University of Southern California Dissertations and Theses

An application of the ridge regression method to a neurological nursing care study

(USC Thesis Other)

An application of the ridge regression method to a neurological nursing care study

PDF

Download

Open document

Flip pages

Download a page range

Download transcript

Copy asset link

Request this asset

Transcript (if available)

Content AN APPLICATION OF THE RIDGE REGRESSION METHOD TO A NEUROLOGICAL NURSING CARE STUDY by Patricio Mendoza Velez A Thesis Presented to the FACULTY OF THE GRADUATE SCHOOL UNIVERSITY OF SOUTHERN CALIFORNIA In Partial Fulfillment of the Requirements for the Degree MASTER OF SCIENCE (Biometry) March 197 8 UMl Number: EP54860 All rights reserved INFORMATION TO ALL USERS The quality of this reproduction is dependent upon the quality of the copy submitted. In the unlikely event that the author did not send a complete manuscript and there are missing pages, these will be noted. Also, if material had to be removed, a note will indicate the deletion. UMl EP54860 Published by ProQuest LLC (2014). Copyright in the Dissertation held by the Author. Microform Edition © ProQuest LLC. All rights reserved. This work is protected against unauthorized copying under Title 17, United States Code ProQuest LLC. 789 East Eisenhower Parkway P.O. Box 1346 Ann Arbor, Ml 48106 - 1346 UNIVERSITY OF SOUTHERN CALIFORNIA THE GRADUATE SCHOOL UNIVERSITY PARK LOS ANGELES. CALI FORN1 A 90 0 0 7 & '78 This thesis, 'written by Patricio Mendoza Velez under the direction of h.À-3..Thesis Com m ittee, and appro'oed by a ll its members, has been p re sented to and accepted by the D ean o f T h e G raduate School, in p a rtia l fu lfillm e n t of the requirements fo r the degree of MASTER OF SCIENCE Dean. Date. THESm COMMITTEE Chairman ACKNOWLEDGMENTS | I I would like to extend my appreciation to my committee | chairwoman. Dr, Linda Chan, for her patience, advice and j assistance in the preparation of Lhis paper; to Dr. Stanley Azen who greatly accelerated my progress in completing this paper; and to Dr. Harland Sather who initially introduced me to this type of analysis and provided valuable discus sions in the past. I am indebted to Dr. Jeanne Berthold for allowing me to participate in her study in 197 5 and for availing me of her considerable experience in data analysis. I am also grateful to her assistant Ms.^. Barbara Storm for her co operation and assistance in providing me the data needed and stimulating discussions about the data. Lastly I am thankful to my wife, Othelia, for her pa tience in typing the several versions of this paper and to my son, Joseph, who was a constant source of encouragement to me throughout this sometimes disheartening process of thesis writing. 11 TABLE OF CONTENTS ACKNOWLEDGMENTS ........................................... ii LIST OF TABLES ... . iv LIST OF FIGURES ........................................... V Chapter I 1 I. INTRODUCTION 1 i I II. STATISTICAL METHODS 3 | Linear Regression ' Ridge Regression ' Index of Stability of | Relative Magnitude (ISRM) j III. STUDY D A T A 7 ! I IV. RESULTS 9 j V. DISCUSSION 11 j REFERENCES 13 ; I APPENDICES . . . ; : ............................................ 2 4 ! Ill LIST OF TABLES Table Page 1. Summary Statistics for Neurological Nursing Care Study Variables .................... 15 2. Intercorrelations among the Variables ............ 16 3. Least-squar.es Estimates (k=0) and Variance Inflation Factors ...................... IV 4. Search for Optimal k ....................... 18 5. Ridge Regression Estimates (k=0.2) and Variance Inflation Factors ................. 19 IV ; LIST OF FIGURES I I I Figure Page : 1. Hoerl-Kennard Ridge Trace (Vinod, 197 6) 20 i 2. Ridge Trace on the m-Scale (Vinod, 197 6) 21 j 3. Hoer1-Kennard Ridge Trace i (Nursing Care Data) ............................. 