MULTICOLLINEARITY IN TRANSPORTATION MODELS by SHEUNG-LING CHAN B . A . , U n i v e r s i t y of Hong Kong, 1965 A THESIS SUBMITTED IN PARTIAL FULFILMENT OF ' THE REQUIREMENT FOR THE DEGREE OF MASTER OF ARTS l i n the School of Community & Regional Planning We accept t h i s t h e s i s as conforming t o the r e q u i r e d standards THE UNIVERSITY OF BRITISH COLUMBIA APRIL, 1970. In presenting t h i s thesis in p a r t i a l fulfilment of the requirements for an advanced degree at the University of B r i t i s h Columbia, I agree that the Library s h a l l make i t f r e e l y available and study. for reference I further agree that permission for extensive copying of t h i s thesis for scholarly purposes may be granted by the Head of my Department or by h i s representatives. It i s understood that copying or publication of t h i s thesis for f i n a n c i a l gain s h a l l not be allowed without my written permission. School of Community and Regional Planning The U n i v e r s i t y of B r i t i s h Columbia Vancouver 8,' Canada t • Date: April 1970. i ABSTRACT This thesis explores using multiple regression some of the limitations and implications of analysis in transportation models. Specifically i t investigates how the problem of m u l t i c o l l i n e a r i t y , which r e s u l t s from using intercorrelated variables i n t r i p generation models, affects the v a l i d a t i o n of hypotheses, adversely discovery of underlying relationships and p r e d i c t i o n . The research methodology consists of a review of the l i t e r a t u r e on t r i p generation analysis and a t h e o r e t i c a l exposition on m u l t i c o l l i n e a r i t y . Secondly, t r i p generation data for Greater Vancouver empirical a n a l y s i s . (1968) i s used for Factor analysis and multiple regression techniques are employed. The r e s u l t s demonstrate that m u l t i c o l l i n e a r i t y i s both an explanatory and prediction problem which can be overcome by a combined factor analytic v t and regression method. This method i s also capable of i d e n t i f y i n g and incorporating causal r e l a t i o n s h i p s between land use and t r i p generation into a single model. explanatory, It i s concluded that the d i s t i n c t i o n between the analytic and predictive a b i l i t i e s of a regression model i s a r t i f i c i a l , and that greater emphasis on t h e o r i z i n g in model-construction needed. . • • y. ' is ii . TABLE OP CONTENTS PRELIMINARY PAGES PAGE Abstract i Table of Contents i i Acknowledgements vi CHAPTER I. INTRODUCTION ,1.1 1 J u s t i f i c a t i o n f o r Research 1.2 The Problem and General Hypothesis 3 1.3 Postulates 5 1.4 Methodology 5 1.5 Source of Data 6 1.6 L i m i t a t i o n of Data 8 1.7 Organization of the Chapters to Follow 1.8 D e f i n i t i o n s II. 1 11 11 TRIP GENERATION ANALYSIS - AN OVERVIEW 21 2.1 T r i p Generation i n the Transportation Planning Process 21 2.2 Factors I n f l u e n c i n g T r i p Generation 23 2.3 Approaches t o T r i p Generation A n a l y s i s . . . . . . 26 2.4 Some Considerations i n Using M u l t i p l e Regression Analysis.....' I I I THEORETICAL EXPOSITION OF MULTICOLLINEARITY AS AN ,. EXPLANATORY AND ANALYTICAL PROBLEM 3.1 Model A t t r i b u t e s ' 3.2 A Non-mathematical Summary o f the T h e o r e t i c a l Implications of M u l t i c o l l i n e a r i t y . . . . . 33 39 39 42 Ill PAGE 3.3 A S t a t i s t i c a l E x p o s i t i o n of M u l t i c o l l i n e a r i t y 3.4 Conclusion 43 57 I V . EMPIRICAL VERIFICATION OF HYPOTHESES 60 4.1 Summary of E m p i r i c a l Findings 60 4.2 Formulation of Three Operational H y p o t h e s e s . . . . . . . 61 4.3 V a l i d a t i o n of Hypothesis 1 62 4.4 V a l i d a t i o n of Hypothesis 2 68 4.5 V a l i d a t i o n of Hypothesis 3 79 4.6 Conclusion 91 V . TRANSPORTATION MODELS - A PERSPECTIVE VIEW 94 5.1 Summary of Research Findings 94 5.2 U t i l i t y of Transportation Models 97 5.3 I m p l i c a t i o n s f o r Model B u i l d i n g i n Transportation Studies 101 5.4 Conclusion 103 BIBLIOGRAPHY .* 108 APPENDICES APPENDIX A . A l i s t of V a r i a b l e s Used i n This Study Il8 , APPENDIX B. S t a t i s t i c a l Test of A u t o c o r r e l a t i o n f o r Model 2 by Using the " C o n t i g u i t y Measure f o r k-Color Maps " Technique 119 APPENDIX C . Method of Using Model 3 f o r P r e d i c t i o n 123 APPENDIX D. Input Data and The C o r r e l a t i o n Matrix APPENDIX E . M u l t i p l e Regression Outputs of Model 1. APPENDIX F . Factor A n a l y s i s Outputs f o r the T r i p Generation Data APPENDIX G. M u l t i p l e Regression Outputs of Model 2. APPENDIX H . M u l t i p l e Regression Outputs of Model 3. iv LIST OF TABLES PAGE TABLE I . TRIPS PER DWELLING UNIT CROSS-CLASSIFIED WITH HOUSEHOLD SIZE AND AUTO-OWNERSHIP. TABLE I I . 29 EXAMPLES OF MULTIPLE CORRELATIONS IN A THREEVARIABLE PROBLEM WHEN INTERCORRELATIONS VARY. 55 TABLE I I I . SIMPLE AND PARTIAL CORRELATIONS OF MODEL 1. 64 TABLE I V . SIMPLE AND PARTIAL CORRELATIONS OF MODEL 2. 77 TABLE V . LOSS AND GAIN OF COMMUNALITIES IN MODEL 2 COMPARED WITH MODEL 1. TABLE VT. 78 A LIST OF POSSIBLE EXPLANATORY FACTORS OMITTED BY MODEL 2. . ' \ • .- • 84 LIST OF ILLUSTRATIONS PAGE FIGURE 1. TRAFFIC DISTRICTS OF GREATER VANCOUVER FIGURE 2.. GEOMETRIC INTERPRETATION OF MULTIPLE REGRESSION IN A THREE VARIABLE PROBLEM WITH NO INTERCORRELATION BETWEEN INDEPENDENT VARIABLES 44 FIGURE 3. GEOMETRIC INTERPRETATION OF MULTIPLE REGRESSION • IN A THREE VARIABLE PROBLEM WITH INTERCORRELATION BETWEEN INDEPENDENT VARIABLES . 45 GEOMETRIC INTERPRETATION OF MULTIPLE REGRESSION IN A THREE VARIABLE PROBLEM WITH NEAR PERFECT INTERCORRELATION BETWEEN INDEPENDENT VARIABLES 47 GRAPH SHOWING MULTIPLE R 2 .. OFr23 56 FIGURE FIGURE 4. 5. 0 7 , AS A FUNCTION FIGURE 6. COMPOSITION OF SEVEN MAJOR FACTORS OUT OF 29 VARIABLES. .. 70 FIGURE 7. LOCATION OF THE FIRST AND SECOND COMPONENT VECTORS FOR THE VARIABLES IN TWO-DIMENSIONAL SPACE. 73 LOCATION OF THE SECOND AND THIRD COMPONENT VECTORS FOR THE VARIABLES IN A TWO-DIMENSIONAL SPACE. • 75 LOCATION OF THE THIRD AND FOURTH COMPONENT VECTORS FOR THE VARIABLES IN A TWO-DIMENSIONAL SPACE. 76 FIGURE 10. OBSERVED AND CALCULATED VALUE OF Y FOR MODEL 1 80 FIGURE 11. OBSERVED AND CALCULATED VALUE OF Y FOR MODEL 2 8l FIGURE 12. MAP SHOWING DISTRIBUTION OF RESIDUALS FOR MODEL 2 83 FIGURE 13. FACTOR SCORE DISTRIBUTION FOR FACTOR I I (EMPLOYMENT) 85 FIGURE FIGURE 8. 9. FIGURE 14. FACTOR SCORE DISTRIBUTION FOR FACTOR I I I FIGURE 15. OBSERVED AND CALCULATED VALUE OF Y FOR MODEL 3 (DENSITY) 86 89 ACKNOWLEDGEMENTS Thanks are due to Professor P. 0. Roer for advice and supervision i n the preparation of t h i s t h e s i s . I am also much indebted to D r . N . d . Cherukupalle for i n s p i r a t i o n and encouragements. In a d d i t i o n , I am grateful to N. D. Lea & Associates, Vancouver, for permission to use part of t h e i r Burnaby Transportation Study data. F i n a n c i a l assistance from the Mellon Foundation i s acknowledged. F i n a l l y , s p e c i a l words of thanks must be extended to my colleague, How-Yin Leung for h i s c r i t i c i s m s and help i n proof-reading the manuscript. 1 CHAPTER I INTRODUCTION M u l t i p l e regression i s one o f the most widely used techniques i n data a n a l y s i s and model b u i l d i n g ; i t i s often abused due t o a lack o f understanding o f i t s basic assumptions. This chapter introduces the problem o f m u l t i c o l l i n e a r i t y which r e s u l t s from v i o l a t i o n o f the assumption that p r e d i c t o r s i n the regression equation are independent. The d i s c u s s i o n i s conducted w i t h i n the context o f t r a n s p o r t a t i o n models. 1.1 J u s t i f i c a t i o n f o r Research Estimation o f t r a v e l demand i s an important and i n t e g r a l part of the t r a n s p o r t a t i o n planning process. generation Among i t s various phases, t r i p and modal s p l i t procedures have g e n e r a l l y r e l i e d h e a v i l y on s t a t i s t i c a l methodology such as m u l t i p l e r e g r e s s i o n . These procedures require a sound knowledge o f the s t r u c t u r a l r e l a t i o n s h i p contained i n the basic data s e t . Yet a survey of the l i t e r a t u r e i n t h i s f i e l d " i n d i c a t e s that major e f f o r t has so f a r been i n the d i r e c t i o n o f s t a t i s t i c a l e f f i c i e n c y and s e l e c t i o n o f optimal r e l a t i o n s h i p s between variables. L i t t l e effort has been devoted to understanding the inferences concerning t r a v e l behaviour that are i m p l i c i t i n these procedures. 1 Stated simply, most model-builders are overly concerned with obtaining a high c o r r e l a t i o n c o e f f i c i e n t , and hence a good f i t of data, and less attention has been paid to the a n a l y t i c a l and explanatory powers of the model. The general view held i s that prediction i s not necessarily dependent on explanation. To the extent that the function of a model i s purely p r e d i c t i v e , as opposed to those models that seek to explain certain phenomena or to establish r e l a t i o n s h i p , a high c o r r e l a t i o n i s seen as an end in causal itself. 2 Reacting to the above a t t i t u d e , Muller and Robertson cautioned that multiple regression equations with a high c o r r e l a t i o n c o e f f i c i e n t , but containing i l l o g i c a l r e l a t i o n s h i p s , a r e statistically unstable. This i s self-evident as regression, and other mathematical models for that matter, i s only as accurate and as useful as the v a l i d i t y of the assumptions that are made and the s t a t i s t i c a l result obtained. significance of the It i s e n t i r e l y possible to produce r e s u l t s meeting a l l of the various s t a t i s t i c a l c r i t e r i a and yet offer no explanation of the causative r e l a t i o n s h i p s . In order to forecast, such a causal 3 relationship is essential. The use of i n t e r c o r r e l a t e d variables ' being recognized, by Alonso and Harris obscuring the causative r e l a t i o n s h i p s . 5 in regression i s i n c r e a s i n g l y among others, as a problem Unfortunately variables in the urban context used for transportation models are more often than not s p a t i a l l y d i s t r i b u t e d i n a correlated fashion, e . g . car ownership i s correlated with income, income with density, density with distance from C . B . D . , e t c . Thus, i f a l l these* i n t e r c o r r e l a t e d variables were 3 used in the model, i t becomes extremely d i f f i c u l t to determine which are the causal factors r e l a t e d to urban t r a v e l . In other words, it is not known then'whether t r i p s are a function of a l l these variables working independently or whether they are i n t e r a c t i n g and i n effect overlapping. Moreover, the existence of c o l l i n e a r i t y among variables casts many doubts on conventional s t a t i s t i c a l analyses and creates 6 severe operating problems. It i s f e l t therefore, that as trans- portation planners are more and more r e l y i n g on regression as a t o o l for planning, i t i s perhaps timely to place i n perspective t h i s issue which bears on the r e l i a b i l i t y of the model as an explanatory and p r e d i c t i v e device. 1.2 The Problem and General Hypothesis , The multiple regression model usually takes the form of Y = a + b,x, + b „ x „ + b~x 0 + •LI 22 5 5 +bx m m In specifying the model i n t h i s form, i t i s assumed that the. various independent variables make independent and additive contributions to the prediction of the variances observed i n the dependent variable Y . I f the assumption of independence i s v i o l a t e d , then the 7 problem of c o l l i n e a r i t y i s introduced . (in the case of two correlated independent v a r i a b l e s ) , or m u l t i c o l l i n e a r i t y (in the case of three or more correlated v a r i a b l e s ) . , /' The term m u l t i c o l l i n e a r i t y i s the name given to the general problem which a r i s e s when some or a l l of the explanatory variables i n an equation are so h i g h l y correlated that i t becomes very d i f f i c u l t , 4 i f not impossible, to separate t h e i r i n d i v i d u a l influences and obtain 8 a reasonably precise estimate Of t h e i r e f f e c t s . Secondly, since the variables are highly correlated, they reinforce each other's r e l a t i o n ship with the c r i t e r i o n , or suppress the true contribution of other variables i n the equation. In the former case, the tendency i s towards d i s t o r t i n g the value of the multiple c o r r e l a t i o n beyond i t s true proportion; i n the l a t t e r , coefficient some variables of explanatory value may never be able to enter the equation due to the predominance of c o l l i n e a r s e t s . As previously stated, t h i s problem i s p a r t i c u l a r l y prevalent in transportation models due to the type of variables employed. objective of t h i s study, therefore, i s to investigate The how m u l t i - c o l l i n e a r i t y affects the performance of transportation models, in respect to v a l i d a t i o n of hypotheses, discovery of underlying r e l a t i o n ships and p r e d i c t i o n . Methods for overcoming the problem w i l l be suggested in the course of i n v e s t i g a t i o n . Moreover, the implications of p r e d i c t i o n versus explanation i n model-building w i l l be discussed. The following general hypothesis i s developed as a focus for the research: "When c o l l i n e a r i t y exists i n a regression model, explanatory and a n a l y t i c a l powers are decreased, despite the apparently good predictive power shown by a high multiple c o r r e l a t i o n coefficient." / . 5 1.3 Postulates The i n v e s t i g a t i o n i s based on the f o l l o w i n g assumptions: 1) A good s t a t i s t i c a l f i t does not assure a good p r e d i c t i v e 9 model. A model's strength l i e s e s s e n t i a l l y i n the sound- ness of i t s t h e o r e t i c a l base. 2) Only v a r i a b l e s that can be supported by i n t u i t i v e l y sound arguments should be used i n regression a n a l y s i s . I t makes l i t t l e sense t o throw a l l p o s s i b l e v a r i a b l e s i n t o the pot i n a shotgun approach merely t o obtain a high c o r r e l a t i o n coefficient. 1 0 3) M u l t i p l e regression models are b a s i c a l l y concerned with postulates of cause and e f f e c t . Hence t h e i r v a l i d i t y as f o r e c a s t i n g t o o l s must r e l y on causative r e l a t i o n s h i p s . 1.4 1 1 Methodology A twofold s t r a t e g y i s adopted. F i r s t l y , a t h e o r e t i c a l exposition on c o l l i n e a r i t y or m u l t i c o l l i n e a r i t y based on l i t e r a t u r e research i s given. Hopefully t h i s w i l l throw l i g h t on why and how c o l l i n e a r i t y a f f e c t s the v a l i d i t y and u t i l i t y of the model. Secondly, the general hypothesis i s t o be v e r i f i e d e m p i r i c a l l y by: 1) Using m u l t i p l e regression a n a l y s i s t o examine, i n depth, a t y p i c a l example of t r a n s p o r t a t i o n models on t r i p ; generation, w i t h s p e c i a l emphasis on the undesirable p r o p e r t i e s a s s o c i a t e d with c o l l i n e a r i t y . 6 2) Factor analysis of the data to extract underlying dimensions, and to see i f the model has incorporated the s i g n i f i c a n t factors into the equation.' 3) Formulating a new multiple regression model to eliminate c o l l i n e a r sets and compare r e s u l t s . Data analysis i s c a r r i e d out by UBC IBM 360 d i g i t a l computer. 12 computer programs used are the TRIP 13 and FACTO for regression analysis and the l a t t e r for factor 1.5 The packages, the former analysis. Source of Data The t r i p generation model studied here and i t s associated data has been obtained through the courtesy of N. D. Lea & Associates, Vancouver. The.data was c o l l e c t e d for thirty-two t r a f f i c districts of Greater Vancouver i n 1968, p a r t l y through a telephone survey and p a r t l y from census information (See Figure 1 ) . A t o t a l of twenty-nine variables are used for computation in t h i s thesis and a l l variables 14 are measured on i n t e r v a l s c a l e s . It should be pointed out that the o r i g i n a l data from N. D. Lea & Associates consisted of ten dependent variables and sixty-nine independent v a r i a b l e s . The former i s a f i n e r breakdown of the nature 15 of t r i p s . The sixty-nine independent variables include the twenty- 16 nine variables transgenerated selected, in t h i s study, the rest being complex or 17 variables. ' By step regression, • • ' • ' • / ' a t o t a l of about f o r t y - equations were developed and f i n a l l y nine were selected, ' . one for each • FIG. 1 TRAFFIC DISTRICTS OF GREATER VANCOUVER 8 dependent variable except t o t a l t r i p production. The model developed for t o t a l t r i p generation ( t o t a l persons t r i p s excluding walk t r i p s per day) i s selected for d e t a i l e d examination here for two reasons: . 1 ) It i s representative generation. of transportation models for t r i p Hence findings w i l l generally be applicable to other models in the f i e l d . 2) It i s a convenient example because data i s available F a m i l i a r i t y with l o c a l conditions f a c i l i t a t e s locally. interpretation and v i s u a l i z a t i o n of the issues involved. 