comparison scales described on page 44. Essentially, this technique involved transforming the data to a linear form and proceeding with a least squares linear fit. For example, if the function fitted is ip = a Log S + b, the procedure takes the log of each S value, then fits a linear regression to and Log S based on the least squares criterion. For the pair comparison data, the above procedure was satisfactory in terms of producing a reasonable fit for each function attempted. However, the standard error of estimate for the constant sum data indicated that the fit for functions dependent on log transormations, e.g., exponential, log, and power, were unreasonably poor 159 l6o compared to a simple linear function. At this point, another technique for fitting these functions was introduced. A non-linear least squares estimation technique using an iterative approach was attempted. This technique, which does not employ log transformations, etc., produced much better fits for the constant sum functions dependent on log transformations. An additional check of the problem was introduced by applying this technique to a sample of the pair com parison scales. In contrast to the results found for the constant sum scales, the fits for the pair comparison scales were nearly identical to the initial approach based on transforming the data to a linear form consis tent with the function specified. Hence, separate methods for curve fitting had to be employed for the two sets of scales. A probable explanation for these findings based on the power function is given below: Givens = a (S + K)n — > Log ^ = n log (S + K) + log a assume a, K and n are given (see Table 10) Consider ^ = n a (S + K)n ^ + o (A S) 161 let Log ^ f = Log f & i = n + o (4 S) A S S + K Now consider the ratio a o > 4 Y which we would like bounded, I.e., A ^ should be equivalent to some constant of A ^ but n S + K -fl=t j = & \ j ) na (S + K)J = constant if and only if is bounded below or 4 j o . j. a y . Given ^ Y* , A ^ will be bounded above if we choose ^ to be the reciprocal of A ^ ; that is A (jj < specified bounds (b). Then, the lower bound of (jJ can be taken as a constant times (b). The preceding argument is illustrated in the following example: V .00 .01 .50 i fL.OO 2.00 log V -OO 5.395-10 9.307-10 JO. 00 .69 A Y> .01 .49 .50 1 ■ ■ l L - ■ J 1.00 1.00 A <f' 0 ° 3.91 .69 j .69 .40 i .... . 1 ■ 1 continued ^ A A '----- y/ 3.00 A.00 5.00 6.00 ' 7.00 8.00 9.00 log i/> 1.,09 1.39 1.61 1.79 1.95 2.08 2.20 1.00 1..00 1.00 1.00 1.00 1.00 1.00 < .29 .22 .18 .15 .13 .12 .11 It is evident from this example that the ratio of a approaches infinity when p goes to zero. Actually, in this example, the ratio becomes larger than 1 whenever ip is less than one. Consequently, the ratio is not bounded below and for this type of problem a logarithmic transformation will produce a very poor fit in this lower region. A log transformation is appropriate and will have comparable fluctuation of 4 ^ for A S, if, in this example, the range of \p is between .5 and 10.0. 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Creator Eyman, Richard Kenneth (author)

Core Title The Effect Of Subject Sophistication On Ratio And Discrimination Scales

Degree Doctor of Philosophy

Degree Program Psychology

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

Tag OAI-PMH Harvest,psychology, experimental

Format dissertations (aat)

Language English

Contributor Digitized by ProQuest (provenance)

Advisor Guilford, Joy P. (committee chair), Cliff, Norman (committee member), Meyers, Charles Edward (committee member)

Permanent Link (DOI) https://doi.org/10.25549/usctheses-c18-191004

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Legacy Identifier 6605478.pdf

Dmrecord 191004

Document Type Dissertation

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Rights Eyman, Richard Kenneth

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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...

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