22 4. Ridge Trace on the m-Scale j (Nursing Care Data) 23 i V CHAPTER I INTRODUCTION Researchers have traditionally used the method of least-squares for obtaining estimates of regression para meters in linear models. Such estimates have the theoreti cal property of being unbiased with minimum variance among all linear estimators (Scheffe, 1959). In situations where the independent variables in the linear model are highly correlated, these estimates may have a large error, i.e., the expected value and the variance of the squared distance from the estimate to the parameter may be exceptionally large. To measure the impact of the simple correlations of the independent variables on the variance of the estimates of the regression parameters, Hoerl and Kennard (1970) introduced the variance inflation factor (VIE). Its use has also been demonstrated by Marquaidt and Snee (197 5). For each parameter in the model, the VIF is the diagonal element of the inverse of the simple correlation matrix. I When VIF = 1 there is no influence of the correlation of I I the estimates. When VIF > 1, there is an influence, and I the magnitude of VIF reflects the "amount" of influence. I I In a linear model, if the VIFs of some of the least- I 1 ; squares estimates are large, then it is reasonable to con- !sider a biased estimation procedure instead of the usual j i : least-squares procedure. To this end, Hoerl and Kennard ! (197 0) introduced the method of ridge regression which involves finding biased estimates of the parameters in order to reduce the effects of the correlations among the independent variables. The technique of ridge regression introduces into the linear model a parameter k (k ^ ^) which measures the amount of bias in the resulting esti mates. The statistical problem then becomes one of find ing an optimum k which yields the smallest mean square error in the parameters. Many methods have been developed to select an optimum, but Vinod (197 6) suggested a method involving the use of the Index of Stability of Relative Magnitude (ISRM) for finding k. This paper demonstrates the use of the method of ridge regression to analyze a set of data obtained from a neurological nursing care study known to have highly correlated independent variables. ISRM will be used to find the optimum k. Results are compared to those obtained using least-squares. _____2 CHAPTER II STATISTICAL METHODS Linear Regression For the procedures discussed in this section, we assume that the independent variables are standardized by subtracting the sample mean and dividing by the sample standard deviation. The method of least-squares is used to generate estimates 3 for 3 in the linear model Y = X3 + e (1) where Y is the (n x 1) observation vector, X is the (n x p) standardized data matrix of rank p, 3 = (3i,...,3 )' is the ~ X p (p X 1) parameter vector, and e = (e^,...,e^)' is the 2 (n X 1 ) error vector. Here E(e) = 0 and E(ee') = a I (I is the n X n identity matrix of rank n). The least-squares estimate 3 = (3w*..y3 )' is known to be unbiased and has ~ X p the smallest variance among all linear unbiased estimators of 3. This solution is given by 3 = (X'X)"^X'Y (2) where X'X is the correlation matrix. The average value of the squared distance from g to ,3 is ' ; E(0^) = E(6-6)'(6-e) = 2 (3) ! - - - - i=l i j I where 6 is the distance from 3 to 3 and are the char- i ; I iacteristic roots of X'X. If the minimum root A . is close 1 1 imn ‘ ' I 1 2 2 f to zero, then E(6 ) = a /. will become very large. This ! min I situation occurs when the correlations among the independent variables are large. | The variance inflation factor (VIF) of the estimate ^ ”1 3 is then given by the ith diagonal element