1.6 Limitation of the Data Since t h i s i s a r e a l data, i . e . a l l information i s grouped on the basis of geographical u n i t s , three l i m i t a t i o n s are recognized: a) The problem of autocorrelation, i . e . measurements obtained i n one area are not e n t i r e l y independent of those obtained i n other areas. Certain population and land use c h a r a c t e r i s t i c s between contiguous areas may exhibit greater s i m i l a r i t y than non-contiguous areas. I f t h i s i s indeed so, then one of the assumptions correlation analysis, that residuals of from regression are mutually . independent random variables w i l l be v i o l a t e d . Statistical tests are c a r r i e d out on the r e s i d u a l d i s t r i b u t i o n of Model 2 (See Figure 12) using the "contiguity measures for i k-colour l8 maps technique". The r e s u l t reveals that there i s no • . / s i g n i f i c a n t autocorrelation i n t h i s set of data. Computations can be found in Appendix B . b) The t r a f f i c d i s t r i c t s are not uniform i n s i z e . This gives r i s e to the problem i n i d e n t i f y i n g or e l i m i n a t i n g differences in parameters which may be a t t r i b u t e d merely t o differences in s i z e of a r e a l u n i t s from those differences which are owing t o 19 "truly different" relationships. For example, when comparing two d i s t r i c t s of equal s i z e , the absolute number of people r e s i d i n g w i t h i n them r e f l e c t s the i n t e n s i t y of r e s i d e n t i a l use. However, f o r d i s t r i c t s of unequal s i z e s , there i s no r e a l b a s i s for comparison unless rate v a r i a b l e s are used. As can be seen l a t e r on i n t h i s s e t . o f data, the v a r i a b l e " A r e a " i s found to e x p l a i n a s i g n i f i c a n t amount of v a r i a t i o n i n t r i p generation (See page 6 7 ) . This r e v e a l s that the a n a l y s i s u n i t s f o r t h i s study are d i v i d e d i n such a way that, the s m a l l d i s t r i c t s are found w i t h i n the urban areas w i t h the large d i s t r i c t s at the metropolitan f r i n g e ; "Area" i n e f f e c t becomes a proxy for distance from C . B ; D . and to some extent r e f l e c t s degree of u r b a n i z a t i o n of the d i s t r i c t . Hence a d d i t i o n a l care must be exercised i n i n t e r p r e t i n g outputs under these circumstances. c) The h i g h l y aggregated data on a d i s t r i c t b a s i s f o r t h i s set poses some problems of i n t e r p r e t a t i o n and a p p l i c a t i o n . In g e n e r a l , geographical aggregation of data i s not as e f f i c i e n t • 20 as i t may be. As Fleet and Robertson pointed out, the underlying assumption of a r e a l aggregation i s that contiguous households e x h i b i t some s i m i l a r i t y In f a m i l y and t r a v e l characteristics. The degree t o which these u n i t s are not 10 homogenous results i n a loss of disaggregated d e t a i l . An example of a d e t a i l e d household c h a r a c t e r i s t i c that does not "show up" i n explaining the zonal t r i p generation i s familyincome, which i n t u i t i v e l y would d i r e c t l y r e f l e c t in household c h a r a c t e r i s t i c s trip-making. differences and i n p a r t i c u l a r , differences in However,, when t h i s information i s averaged for an a r e a l unit composed of a number of households and r e l a t e d to the number of t r i p s generated by that u n i t , almost a l l these differences are l o s t . "weak" r e l a t i o n s h i p s . making s t a t i s t i c a l This has resulted in the seemingly Therefore there i s an inherent danger i n inference from highly aggregated data concerning disaggregated r e l a t i o n s h i p s . S i m i l a r l y , because t r i p generation data for t h i s study has been c o l l e c t e d at the t r a f f i c d i s t r i c t l e v e l , the apparently good r e s u l t s of analysis (the extremely high c o r r e l a t i o n c o e f f i c i e n t s obtained) are misleading. This i s i n t u i t i v e l y obvious since . the larger the u n i t , the more t o t a l v a r i a t i o n w i l l be lost within the u n i t s . L i t t l e of the t o t a l v a r i a t i o n i s a c t u a l l y l e f t to explain between the u n i t s , thus allowing a high proport i o n of the between-group-variance to be unaccounted f o r . As such i t i s s t a t i s t i c a l l y incorrect to use equations developed at the d i s t r i c t l e v e l to calculate t r i p s generated at the smaller zonal l e v e l because another set of variables may do a better job at t h i s l e v e l . 11 Since the primary i n t e r e s t of t h i s study i s t o i n v e s t i g a t e trip generation c h a r a c t e r i s t i c s at an a r e a l l e v e l , the inferences drawn w i l l not be a p p l i c a b l e to i n d i v i d u a l and household t r a v e l behaviour. 1.7 Organization of the Chapters t o Follow Chapter I I presents an overview of current p r a c t i c e s and develop- ments i n t r i p generation a n a l y s i s based on l i t e r a t u r e r e s e a r c h . Chapter I I I contains a d i s c u s s i o n on model a t t r i b u t e s and a s t a t i s t i c a l e x p o s i t i o n on the problem of m u l t i c o l l i n e a r i t y . Chapter IV attempts to v a l i d a t e , e m p i r i c a l l y , the general hypothesis through the t e s t i n g of three o p e r a t i o n a l hypotheses and the development of an a l t e r n a t i v e model. F i n a l l y , Chapter V deals with the planning i m p l i c a t i o n s of the f i n d i n g s and concludes w i t h a summary and suggestions f o r future . . . . > research. ' \i 1.8 ' - Definitions The f o l l o w i n g i s a b r i e f resume of the terms used i n the t e x t . More rigorous and t e c h n i c a l expositions on regression and f a c t o r a n a l y s i s can be found i n standard textbooks on these s u b j e c t s . 21 ' Linear M u l t i p l e Regression A n a l y s i s Using the least-squares p r i n c i p l e , m u l t i p l e regression i s a technique for measuring the influence of some independent v a r i a b l e s on a dependent v a r i a b l e ( c r i t e r i o n ) . (predictors) In the context of t h i s study, 12 the aim of linear multiple regression i s to obtain from land use, t r a f f i c and population data an equation of the form: t; Y = a + b-jX^ + bgXg + 33 B where Y i s the zonal measure of t r a v e l . zonal land use and population factors, + X b nXn A l l the x ' s are the independent each of which has a separate influence on Y with per unit effects given by b^, b^, b^, e t c . Since not a l l of the numbers of t r i p s per zone may be explained by the x ' s in the equation, ' a ' i s a number put in to represent the unexplained part of the value of Y . the equation. It i s often referred to as the 'constant 1 of A large constant for t h i s reason i s undesirable as it indicates the p o s s i b i l i t y of presence of other explanatory variables not taken i n t o account. The b-coefficients are c a l l e d regression c o e f f i c i e n t s which in the case of standardized variables are c a l l e d B coefficients. the different Beta c o e f f i c i e n t s indicate the r e l a t i v e weights of independent\,variables. Standardized Variables Variable values are transformed into standard or Z scores rather than in raw scores. The Z score expresses the measurement of a variable for an i n d i v i d u a l i n terms of i t s deviation from the mean value of the distribution. The formula i s Z = x ± - x ' 13 F Probability A measure of whether the regression c o e f f i c i e n t i s s i g n i f i c a n t to the regression equation. Generally when the F P r o b a b i l i t y i s greater than 0.05 the regression c o e f f i c i e n t i s not s i g n i f i c a n t at the 5$ l e v e l . Standard E r r o r of Estimate (Residual Standard Deviation or Root Mean Square E r r o r ) I t i s u s u a l l y denoted by S . I t i s a summary of a l l the squared d i s - crepencies of a c t u a l measurements from the p r e d i c t e d measurements. A general measure of the value of a regression equation i s the standard e r r o r of estimate as a percentage of the mean value of the dependent v a r i a b l e ; a good equation has a s m a l l standard e r r o r of estimate which i s a small percentage of the mean, and v i c e v e r s a . Correlation Coefficient I t i s possible to measure the degree of a s s o c i a t i o n between two v a r i a b l e s by means of a s t a t i s t i c known as c o e f f i c i e n t of simple correlation. I t i s g e n e r a l l y represented by ' r / , which can assume values i n the range +1 o n l y . The c l o s e r ' r ' i s to the r e l a t i o n s h i p between the two v a r i a b l e s . +1, the stronger As a measure of c o r r e l a - t i o n between one dependent v a r i a b l e and more than one independent v a r i a b l e , the s t a t i s t i c i s known as c o e f f i c i e n t of m u l t i p l e c o r r e l a t i o n or ' R ' . There i s a further property of the c o e f f i c i e n t of c o r r e l a t i o n which i s u s e f u l i n i n t e r p r e t i n g the r e s u l t s given by the regression e q u a t i o n . The percentage of the t o t a l v a r i a t i o n i n the dependent v a r i a b l e which i s ' e x p l a i n e d ' by an independent v a r i a b l e i s approximately equal t o 14 one hundred times the square of ' r ' . This s t a t i s t i c i s known as the p c o e f f i c i e n t of determination or r_ i n case of simple c o r r e l a t i o n , and 2 R_ for multiple c o r r e l a t i o n . It should be noted that the tendency for one variable to vary with another, as shown by i t s ' r ' , i s no evidence of any causal r e l a t i o n s h i p , since i t may be that both variables are influenced by other variable(s) not examined. F i s h e r ' s Transformation A method to transform the value of the c o r r e l a t i o n coefficient regression equation into a s t a t i s t i c known as . Fisher's of a ' z ' so that the t e s t i n g of whether ' r ' i s s i g n i f i c a n t l y different from zero or to compare the difference between two r ' s 't1 test. can be c a r r i e d out by The p r i n c i p l e involves approximating * r ' i n t o a normal sampling population regardless of the size of sample and population. P a r t i a l and Simple C o r r e l a t i o n s ^ A simple c o r r e l a t i o n expresses the r e l a t i o n s h i p between two variables under consideration, not holding constant any other v a r i a b l e s . There- fore, i f there i s any c o l l i n e a r i t y between the explanatory variable and other Independent v a r i a b l e s , this relationship also. then the simple 'r' will incorporate The p a r t i a l . ' r 1 expresses the r e l a t i o n s h i p between the Independent variable under consideration and the dependent . v a r i a b l e , holding the effect of other independent variables constant. Factor A n a l y s i s A generic term f o r a v a r i e t y of procedures developed f o r the a n a l y s i s of i n t e r c o r r e l a t i o n s w i t h i n a set of v a r i a b l e s . The most common type of f a c t o r a n a l y s i s i s p r i n c i p a l component a n a l y s i s ; i t collapses large masses of data i n t o basic underlying dimensions, and i s capable of e l i m i n a t i n g c o l l i n e a r sets of v a r i a b l e s w i t h i n a set of data to produce an underlying set of independent or orthogonal f a c t o r s . Factor Loading The square root of the t o t a l variance of a v a r i a b l e accounted f o r by the f a c t o r s . In other words, i t i s the c o r r e l a t i o n c o e f f i c i e n t between the v a r i a b l e under consideration and the f a c t o r s . Communality The proportion of common variance of a v a r i a b l e accounted f o r by a 2 factor. I t can be regarded as the R between the v a r i a b l e under consideration and a f a c t o r . Factor Score The score that an i n d i v i d u a l obtains f o r a p a r t i c u l a r f a c t o r . It is c a l c u l a t e d from the scores the i n d i v i d u a l gets i n a set of v a r i a b l e s c o n t r i b u t i n g t o that f a c t o r by r e g r e s s i o n . and o r t h o g o n a l i z e d . U s u a l l y i t i s standardized (See Appendix C) Model An experimental design based on a t h e o r y . Being a s i m p l i f i e d representation of r e a l i t y , i t i s frequently truncated t h e o r i e s , s a c r i f i c i n g richness and completeness f o r o p e r a t i o n a l purposes. 23 16 T r i p Generation A term commonly used to describe the number of t r i p s s t a r t i n g or ending in a p a r t i c u l a r area in r e l a t i o n to the land use and/or socio-economic characteristics of that area. T o t a l T r i p s Generated in a.Zone It i s the number of person or vehicle t r i p s , by a l l modes of made by residents to and from a zone. transport, It does not include walk t r i p s and t r i p s by t a x i and trucks. T o t a l T r i p s Attracted to a Zone Refers to the number of person or vehicle t r i p s , by a l l modes of trans- p o r t , but excluding walk t r i p s and t r i p s by t a x i and trucks, ending i n a zone. Prediction Conditional statements about future developments - statements which are 24 conditioned by varying assumptions of p o l i c y and external c o n d i t i o n s . 2S Measurement and S p e c i f i c a t i o n Errors Two general categories of error can be distinguished i n any experimental design. The f i r s t i s measurement e r r o r . It includes data c o l l e c t i o n e r r o r s , errors of s c a l i n g and sampling e r r o r s . For the roost part, the model-builder i s unable to c o n t r o l these errors unless he i s responsible for the design of the data c o l l e c t i o n survey. S p e c i f i c a t i o n error a r i s e s from a misunderstanding or a purposeful s i m p l i f i c a t i o n in the model of the phenomenon we are t r y i n g to 17 r e p r e s e n t e.g. the r e p r e s e n t a t i o n o f a n o n - l i n e a r r e l a t i o n by a expression, omission linear of s i g n i f i c a n t v a r i a b l e s , i n c l u s i o n of i n t e r - c o r r e l a t e d v a r i a b l e s as w e l l as the f a i l u r e t o c o r r e c t l y e v a l u a t e a • 25 model. design. These can be more e a s i l y c o n t r o l l e d w i t h a good r e s e a r c h 18 Footnotes ^Christopher R. Fleet and Sydney R. Robertson, " T r i p Generation In The Transportation Planning Process", Highway Researcb Record, No. 240 (1968), p . 1 1 . U . S . Department of Transportation/Federal Highway A d m i n i s t r a t i o n , Bureau of P u b l i c Roads, Guidelines f o r T r i p Generation A n a l y s i s , (June 1967), p.109. 3 I b i d , p.25. ^ W i l l i a m Alonso, " P r e d i c t i n g Best With Imperfect Data", Journal of The American I n s t i t u t e of Planners, V o l . 34, No. 3, (1968), p.249. 5 B r i t t o n H a r r i s , "New Tools f o r P l a n n i n g " , Journal of The American I n s t i t u t e of Planners, V o l . 31, No. 2 (1965), p.95. 6 I b i d , p.95. ^ N . d . Cherukupalle,"Regression A n a l y s i s - I n t e r p r e t a t i o n of Computer Outputs, Etc.", Planning 508 Course Notes, October 1969, School of Community & Regional Planning, U . B . C , p . l . 8 J . Johnston, Econometric Methods, New York, McGraw H i l l I n c . , (1963), p.201. ^K. Rask Overgaard, "Urban Transportation P l a n n i n g : T r a f f i c E s t i m a t i o n " T r a f f i c Quarterly, ( A p r i l 1967), p.202. 1 0 K. 1 1 12 l3 J e f f r e y M. Zupan, "Mode Choise: Implications f o r P l a n n i n g " , Highway Research Record, No. 251 (1969), p . l 4 . Rask Overgaard, op. c i t . , p.202. James H . B j e r r i n g , J . R. H . Dempster and Ronald H . H a l l , U . B . C . TRIP (Triangular Regression Package), (The U n i v e r s i t y of B r i t i s h Columbia Computing Centre, January 1968). James H . B j e r r i n g , U . B . C . FACTO - Factor A n a l y s i s Program, (The U n i v e r s i t y of B r i t i s h Columbia Computing Centre, May 1969). 19 14 An i n t e r v a l - s c a l e deals with quantative measurements i n e q u a l i t y of u n i t s , which means the same numerical distance i s associated w i t h the same e m p i r i c a l distance on some r e a l continuum such as length and weight. 15 The ten dependent v a r i a b l e s for the N . D. Lea Burnaby Transportation Study a r e : home-based work a t t r a c t i o n , home-based other a t t r a c t i o n , non-home-based d e s t i n a t i o n s , t o t a l a t t r a c t i o n , home-based work p r o d u c t i o n , home-based other p r o d u c t i o n , non-home-based p r o d u c t i o n , non-home-based o r i g i n s , t o t a l production and t o t a l t r i p generation. 16 See Appendix A . 17 Basic V a r i a b l e s are v a r i a b l e s c o l l e c t e d i n the survey. Complex and transgenerated v a r i a b l e s are. those obtained by combining b a s i c v a r i a b l e s i n various manner, e , g . a d d i t i o n , s u b t r a c t i o n , m u l t i p l i c a t i o n and d i v i s i o n , logarithm, cosine, and s i n e , e t c . , e . g . In d e n s i t y i s derived from In ( t o t a l p o p u l a t i o n / a r e a ) . 18 „ For d e t a i l s on t h i s method, see Michael F . Dacey, A Review on Measures of C o n t i g u i t y f o r Two and k-colour Maps", S p a t i a l A n a l y s i s (New York: Prentice H a l l I n c . , 1968), E d . by B r i a n J . Berry and D. F . Marble, pp.479-490. 19 E . N . Thomas and D. L . Anderson, " A d d i t i o n a l Comments on Weighting Values i n C o r r e l a t i o n Analysis of A r e a l Data", l o c . c i t . , pp.431-445. 20 Christopher R. Fleet and Sydney R. Robertson, op. c i t . , p . 13. A d a p t e d from: M. A . T a y l o r , Studies of T r a v e l i n Gloucester, Northampton & Reading, Road Research Laboratory Report, L . R . 141 ( M i n i s t r y of Transport, Great B r i t a i n , 1968), pp.142-149. 22 N . d . Cherukupalle, op. c i t . , pp.4-5. 23 21 B r i t t o n H a r r i s , "The Use of Theory In The Simulation of Urban Phenomena"> J o u r n a l of the American I n s t i t u t e of Planners, V o l . 32, No. 5 (September 1966), p.265. . . - . 20 Britton Harris, 25 "New Tools f o r P l a n n i n g " , op. c i t . , p.9L W i l l i a m Alonso, op. c i t . , p.248. ) / I 21 CHAPTER I I TRIP GENERATION ANALYSIS - AN OVERVIEW This chapter provides an overview of current p r a c t i c e s and developments i n t r i p generation a n a l y s i s . P a r t i c u l a r reference is being made to consideration and l i m i t a t i o n s i n using the m u l t i p l e regression 2.1 technique. T r i p Generation In The Transportation Planning Process Decisions on t r a n s p o r t a t i o n f a c i l i t i e s i n urban areas are made everyday. Each d e c i s i o n has complex i m p l i c a t i o n s . f o r the e n t i r e urban community. To a i d i n making these decisions,, e f f e c t i v e and accurate forecasts of t r a v e l demands are necessary. These forecasts are g e n e r a l l y made w i t h i n the framework of an urban t r a n s p o r t a t i o n study which i s a systematic process s e r v i n g as a b a s i s on which to p l a n , design and evaluate t r a n s p o r t a t i o n systems. The t r a n s p o r t a t i o n planning process i s g e n e r a l l y considered t o consist of the f o l l o w i n g : population and economic s t u d i e s , land use s t u d i e s , t r i p generation, t r i p d i s t r i b u t i o n , modal s p l i t , t r a f f i c assignment and system < evaluation."