of (X'X) , i = 1,...,p. Ridge Regression In the ridge regression, a small positive quantity k, representing the amount of bias, is added to each diagonal element of X'X before inverting. Thus, the ridge regres sion estimate of 3r denoted by 3* = (3J,...,3*)' is ex pressed as 3* = (X'X," + kl) ”^X'Y. (4) The properties of 3* are described in Hoerl and Kennard (1969) and an application of ridge regression is found in Marquardt and Snee (197 5). For a summary of the properties of ridge regression estimates, the reader is referred to Marquardt (19 70). The VIF for ridge regression estimates are now the diagonal elements of (X'X + kl) ^. To obtain an optimum k for each 3^, one graphs a : ridge trace which is a plot of 3* (vertical axis) as a i i function of k (horizontal axis). Optimum k occurs when the , ridge trace stabilizes (See Figure 1). I In general, there is an optimal value of k for any , ! ' 3j^. However, in practice it is not easy to find this k ; since there may appear to be a wide range of admissable | I values for large k, i.e., the ridge trace appears to be ^ stable for large values of k (See Figure 1). I I To ameliorate this situation, Vinod (1976) introduced | j a constant m, termed "multicollinearity allowance", to , replace k. This is defined as I I P I m = p - E ------ . (5) L i=.l k The new constant m does not have the undesirable property of k so that plotting m on the horizontal axis of the ridge trace instead of k will not give an appearance of greater stability for larger m (See Figure 2). Index of Stability of Relative Magnitude (ISRM) Vinod (197 6) also introduced an index for selecting optimum m. This index, called the Index of Stability of Relative Magnitude (ISRM) is a numerical measure of the amount of deviation of the correlation matrix from the identity matrix. In symbols P 6 ^ D 2 ISRM = E ( ^ • E -1) ^ (6) i=l ^i S - p where S = E ---------- and 9 = . i=l {A^ + k)2 A^ + k For the derivation of ISRM, the reader is referred to Vinod (1976). An important advantage of ISRM is that for most ex amples, ISRM yields a narrow range of desirable values for m. In practice, one can easily compute ISRM for each m (0 m p) and choose m where ISRM is smallest. Note that k can then be obtained from Eq. 5, and hence the ridge regression estimates from Eq. 4. CHAPTER III STUDY DATA The above methods are demonstrated using a set of data obtained from a nursing care study performed in the Neurological Unit at Rancho Los Amigos Hospital, Downey, California (See Berthold, 1975). The dependent variable, Y, is the average observed (direct) nursing care hours per day per patient. The seven independent variables are: = index of physical care need = index of emotional stability Xg = index of rehabilitation program need X^ = composite of activities of daily living Xg = index of functional capacity developed by Kenny Rehabilitation Institute X^ = index of functional capacity developed by Katz X^ = index of functional capacity developed by Barthel The first three variables are direct measurements of pa tient care needs. Their scale ranged from 0 to 100; the higher the scale, the higher the need. The fourth variable is a composite variable representing the average of daily living activity scores. The last three variables Xg, X^ and X^, represent three different measures of functional capacity of patients. At Rancho Los Amigos Hospital, data is collected on variables X^,...,X^ in order to comply to the State of California ruling (Title 22, Social Security, Register 76, No. 41, October 9, 1976) which mandates "...a method of determining staffing requirements based on assessment of patient needs. This assessment shall take into considera tion at least the following: (1) ability of the patient to care for himself; (2) his degree of