*" 22 T r i p generation i s the term commonly used to denote the study of amounts of person and vehicular t r a v e l . This phase i s intended to prepare forecasts of t r a v e l demand, usually by small areas c a l l e d traffic zones. The r e s u l t i s , i n essence, a s p a t i a l d i s t r i b u t i o n on frequency of trip-making, defined at one end of the t r i p and s t r a t i f i e d " 2 by the types of t r i p s being made. The t r a d i t i o n a l linkage between land use and t r a v e l i s introduced in t h i s phase when the number of t r i p s that begin or end in a given zone can be r e l a t e d to the and socio-economic c h a r a c t e r i s t i c s ends form the measure of of that zone. activities The generated ' t r i p production' and ' t r i p a t t r a c t i o n ' in t r i p d i s t r i b u t i o n and modal s p l i t models. trip used The r e s u l t i n g t r a v e l patterns are then assigned to the highway or t r a n s i t network i n the traffic assignment stage. and transportation Many alternative plans of both land use systems can then be evaluated i n the system analysis phase. It can be seen that\<the t r i p generation phase i s a c r u c i a l step i n bridging the gap between land use and t r a v e l . A l s o , apart from producing the number of t r i p s per zone as inputs for subsequent analysis i n the transportation planning process, i t holds the key to an understanding of the varied interacting r e l a t i o n s h i p between t r a v e l and the surrounding environment. An insight i s a prerequisite transportation and land use p o l i c i e s . to the determination of 23 2.2 Factors Influencing T r i p Generation Travel i s uniquely human and subject to a l l the complexities and variations in human behaviour. The basic trip-making unit i s the i n d i v i d u a l , whose behaviour i s conditioned by h i s own c h a r a c t e r i s t i c s and by those of the household in which he l i v e s . Therefore, t r i p - making on a person or household basis i s governed by the socio-economic factors of the home. Throughout the years, many transportation studies have consistently found that variables such as car ownership per household, family s i z e , income per dwelling unit and occupation of the head of household are capable of explaining t r i p generation satisfactorily. The general conclusion i s that high income f a m i l i e s , who are also often multi-car f a m i l i e s , make more t r i p s than low income f a m i l i e s , which, on the other band, often own no car .and must r e l y on public transportation and thus generate fewer automobile trips. 5 S. T . Wong in a t r i p generation analysis of Chicago data, found that t o t a l d a i l y r e s i d e n t i a l t r i p s per occupied dwelling unit are dependent on such household c h a r a c t e r i s t i c s as car ownership, choice of mode, t r i p 6 purpose, age of trip-maker and distance from C . B . D . M. A . Taylor came to s i m i l a r conclusion in h i s analysis of Gloucester, Northampton and Reading data. He found that the t r i p rate per person, or per household, i s r e l a t e d to the socio-economic index, as w e l l as t r a v e l time to the town centre. Recent studies have indicated that household size is 7 emerging as a more important variable than car ownership. This i s h i g h l y probable i n view o f the f a c t ubiquitous that automobile ownership i s becoming more i n North A m e r i c a . More o f t e n than n o t , t r i p g e n e r a t i o n e s t i m a t e s are an a r e a 1 l e v e l . traffic The geographic zone, or the t r a f f i c d i s t r i b u t i o n and assignment district. variation is lost (see generated Due t o the p r o c e s s o f from a zone, as much o f the on p p . 9 - 1 0 ) . less l i k e l i h o o d of asserted o f the h o u s e h o l d s i n c e the z o n a l or d i s t r i c t A l l measures r e l a t i v e e.g. total unit. correlated t h e y are not factors trips level: t o the s i z e o f p o p u l a t i o n i n the number o f d w e l l i n g u n i t s , activities that trips In the main, t h r e e t o t a l p o p u l a t i o n , t o t a l number o f c a r s , b) C h a r a c t e r i s t i c s The f a c t o r s which measure the g r o s s amount o f c o n s i d e r e d important i n d e t e r m i n i n g the t o t a l number o f a) the They are more l o g i c a l l y dependent on human a c t i v i t y w i t h i n the a n a l y s i s generated at variables significant cannot be e x p e c t e d t o be with socio-economic c h a r a c t e r i s t i c s p o p u l a t i o n and l a n d use further trip detailed i n the form o f t o t a l number o f produced per zone or per d i s t r i c t dimensionslly compatible. Taylor the aggrega- together. the i n d i v i d u a l o r h o u s e h o l d l e v e l b e i n g e a r l i e r discussion generally n e c e s s a r y f o r the by g r o u p i n g heterogeneous households measures o f t r i p g e n e r a t i o n are is are no longer the s i g n i f i c a n t h i g h e r the l e v e l o f a g g r e g a t i o n , information at This is processes. t i o n , household c h a r a c t e r i s t i c s in explaining t o t a l t r i p s l e v e l of a g g r e g a t i o n conducted at total tract, labour force, etc. and i n t e n s i t y o f l a n d use t o be found i n a u n i t i s - the amount o f a l s o an e x p l a n a t o r y factor 25. for t r i p generation. variables acre. It i s usually stated i n terms of density such as dwelling units per acre or employees per Variations in i n t e n s i t y have d i s t i n c t impact on the number and type of t r i p s that are produced. In a d d i t i o n , the type of land use a c t i v i t y , be i t r e s i d e n t i a l , commercial, i n d u s t r i a l or i n s t i t u t i o n a l , gives r i s e to different rates, e.g. trip i t was found in Chicago that r e s i d e n t i a l use accounted for over 50$ of a l l t r i p s generated, followed by ,/ 9 commercial and manufacturing uses. The number of employees in each a c t i v i t y or commercial f l o o r area are t y p i c a l measures of t h i s factor. c) Location of land use activities: This factor refers to the s p a t i a l d i s t r i b u t i o n of land uses within the study area, e . g . an area of mixed land uses w i l l generate more walking t r i p s and less vehicle t r i p s , whereas a predominantly r e s i d e n t i a l area with no shopping nearby w i l l generate more vehicle t r i p s . facilities A study of traffic 10 characteristics i n suburban r e s i d e n t i a l areas in Washington • found that areas with, extensive shopping f a c i l i t i e s nearby generated four times more pedestrian t r i p s than a s t r i c t l y r e s i d e n t i a l neighbourhood. However, the l a t t e r generates s l i g h t l y more vehicle t r i p s than the former area. from C.B.D. i s one of the measures of t h i s / factor. Distance In conducting comparative studies between c i t i e s , which i s an even higher l e v e l of aggregation than the t r a f f i c factors district, such as size of population and area, urban form and density, and the economic l e v e l of the average resident, as manifested by car ownership, are the major explanatory v a r i a b l e s . Findings from O.D. data i n thirteen c i t i e s (Chicago, D e t r o i t , Washington, P i t t s b u r g , S t . Louis, Houston, Kansas C i t y , Phoenix, N a s h v i l l e , S t . Landerdale, Chattanooga, Charlotte and Reno) substantiated t h i s . 1 1 The r e s u l t also revealed that as density increases there i s an increase i n t o t a l person-trips, a decrease In person-trips i n vehicles and an increase i n t r a n s i t t r i p s . Compactness of an urban area could therefore be construed as a means of minimizing urban travel. In conclusion, t r i p generation i s dependent on population, land use and socio-economic f a c t o r s . However,, i t i s emphasized that the r e l a t i o n s h i p s developed from grouped data are sensitive to the size of the zones and the degree of i n t e r n a l homogeneity achieved i n drawing t h e i r boundaries. Undoubtedly t h i s i s an important factor i n explaining why different variables become s i g n i f i c a n t at various l e v e l of zonal aggregation. 2.3 I > • . ' . Approaches to T r i p Generation Analysis Throughout the h i s t o r y of transportation planning, various techniques, each with Increasing s o p h i s t i c a t i o n , have been employed to 27 quantify and analyze t r a v e l patterns of urban dwellers. A l l the techniques developed are e s s e n t i a l l y based on the assumption that people are p r e d i c t a b l e , . i . e . there i s a l o g i c a l and orderly pattern such that mathematical formulae can be developed to express t r a v e l behaviour. Another important concept inherent in these procedures i s that t r a v e l occurs only as the consequence of persons being unable to f u l f i l l a l l desires at a common l o c a t i o n , I . e . when a l l functions cannot be incorporated into a single l o c a t i o n . Different functions, 12 or land uses, which are s p a t i a l l y separated i n i t i a t e person t r i p s . A t h i r d assumption i s that the r e l a t i o n s h i p between t r i p s and land use and socio-economic variables i s stable over time. Below i s a b r i e f resume of the techniques used i n t h i s f i e l d : 1) Growth Factor Method It was much used p r i o r to 1950 to obtain an estimate of the future t r i p generation of a zone. The present number of t r i p s i s m u l t i p l i e d byVa growth f a c t o r , representing the product of the r a t i o s between the future and the present population, car 13 density and car u t i l i z a t i o n . In essence i t i s an extra- polation technique. 2) Land Area T r i p Rate Analysis Since e a r l y 1950, a n a l y t i c a l techniques have been used i n an attempt to quantify urban t r i p volumes i n terms of the land uses associated with t r i p ends. E x i s t i n g land uses are categorized by type of a c t i v i t y , location and i n t e n s i t y of i use, such as r e s i d e n t i a l , manufacturing, commercial, 28 t r a n s p o r t a t i o n , p u b l i c b u i l d i n g and p u b l i c open space, e t c . T r i p r a t e s are c a l c u l a t e d r e l a t i n g observed number of t r i p s per acre of land to the land use c a t e g o r i e s . land use c l a s s i f i c a t i o n s are. used as an'end'or variable. In other words, classifactory Another set of generation figures may be obtained by r e l a t i n g the number of t r i p s to the f l o o r a r e a . Many European studies have estimated future t r i p generation from . the number of residents and employees i n the zones. This may be regarded as a s p e c i a l case of the land use method w i t h only two land use categories being considered, namely r e s i d e n t i a l and employment a c t i v i t i e s . P r o j e c t i o n s f o r the future are obtained by applying these t r i p generation r a t e s per u n i t of area i n a given p e r i o d of time to the future land use p a t t e r n . 1 ^ 15 3) Cross C l a s s i f i c a t i o n A n a l y s i s Much of the e a r l y work on t h i s was undertaken by the Puget Sound Regional Transportation Study. or d i s t r i b u t i o n - f r e e technique. I t i s a non-parametric Essentially, ' n ' number of v a r i a b l e s are s t r a t i f i e d i n t o two or more appropriate c r e a t i n g an ' n ' dimensional m a t r i x . groups, Observations on the dependent v a r i a b l e are then a l l o c a t e d to the various c e l l s of the m a t r i x , based on the values of the s e v e r a l independent v a r i a b l e s , and then averaged to obtain the t r i p rate per d w e l l i n g u n i t w i t h c e r t a i n socio-economic characteristics. The f o l l o w i n g table i s produced by t h i s technique. TABLE I TRIPS PER DWELLING UNIT CROSS-CLASSIFIED WITH HOUSEHOLD SIZE AND AUTO OWNERSHIP Average T o t a l Person Trips Per d . u . No. of Persons Per d . u . No. of autos owned per d . u . Weighted 3 & Over 0 Average 1 1.03 2.68 4.37 2 1.52 5.13 7.04 2.00 4.38 3 3.08 7.16 9.26 10.47 7.46 4 3.16 7.98 11.56 12.75 9.10 5 3.46 8.54 12.36 17.73 10.16 6 - 7 7.11 \9.82 12.62 16.77 11.00 8 & Over 7.00 9.66 17.29 22.00 12.24 1.60 6.62 10.53 13.68 6.58 Weighted Average Source: 1.72 — 1962 O.D. data by the Madison Area Transportation Study, Madison, Wisconsin. / 1 30 S i m i l a r l y , t r i p s per d w e l l i n g u n i t may be cross-classified w i t h other v a r i a b l e s considered by the analyst t o be p o s s i b l e i n d i c a t o r s of t r a v e l demand, e . g . family income, stage i n family l i f e - c y c l e , e t c . Once the important i n d i c a t o r s of household t r a v e l are i s o l a t e d , forecasts of d w e l l i n g u n i t s by car ownership and f a m i l y s i z e c h a r a c t e r i s t i c s are a p p l i e d t o the base year t r i p rate matrix above. A straightforward approach would involve estimating percentages of the t o t a l future number of d w e l l i n g u n i t s , by zone, that are to f a l l i n t o each c e l l i n the m a t r i x . expected T o t a l t r i p production for a zone would then be determined by applying the appropriate t r i p r a t e to the number of d w e l l i n g u n i t s and summing i n d i v i d u a l t r i p estimates. Por example, i f a zone i s expected to contain 500 d w e l l i n g u n i t s i n the design year w i t h 50$ having a family of three and owning one automobile, t h e i r share of the t o t a l person t r i p production estimate would be: 7.16 t r i p s / d . u . X 250 d . u . = 1,790 t r i p s The remaining 50$ of households would be apportioned among the appropriate t r i p r a t e c e l l s of the matrix i n a s i m i l a r way. The grand t o t a l would give the design year estimate of zonal t r i p productions. The same technique could be used f o r t r i p production estimates by purpose.. This method has the advantage of being able t o detect curvilinear relationship. Since i t need not assume n o r m a l i t y 31 in the data, nominal and o r d i n a l data can be handled as w e l l . The approach i s somewhat tedious and more d e t a i l e d than the r e l i a b i l i t y of the data or the s t a t i s t i c a l v a l i d i t y of the r e l a t i o n s h i p would warrant. Moreover, the f i n e r the s t r a t i f i c a t i o n , the larger the sample r e q u i r e d . Further, there i s no simple way of measuring the amount of v a r i a t i o n in the dependent variable explained by the independent variable under consideration. 4) Multiple Regression Analysis Multiple regression i s by far the most popular technique currently employed i n t r i p generation a n a l y s i s . With the a i d of computers, the development of t r i p generation models becomes a r e l a t i v e l y fast 'pre-packaged' process. By t h i s process future t r i p generation i s determined from a regression equation using such explanatory variables as car density, distance from C . B . D . , r e s i d e n t i a l density, income, e t c . variables, With a proper combination of i t i s often possible to develop from the survey data an expression for t r i p generation which i s significantly, in a s t a t i s t i c a l correlated sense, with the observed 16 number of t r i p s . In most applications of regression analysis, the assumptions of l i n e a r i t y , normality and homogeneity of variance of a given set of data are accepted without statistical 17 verification. The different procedures used to develop an estimating equation are enumerated below: 32 a) E a r l i e r model-builders attempted to search for independent variables that were i n d i v i d u a l l y correlated with the dependent v a r i a b l e . Multiple regression equations were then established consisting of various combinations and permeations' of these v a r i a b l e s . Those f i n a l l y selected were more often than not those having the highest correlation coefficient. Another method i s the manual "tear-down."'method where a l l variables and combinations of variables are included i n i t i a l l y and then eliminated by inspection of t h e i r simple c o r r e l a t i o n coefficients. One of the variables i n a highly correlated pair is eliminated and the regression calculations are then repeated; the F-ratios with and without the eliminated variable are compared as a check on the variable's significance. b) The former methods are now replaced by step regression. Two types of step regression programs are a v a i l a b l e . first The i s the ' b u i l d - u p ' method, i . e . a battery of independent v a r i a b l e s , whether basic, complex or trans- generated, are fed into the computer which selects the variable best correlated with.the c r i t e r i o n , one at a time, and adds i t to the equation, with the object of obtaining the highest R. The stepwise addition of variables / continues u n t i l the s p e c i f i e d F - r a t i o of remaining variables i s no longer s i g n i f i c a n t for i n c l u s i o n . 33 The second method which i s less widely used i s c a l l e d the 'tear-down' method, i t successively deletes variables from an equation that at f i r s t contains a l l possible variables. The s p e c i f i e d F r a t i o i s the c r i t e r i o n used at each step for dropping out a v a r i a b l e . 2.4 Some Considerations i n Using Multiple Regression Analysis Due to the fact that multiple regression analysis i s i n v a r i a b l y performed with the computer using prepacked programs, there i s an inherent danger for the analyst to become more and more dissociated from the data he i s analyzing. Consequently i t i s emphasized that the f i r s t task in analysis must be to e s t a b l i s h a t h e o r e t i c a l framework through conceptualization of the relationships to be investigated. Careful formulation of the problem and hypothesis enables the analyst to completely c o n t r o l the process, instead of leaving the job of finding r e l a t i o n s h i p s e n t i r e l y i n the hands of the computer. Identifying and defining relationshipsfbetween t r a v e l demand and the urban environment not only a s s i s t s i n the selection of independent variables consistent with the hypothesis put foward, but also helps eliminate those associated with the dependent variable simply by chance. The use of i n t e r c o r r e l a t e d independent variables should be reviewed c r i t i c a l l y p r i o r to computation. The implication of t h i s w i l l be dealt with at length i n Chapters and IV. / III 34 Another point worthy of note i s that the analyst should have an idea before-hand of the degree to which the equation produced can be expected to f i t the data. In other words, the amount of accuracy achievable by improving the multiple R and the standard error of estimate (S ) i s governed by the standard error of the mean (S- = — ) of the c r i t e r i o n . This s t a t i s t i c which indicates the sampling accuracy of the data being ' f i t t e d ' sets an upper l i m i t to which the analyst should attempt to improve the S S of the regression equation. Por y.x to be pushed to greater accuracy than S- i s spurious. Therefore, y.x when S x i s approaching or equal to the value of S-, the regression x y.x analysis can be terminated as further 'fine tuning' only r e s u l t s i n '18 false precision. When computation i s completed, the models developed should be evaluated both s t a t i s t i c a l l y examining s t a t i s t i c a l and a n a l y t i c a l l y . The former involves measures of r e l i a b i l i t y and v a l i d i t y of the equation such as coefficient of multiple c o r r e l a t i o n and determination, standard error of estimate; , standard error of the coefficients and the d i s t r i b u t i o n of r e s i d u a l s . regression Section 1.8 in Chapter I gives a d e t a i l e d non-mathematical account of the meaning of these terms. Note that a l l four should be used simultaneously as each provides a measure of different aspects of the estimating equation. The t r a d i t i o n a l over-emphasis on the coefficient of multiple c o r r e l a t i o n should be avoided. The s a t i s f a c t i o n of a l l these s t a t i s t i c a l tests does not eliminate the need to evaluate the equation for reasonableness. As mentioned before, equations are l i k e l y to be v a l i d i f formulated on a reasoned 19 hypothesis. Those e x h i b i t i n g no causal or an i l l o g i c a l r e l a t i o n s h i p should be discarded i n favour of the ones w i t h good explanatory and a n a l y t i c a l powers. The supporting arguments f o r t h i s a r e : 1) M u l t i p l e regression models are e s s e n t i a l l y p r e d i c t i v e i n function. Therefore, they must be capable of r e f l e c t i n g both r e a l world phenomena and to s p e c i f y a causal sequence among variables. The very form of the equation ( e . g . one u n i t of change i n X w i l l cause say, two u n i t s of change i n Y) d i c t a t e s c a u s a l i t y as a necessary condition f o r i t to have any v a l i d i t y . 2) Regression models assume s t a b i l i t y of r e l a t i o n s h i p over t i m e . The model i s a v a l i d f o r e c a s t i n g t o o l only i f the r e l a t i o n s h i p on which i t i s based can be shown to be s t a b l e . Whether or not these parameters can be expected to e x h i b i t secular s t a b i l i t y depends l a r g e l y on the extent to which the model includes s t r u c t u r a l r e l a t i o n s h i p s . 2 ^ Put i n another way, the r e l a t i o n s h i p s are more l i k e l y to be stable i f the v a r i a b l e s cover those basic motivating f a c t o r s of urban t r a v e l that are not l i k e l y to change w i t h time or from one c i t y to another. In the process of p r e d i c t i n g the value of the dependent v a r i a b l e i n some design year, the f o r e c a s t i n g of independent v a r i a b l e s , u s u a l l y from other sources, i s a p r e - r e q u i s i t e . The model should be evaluated i n terms of whether the v a r i a b l e s used are easy t » p r o j e c t . Those / whose future estimates are not a v a i l a b l e or cannot be forecasted should be o m i t t e d . 36 A common dilemma facing the analyst i s the question of whether the improvement to the equation by adding another variable i s enough to offset the a d d i t i o n a l effort trade-off situation. in forecasting it. This i s l a r g e l y a On the one hand, over-simplified models, though • operationally f e a s i b l e , may be t h e o r e t i c a l l y so crude that they have little validity. On the other hand, over-complicated models based on obscure variables can be equally hazardous. What strategy to adopt w i l l have to be resolved on an i n d i v i d u a l basis, although i t often advisable to choose somewhere between the two extremes. is 37 Footnotes . ^Christopher R. Fleet and Sydney R. Robertson, ' T r i p Generation i n The Transportation Planning Process", Highway Research Record, No. 240 (1968), p.12. 2 Ibid. 3 U . S. Department of Transportation/Federal Highway A d m i n i s t r a t i o n , Bureau of P u b l i c Roads: Guideline f o r T r i p Generation A n a l y s i s , (June 1967), P.4. ^Loc. c i t . , p.8. 5 S . T . V/ong, "A M u l t i v a r i a t e A n a l y s i s of Urban T r a v e l Behaviour i n Chicago", Transportation Research, Volume 3 (1969), p.36. M. A . T a y l o r , Studies of T r a v e l i n Gloucester, Northampton and Reading, Road Research Laboratory Report, LR l 4 l , ( M i n i s t r y of Transport, Great B r i t a i n , 1968), pp.153-155. K. Rask Overgaard, "Urban Transportation P l a n n i n g : T r a f f i c Quarterly ( A p r i l 1967), p.203. Traffic Estimation" Q M. A . T a y l o r , op. c i t . , pp.153-155- ^ U . S. Department of Transportation/Federal Highway A d m i n i s t r a t i o n , Bureau of P u b l i c Roads, op. c i t . , p.17. 10 W . L . Mertz, "A Study of T r a f f i c C h a r a c t e r i s t i c s i n Suburban R e s i d e n t i a l A r e a s " , P u b l i c Roads, Volume 29 (August 1957), p.210. ^''"Herbert S. Levlnson and F. Houston Wynn, "Some Aspects of Future Transportation i n Urban A r e a s " , Highway Research Board B u l l e t i n , No. 326 (1962), p.16. 12 Louis E . Keefer, P i t t s b u r g Transportation Research L e t t e r , Volumes 2-4 (May I960), pp.12-13. 13 K. Rask Overgaard, op. c i t . , p.201. / • . , 38 14 1 5 U. S. Department of Transportation/Federal Bureau of P u b l i c Roads: op. c i t . , p . l . Highway A d m i n i s t r a t i o n , I b i d , p.19. Data of Table I i s based on 1962 O.D. survey data supplied by the Madison Area Transportation Study, Madison, Wisconsin. 16 K. Rask Overgaard, op. c i t . , p.202. ^ W . R. J e f f e r i e s and E . C. C a r t e r , " S i m p l i f i e d Techniques for Developing Transportation Plans - T r i p Generation i n Small Urban Areas", Highway Research Record, No. 240 (1968), p . 7 3 . 18 Christopher R. Fleet and Sydney R. Robertson, op. c i t . , p.19. ^ M . A . T a y l o r , op. c i t . , p . 165. 20 21 J . H . Niedercorn and J . F. Ka i n , "Suburbanization of Employment and Population 1948-1975", Highway Research Record, No. 38 (1963), p.37. T . B. Deen, V/. L . Mertz and N . A. I r w i n , " A p p l i c a t i o n of a Modal S p l i t Model t o T r a v e l Time Estimates f o r the Washington A r e a " , Highway Research Record, No. 38 (1963), p.98. 39 CHAPTER I I I THEORETICAL EXPOSITION OP MULTICOLLINEARITY AS AN EXPLANATORY AND ANALYTICAL PROBLEM This chapter puts reward the idea that a regression model Should have three necessary a t t r i b u t e s . Within the framework of these a t t r i b u t e s the t h e o r e t i c a l i m p l i c a t i o n s of m u l t i c o l l i n e a r i t y to modelb u i l d i n g i s examined. Due to the mathematical nature of the e x p o s i t i o n , a b r i e f summary i s provided i n Section 3.2 f o r the non-mathematical reader. Those i n t e r e s t e d i n pursuing the s t a t i s t i c a l proofs and t h e o r i e s .can turn to Section 3.3 f o r d e t a i l s . E m p i r i c a l examples w i l l be represented i n Chapter I V . 3.1 Model A t t r i b u t e s Before proceeding onto the v e r i f i c a t i o n of the general hypothesis, i t i s necessary to discuss the meaning of a n a l y t i c a l , explanatory and p r e d i c t i v e power of the model. C h r i s t 1 enumerated s e v e r a l desirable p r o p e r t i e s which a model should possess. They a r e : relevance, s i m p l i c i t y , accuracy of c o e f f i c i e n t s , 40 t h e o r e t i c a l p l a u s i b i l i t y , explanatory a b i l i t y and forecasting ability. In fact relevance and accuracy are implied i n i t s explanatory and forecasting a b i l i t i e s . explanatory, In sum, a model should possess three a t t r i b u t e s a n a l y t i c a l and predictive powers. Explanatory a b i l i t y means that a model should be able to explain the behaviour of variables under examination. Consistency and relevancy are the main ingredients i n t h i s because they aim at eliminating redundant variables that do not contribute to the explana- tion of a given phenomenon, but are included only because of high c o r r e l a t i o n to the c r i t e r i o n . An equation i s considered b e t t e r , other 2 things being equal, the wider the range of data i t can e x p l a i n . Therefore, a model i s a better explanatory model i f i t i s able to extract the pertinent underlying dimensionalities of the available data. A n a l y t i c a l power means that a model should be able to causal r e l a t i o n s h i p s , where possible, establish and enable the t e s t i n g of hypothesis through deductive reasoning. specifi A model after a l l i s no more than a formal statement of the outcome of analysis by which theories can be conceptualized and formulated. Planning Is future-oriented, that can forecast the future. model i s a function o f : , v , ; : : therefore, the analyst wants models The predictive a b i l i t y of a regression 41 1) I t s a b i l i t y to f i t the data (as shown by R and S y.x ), 2) I t s t h e o r e t i c a l p l a u s i b i l i t y , 3) Ease i n obtaining r e l i a b l e forecast of independent variables. The p r i n c i p l e of t h e o r e t i c a l p l a u s i b i l i t y may urge the analyst to b u i l d a model based on a comprehensive theory, thus more often than not r e s u l t i n g i n a complex model. c a l l s for simple models. On the other hand, the l a s t p r i n c i p l e S i m p l i c i t y may r e f e r e i t h e r to the f u n c t i o n a l form or the number of explanatory v a r i a b l e s included i n the r e l a t i o n ship. Although s i m p l i c i t y i t s e l f i s a desirable feature, the model 3 must a l s o be a p l a u s i b l e one. Bearing i n mind that a complex model minimizes s p e c i f i c a t i o n e r r o r due to omission of v a r i a b l e s , but Increases measurement e r r o r , and that a simple model does the exact 4 opposite, an optimal combination must e x i s t . I t i s p o s s i b l e , up to a p o i n t , to gain advantage of s p e c i f i c a t i o n without i n c r e a s i n g the measurement e r r o r . substantially Therefore, i n determining the p r o j e c t i o n i m p l i c a t i o n of the model, the analyst must scrupulously examine not merely the s t a t i s t i c a l measures a p p l i e d to c a l i b r a t i o n , but a l s o the model structure i t s e l f to discover p o s s i b l e and c o n t r a d i c t i o n s . inconsistencies I t i s possible, t h a t , by reformulating a model, 2 i t s R may be lowered f o r the p e r i o d of c a l i b r a t i o n , but that t h i s , for t h e o r e t i c a l reasons, w i l l increase the confidence i n i t s p r e d i c t i v e 5 accuracy. 42 3.2 A Non-Mathematical Summary of the T h e o r e t i c a l Implications of Multicollinearity. M u l t i c o l l i n e a r i t y has the f o l l o w i n g undesirable e f f e c t s on the regression model: 1) C o l l i n e a r i t y causes d e t e r i o r a t i o n i n the least-square estimating procedure i n the regression system. When two independent v a r i a b l e s are i n t e r c o r r e l a t e d , one of them i s superfluous. (See Sections 4.3 and 4.4). Such redundancy i s c o n t r a r y . t o the p r o p e r t i e s o f consistency and relevancy i n the model . construct. 2) C o l l i n e a r i t y i s a source f o r compounding e r r o r s of the data s e t , both during the sampling and f o r e c a s t i n g p e r i o d s . Large standard e r r o r s of the beta c o e f f i c i e n t s u s u a l l y r e s u l t . Since the values of the beta c o e f f i c i e n t s become extremely unstable and h i g h l y susceptible to sampling e r r o r , c o n f l i c t i n g conc l u s i o n s regarding the behaviour of a v a r i a b l e from d i f f e r e n t samples of the same population can be drawn. to hypothesis t e s t i n g . This i s hazardous This i s demonstrated e m p i r i c a l l y i n Section 4.3. 2 3) C o l l i n e a r i t y tends t o make the value of R and R very u n r e l i a b l e and indeterminate. Hence, a degree of f i t obtained under t h i s • • / • ' 43 c o n d i t i o n o f t e n amounts t o f a l s e p r e c i s i o n and s e l f - d e l u s i o n . (See 3.3 Section 4.4) Problem o f M u l t i c o l l i n e a r i t y - A S t a t i s t i c a l Using the t h r e e a t t r i b u t e s as c r i t e r i a Exposition f o r a s s e s s i n g the u t i l i t y o f models, a d i s c u s s i o n on t h e t h e o r e t i c a l i m p l i c a t i o n s u n d e r l y i n g t h e problem o f m u l t i c o l l i n e a r i t y i s i n o r d e r h e r e . The f o l l o w i n g w i l l show how and why m u l t i c o l l i n e a r i t y v i o l a t e s the p r o p e r t i e s o f a good model construct. 1) Fitting A Line Instead In the case o f simple o f A Plane r e g r e s s i o n , s t a t i s t i c a l f i t t i n g o f data p o i n t s amounts t o drawing a l e a s t - s q u a r e scatter. line through the F o r m u l t i p l e r e g r e s s i o n models o f 'n' v a r i a b l e s , t h i s amounts t o f i t t i n g a n - d i m e n s i o n a l r e g r e s s i o n s u r f a c e t o a l l the data p o i n t s . Geometrically, i n the t h r e e v a r i a b l e Y = f (X.^X^) and when X^ and X^ a r e not c o r r e l a t e d , p o i n t s w i l l be w i d e l y s c a t t e r e d on the X^Xg p l a n e . equiprobability ellipse w i l l t h e n become a c i r c l e . the-data 99$ A The r e s u l t a n t r e g r e s s i o n s u r f a c e w i l l be a t h r e e d i m e n s i o n a l through the s c a t t e r . (See F i g u r e 2 ) . case, However, when X plane and X 1 are c o r r e l a t e d , then the r e g r e s s i o n p l a n e becomes an ellipse, f l a t t e n e d i n one o f i t s dimensions.; When X^ (See F i g u r e 3). 2 46 and X^ are near p e r f e c t l y correlated, i . e . a l i n e a r function exists between them. Y = a + Therefore, b ^ + b2x2 Xj_= c + dX 2 The geometric interpretation of least square f i t t i n g i n t h i s case i s i n t e r e s t i n g and r e v e a l i n g . It means that the scatter of points i n the X^X plane must l i e e x c l u s i v e l y on the straight l i n e X-j_ = c + dXg; the.Y value then gives r i s e merely to a v e r t i c a l scatter of points ( i . e . i n the Y d i r e c t i o n ) above and below a single straight line i n a three dimensional space. An attempt to f i t an equation to the data involves i n s e r t i n g a plane i n a three dimensional scatter of p o i n t s , but i n t h i s case, the scatter i s r e a l l y only two dimensional, for the complete lack of scatter in the X-jXg plane means that a l l the sample points l i e i n . a plane p a r a l l e l to the Y-axis and which contains the l i n e X^ = c + dX^. The regression plane then becomes a l i n e . (See Figure 4) Frisch termed t h i s phenomenon as " p - f o l d f l a t t e n e d " regression when such c l u s t e r i n g occurs i n a n-dimensional regression (n ) 2). In the case quoted above, n = 3, p =.1. regression i s "one-fold f l a t t e n e d " . surface Therefore, the It i s not a true three variable multiple regression problem but should be a two variable or simple regression problem. The variable X^ has nothing to 48 do i n the l i n e a r regression system. From the standpoint of l i n e a r regression i t i s a superfluous v a r i a b l e drawn i n t o observation and the whole system of regression coefficients can i n fact be considered a r t i f i c i a l . The s i g n i f i c a n c e of t h i s l i e s i n that i n c l u s i o n of c o l l i n e a r sets i s contrary t o the properties of consistency and relevancy that are desirable i n the model c o n s t r u c t . This i s obvious as the r e s u l t i n g model Includes redundant v a r i a b l e s that do not. ' e x p l a i n ' v a r i a t i o n i n the c r i t e r i o n . Moreover, t h e i r presence may preclude the i n c l u s i o n of relevant v a r i a b l e s that have been overlooked. 2) Indeterminacy of Beta C o e f f i c i e n t s U s u a l l y Accompanied by a Large Standard E r r o r When the c o r r e l a t i o n between the independent v a r i a b l e s i s h i g h , • the sampling e r r o r of the p a r t i a l slopes and p a r t i a l c o r r e l a t i o n s w i l l be quite l a r g e . As a r e s u l t there w i l l be a number of d i f f e r e n t combinations of regression c o e f f i c i e n t s , and hence p a r t i a l c o r r e l a t i o n s , which gives almost e q u a l l y good f i t s t o the e m p i r i c a l d a t a . The f o l l o w i n g example w i l l serve t o 7 i n d i c a t e the problem i n v o l v e d . Let Y = a + b_X + b^Z + e • 1 2 1 49 Suppose X and Z are p e r f e c t l y r e l a t e d according to the equation X = c + dZ P u t t i n g i n numerical values f o r the c o e f f i c i e n t s , = 5, (2) X 3 (1) - (3) = for a = 6, 3, c = 1 and d = 2, we have Y = 6 + 5X X = 1 + 2Z 3X = 3 + 6z Y = 3 + 8x + 3Z + e (1) . . (2) (3) - 3Z + (1') e x Eqn. ( 1 ' ) i s therefore mathematically the same as eqn. (l). But there are obviously an i n f i n i t e l y large number of such equations that gives e q u a l l y good f i t t i n g to the data. There- fore there i s no way of determining the c o e f f i c i e n t s u n i q u e l y . However, i f an e r r o r term e^ were t o be added t o eqn. (2), a unique s o l u t i o n w i t h l e a s t squares can then be obtained as eqn. (4) now i s mathematically d i s t i n c t from eqn. Y = 6 + 5X + 3 Z + X = 1 + 2Z + (2') X 3 1 3X = 3 + 6Z + 3e (3') Y = 3 + 8X - 3Z (1) - (l): i * e (1) e . . 2 + e x (3') - 3e . . . . . (4) But such a s o l u t i o n w i l l not render the regression model desirable p r o p e r t i e s . (2 1 ) This i s i n t u i t i v e l y obvious as the 50 entire estimate of parameters hinges on the error term e^ which means that with s l i g h t modifications of the magnitude that could e a s i l y be due to sampling or measurement error, one might obtain estimates which d i f f e r considerably from the 8 original set. This may lead to erroneous conclusions about the hypotheses to be t e s t e d . Examination of the formula for standard ( error coefficients shows that the higher the correlation between independent v a r i a b l e s , the of beta the greater the standard error of coefficient. 1 - R 12.23 2 1.234 ,m 2.34 where m = number of variables N = number of R i n the equation of determination:;. In the three variable case: 12.23 ) (N - m) ,m observations = multiple c o e f f i c i e n t 1.23 / m 51 When v a r i a b l e 2 and 3 are h i g h l y c o r r e l a t e d the denominator of the equation becomes very small and hence the standard e r r o r w i l l increase c o n s i d e r a b l y . When one considers the c o r r e l a t i o n matrix of v a r i a b l e s , one can i n fact think of them as variances and covariances matrix for the same set o f . v a r i a b l e s when they are standardized. The i n t e r c o r r e l a t i o n of independent v a r i a b l e s i s none other than t h e i r covariance. The covariance i n d i c a t e s the degree to which two v a r i a b l e s are l i k e l y to e r r i n the same or d i f f e r e n t d i r e c t i o n s because of sampling f l u c t u a t i o n s . I f covariance of two v a r i a b l e s i s p o s i t i v e , t h i s means that an overestimate of one w i l l lead to an overestimate of the other, and the same for underestimates. I f t h e i r covariance i s negative, the overestimate of one w i l l be accompanied by an underestimate of another, and v i c e v e r s a . I f t h e i r covariance i s near z e r o , then there i s no1 c o r r e l a t i o n between the v a r i a b l e s , the-overestimate or. underestimate of one bears no r e f l e c t i o n on the other. The importance of t h i s concept i s that i f X^ and Xg are two c o r r e l a t e d v a r i a b l e s that have been included i n the model,'the change i n X^ i n the sampling p e r i o d i s always accompanied by the change i n X^. This being so, one could never discover the c o e f f i c i e n t of e i t h e r . X^ or X^, and a l l we could t e l l is when X^ (and hence X^) changes by one u n i t , Y u s u a l l y changes, say, by 0.8 u n i t i n the same d i r e c t i o n . We could not r u l e out 52 the p o s s i b i l i t y that b^ (the b coefficient (the b coefficient for X^) i s 0 and bg for X^) i s 0.8, or that t h e i r beta are r e s p e c t i v e l y 0.8 and 0; or 0.4 and 0.4. be overcome even by taking large samples. values This problem cannot The only recourse is Q to choose the correct model to begin with.