illness; (3) requirements for special nursing activities; (4) skill level of personnel required in his care; (5) placement of the patient in the nursing unit." Data were collected on 10 3 patients for the neurologi cal training program using the standard work sampling method, and a non-participant observation technique. For details of the method of data collection see Berthold (1975). The computer program for the ridge regression was written by Ketola (1977) -- see Appendix I, and the program for the selection of k was written by the author — see Appendix II. CHAPTER IV I RESULTS ! I Table 1 summarizes the means and standard deviations ! I for the dependent and seven independent variables. Table ' i 2 presents the correlation matrix among the eight variables.| Correlations larger than 0.2 were significant at P < 0.0 5". ; i Note the extremely large correlations among the four indicesj of functional capacity (X^ to X^). | Table 3 presents the least-squares estimates of the ! seven parameters along with the residual sum of squares | I and variance inflation factors. The influence of the cor- | relations was largest for the estimates of 3^, 3g, 3g and 3y. The correlations have inflated the variance of the least-squares estimate of 3^ about 23 times, 3^ about 17 times, 3g about 6 times, and 3-y about 14 times. The vari ances of the estimates for X^, X 2 and Xg are slightly in flated, however. Table 4 presents the results of the search for optimal k by finding the minimum ISRM. The minimum occurred at k = 0.20. These results are depicted in Figures 3 and 4 9 which are ridge traces of the estimates as functions of k and m, respectively. Table 5 presents the ridge regression estimates and variance inflation factors at k = 0.2. The total VIF was reduced from 65.6 to 14.7 and the residual sum of squares increased from 0.6767 to 0.7070 (compare Tables 3 and 5). 10 { CHAPTER V DISCUSSION I These results demonstrate the use of ridge regression | I I jon a set of highly correlated data. As seen from Tables j '3 and 5, there is a large decrease in Total VIF and a small . 1 increase in residual sum of squares. The user of ridge regression must decide whether this tradeoff is acceptable for the purpose of estimation. | I 1 Among the various statistical packages which are | available, neither SPSS, BMDP or SAS have ridge regression programs. However, it is possible to modify the available linear regression programs to perform ridge regression (See Appendices I and II). Biased estimation is only one of the tools used in the analysis of a set of data. The researcher must understand the technical background of the problem, the definition of the candidate variables and the nature of appropriate models before he considers a biased estimation procedure such as \ i ridge regression. j Theoretical questions which arise regarding the use | I . . ___________________________________________________________________ of ridge regression include i) the estimation of the amount ' : of bias, ii) the variation in the standard errors of the i I : I ridge regression estimates, iii) the use of stepwise pro- ( ■ I cedures for selecting predictors of the dependent variables,' I ! i Î I and iv) the procedures for statistical inference using ridge I • j regression estimators. These questions suggest further re- i ' I I search possibilities. For discussion and applications see Brown (1977), Garbade (1977), Gorman and Thomas (1966), Hoerl (1962), Marquardt (1970), Suich and Derringer (1977) and Swindel (197 4). 12 I REFERENCES t i ! I I I 1 j I Berthold, J.S., Nursing Service Responsibilities and Patient ! I J Care in a Rehabilitation Facility, Unpublished Annual 1 i ; I Report, 197 5. j i 1 Brown, P.J., Centering and Scaling in Ridge Regression, Technometrics, 19, 35-36, 1977. Garbade, K., Two Methods for Examining the Stability of Regression Coefficients, Journal of the American Statis tical Association, 12, 54-63, 1977. Gorman, J.W. and Toman, R.J., Selection of Variables for Fitting Equations to Data, Technometrics, 9^, 27-51, 1966. Hoerl, A.E., Application of Ridge Regression to Regression Problems, Chemical Engineering Progress, 58, 54-59, 1962. Hoerl, A.E. and Kennard, R.W., Ridge Regression: Applica tion to Nonorthogonal Problems, Technometrics, 12, 6 9-82, 1970. Ketola, J., Ridge Regression for Nonorthogonal Data, Perspective (UCLA), 35-48, 1977. Marquardt, D.W., Generalized Inverse, Ridge Regression, Biased Linear Estimation and Nonlinear Estimation, Technometrics, 12, 591-612, 1970. 13 -Marquardt, D.W. and Snee, R., Ridge Regression in Practice, ' The American Statistician, 29, 3-20, 1975. ;Scheffe, H., The Analysis of Variance, 1959. I jSuich, R. and Derringer, G., Is the Regression Equation I I Adequate? One Criterion, Technometrics, 19, 213-216, I 1977. I ISwindel, B.F., Instability of Regression Coefficients, I The American Statistician, 28, 63-65, 1974. 1Vinod, H., Application of New Ridge Regression Methods to a Study of Bell System Scale Economics, Journal of the American Statistical Association, 71, 835-841, 1976. 14 TABLE 1 SUMMARY STATISTICS FOR NEUROLOGICAL NURSING CARE STUDY VARIABLES (n = 103) Variable Mean + sd X, 61.1 + 23.4 1 — X^ 45.9 + 22.6 Xg 55.9 + 30.9 X. 4.62 + 2.09 4 — Xg 5.96 + 4.87 X_ 84.9 + 25.9 6 — X„ 31.6 + 26.5 / — 3. 52 + 2. 18 15 TABLE 2 INTERCORRELATIONS AMONG THE VARIABLES X2 X3 ^4 %5 ^ 6 X, Y X, 1.00 .40 .20 .75 -.71 . 60 - . 6 8 .44 X2 1 . 0 0 .37 .24 — .24 . 22 -.24 .32 ""s 1 . 0 0 - . 0 2 - . 0 1 . 0 1 .00 .03 ^4 1 . 00 -.97 .83 -.92 .59 ^5 1 . 0 0 — .82 . 92 — .49 1 . 0 0 -.92 .46 X? 1 . 0 0 -.49 1.00 16 TABLE 3 LEAST-SQUARES ESTIMATES (k = O) AND VARIANCE INFLATION FACTORS Variance Least-squares Estimate VIF X. X. X. X X, X X. -.0007 .2115 -.0260 .6940 .2649 .0685 .0190 Residual Sum of Squares = 2. 92 1.33 1.29 22.57 17.36 6.39 13.74 Total = 65.60 0.6767 17 TABLE 4 SEARCH FOR OPTIMAL k k m ISRM Total VIF .00 . 0 0 12. 39 65.6 .02 .89 6.73 44.5 . 04 1. 39 4.22 34.8 . 06 1.74 2.92 29.0 .08 2 . 0 0 2 . 2 0 25.1 .10 2 . 2 2 1.80 2 2 . 2 .12 2.41 1.57 2 0 . 0 .14 2.56 1.45 18.3 .16 2.71 1.38 16.9 .18 2.83 1.34 15.7 .20 2 . 94 1.329 14.7 .22 3.04 1.331 13.9 .24 3.14 1. 34 13.1 .26 3.23 1.35 12.4 .28 3.31 1.37 1 1 . 8 .30 3.39 1. 39 11. 3 CO 7. 00 * 7.0 * Undefined 18 TABLE 5 RIDGE REGRESSION ESTIMATES (k = 0.2) AND VARIANCE INFLATION FACTORS Variable Ridge regression Estimate VIF X. X. X X, X X. 0775 1669 0349 1893 0735 0768 0724 1.59 1.01 0.96 3.15 2.86 2.18 2.98 Total = 14.73 Residual Sum of Squares = 0.7070 19 RR coefficient» .75 - .70 65 .60 - 55 5 0 45 40 35 .30 .25 .20 . - Î 5 , i o .2 .3 .0 .7 .5 ,4 6 .8 .9 1.0 CONSTANT ADDED TO THE DiAGONAL OF CORRELATION MATRIX, k FIGURE 1 Hoerl-Kennard Ridge Trace (Vinod, 1976) 20 RR coefficients 75 .70 65 - 60 55 50 .45 40 35 .30 .25 20 .15 .05 0.5 0.0 1.0 1.5 2.0 2.5 3.0 MULTICOLLINEARITY ALLOWANCE, m FIGURE 2 Ridge Trace on the m-Scale (Vinod, 1976) 21 RR coefficients Constant added to the diagonal of correlation matrix,' k FIGURE 3 Hoerl-Kennard Ridge Trace (Nursing Care Data) 22 RR c o e f f ic ié ir it - Multicollinearity allowance, m FIGURE 4 Ridge Trace on the m-Scale (Nursing Care Data) 23. APPENDIX I REVISED KETOLA*S SAS RR PROGRAM (Ketola, 1977) 1 2 NOTE . NO TE : 210 NOTE ■ NO TE 2 1 1 21 2 2 13 2 1 4 21 5 21 6 217 21 a 2 1 9 220 22 1 222 223 224 22 5 226 227 228 229 230 23 1 232 233 234 23 5 236 237 238 239 240 24 1 24 2 243 244 245 246 247 248 249 250 251 DATA RÎDANA : INPUT IX1-X7 Y) c a r d s ; (3 17 3»P8.3 3 63 2*48.3 -¥2 3 17 2*48.3 S 41 48,31; DATA SET WORK.RIDANA HAS 103 OBSERVATIONS AND 8 