- 7 One could argue that i f the aim i s not p r i m a r i l y to estimate parameters in the regression equation, but instead to forecast the value of the dependent v a r i a b l e , then the i n a b i l i t y to determine the true separate value for beta c o e f f i c i e n t s not be problematic. The answer i s both yes and no. will One must r e a l i z e that the whole basis for prediction i s the assumption that the r e l a t i o n s h i p observed between independent and dependent variables w i l l remain constant. I f the j o i n t d i s t r i b u t i o n of the independent variables between themselves and also with the dependent variable stays the same in the forecasting p e r i o d , then there i s no disadvantage in m u l t i c o l l i n e a r i t y . This i s , however, subject to the following q u a l i f i c a t i o n s ; 1) That the standard error of the beta coefficient be great. will This means that there i s less f a i t h in the prediction. 2) That the high c o r r e l a t i o n may y i e l d a higher multiple R than warranted. This w i l l render the estimate unrealistic. I f the sampling r e l a t i o n s h i p of the independent variables with the c r i t e r i o n i s much altered during the forecasting period, 53 v a r i a t i o n of one necessitates the v a r i a t i o n of the o t h e r . Hence the p r e d i c t e d value w i l l have a greater margin of e r r o r . However, i f the independent v a r i a b l e s are not s i g n i f i c a n t l y c o r r e l a t e d , the change i n the r e l a t i o n s h i p of one w i t h the c r i t e r i o n need not a f f e c t the o t h e r s . In t h i s way, the margin of e r r o r i s minimized and a more r e l i a b l e p r e d i c t i o n can be obtained. 3) E f f e c t on M u l t i p l e C o e f f i c i e n t of Determination There are two c o n f l i c t i n g e f f e c t s that m u l t i c o l l i n e a r i t y has 2 on the m u l t i p l e R , namely: 2 a) M u l t i p l e R increases as the s i z e of i n t e r c o r r e l a t i o n of independent v a r i a b l e s decreases. 2 b) M u l t i p l e R increases when the i n t e r c o r r e l a t i o n of independent v a r i a b l e s i s h i g h . This can be better understood by looking at the formula f o r m u l t i p l e c o e f f i c i e n t of determination i n . a three v a r i a b l e case. H\. 2 3 4 + -?3 - .. o r R „2 1.23 = r 2 12 •+• , . . . (5) " ' 23 1 r 0 l 3 two independent v a r i a b l e s i s z e r o . , i f the i n t e r c o r r e l a t i o n of . . . . (6) If the c o r r e l a t i o n r of eqn. (5) i s is because o f i t s approaches numerator z e r o , which has a tendency t o make R ^ 23 On the o t h e r hand, t h e r e large, z e r o , the t h i r d term i n the is role a distinct larger. gain i n having r ^ In the d e n o m i n a t o r . 1.0, the denominator approaches z e r o . If very r Even though 2 the numerator may become s m a l l , under t h e s e c o n d i t i o n s R can 2 be q u i t e large. A large R is thus o b t a i n e d by h a v i n g r' e i t h e r v e r y s m a l l or v e r y l a r g e . This is because i t s role 2 the numerator o n l y d e c r e a s e s R i n a l i n e a r manner, but 2 r o l e i n the denominator i n c r e a s e s R In o r d e r t o v i s u a l i z e the e f f e c t its 10 exponentially. of i n t e r c o r r e l a t i o n of the 2 predictors following on m u l t i p l e R , F i g u r e 5 i s p l o t t e d based on the table. in 55 TABLE I I EXAMPLES OFfMULTIPLE CORRELATIONS IN A THREE VARIABLE PROBLEM WHEN. INTERCORRELATIONS VARY Example r r i2 r R2 13 23 1.23 R 1.23 1 0.4 0.4 0.0 0.3200 0.57 2 0.4 0.4 0.4 0.2286 0.48 3 0.4 0.4 0.9 0.1684 0.41 4 0.4 v, 0.2 0.0 0.2000 0.45 5 0.4 0.2 0.4 0.1619 0.40 6 0.4 0.2 0.9 0.2947 0.54 7 • . ' 0.4 0.0 0.0 0.1600 0.40 8 0.4 0.0 0.4 0.1905 .0.44 9 0.4 0.0 0.9 0.8421 0.92 Source: . J . R. G u i l f o r d , "Fundamental S t a t i s t i c s i n Psychology and Education, (New York: "McGraw H i l l I n c . , 1965), p.404.. 56 FIG. 5 GRAPH SHOWING MULTIPLE R AS A FUNCTION OF r 1.23 R • 23 1.23 l.Or / / 0.8 / / / / / 0.6h / / (3) 0.4 0.2 0.2 (1) 0.6 0.4 as a function of r when r 0.8 &r 1.0 23 k are the same (0.4). / 2 (2) R as a function of r when J- • £J 2 ^ 1 23 a S a fr"10"*'*011 °* the dependent variable, r (0.4) and r (0.2) are unequal. 1 <£ ..•13 one variable i s not correlated with r = 0.4, r = 0.0 12... 13.. w n e n 57 From the graph i t . can be seen that 1) When r 12 and r 13 are the same, increase i n r • decrease multiple R 2) When r 23 . and r „ are unequal, increase i n r 12 13 will will 23 2 decrease multiple R , up to a point, but once r 1.23 23 2 i s above 0.6, then multiple R ^ ^3 increases s t e a d i l y . 3) When r and r 12 are very unequal, e s p e c i a l l y i f one 13 of them has no c o r r e l a t i o n with the c r i t e r i o n , then increase i n r R up to 0.24 w i l l decrease multiple 2 9 1.23 , once above that, R 1.23 increases very r a p i d l y towards u n i t y . Therefore, there i s a d i s t i n c t disadvantage i n having correlated independent variables because i t tends t o make the p value of R highly unreliable and indeterminate. By the same 2 token,a large R .obtained by having r ^ very small i s a more r e l i a b l e estimate, than having r very large because of the foregoing explanation. 3.4 Conclusion The preceding attempts to show that prediction i s not independent of other model a t t r i b u t e s . Explanatory/and a n a l y t i c a l a b i l i t y can only be a t t a i n e d by a c o n s i s t e n t and l o g i c a l model c o n s t r u c t . Harris un- compromisingly f a v o r s the a n a l y t i c approach o f t h e o r i z i n g i n model . construction, enlightened and Blalock asserts that t i o n i s the key by an adequate i n d u c t i v e understanding 1 1 " u n d e r s t a n d i n g " o f the phenomenon under examina- 12 to accurate p r e d i c t i o n . v Both p o s i t i o n s are w e l l founded. 59 Footnotes 1 C. F . C h r i s t , Econometric Models and Methods, (New York: John Wiley & Sons I n c . , 2 Ibid, p.5. 1966), p . 4 . ^Walter Y. Oi and Paul W. Shouldiner, Demands", "An Analysis of Urban Travel (Transportation Centre, Northwestern U n i v e r s i t y , 1962), p.73. \ i l l i a m Alonso, "Predicting Best With Imperfect Data", Journal of the American Institute of Planners, Volume 34, No. 3. (1968), pp.248-251. ^Britton H a r r i s , "Quantitative Models of Urban Development: Their Role in Metropolitan Policy-Making", Issues in Urban Economics, Edited by Harvey S. Perloff & Lowdon Wingo J r . , (Johns Hopkins Press, 1968), p.383. ^R. F r i s c h , ^Correlation and Scatter in S t a t i s t i c a l V a r i a b l e s , (University of Oslo, 1934), p.57. 7 3. M. Blalock J r . , "Correlated Inc H. M. Blalock J r . , "Correlated Independent V a r i a b l e s : The Problem of M u l t i c o l l i n e a r i t y " , S o c i a l Forces, Volume 42 (December 1963), pp.233-234. Ibid. 9 ' C. F . C h r i s t , op. c i t . , p.389. 8 ^J. P. G u i l f o r d , Fundamental S t a t i s t i c s in Psychology and Education, (New York: McGraw H i l l I n c . , 11 B r i t t o n H a r r i s , op. c i t . , 12 'H. M. Blalock H. M. Blalock J r . , I960), p.274. 1965), p.404. p.38l. Social S t a t i s t i c s , (New York: McGraw H i l l / Inc., 6o CHAPTER IV EMPIRICAL VERIFICATION OF HYPOTHESES 4.1 Summary of Empirical Findings This chapter v e r i f i e s the general hypothesis through the testing of three operational hypotheses. model with c o l l i n e a r variables By f i r s t examining a t r i p generation (Model 1), i t i s shown that: a) Within the c o l l i n e a r set, one variable i s a linear trans- formation of the other and i s redundant. b) The large standard error of the constant confirms that c o l l i n e a r i t y i s a; source of compounding e r r o r . . c) Simple and p a r t i a l c o r r e l a t i o n coefficients exhibit remarkable discrepancies i n the equation. C o n f l i c t i n g conclusions can be reached for the relationships among v a r i a b l e s . d) The high R of t h i s model implies greater accuracy than j u s t i f i e d by the input data. Secondly, the data i s subjected to factor analysis with a view to obtaining a set of orthogonal f a c t o r s . When the r e i n f o r c i n g effect of the c o l l i n e a r variables i n Model 1 i s eliminated, the R i s s i g n i f i c a n t l y lowered,as i n Model 2 which incorporates b a s i c a l l y the 61 same f a c t o r s . C o l l i n e a r i t y makes R very u n r e l i a b l e and i t loses., much of i t s value as a s t a t i s t i c a l measure of the strength of the model. I t i s a l s o noted that the orthogonal factors are more e f f i c i e n t i n detecting the t r a f f i c - l a n d use r e l a t i o n s h i p s i n the d i s t r i c t s than are the o r i g i n a l v a r i a b l e s , which are subject to the subtle influence of many i n t e r a c t i n g f a c t o r s . T h i r d l y , i t i s discovered that the omission of land use dimensions have r e s u l t e d i n large r e s i d u a l s i n Model 2 . developed. An a l t e r n a t e model i s I t incorporates land use factors that not only give the model a b e t t e r t h e o r e t i c a l c o n s t r u c t , but which are a l s o capable of producing a good f i t of data. This f i n d i n g appears t o i n d i c a t e that there i s no a r t i f i c i a l d i s t i n c t i o n between a model's a n a l y t i c a l , explanatory and p r e d i c t i v e f u n c t i o n s . A l l t h r e e , i n f a c t , must go hand-in-hand i n order to produce an o p e r a t i o n a l model. 4.2 Formulation of Three Operational Hypotheses This s e c t i o n attempts to t e s t , e m p i r i c a l l y , the General .Hypothesis: "when c o l l i n e a r i t y e x i s t s i n a regression model, explanatory and a n a l y t i c a l power are decreased, despite the. apparently good p r e d i c t i v e power shown by a high m u l t i p l e c o r r e l a t i o n c o e f f i c i e n t . " / 62 Three operational hypotheses are formulated for t h i s purpose, namely: H^: When c o l l i n e a r i t y e x i s t s , the true contribution of some independent variables may be exaggerated, obscured or suppressed. H^: When h i g h l y correlated independent variables exist in a model, 2 the multiple R i s an unreliable estimate of the true r e l a t i o n - : ship between the predictors and the c r i t e r i o n . H^: When h i g h l y correlated independent variables are included in a model, other s i g n i f i c a n t explanatory variables may be omitted due to the predominance of c o l l i n e a r s e t s . 4.3 V a l i d a t i o n of Hypothesis 1 To v e r i f y hypothesis 1, multiple regression analysis to the t r i p generation model derived by step regression Vancouver. for Greater This w i l l henceforth be referred as Model 1. t h i s computer output appear in Appendix E) i s applied (Details of The equation takes the form: T o t a l T r i p s Generated = 338.7013 - (594.0954) O.65I x (Labour Force) (0.1574) + X(Dwelling Units with Car) 2.5415 (0.2666) - 79.4897 X(Area) (26.5996) R2 = 0.9647, R = 0.9822, / S 1842 y.x .(The figures i n parentheses denote standard errors of regression coefficients.) . ; 63 The r e g r e s s i o n a n a l y s i s r e v e a l s t h a t 9 6 $ o f the v a r i a n c e i n " T o t a l T r i p s G e n e r a t e d " i s e x p l a i n e d by "Labour F o r c e " , U n i t s w i t h C a r " and "Area". The i n t e r c o r r e l a t i o n between F o r c e " and " D w e l l i n g U n i t s w i t h C a r " i s 0 . 9 7 5 significant "Dwelling "Labour , but t h e r e i s no c o r r e l a t i o n between Area and t h e o t h e r two independent variables. In view o f the presence o f a c o l l i n e a r s e t i n t h e model, a simple r e g r e s s i o n i s performed which i n d i c a t e s t h a t t h e second variable i s merely a l i n e a r transformation o f the f i r s t , Labour Force = 1 . 6 5 7 1 r = 0.975' The explanation i s obvious as f o l l o w s , (D.U.W.C.) - 6 6 8 . 6 0 7 4 as both v a r i a b l e s d e s c r i b e aggregated c h a r a c t e r i s t i c s o f t h e household, and both a r e s t a b l e p r o p o r t i o n s o f the s i z e o f z o n a l i n h a b i t a n t s . This a n t i c i p a t e s the r e s u l t o f the f a c t o r a n a l y s i s . A point.worthy o f the c o n s t a n t o f note i s t h a t l a r g e s t a n d a r d e r r o r (338.7013). The c o n f i d e n c e i n t e r v a l f o r the value of t h e i n t e r c e p t a t 9 5 $ l e v e l i s 3 3 8 . 7 0 1 3 ± 664.24. v a l u e o f t h e constant (594.095^0 Therefore, the can be anywhere between - 3 2 5 . 5 3 8 7 and 1,002.9413. 2 The e r r o r i s c o n s i d e r e d u n u s u a l l y l a r g e f o r an e q u a t i o n w i t h a R of 0 . 9 6 4 7 . in r e s u l t s In f a c t i t would be o f i n t e r e s t t o compare t h e d i f f e r e n c e i f t h e r e g r e s s i o n l i n e i s f o r c e d through z e r o , i . e . w i t h the c o n s t a n t eliminated. / 64 Another p o i n t o f i n t e r e s t i s t h a t t h e s t a n d a r d e r r o r o f e s t i m a t e ' T o t a l T r i p s G e n e r a t e d " i n t h i s model i s 1842, for than t h e s t a n d a r d e r r o r o f t h e mean o f the sample. w h i c h i s even l e s s (The sample s i z e . c i s 32, and t h e s t a n d a r d d e v i a t i o n i s 9314. T h e r e f o r e , S- = x 9314 */32 = 1892). As p r e v i o u s l y p o i n t e d o u t i n . S e c t i o n 2.4 when the e s t i m a t i n g = y / X (p.34), e q u a t i o n i s pushed t o g r e a t e r a c c u r a c y t h a n j u s t i f i e d by the input data, t h i s r e s u l t s i n f a l s e p r e c i s i o n . Hence 2 the v e r y h i g h R o b t a i n e d can be c o n s i d e r e d s p u r i o u s because t h e degree o f " f i t " i s c l o s e r t o t h e t o l e r a n c e with the input data. l i m i t s t h a n those associated A l t h o u g h the d i f f e r e n c e ' i n magnitude i s not b i g , i t demonstrates t h a t t h i s phenomenon can o c c u r b y t h e i n c l u s i o n o f • collinear variables. A comparison o f t h e s i m p l e and p a r t i c a l c o r r e l a t i o n c o e f f i c i e n t s o f the dependent and independent v a r i a b l e s i s r e v e a l i n g . (See T a b l e I I I ) TABLE I I I ' SIMPLE AND PARTIAL CORRELATIONS.OF MODEL 1. Simple C o r r e l a t i o n Variables Labour Force r r 0.9209. 0.848-1 Partial Correlation 2 v. . r -O.6158 0.3792 Partial r changes sign* 0.8744 0.7645 Partial r lower* -0.4917 0.2419 Partial r higher* 2 • DUWC 0.9700 . Area •0.0242 0.9409' .0.00059 - Remarks * T h e i r d i f f e r e n c e s a r e s i g n i f i c a n t a t 0.01 level. 65 The simple correlation coefficients represent the effects of one independent variable on the dependent v a r i a b l e , with the effects of other variables allowed to vary at the same time. Partial c o r r e l a t i o n coefficients represent the effects of one independent variable on the dependent v a r i a b l e , holding constant the effects of other v a r i a b l e s . The conclusion to be drawn from the simple correlation coefficients are: 1) "Labour Force" i s very highly correlated with "Total Trips Generated" in a p o s i t i v e d i r e c t i o n . It explains about 85$ of the l a t t e r ' s variance. 2) "Dwelling unit with Car" i s also very highly correlated with Trips Generated" in a p o s i t i v e d i r e c t i o n . l a t t e r * s variance. "Total It explains 9k% of the (Column 3 of Table i l l ) 3) "Area" i s s l i g h t l y correlated with "Total T r i p s Generated" i n a positive direction. It explains p r a c t i c a l l y nothing of the l a t t e r ' s variance, 'i However, when the influence of other variables i s being p a r t i a l l e d out, the conclusion to be drawn from the p a r t i a l c o r r e l a t i o n c o e f f i c i e n t s are: 1) "Labour Force" is moderately correlated with ' T o t a l Trips Generated" in a negative d i r e c t i o n . In other words, the larger the Labour Force, the fewer the t r i p s generated. The explanation seems to l i e in the fact that i n t h i s set of data, negatively correlated w i t h : per Dwelling U n i t " , "Time to C . B . D . " , "Labour Force" i s "Cars per Dwelling U n i t " , "Population "Percentage of Dwelling Units with C a r " , "Area" and "Income per Dwelling U n i t " . 66 This suggests that d i s t r i c t s w i t h a large labour force tend to have lower car ownership, lower income per d w e l l i n g u n i t , and fewer persons per household. These d i s t r i c t s are a l s o close to the C.B.D. and have small areas. These c h a r a c t e r i s t i c s point to d i s t r i c t s w i t h higher d e n s i t y d w e l l i n g s , lower socio-economic s t a t u s , s i n g l e person households and areas of mixed land u s e s . Area of mixed land uses and high d e n s i t y u s u a l l y generates fewer v e h i c l e t r i p s because of a v a i l a b i l i t y of employment, shopping ,1 and entertainment f a c i l i t i e s nearby. Thus, the negative, r e l a t i o n s h i p between ' T o t a l T r i p s Generated" and "Labour F o r c e " appears p l a u s i b l e . In a d d i t i o n , "Labour Force" r e a l l y does not e x p l a i n very much of the variance of the c r i t e r i o n - o n l y 37.92$ as opposed to the somewhat exaggerated estimate shown by the simple c o r r e l a t i o n c o e f f i c i e n t . The reason f o r the discrepancy between the simple and the p a r t i a l c o r r e l a t i o n c o e f f i c i e n t s is that the strong p o s i t i v e r e l a t i o n s h i p between "Labour F o r c e " and "Dwelling Unit w i t h C a r s " , which i s p o s i t i v e l y c o r r e l a t e d w i t h "Total T r i p s Generated", obscures the true negative r e l a t i o n s h i p cited. Hence h i g h l y misleading conclusions can be drawn by examining simple c o r r e l a t i o n c o e f f i c i e n t s alone i n a model with colllnear sets. 2) C o r r e l a t i o n of "Dwelling U n i t s w i t h Car" and " T o t a l T r i p s ' Generated" i s lowered when the r e i n f o r c i n g e f f e c t of "Labour F o r c e " i s removed. 3) "Area" i s shown to explain a much larger portion of the variance of "Total Trips Generated" than warranted by the simple c o r r e l a - tion coefficients. This indicates that i t s true effect on the c r i t e r i o n has e a r l i e r been suppressed due to the dominance of the c o l l i n e a r s e t . At f i r s t glance the outcome appears un- reasonable because the absolute area of a d i s t r i c t has no bearing on the number of t r i p s generated. a r b i t r a r y lines on the map. Boundaries are but However, close examination reveals that the t r a f f i c d i s t r i c t s are set up i n such.a way that a l l large t r a c t s happen to be r u r a l areas outside the C i t y of Vancouver. Therefore, "Area " becomes a proxy for distance from C.B.D. and to some extent represents the degree of urbanization p of the t r a c t . In the l i g h t of t h i s , the higher p a r t i a l r then appears p l a u s i b l e . Results of the multiple regression analysis indicates that Model 1 has a few. undesirable p r o p e r t i e s : 1) It does not explain the underlying r e l a t i o n s h i p s among variables in t h e i r true perspective, as evidenced by the discrepancies i n the simple and p a r t i a l c o r r e l a t i o n c o e f f i c i e n t s . t e s t i n g of hypotheses a d i f f i c u l t 2) This makes the task. The high i n t e r c o r r e l a t i o n of "Labour Force" with "Dwelling Units with Car" produces a 'one-fold f l a t t e n e d ' regression.system. of the two variables appears superfluous. , One 68 3) There i s a p o s s i b i l i t y that other important dimensions that could e x p l a i n t r i p generation have not been entered i n t o the model due to the predominance of the c o l l i n e a r s e t . This a n a l y s i s , t h e r e f o r e , supports Hypothesis 1 which states that: "When c o l l i n e a r i t y e x i s t s , the true c o n t r i b u t i o n of some independent v a r i a b l e s may be exaggerated, obscured or suppressed." Under such circumstances, i t i s d i f f i c u l t to decide from the model which are the causal f a c t o r s f o r t r i p g e n e r a t i o n . 4.4 V a l i d a t i o n of Hypothesis 2 A. Factor A n a l y s i s In view of the above f i n d i n g s , the set of data was subjected to f a c t o r a n a l y s i s by the p r i n c i p a l component method. was employed to obtain a simple s t r u c t u r e . Varimax r o t a t i o n Since t r i p generation i s a function of land use and socio-economic c h a r a c t e r i s t i c s of the p o p u l a t i o n , i t might be i n t e r e s t i n g to determine the underlying dimensions that explains i t . P r i n c i p a l component a n a l y s i s i s I d e a l for handling such a problem, i t e l i m i n a t e s a l l redundant f a c t o r s w i t h i n a set of v a r i a b l e s and produces and u n d e r l y i n g set of orthogonal 2 factors. Out of a t o t a l of twenty-nine v a r i a b l e s , f a c t o r a n a l y s i s only produces seven major f a c t o r s . The loadings of v a r i a b l e s on the f a c t o r s are c l e a r , i . e . each loads very h i g h l y on one major f a c t o r alone, w i t h no v a r i a b l e loading h a l f - a n d - h a l f on two f a c t o r s . A 69 diagrammatic representation of the composition of these seven major factors i s shown i n Figure 6 . (Detailed r e s u l t s of the factor analysis are shown i n Appendix F ) . Factor I ( S i z e ) : This factor accounts for 43$ of the variance, i s composed of variables descriptive of the size of population. expected, it As "Labour Force" and "Dwelling Units with Car" are collapsed into t h i s factor, meaning that they i n fact explain the same dimension in the data. A l l variables load p o s i t i v e l y on t h i s f a c t o r . One point of interest i s that variables for single family dwellings are picked up here whereas those for multiple family dwellings are picked up i n Factor III (Density). This shows that single family dwelling variables are good approximations of the t o t a l population since t h i s is the predominant North American way of l i f e . The same may not apply to c i t i e s where apartment l i v i n g i s more prevalent, such as i n Asiatic c i t i e s . Factor II (Employment): ^ A l l employment variables are collapsed into t h i s dimension accounting for 25$ of the variance. load p o s i t i v e l y on the factor with two exceptions: U n i t " and "Percentage of Dwelling Unit with C a r " . These variables "Cars per Dwelling The explanation l i e s i n that commercial and i n d u s t r i a l areas have lower car ownership as a r e s u l t of lack of r e s i d e n t i a l u n i t s . Fig. 6 Factor COMPOSITION OF SEVEN MAJOR FACTORS OUT OF 29 VARIABLES I Size Factor I I Employment fo 25 % 12 ? Percentage of trace: 1. total population 2. pop. single family <. labour force t o t a l 5.labour force,sing.fam. 7.dwelling u n i t , t o t a l S.sinple family d.u. 10.d.u. with car 11 .s.f'.d.u. with car 13.cars total 19. gross income 20. bus mil os 1 Note: F. IV Income Density Variance e x p l a i n e d 43 F. I l l 94.7 F. V Are a 4 fo F. VI Student 5 ,o F. VII Household Size 2.7/o of„ 18.time to 17.student 15.popul. 3.pop. multi. farm* 22.income 14.car per d.u.* C.H.D. 4 - 6 pm. per d.u. 1 6 . f d.u. with car* 6.lab. force, m.f.* pe r d.u. 23. t o t a l employment 9.multi.fan. d.u.* 24. employ, public 12.m.f.d.u. witli car* 25. emp. industry 29.population density* 26. emp. service 27-emp. entertainment 28.emp. density * denotes negative factor loadings o 71 Factor I I I (Density): This f a c t o r includes a l l v a r i a b l e s f o r multiple f a m i l y d w e l l i n g s and " P o p u l a t i o n D e n s i t y " , and a c c o u n t s f o r 12% o f the v a r i a n c e . A l l v a r i a b l e s l o a d n e g a t i v e l y on t h i s f a c t o r , this means t h a t t r a c t s w i t h p o s i t i v e l o a d i n g s on i t w i l l be low d e n s i t y a r e a s whereas those w i t h n e g a t i v e F a c t o r IV (income): l o a d i n g s w i l l have h i g h e r d e n s i t i e s . T h i s f a c t o r • i n c l u d e s o n l y one v a r i a b l e , "Income per D w e l l i n g U n i t " , and a c c o u n t s f o r 5/o o f t h e v a r i a n c e . Strangely enough "Gross Income" i s n o t p i c k e d up i n t h i s f a c t o r b u t i n F a c t o r I , showing t h a t i t i s a b e t t e r measure o f s i z e t h a n a c t u a l economic status of the t r a c t . Factor V (Area): T h i s f a c t o r c o n s i s t s o f "Time t o C.B.D." and "Area". The former v a r i a b l e l o a d s p o s i t i v e l y here because t h e l a r g e r t h e a r e a l u n i t s , t h e l o n g e r t i m e i t t a k e s t o r e a c h C.B.D. This f a c t o r a c c o u n t s f o r 4$ o f t h e v a r i a n c e . Factor VI (Student): I t i s composed o f o n l y one v a r i a b l e , "No. o f Students'' i n t h e d i s t r i c t a t 4-6 p.m. the T h i s f a c t o r a c c o u n t s f o r 3$ o f variance. F a c t o r V I I (Household S i z e ) : The o n l y v a r i a b l e t h a t l o a d s i n t h i s f a c t o r i s "Population per Dwelling U n i t " . explained by t h i s dimension. 2.7$ o f t h e v a r i a n c e i s Three graphs have been p r e p a r e d t o i l l u s t r a t e variables it i n one f a c t o r i s hoped t h a t regression in justaposition relationships analysis can be Each arrow r e p r e s e n t s a v e c t o r of v a r i a b l e s for a particular variable The c l o s e r more c o l l i n e a r the v a r i a b l e s vectors i.e. approaches t h e y are versus Factor I V . The longer the arrow, the h i g h e r the factor. the v e c t o r s sets are. If are lie i n a twoloading. close to t o one a n o t h e r , the angle the between two zero, independent and o r t h o g o n a l . v e r y l i t t l e on F a c t o r I I , l o a d i n g h i g h l y on F a c t o r I and v i c e v e r s a . much employment o p p o r t u n i t y i n r e s i d e n t i a l (Employment D e n s i t y ) f u r t h e r T h i s means t h e r e areas. substantiates this is fact Variables V a r i a b l e l 8 (Time t o C . B . D . ) both o f the two f a c t o r s loads 14 loads but p o s i t i v e l y on F a c t o r I meaning t h a t employment a r e a s have low c a r ownership but r e s i d e n t i a l high car ownership. not because i t (Cars per D w e l l i n g U n i t ) and 16 (% o f D w e l l i n g U n i t w i t h C a r ) n e g a t i v e l y on F a c t o r I I , load V a r i a b l e 28 n e g a t i v e l y on F a c t o r I but p o s i t i v e l y on F a c t o r I I . showing t h a t areas have l o a d s n e g a t i v e l y on the f u r t h e r away from C . B . D . , the lower the p o p u l a t i o n and employment o p p o r t u n i t i e s . Boston. the 9 0 ° , the c o r r e l a t i o n between them approaches Figure 7 reveals that v a r i a b l e s pattern i s so, on F a c t o r I v e r s u s and F a c t o r I I I V a r i a b l e s c o n t r i b u t i n g h e a v i l y t o one f a c t o r w i l l axis of that In d o i n g discovered. Factor II versus Factor III dimensional space. with another. not o t h e r w i s e r e v e a l e d by m u l t i p l e F i g u r e s 7, 3 and 9 show l o a d i n g s Factor I I , the b e h a v i o u r o f This t r u e f o r Vancouver but may not a p p l y f o r c i t i e s l a n d use such as 73 FIG. 7 LOCATION OF TIE FIHST & SECOND COMPONENT VECTORS FOR TI-D-J VARIABLES IN TtfO-DI MENS IONAL SPACE FACTOR II 14 16 74 Figure 8 shows the loading of variables Factor I I I . Variables loading highly negatively on Factor I I I , on Factor II versus p o s i t i v e on Factor II load meaning that i n areas of high employment opportunities, there are more multiple family dwelling u n i t s . This factor manages to pick up areas of mixed land uses. Figure 9 shows the loadings of variables Factor IV. Note that variables load negatively on Factor IV. generally have lower income. on Factor III versus loading negatively on Factor III also This implies high density areas Variable 19 (Gross Income) and 29 (Population Density) load negatively on Factor III but p o s i t i v e l y on Factor IV showing that gross income diminishes with lower density although Income per Dwelling Unit may l i k e l y be higher i n the latter areas. I f the three graphs are superimposed, there i s v i r t u a l l y no overlap in the position of component vectors i n the factor This further confirms that the resultant space. factors are d i s t i n c t and uncorrelated dimensions of the data. 3. Regression on Two Factors: Size & Area Results of the factor analysis indicates that the i n c l u s i o n of both "Labour Force" and "Dwelling Units with Car" in Model 1 i s statistically same t h i n g . and t h e o r e t i c a l l y incorrect because they explain the A new regression model (Model 2) i s formulated by regressing Factor I (Size) and Factor V (Area) on "Total T r i p s Generated". Now that these two factors are orthogonal, r e i n f o r c i n g 76 FIG. 9 LOCATION OF Till:: THIRD & FOURTH COMPONENT VECTORS FOR fill-: VARIABLES IN f.vO-DIiUiNSIOXAL SPACE FACTOR IV 25 -1.0 77 effect of c o l l i n e a r sets i s e l i m i n a t e d and the r e s u l t a n t l i k e l y t o be more r e a l i s t i c . significant difference T h i s can be seen i n the between the simple and p a r t i a l shown i n T a b l e I V b e l o w . estimate lack of correlations The new model t a k e s the form o f T o t a l T r i p s Generated Per Day = R2 = 0.9157X(Size) 0.8275 , is : - 0.0588X(Area) R = 0.9097 TABLE IV SIMPLE AND PARTIAL CORRELATIONS OF MODEL 2 Simple C o r r e l a t i o n r r 12 = 0.9078 r i2 = °' Partial Correlation r ^ - 8 2 4 1 - -0.0431 r 2 = 0.0016 13 13 r Model 1 which e s s e n t i a l l y 13.2 0.9095 r2 = 0.8272 12.3 = -0.1399 r2 = 0.0i95 13.2 has o n l y the same two dimensions Remarks No s i g n i ficant difference between the s i m p l e and partial 'r's at 0.05 level as 2 Model 2 y i e l d s a much h i g h e r R latter. The l o w e r i n g o f R i s o f 0.9647 compared w i t h 0.8275 f o r significant at 0.01 l e v e l . the However, one may a.rgue that the lowering of R i s not attributed to elimination of c o l l i n e a r i t y , but due to loss of information in the process of analysis. A table i s therefore computed to find out i f t h i s is factor true. TABLE V LOSS AND GAIN OF COKMUNALITIES IN MODEL 2 COMPARED WITH MODEL 1 Variables in Model 1 Factor I (Size) Factor V (Area) 4.Lahour Force 0.82 0.0025 10.D.U.W.C. 0.91 — 21.Area 0.015 0.866 Comrnu nalities R Loss of Information 1 - R 0.8225 0.1775 0.91 0.09 0.881 ' 0.119 Gain of Information* Other Variables 1.Pop.Total 0.93 — 2.Pop.S.F. 0.99 5.Lab.Force S . F . 0.93 0.93 0.0006 0.9906 0.9906 0.98 0.0008 0.9808 O.9808 7.Dwelling Unit Tota 1 0.82 — 0.82 0.82 8.S.F.D.U. 0.99 0.001 0.991 0.991 l l . S . F . D . U . with Car 0.99 0.0016 0.9916 0.9916 13-Cars Total 0.91 0.0002 0.9102 0.9102 iS.Time to C.B.D. 0.05 0.343 0.393 0.393 19.Gross Income 0.596 0.0006 0.5966 0.5966 20.Bus Miles 0.523 0.178 0.701 0.701 * Only those with high contributions are presented. The above table shows that in fact the loss of information in Model 2 compared with Model 1 is more than compensated by the communalities contributed by other variables to the two factors in the 2 equation. Therefore, the R of 0.8275 i s a l i b e r a l estimate of the two dimensions that are present in Model 1. Hence Hypothesis 2 which states that "When highly correlated 2 independent variables e x i s t in a model, the multiple R i s an u n r e l i a b l e estimate of the true r e l a t i o n s h i p between the predictors and the 2 c r i t e r i o n " , i s v a l i d a t e d by comparing the R of Models 1 and 2 . The 2 significance of t h i s finding i s that the high R in Model 1 i s un- r e l i a b l e and implies a degree of f i t not warranted by the data. 4.5 V a l i d a t i o n of Hypothesis 3 A. Search for Missing Factors To f i n d out i f there are any missing dimensions in Models 1 and 2, t h e i r residuals are p l o t t e d (See Figures 10 and 11). Residuals of Model 1 gives s l i g h t l y better f i t of the data than Model 2 (See Appendices E and G for r e s i d u a l values). high R in the former. This Is attributable to the Both models give poor estimate of D i s t r i c t 3 (Point Grey) and 30 (North Surrey). In a d d i t i o n , while residuals D i s t r i c t s 13 (North Vancouver City) and 29 (Newton) are r e l a t i v e l y large for Model 1, the same applies to D i s t r i c t s 2 (West End), 4 ( K i t s i l a n o , Fairview and Shaughnessy), (New Westminster) for Model 2 . 8 (South Vancouver) and l o for 8o PIG. 10 OBSERVED & CALCULATED VALUE ( STANDARDIZED A Y OF Y VARIABLES) F O R MQDETJ 1 8l PIG. 11 OBSERVED & CALCULATED VALU1J OF Y FOR MODEL 2 o A Y T2.5 (STANDARDIZED VARIABIJ;S) 8 o 3 • 82 The residuals 12). for Model 2 are again p l o t t e d on a map (See Figure Their d i s t r i b u t i o n reveals an i n t e r e s t i n g p a t t e r n . of Vancouver C i t y (West of Cambie) and V/est Vancouver are The west side consistently being over-estimated whereas the east and south portions of the metropolitan area are generally being underestimated, excepting New VJestminster. The large p o s i t i v e residuals are found at D i s t r i c t 1 ( C . B . D . ) , D i s t r i c t 2 (West End), D i s t r i c t 3 (Point Grey), D i s t r i c t 4 ( K i t s i l a n o , Fairview and Shaughnessy), and D i s t r i c t 16 (New Westminster). A possible explanation i s that these are areas of mixed land uses; the omission of Employment and Density Factors r e s u l t s in an overestimate of vehicle t r i p s generated based only on the Size and Area Factors. As previously pointed out, employment opportunities and higher density within the t r a c t s decreases the number of t r i p s generated because of the a v a i l a b i l i t y of jobs, shops and entertainment nearby. Surrey). The largest negative r e s i d u a l occurs in D i s t r i c t 30 (North One suspects that an underestimate here can be explained by the omission of the Density and Household Size Factors. First, families further away from the c i t y tend to be larger in size and hence the higher frequency in trip-making. A l s o , i n areas of lower density, more t r i p s per dwelling unit are generated because of more extensive t r a v e l requirements to s a t i s f y employment, shopping' and entertainment needs. / FIG. 12 MAP SHOWING DISTRIBUTION OF RESIDUALS FOR MODEL 2 84 Since i t i s suspected that the i n c l u s i o n o f land use f a c t o r s such as employment and d e n s i t y w i l l provide a b e t t e r understanding and estimate of t r i p generation, the scores of these two f a c t o r s have been mapped. (See Figures 13 and 14). The purpose i s t o see whether t h e i r f a c t o r score d i s t r i b u t i o n coincides w i t h areas of poor estimate. A rule of thumb i n the search f o r a d d i t i o n a l explanatory f a c t o r s I s t o look for the ones w i t h low or negative scores f o r areas w i t h large p o s i t i v e r e s i d u a l s , and the opposite f o r areas w i t h large negative r e s i d u a l s . By doing so i t was hoped that, the value,of the r e s i d u a l s of Model 2 could be minimized. 3 A f t e r d e t a i l e d examination of the f a c t o r score d i s t r i b u t i o n , the f o l l o w i n g t a b l e was a r r i v e d a t : TABLE VI A LIST OF POSSIBLE EXPLANATORY FACTORS OMITTED BY MODEL 2 Area of Poor Estimate Residuals D i s t r i c t 1. 0,6645 VII. VI -0.39595 0.13083 District 2 O.8767., . III -4.20216 District 3 1.0566'. II -0.2656 2 District.4 0.8668' III IV -2.82124 -0.41607 District 8 -O.6743.. III 1.00236 D i s t r i c t 16 0.6391'-- III D i s t r i c t 30 -0.5274'- Possible Explanatory Factors VII Factor Score -0.5523 2.97536 FIG. 13 FACTOR SCORE DISTRIBUTION FOR. FACTOR II (EMPLOYMENT) FIG. U FACTOR SCORE DISTRIBUTION FOR FACTOR 111 (DENSITY) 87 B. Development o f an A l t e r n a t e Model U s i n g the above r e s u l t s as a guide, a number of r e g r e s s i o n u s i n g both f a c t o r s and v a r i a b l e s were t r i e d . The o b j e c t was t o d e v e l o p an a l t e r n a t i v e model capable o f i n c o r p o r a t i n g c a u s a t i v e t r i p generation factors of i n a d d i t i o n t o meeting a l l the s t a t i s t i c a l measures o f predictive efficiency. using v a r i a b l e s only, were i n c a p a b l e equations I t was found t h a t w h i l e models developed by In g e n e r a l , s a t i s f i e d the s t a t i s t i c a l t e s t s , t h e y o f e x p l a i n i n g a wide range o f data and t o i n c l u d e t h e land u s e - t r a v e l r e l a t i o n s h i p s . Hence t h e y were d i s c a r d e d i n favour of models d e v e l o p e d from f a c t o r s . The f o l l o w i n g model, d e s i g n a t e d s a t i s f a c t o r y one. of population trial, Model 3, i s c o n s i d e r e d I t shows t h a t t r i p g e n e r a t i o n ^Figures y.x = 0.2489 II - - 0.2718 X ( F a c t o r (0.0462) I I I - Density) + 0.1776 x ( F a c t o r (0.0447) VT - S t u d e n t ) R = 0.973 ' ' R indus- I - Size) + 0.2675 x ( F a c t o r (0.0603) i n p a r e n t h e s e s denote t h e . s t a n d a r d coefficients. (commercial, function of density, i . e . T o t a l T r i p s Generated - 0.9904 x(Factor V (0.0478)* S i s a positive function s i z e , i n t e n s i t y o f l a n d use a c t i v i t y i n s t i t u t i o n a l and s c h o o l ) and a n e g a t i v e the most 2 Employment) = 0.946 errors o f r e g r e s s i o n 3oth the R and S y.x (See Figure 15 and Appendix H) are s i g n i f i c a n t improvements over Model 2. l o g i c a l and causative. Moreover, the r e l a t i o n s h i p expressed i s The method of using t h i s model f o r p r e d i c t i o n i s i n Appendix C . Up t o the present, many t r a n s p o r t a t i o n studies have postulated that t r i p generation i s a function of land use. Despite t h i s , so f a r few regression models have been developed t o incorporate this r e l a t i o n s h i p i n a comprehensive manner, apart from the land-area t r i p rate method which employs-land use.as "end" v a r i a b l e s . One explanation being that model-builders are content with securing a. high R using a minimum number of v a r i a b l e s i n order t o make the model o p e r a t i o n a l , so that the t h e o r e t i c a l s t r u c t u r e i s s a c r i f i c e d . In using f a c t o r a n a l y s i s t o e x t r a c t the "hidden dimensions" of the data, not only can a more i n t e l l i g e n t s e l e c t i o n of f a c t o r s be made, but the danger of i n c l u d i n g c o l l i n e a r v a r i a b l e s i s a l s o eliminated.