VARIABLES. THE DATA STATEMENT USED 0.51 SECONDS AND 9OK. PRCC CGRR 0L'TP=CGRREL; . DATA SET- WORK .C0RR5L HAS 1 I OBSERVATIONS AND 10 VARIABLES. THE p r o c e d u r e CORR USED 1.02 SECONDS AND 136K AND PRINTED PAGES 1 PROG MATRIX : NSUBJ=103; N=J(1); FETCH d a t a : N=4:NROW I DATA > : J = NCCL(DATA) - 1 : 13 = i:j ; R = DATA! N , •*) ; SX = OlAG(REGIP(d a t a (2,IJ))); SY = RECIP(0ATAI2.J4I)); RX=RI I J, t J ) ; RY = R (IJ,J+1); EIGEN y E RX; GR X= E*D I AG {REGI P IFU Z 2 I M ) ) ) E • ; PRINT M E RX; PRINT GRX; e OLS=GPX*PY; SSE = 1-RY* =GRX*PY; .MSe = 55E4RECIP( NSUBJ-J-1 1 : PRINT B_OLS; tb_o l s =s x*8_c l s*s y ; p r i n t t s_o l s ; ELO = -MSE^TRAGE C GRX) PRINT ELO MSE: Q=ssQ( a_0Ls 1-elo; ■ .. . p r in t-o ; IF Q<=0 THEN 00 ; STOP; END: K=,o1 ; it=o ; 1_=E* *RY ; LOOP: Kj=K* j< J .t1 ; IT=IT+i; IF IT>25 then GOTO GOTK; F=SSO (L4REGIPI M+.<J ) 1-a ; IF A0S(F)<1.E-5 t h e n GO TO GOTK; RMK= ( M+K J) ( M + K J ) # ( M+K J ) : 0F=2*SUM(L«LYRECIP{RVK)): ■ CF = F*R£CIP< OF) ; K=K+GF: GO TO l o o p ; GOTK: a:< = E*Ol AG (RECIPI M+KJ ) ) 4E • *RY ; ' 8K3=SSQ(BKi: PRINT 8K8: T3K=SX*BK*SY; PRINT TSK: QMSE=( l-(RY * = ( £*0I AG{REG IP{M + KJ) )*E' ) *Ry ) )*REG IP CNSU3J-J-I ) : PRINT g m s e ; TO 2 PRINT RT: K ; 24 APPENDIX I (Cont'd) 252 Q<=ssa(3_0LS) I Ic; K-0; 2 53 OSK^ ( 9_0l_S • ) 1 i 0 Î I SSE ; 254 LL: K=<+,0 1 ; 25 5 K J , i ) ; 25 6 GRX=E*DÏAG(RECIPI M>KJ ) ) *E * Î 257 S_R=GPX4RY; , 2 55 SS£=1-RY’4GPX4PY; 259 S0'<-SSO( B_P > ; T - 26 0 n<*oK//(S9<I!<); 261 aSK=08<//( (8 P' ) I I< 1 i 5SE1 ; ■ ■ . ^ - 26 2 PRI NT K ; 263 PRINT GRX; 264 ■IF K<.5 THFN GOTO LL: 26 5 GK.=aK I I J ( NROw< OX) , 1 . ai : 266 .OUTPUT C< 0UT=a8: 267 OUTPUT 03X 0UT=3HTAS; NOTE: 04TA SET WORK.26 HAS 52 OBSERVATIONS AND 4 VARIABLES. NOTE: DATA SET WORK.BETAS HAS 52 OBSERVATIONS AND !0 VARIABLES. note: the p r o c e d u r e MATRIX USED 6.11 SECONDS AND 140X AND PRINTED PAGES 3 TO 25. 26 9 PP.OC s c a t t e r 0ATA=9S: plot C0L2*C0L1='*« C0L2*C0L3 = ‘Q’ / OVERLAY: 269 title RIDGE TRACE: LABEL COL I =*B(X1 ' * 3(X ) ’ COL2=X VALUE COL3 = Q: n o t e : the p r o c e d u r e s c a t t e r used 0.56 SECONDS AND 142X AND PRINTED PAGE 26. 270 PROC SCATTER DATA=BETASI PLOT C0L3ACOL1 = •I • COLS^COLP*•2 * COLB*COL3** 3 * 271 COLS*COL4 = '4 ‘ C0U8*CnL5=-'5 ’ COL 8 *COL 6= « 6 ’ COL.8*COL7= • 7*/OVERLAY ; 27 2 TITLE RIOg E TRACE 300-1-085; LABEL C0Ll*9t(X) C0L2=82(X) C0L3=33(K1 273 C0L4=B4(K ) C0L5 = 95<K) COL6=86(X) C0L7=37(X) C0L3=X C0L9=SSS: NOTE: THE PROCEDURE SCATTER USED 0.91 SECONDS AND 144< AND PRINTED PAGE 27. 27 4 PPOC PRINT DATA = SETAS: , , NOTE: THE PROCEDURE PRINT USED 0.66 SECONDS AND ilOX AND ORINTED PAGE 23. n o t e : SAS USED I44X MEMORY. NOTE: BARR, GOODNIGHT, SALL AND HELWIG SAS INST ITUT5 INC. o.O. BOX 10066 RALEIGH, N.C. 2 760 3 25 APPENDIX II PROGRAM THAT COMPUTES k FOR WHICH ISRM IS MINIMUM . 1 0 PAT ; 2 1 0 OCL 3 1 0 OCL 4 1 0 OCL 5 1 0 DCL 6 I c dcl 7 1 0 DCL 3 1 0 DCL 9 I 0 DCL 10 1 0 p=7: 16 1 0 GET ; 17 1 0 DO W ; ■ 13 1 1 I SRM 1 19 1 1 2 1 1 2 22 1 2 ■ 23 1 1 25 I 2 26 I. 2 27 1 2 28 1 2 29 3 0 1 T PUT 32 1 1 IF I 33 ■ 1 I ELSE 34 1 1 PUT S 35 1 1 K=X + 36 I 1 END ; 37 1 0 - END PROCEDURE OPTrCNS(MAIN); E(7> FIXED<7,5) :N[T<(7)0); P F I XED ; DENCM DECIMAL FLOAT (.7); ISRM.DECIMAL FL0ATI7); K DECIMAL float(7); LMBO DECIMAL FLOAT (7).; SEAR DECIMAL FL0AT(7); OLD d e c i m a l FLCAT(7); OENGM=0; ISRV=0; K=0: S8AR=0 1=1 TO 7)); 0LD=7 .OOOOCE4 7C LIST!(8(1) DO E C<<=.3); AR.FACTORl.DENOM,RESULT=0; 00 1=1 TO 7 ; LMaD = B(I); SSAR=(LM30/((LM8D+X)**2)l+SSAR: END : DO 1=1 TO 7 ; LMSD=0{ I)î FACT0R1= (P*( (LMEO/( LMBO + Kl ) 7*2 ) ) , * DENOM=:(S3AR7LMaO ) ; RESULT*( {(FACTOR 1/DENCM )-1 )**2): rSRM=ISRM+R£SULT; END ; ■ LISTCO: PUT LI ST(ISRM); :rm <=old t he n c l d= is. r m ; PUT E0IT('=T=I5RM STARTS TO INCREASE HERE )( COL (50), A); 1 PAT 26

Linked assets

University of Southern California Dissertations and Theses

Conceptually similar

PDF

A comparison of findings from City of Hope's Mobile Screening Unit program with data from the Los Angeles County Cancer Surveillance Program

PDF

Cytological studies on the infection of synchronized human tumor cells by Adenovirus type 1

PDF

A course of study in orthopedic nursing

PDF

Detection of marine neurotoxins

PDF

The application of an approximately optimal nonlinear stochastic control and estimation algorithm to a pharmacokinetic problem

PDF

A matrix exponential approach to solving systems of linear differential equations with application to pharmacokinetic models

Studies on the toxin of <italic>Clostridium perfringens</italic>

PDF

Studies on the toxin of Clostridium perfringens

PDF

The extended Kalman filter as a parameter estimator with application to a pharmacokinetic example

PDF

Innervation of the human tooth

PDF

Semi-linear regression problems with an application in pharmacokinetics

PDF

The effects of a previous vitrectomy on retinal detachment patients and a method for computing a stratified kappa statistics

PDF

Two studies in the analysis of ophthalmic data sets

PDF

Focus group debriefings: Front-line practitioners' views of the long term care system in Los Angeles County

PDF

A study of the effects of geriatric experience in health professional on feelings about the elderly

PDF

Searching for an identity: A study of residential care facilities for the elderly in California

PDF

An analysis of how auspices and administrator preparation affect who is served by board and care homes for the elderly

PDF

Course of study on resistive exercise for the secondary school

The influence of certain water-softening processes on <italic>Escherichia coli</italic>

PDF

The influence of certain water-softening processes on Escherichia coli

PDF

The use of complaints against nursing homes as an index of quality care

PDF

The utilization of volunteers in nursing homes: Staff attitudes, their effects, determinants and response to intervention

Asset Metadata

Creator Velez, Patricio Mendoza (author)

Core Title An application of the ridge regression method to a neurological nursing care study

Degree Master of Science

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

Tag Biological Sciences,Health and Environmental Sciences,OAI-PMH Harvest

Format application/pdf (imt)

Language English

Contributor Digitized by ProQuest (provenance)

Permanent Link (DOI) https://doi.org/10.25549/usctheses-c37-164121

Unique identifier UC11649761

Identifier EP54860.pdf (filename),usctheses-c37-164121 (legacy record id)

Legacy Identifier EP54860.pdf

Dmrecord 164121

Document Type Thesis

Format application/pdf (imt)

Rights Velez, Patricio Mendoza

Type texts

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

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

Repository Name University of Southern California Digital Library

Repository Location USC Digital Library, University of Southern California, University Park Campus, Los Angeles, California 90089, USA