^ Therefore, Model 3 can be considered as a step towards i n j e c t i n g a stronger s t r u c t u r a l r e l a t i o n s h i p i n t o the equation rather than being s a t i s f i e d merely with i t s a b i l i t y t o f i t the data. In a d d i t i o n , the land use f a c t o r s are not used as 'end' or exogenous v a r i a b l e s but has become endogenous. The model i s , therefore, more dynamic and responsive t o t r a n s p o r t a t i o n - l a n d use p o l i c y i m p l i c a t i o n s . FIGURE 1 5 OBSERVED & CALCULATED VALUE OF Y FOR MODEL 3 90 In order to p r e d i c t the future t r i p generation, forecasting independent v a r i a b l e s such as car ownership for t r a d i t i o n a l regression models has to be derived i n two steps. F i r s t l y land use activities have to be projected to the design year based upon which population estimates and hence the number of cars i n a zone can be estimated. 3y making land use f a c t o r s endogenous i n Model 3, one step i n the f o r e c a s t i n g process i s eliminated as land use p r o j e c t i o n s becomes the d i r e c t input, thus minimizing some measurement e r r o r s . Despite the f a c t that Model 3 has more v a r i a b l e s , i t .is believed that i t has gained s u f f i c i e n t advantage In s p e c i f i c a t i o n without introducing s i g n i f i c a n t a d d i t i o n a l measurement e r r o r s to the model. In a d d i t i o n , i t s b e t t e r t h e o r e t i c a l base coupled with s t a t i s t i c a l e f f i c i e n c y enables us to have more confidence i n i t s p r e d i c t i v e power. However, at t h i s stage i t i s unable to demonstrate q u a n t i t a t i v e l y the r e l a t i v e reduction or gain i n the two types of e r r o r s introduced by the added complexity of f a c t o r a n a l y s i s . . I t i s f e l t that f u r t h e r research t h i s issue may into be of i n t e r e s t . A comparison of Mode Is 1 and 3 demonstrates that the former has indeed omitted some s i g n i f i c a n t land use explanatory v a r i a b l e s (such as employment and density) i n the estimating equation. Consequently ' Hypothesis 3 which states that 'When h i g h l y c o r r e l a t e d v a r i a b l e s are included i n a model, o t h e r . s i g n i f i c a n t explanatory v a r i a b l e s may omitted due to the predominance of c o l l i n e a r s e t ' i s v a l i d a t e d . be 91 4,6 Conclusion This chapter substantiates the theme of the preceding chapter, that m u l t i c o l l i n e a r i t y i s i n fact an explanatory and a n a l y t i c a l problem in model^construction. By rigorous s t a t i s t i c a l analysis of the empirical data c o l l e c t e d in Vancouver, It is shown that the presence of c o l l i n e a r set of variables has a number of undesirable effects on the performance of the model, such as exaggerating, obscuring and suppressing plausible relationships which make the testing of hypotheses difficult. In a d d i t i o n , one has less f a i t h in such a model as a predictive t o o l because of the lack of l o g i c a l theory i n i t s Through the v a l i d a t i o n of the three operational hypotheses, general hypothesis: construct. the "When c o l l i n e a r i t y e x i s t s in a regression model, explanatory and a n a l y t i c a l powers are decreased, despite the apparently good predictive powers shown by a high multiple c o r r e l a t i o n coefficient", can be accepted as generally a p p l i c a b l e . Also, this "finding bears, truth on the philosophy that even for models that are b u i l t for p r e d i c t i o n , they must also be concerned with explanation. The popularly-held view of the dichotomy between predictive and explanatory models appears fallacious. Another outcome of the data analysis i s the development of an ' alternate model as a step towards giving t r i p generation models a more s o l i d t h e o r e t i c a l framework. In t h i s process, i t i s found that the t o t a l t r i p s generated per areal unit is a d i r e c t function of measures of population s i z e , i n t e n s i t y and c h a r a c t e r i s t i c s of land use in the t r a c t , whereas socio-economic c h a r a c t e r i s t i c s do not come Into play at the zonal l e v e l . The low c o r r e l a t i o n between "Total T r i p s 92 Generated" and v a r i a b l e s such as "Car Ownership per Dwelling U n i t " (-0.0382), "Persons per Dwelling U n i t " (-0.1457), "% Dwelling Unit with C a r " (-0.0393) and "Income per H o u s e h o l d " (0.0697) supply ample proof of t h i s statement. Hopefully t h i s suggests Where research efforts should be d i r e c t e d i n b u i l d i n g t r i p generation models at the zonal level. / 93 Footnotes 1 A. M. Voorhees, Transportation Planning and Urban Plan: Canada, Volume 4, No. 3. (1965), p.101. Development", p Shue Tuck Wong, "A M u l t i v a r i a t e S t a t i s t i c a l Model f o r P r e d i c t i n g Mean Annual Flood i n New England", Annals, A s s o c i a t i o n of American Geographers, Volume 53 (1963), pp.298-311. • 3 Factor scores f o r seven f a c t o r s are i n Appendix F. 1 9h CHAPTER V TRANSPORTATION MODELS - A PERSPECTIVE VIEW 5.1 Summary of Research Findings The. foregoing i n v e s t i g a t i o n , -based both on s t a t i s t i c a l and empirical r e s u l t s , theory has shown that i n formulating multiple regression models for transportation planning purposes, the use of i n t e r c o r r e l a t e d predictors not only gives r i s e to s p e c i f i c a t i o n e r r o r , but also to spurious inferences and t o spurious p r e d i c t i o n s . This renders the model less e f f e c t i v e as a p r e d i c t i v e and analytic t o o l . One of the possible ways to overcome t h i s problem i s by subjecting a l l input variables to a factor analysis to determine the underlying dimensions of the data set as w e l l as to eliminate redundant or • confounding v a r i a b l e s . By experimenting with t r i p generation data for Vancouver, the r e s u l t i s s u f f i c i e n t l y promising to warrant wider use in the transportation,'planning process. of t h i s approach can be summarized as: 1) M u l t i c o l l i n e a r i t y i s . eliminated.) 2) The sharp reduction of variables factors The more s a l i e n t contributions into smaller number of a s s i s t i n organizing huge masses of data, into .manage- able size for further analysis. 95 3) The factors themselves form meaningful constructs that give further insight into the trip-generation-land use r e l a t i o n s h i p . 4) The factor scores have more desirable s t a t i s t i c a l properties (e.g. greater r e l i a b i l i t y ) than those of single variables i n isolation. As the factors combine information from several v a r i a b l e s , predictive accuracy tend to increase due to the gain i n s p e c i f i c a t i o n . In a d d i t i o n , the explanatory and a n a l y t i c powers of the model are enhanced. In the land area t r i p rate a n a l y s i s , i t i s shown that different land uses adequately i s o l a t e attributes which r e s u l t i n different t r i p generation r a t e s . Oi and Shouldiner, however, f e l t that the absence of any s t a t i s t i c a l significance tests suggests caution i n 1 accepting t h i s assumption. This method is incapable of handling the effects of the i n t e r a c t i o n among different arrangements of land uses on the number of t r i p s generated, and i t s treatment of land use as a non-quantifiable explanatory variable i s somewhat unsatisfactory. On the other hand, the regression technique thus far employed emphasizes on p r e d i c t i o n , hence often use a simple explanatory v a r i a b l e , such as car ownership, in the equation. These simple regressions cannot be interpreted as neat causal r e l a t i o n s h i p s . The i n t e r c o r r e l a - tions among alternative explanatory variables confound the parameter estimates. The neglect of a l l but a single explanatory variable tends to overstate i t s true effect because of "its correlations with other 96 variables. However, even when other variables are included, the o v e r a l l goodness of f i t w i l l not be appreciably improved. This gives r i s e to the d i f f i c u l t y that even when the analyst wants to include more dimensions into the data, s t a t i s t i c a l test of significance w i l l . n o t j u s t i f y their,*--'inclusion, although they may be v a l i d on 2 theoretical'grounds. As a c o r o l l a r y to eliminate multicollinearity> t h i s investigation indicates that the combined factor a n a l y t i c and regression solution seems capable of overcoming the p i t f a l l s evidenced i n the result of Model 3. of both approaches Unlike the t r i p rate method, land use variables are used as explanatory rather than variables. as classificatory Interaction of different patterns of land use on t r i p generation i s taken into account and tests of significance are to the r e s u l t s . Secondly, the pertinent causal relationships are included and the confounding effect out. attached of correlated variables i s ironed A l l explanatory factors are found to be s i g n i f i c a n t . This analysis shows that i n an attempt to i s o l a t e causal r e l a t i o n - ships which conform to some t h e o r e t i c a l framework and yet s a t i s f y statistical c r i t e r i a , the model gains a d d i t i o n a l strength as a predictive t o o l . It also demonstrates that explanation and prediction can and should be combined i n the same analysis because i t y i e l d s more fruitful results. The following sections w i l l place the significance of the findings into the larger framework of model-building i n the transportation planning process. 97 5.2 U t i l i t y of Transportation Models In general, wide v a r i a t i o n i n the a v a i l a b i l i t y - o f resources and data have led to an almost equally wide v a r i a t i o n i n the scope, coverage and complexity of transportation studies. . While a number of combinations of techniques have been t r i e d , the general c r i t i c i s m appears to be the i n a b i l i t y of transportation models to move away from 3 the t u r g i d empiricism of d a t a - f i t t i n g . The Rand Report observed that even under the best of execution and circumstances, most transportation studies have been remarkably mechanic in.conception, e s p e c i a l l y i n establishing the r e l a t i o n s h i p between land use and t r i p generation. 4 W. L . Garrison also remarked t h a t : "I have serious reservations i n my own mind with respect to the role of these models in (transportation) studies. This is because I am unable to express a theory or even provide a simple description of the choice behaviour that these models r e p r e s e n t . " Such shortcomings l a r g e l y r e s u l t from the strong emphasis on prediction of most transportation models. The transportation planner i s often so engrossed with the number emerging from the model that his r e a l objective - to find out the hows and whys of interaction between urban a c t i v i t i e s so that plans can be formulated and evaluated - tend to be lost sight o f . * ;. At t h i s juncture, a c l a r i f i c a t i o n of the u t i l i t y of transportation models i n the planning process i s i n order. E s s e n t i a l l y a transportation 98 model i s so t h a t an e x p e r i m e n t a l d e v i c e t o a b s t r a c t t r a v e l demand and the f u n c t i o n i n g o f the urban system can be observed by v a r y i n g t r a n s p o r t a t i o n and l a n d use systematic and inputs. The knowledge g a i n e d w i l l b a s i s whereby a l t e r n a t i v e p o l i c i e s and p l a n s form a can be proposed evaluated. To a c h i e v e t h i s g o a l , a) patterns the model s h o u l d be capable P r e d i c t i n g what e f f e c t situation effects will of: occur over time i f the i s a l l o w e d t o run on u n a l t e r e d . existing By showing what m o d i f y i n g can be produced by a p a r t i c u l a r d e c i s i o n or p o l i c y o r by a new arrangement o f the elements a f f e c t i n g enables us t o judge between a l t e r n a t i v e s future consequences. In o r d e r t o p r e d i c t , movement, i n the l i g h t o f t h e i r one o f two k i n d s o f knowledge i s may u n d e r s t a n d the dynamics o f an e v e n t , which are the that describe tions, how i t changes. the f u t u r e . needed. T h i s approach c a l l s sufficient h e a v i l y on the condi- t o determine planner's a b i l i t y t o u n d e r s t a n d urban development and t o s y s t e m a t i z e knowledge. this A second avenue c a s e , due t o the is to project We theories T o g e t h e r w i t h the i n i t i a l a knowledge o f t h e dynamics i s it his from p a s t e v e n t s . In l a c k o f a t h e o r y , the e n t i r e p r e d i c t i o n r e s t s on the q u a l i t y o f our knowledge o f the p a s t , thus making a s c r u t i n y o f t h i s q u a l i t y an important m a t t e r . representative prediction. o f the p r e s e n t and the f u t u r e The p a s t must be f o r the purpose No c o n t i n g e n c y can be c o v e r e d u n l e s s i t has of already occurred and been recorded. This method i s often questioned because the r a p i d change that technology and economy has brought about render many past events Of the two ways, the f i r s t irrelevant. i s favoured because f i r s t l y , p r e d i c t i o n should succeed, we s h a l l want to say why. if Part of the payoff of a good prediction i s the insight i t provides into mechanism a n d ' r e l a t i o n s S e c o n d l y , policy alternatives, adequately i f we want to evaluate i t i s e s s e n t i a l that the'model can "explain" movements as they a c t u a l l y occur, and that the process employed for prediction takes proper account of a l l the major factors involved in determining future movements, including land use with which planners are p a r t i c u l a r l y 7 concerned. In addition to p r e d i c t i o n , models are important educational and research devices. Their formulation reveals the importance Of s t r u c t u r a l i n t e r r e l a t i o n s h i p s which otherwise may pass unnoticed or may not be given t h e i r due emphasis. In the construction.of a model, the analyst becomes aware of the sensitive linkages in the research scheme and he i s therefore able to give attention to these areas as required. This sometimes r e s u l t s in r e - formulation of the problem as new thoughts are generated about fundamental factors which might have gone unnoticed except 8 for t h e i r discovery in the model-building process. . /' 100 .In transportation studies, engineers and p l a n n e r s are concerned w i t h the b e h a v i o u r o f households and b u s i n e s s e s t a b l i s h m e n t s making use decisions. o f the t r a n s p o r t a t i o n Such b e h a v i o u r i s system and i n making l o c a t i o n a l the source o f t r a v e l demand. u n d e r s t a n d t h o r o u g h l y the whole c o n s t e l l a t i o n o f d e c i s i o n s by i n d i v i d u a l s and f i r m s , we c o u l d u n d e r s t a n d at the e x t e n t needs. t o which v a r i o u s urban arrangements Such an u n d e r s t a n d i n g i s in I f we made the same time satisfies their v i t a l t o p r o d u c i n g p l a n s and 9 p o l i c i e s b e s t t o s e r v e the p u b l i c i n t e r e s t . models f a c i l i t a t e analysis o f the c o n t e x t the a r e a s w i t h i n which d e c i s i o n s possible present Sound a n a l y t i c o f p o l i c y , by c l a r i f y i n g must be made, thus making more p o i n t e d c r i t i c i s m s o f the p o s t u l a t e s on which p o l i c y i s based. models are Also, i n v a l u a b l e for basic good e x p l a n a t o r y and a n a l y t i c a l research, even i f t h e y do not 10 find practical application. It is sometimes decisions, argued t h a t w h i l e p l a n n e r s are d e s i g n and p l a n n i n g , concerned w i t h 'academic' researchers are concerned w i t h e x p l a n a t i o n and t h e o r y - b u i l d i n g , and t h a t two s e t s o f a c t i v i t i e s are r a t h e r d i f f e r e n t . these From the above i s apparent t h a t p l a n n e r s need a l l the knowledge s u p p l i e d by . researchers i n order to p l a n . In f a c t it is the u n d e r s t a n d i n g o f the p l a n n e d systems which g i v e p l a n n e r s t e c h n i q u e s prediction. Indeed, can be s u b s t i t u t e d • 11 application. it for can even be s a i d t h a t it for "understanding" " p r e d i c t i o n " as b e i n g of more g e n e r a l 101 5.3 Implications f o r Model B u i l d i n g i n the Transportation Planning Process In view o f impending urban growth a d e c i s i o n t o p l a n the ment and I n t e n s i t y o f land use a c t i v i t i e s t r a n s p o r t a t i o n element i s of is an important o n e . not o n l y a prime v a r i a b l e The i n the achievement a d e s i r e d p l a n , but i s a l s o a major c o s t component o f urban growth, a c o s t which depends s i g n i f i c a n t l y on the form o f the itself. However, the s a t i s f a c t o r y i n t o the l a n d use p l a n s w i l l integration of l a n d use p l a n transportation r e q u i r e much more t e c h n i c a l and u n d e r s t a n d i n g than has h i t h e r t o been d e m o n s t r a t e d . a change i n the a t t i t u d e F o r c i n g the c o n s t r u c t is In p a r t i c u l a r , called for. o f t r a n s p o r t a t i o n models i n t o some l a r g e r view t h e o r y - b u i l d i n g r a t h e r than u s i n g them p u r e l y f o r prediction w i l l , expertise towards the development and r e f i n e m e n t o f t r a n s p o r t a t i o n models i n r e l a t i o n s h i p t o l a n d use of arrange- most l i k e l y t u r n out t o be an statistical advantage. 12 The He f e l t l o g i c behind t h i s that relations i n an e x p l a n a t o r y model, we are a s k i n g what are among the measured v a r i a b l e s , what we would expect On i s put forward most s u c c i n t l y by A l o n s o . and whether t h e y conform t o from v a r i o u s t h e o r i e s the o t h e r hand, p r e d i c t i v e models are n u m e r i c a l p r o d u c t t h a t emerges. and p r i o r e m p i r i c a l w o r k . concerned o n l y w i t h the As t h e s e numbers become and f e e d i n t o o t h e r models, t h e y t e n d t o have a l a r g e e r r o r when p r e d i c t e d f o r a f u t u r e considerations, the s t a t e o f the s y s t e m . variables specification From t h e s e i t would seem t h a t a model which seeks t o i n c r e a s e our 102 u n d e r s t a n d i n g by a s k i n g how c e r t a i n v a r i a b l e s is r e l a t e t o each o t h e r i n a sense l e s s s u b j e c t t o some o f the s o u r c e s o f e r r o r than i d e n t i c a l models d e s i g n e d t o p r e d i c t the f u t u r e . w i t h a sound t h e o r y w i l l through i t s d i r e c t use, s t r u c t u r e we are often r e s u l t p r e d i c t i o n , not but by shedding l i g h t on some f a c e t s o f considering; prediction i t s e l f which he c a l l e d " m u l l i n g o v e r " . of d a t a , i n better T h e r e f o r e , a model proceeds in a the fashion By the same t o k e n , f o r a g i v e n the e x p l a n a t o r y model i s more t o l e r a n t quality of complexity of f o r m u l a t i o n than a p r e d i c t i v e m o d e l . It is is not s u r p r i s i n g t h e r e f o r e taking place. predictors use as A decade ago, o f the f u t u r e . to f i n d that s t r e s s was p l a c e d on t h e i r o f the consequences were made t o i n c o r p o r a t e of alternative policies, and e f f o r t s variables which permit such e x p e r i m e n t a t i o n . More r e c e n t l y , experience has been g a i n e d , the p r a c t i t i o n e r s t o p l a y down the a b i l i t y o f models t o p r e d i c t , v a l u e as educational i n t o them p o l i c y of this as craft have and t o s t r e s s i n s t r u m e n t s which s e r v e t o b r i n g t o the s c i o u s n e s s o f those who make d e c i s i o n s the v a r i a b l e s , trend models were viewed p r i m a r i l y as Somewhat l a t e r , conditional predictors a significant tended their con- the complex i n t e r r e l a t i o n s i n c l u d i n g those which can be m a n i p u l a t e d f o r among normative 13 purposes. Thus the downgrading o f the importance o f the f u n c t i o n and the emphasis predictive o f the e x p l a n a t o r y and a n a l y t i c a l v a l u e s of 14 the model i s i n a c c o r d w i t h the v i e w p o i n t b e i n g advanced h e r e . i n h i s e x a m i n a t i o n o f urban r e s i d e n t i a l models a s s e r t e d t h a t that suffer from inadequate not n e c e s s a r i l y theoretical structures, improve p e r f o r m a n c e . better Stegman, for models data would T h e r e f o r e , m a n i p u l a t i n g and , 1 0 3 a d j u s t i n g t h e p a r a m e t e r s o f e q u a t i o n s i n o r d e r t o improve t h e ' f i t ' o f t h e models t o t h e d a t a w i l l n o t make t h e models b e t t e r p r e d i c t o r s . He p u t f o r w a r d t h e view t h a t p r e d i c t i v e models may be more u s e f u l i n p r o v i d i n g p o l i c y - m a k e r s w i t h a g e n e r a l u n d e r s t a n d i n g o f t h e magnitudes, d i r e c t i o n and i n t e r a c t i o n o f t h e f o r c e s a t p l a y i n t h e urban system, than i n p r o v i d i n g a c t u a l p r e d i c t i o n s . As t r a n s p o r t a t i o n models have already reached a high l e v e l o f s o p h i s t i c a t i o n i n the d a t a - f i t t i n g aspect, t h e p l a n n e r ' s f u t u r e c o n t r i b u t i o n l i e s i n t h e improvement o f t h e q u a l i t y and range o f data and t h e t h e o r e t i c a l b a s i s o f t h e model, p a r t i c u l a r l y i n t h e t r e a t m e n t o f f a c t o r s a f f e c t i n g t r a v e l demands w h i c h stem;- from l a n d use c h a r a c t e r i s t i c s . I n o t h e r words, a t h e o r e t i c a l l y sound and s c i e n t i f i c and approach t o system s i m u l a t i o n o f t r a n s p o r t a t i o n l a n d use i s a d v o c a t e d h e r e . I n a d d i t i o n , a f u r t h e r and p o t e n t i a l l y more i m p o r t a n t t r e n d s u g g e s t e d b y W. D. Peters"*"-* - t h a t o f merging environmental studies with transportation studies - merits s p e c i a l consideration. Combined s t u d i e s o f t h i s k i n d can have i m p o r t a n t e f f e c t i n "humanizing" t h e t r a n s p o r t s t u d y p r o c e s s , a p o i n t w h i c h has been s t r e s s e d r e c e n t l y from t h e t r a f f i c e n g i n e e r i n g s i d e by A. M. 16 Voorhees. 5.4 Conclusion M a t h e m a t i c a l models f o r b o t h r e s e a r c h and p r e d i c t i o n have become e s t a b l i s h e d d u r i n g t h e l a s t few y e a r s i n t h e p l a n n i n g p r o f e s s i o n , i n 104 particular tools. i n the t r a n s p o r t a t i o n f i e l d , The v a l u e as i n d i s p e n s a b l e and q u a l i t y o f these models are n o t r e a l l y adequate a t t e n t i o n . I t has been the o b j e c t : I t i s demonstrated t h a t i n u s i n g techniques, places be modelling s e t o f assumptions. I t i s , therefore, t h a t , where m u l t i p l e r e g r e s s i o n models a r e u s e d i n p l a n n i n g , l i m i t a t i o n s and t h e I m p l i c a t i o n s c l e a r l y understood. i s an e x p l a n a t o r y Its linear predictive severe demand on the m o d e l - b u i l d e r because: i t i s a s s o c i a t e d imperative are models as an such as m u l t i p l e r e g r e s s i o n , t h e m a t h e m a t i c a l framework with a h i g h l y r e s t r i c t i v e their given of this thesis t o discuss some o f t h e problems i n v o l v e d by u s i n g t r i p g e n e r a t i o n example. analytic statistical of s t a t i s t i c a l procedures T h i s s t u d y has shown t h a t m u l t i c o l l i n e a r i t y problem t o model c o n s t r u c t i o n and h y p o t h e s i s t e s t i n g . s i g n i f i c a n c e has a l s o been demonstrated. I t would meaningful t o i n v e s t i g a t e i t s p r a c t i c a l s i g n i f i c a n c e i n shaping a c t u a l t r a n s p o r t a t i o n - l a n d use p o l i c i e s . A f u r t h e r o b j e c t i o n t o past uses o f these models i n t r a n s p o r t a - t i o n s t u d i e s i s ' t h e i r t h e o r e t i c a l c o n t e n t , w i t h o u t which t h e y a r e but extrapolation of s i g n i f i c a n t s t a t i s t i c a l regularities. These e x t r a - p o l a t i o n do n o t c o n t r i b u t e v e r y much t o the t h e o r i e s o f urban s t r u c t u r e and development s i n c e t h e y i g n o r e the b e h a v i o u r o f the urban system. Moreover, t h e myopic concern w i t h p r e d i c t i o n has l e d t o the f o r m u l a t i o n o f some q u e s t i o n a b l e time and e f f o r t models. In view o f the l a r g e amounts o f money, i n data c o l l e c t i o n and a n a l y s i s , t h e b u i l d i n g o f models f o r p r e d i c t i o n o n l y i s u n r e w a r d i n g . In f a c t , i t i s argued t h a t 105 the d i s t i n c t i o n between e x p l a n a t o r y and p r e d i c t i v e a b i l i t i e s model i s o n l y a r t i f i c i a l , not r e a l . towards g r e a t e r A suggestion is therefore made emphasis on t h e o r i z i n g i n m o d e l - c o n s t r u c t i o n t o l a y f i r m e r f o u n d a t i o n s upon which s t a t i s t i c a l analysis can be thus moving away from the realm o f t u r g i d e m p i r i c i s m o f prevalent o f the i n most t r a n s p o r t a t i o n The s t a t i s t i c a l curve-fitting studies. t e c h n i q u e s used i n t h i s s t u d y are d e s i g n e d t o t e s t hypotheses based on a d e s i r e s t a n d i n g o f urban t r a v e l b e h a v i o u r . combined f a c t o r - a n a l y t i c based; It is and r e g r e s s i o n largely to gain a better under- found t h a t by u s i n g the method, i t is capable of i d e n t i f y i n g and i n c o r p o r a t i n g the c a u s a l r e l a t i o n s h i p between l a n d use and t r i p g e n e r a t i o n i n t o a s i n g l e model. t o extend the work t o modal s p l i t established a w e l l - d e f i n e d set of causal factors. Research i n t r o d u c e d by the f a c t o r effort analytic i n terms o f g a i n and r e d u c t i o n i n measurement and s p e c i f i c a t i o n e r r o r s . F u r t h e r , the idea o f combining t r a n s p o r t a t i o n and e n v i r o n m e n t a l s t u d i e s t o "humanize" the t i o n ' planning process merits further e x p l o r a t i o n . new a r e a s f o r f u r t h e r interesting models which up t o now has not y e t may a l s o be d i r e c t e d t o the e f f e c t s model f o r p r e d i c t i o n " p u r p o s e s I t w i l l be research. transporta- These may w e l l be 106 Footnotes 1 Walter Y. O i and P a u l W. S h o u l d i n e r , An A n a l y s i s o f Urban T r a v e l Demands,(Published f o r the T r a n s p o r t a t i o n Center a t Northwestern U n i v e r s i t y by Northwestern U n i v e r s i t y P r e s s , 2 Ibid., 1962), p.47. p.51. . '3 The Rand C o r p o r a t i o n , T r a n s p o r t a t i o n f o r Future Communities: (Rm-2824-7F, The Rand August 10, 196l), p.11. Study P r o s p e c t u s , California, C o r p o r a t i o n , Santa A Monica, 4 W. 1975", L. G a r r i s o n , "Urban T r a n s p o r t a t i o n P l a n n i n g Models i n J o u r n a l o f the American I n s t i t u t e o f P l a n n e r s , Volume 31, No. 2 (May 1965), p. 12. 5 "Aaron F l e i s h e r , Proceedings "On P r e d i c t i o n , and Urban T r a f f l e ", ,.Papers and o f the R e g i o n a l S c i e n c e A s s o c i a t i o n , Volume 7 (l96l) P.45. 6 Ibid. 7 Paul Brenikov, "Land U s e / T r a n s p o r t a t i o n L i m i t a t i o n s " , Report School, of Proceedings, Studies — Methods and Town and Country ( U n i v e r s i t y o f Nottingham 10-21 P l a n n i n g Summer September 1969, published under the a u s p i c e s o f the Town P l a n n i n g I n s t i t u t e , B r i t a i n ) , p.25. 8 L o u i s A. Loewenstein, "On S t u d i e s , Volume 3, Britton Harris, "The No. 9 the Nature o f A n a l y t i c Models", Urban 2 (1966), p.113. Use o f Theory i n the S i m u l a t i o n o f Urban Phenomena' . J o u r n a l o f the American I n s t i t u t e o f Planners., Volume 32, (September 1 0 Brian No. 5 1966), pp.269-270. L. J . B e r r y , "The R e t a i l Component o f the Urban Model", J o u r n a l of the American I n s t i t u t e o f P l a n n e r s , Volume 31, No. 2 (May 1965), p.150. ^A. 1? G. W i l s o n , "Models i n Urban P l a n n i n g , A S y n o p t i c Review o f Recent W. r e d i c t iS nt gu d Best i t h Imperfect LitA e lr oa nt suor,e " ," PUrban. i e s , wVolume 5, No. American I n s t i t u t e o f P l a n n e r s , Volume J o u r n1968) a l o f p.250. the 3Data", (November 3.4, No. 3 (1968), p.254. 107 13 Ibid. Ik M i c h a e l A l l e n Stegman, An A n a l y s i s and E v a l u a t i o n o f Urban Residential Models and T h e i r P o t e n t i a l Role i n C i t y P l a n n i n g , u n p u b l i s h e d P h . D . thesis, ^w. ( U n i v e r s i t y of Pennsylvania, D. Peters, Journal 1966). " P l a n n i n g A s p e c t s o f Land Use T r a n s p o r t a t i o n o f the Town P l a n n i n g I n s t i t u t e , Volume 55, No. 2 1969), p p . 5 9 - 6 1 . 16 Paul Brenikov, op. c i t . , p.27. Studies", (February 10'8 BIBLIOGRAPHY Books Berry, Brian J . L . and Marble, D. F . , S p a t i a l Analysis, Prentice H a l l I n c . , 1968). (New York: Blalock, H. M . , Causal Inferences in Non-Experimental Research, (Chapel H i l l : The University of North Carolina Press, , Social S t a t i s t i c s , (New York: McGraw H i l l , 1964). i960). C h r i s t , C. F . , Econometric Models and Methods, (New York: John Wiley & Sons I n c . , 1966). Cooley, V.'. W., and Lohnes, P. R., Multivariate Procedures for the Behavioral Sciences, (New York: John Wiley & Sons I n c . , F r i s c h . R., Correlation and Scatter in S t a t i s t i c a l Variables, 1962). (Oslo: University of Oslo, 1934). Guilford, J . P . , Fundamental S t a t i s t i c s in Psychology and Education, (New York: McGraw H i l l , 1965)• Econometric Methods, J., Johnston, P. 0 . , and Jackson, R. V/. B . , Modern S t a t i s t i c a l Methods: Descriptive and Inductive, (New York: (Chicago: McGraw H i l l , 1963). Johnston, Rand McNally & C o . , 1959). Kendall, M. G . , A Course in Multivariate Analysis, Griffin & Co., King, L e s l i e J . , (London: Charles 1957). S t a t i s t i c a l Analysis in Geography, Prentice H a l l I n c . , (New York: 1969). Martin, B. V . , Memmott, F. W. I l l , and Bone, A . J . , P r i n c i p l e s and Techniques of Predicting Future Demand for Urban Area Transportation, (The M. I. T . Press, 196l). 109 Meyer, J . R., Kain, J . F . , and Wohl, M . , The Urban Transportation Problem, (Cambridge: Harvard University Press, 1965). M i t c h a l l , Robert B . , and Rapkin, Chester, Urban T r a f f i c : of Land Use, ,(New York: A Function Columbia University Press, 1954). O i , Walter Y . , and Shouldiner, Paul W., An Analysis of Urban Travel Demands, (Evanston: published for the Transportation Centre at Northwestern U n i v e r s i t y , by Northwestern University Press, 1962). Perloff, Harvey S., and Wingo, Lowdon J r . , Issues in Urban Economics, (John Hopkins Press, 1968). Warner, S. L . , Stochastic Choice of Mode in Urban T r a v e l : in Binary Choice, (Evanston: published for the A Study Transportation Centre at Northwestern U n i v e r s i t y by Northwestern University Press, 1962). Wilson, F . R., Journey to Work Modal S p l i t , (London: MacLaren & Sons (New York: John Wiley & L t d . , 1967). Wold, H . , and Jureen, L . , Demand Analysis, Sons I n c . , 1953). A r t i c l e s and Periodicals Ackoff, Russel L . , and H a r r i s , B r i t t o n , "Planning, Operation Research and Metropolitan Systems", American Institute of Planners Conference Proceedings (1964), pp.92-96. Adams, V/arren T . , "Factors Influencing Transit and Automobile Use i n Urban Areas", Highway Research Board B u l l e t i n , No. 230 (1959), pp.101-111. ' Alonso, William, "Predicting B e s t With Imperfect D a t a " , Journal of the American I n s t i t u t e o f P l a n n e r s , Volume 34, N o . 3 (1968), pp.248 no Balkus, Kozmas and Jordan, J . David, "Regression Model for Projecting Vehicle-Miles of Travel i n a Metropolitan Region", Highway Research Record, No. 250 (1968), pp.25-34. Bendsten, P. H . , " T r a f f i c Generation", Socio-Economic Planning Sciences, Volume 1 (1967), pp.33-42. Berry, Brian J . L., "The R e t a i l Component of the Urban Model", Journal of the American I n s t i t u t e of Planners, Volume 31, No. 2 (May 1965), pp.150-155. Bevis, Howard W., "A Model for Predicting Urban T r a v e l Patterns", Journal of the American I n s t i t u t e of Planners, Volume 25, No. 2 (May 1959), pp.87-89. Bishop, D. W., "Some Multivariate Analysis Techniques & Their Applications to Recreation Research", Recreation Research, National Conference on Recreation Research Blakeman, T . L . , (University Park, P a . , 1965), pp.177-193. "Planned Demand for Transportation", T r a f f i c Quarterly, (October 1959), pp.464-477. Blalock, H . M. J r . , "Correlated Independent V a r i a b l e s : The Problem of M u l t i c o l l i n e a r i t y " , S o c i a l Forces, Volume 42 (December 1963), -pp.233-237. '•>: , "Evaluating the Relative Importance of V a r i a b l e s " , American S o c i o l o g i c a l Review, Volume 26 (December 1961), pp.866-874. . Blumenfeld, Hans, "Limitations of Simulation of Future Behavior", Plan: , Canada, Volume 5, No. 1 (1964), pp.4-9. "Are Land Use Patterns P r e d i c t a b l e ? " , Journal of the American I n s t i t u t e of Planners, Volume 25, No. 2 (May 1959), pp.61-66. / ' Ill Branch, M e l v i l l e ' C , "Simulation, Mathematical Models, and Comprehensive C i t y Planning", Urban A f f a i r s Quarterly, University, of New York, Volume 1., No. 3 (March 1966), pp.15-38. Brenikov, Paul, "Land Use/Transportation Studies - Methods and L i m i t a t i o n s " , Report of Proceedings, Town and Country Planning Summer School, University of Nottingham, 10-21 September 1969, (Published under the auspices of the Town Planning I n s t i t u t e , Britain), Burch, J . S., pp.24-30. "Traffic Interactance Between C i t i e s " , Highway Research Record, No. 297 (1961), pp.14-17. Colenutt, R. J . , "3uilding Linear Predictive Models for Urban Planning", Regional Studies, Volume 2 . , No. 1(1968), pp.139-143. Curran, P. B . , and Stegmaier, J . T . , "Travel Patterns i n 50 C i t i e s " , Public Roads, Volume 30, No. 5 (October 1958), pp.105-121. Deen, T . B . , Mertz, W. L . , and Irwin, N . A . , "Application of a Modal S p l i t Model to Travel Estimates for the Washington Area", Highway Research Record, No. 38 (1963), pp.97-123. Deutschman, H . D . , and Jaschik, N. L . , "Income & Related and Land Use Planning Implications", Transportation Highway Research Record, No. 240 (1968), pp.52-65. Fleet, Christopher R., and Robertson, Sydney R., "Trip Generation in the Transportation Planning Process", Highway Research Record, No. 240 (1968), pp.11-33. F l e i s h e r , Aaron, "On Prediction and Urban T r a f f i c " , Papers & Proceedings of the Regional Science Association, Volume 7 (196l), pp.43-50. Garrison, W. L . , "Urban Transportation Studies", Papers on Regional S t a t i s t i c a l Studies, Canadian P o l i t i c a l Science Conference on S t a t i s t i c s , pp.1-14. Association (University of Toronto Press, 1964), 112 Garrison, W. L., "Urban Transportation Planning Models i n 1975", Journal of the American Institute of Planners, Volume 31, No. 2 (May 1965), pp.156-157Grecco, W. L., and Breuning, S. M., "A System Engineering Model f o r T r i p Generation & D i s t r i b u t i o n " , Highway Research Record, No. 38 (1963),.. pp. 124-145. H a l l , P., "New Techniques i n Regional Planning: Experience of Transportation Studies", Regional Studies, Volume 1, (1967), pp.17-21. Hamburg, J . R., and Creighton, R. L., "Predicting Chicago's Land Use Pattern", Journal of the American Institute of Planners, Volume 25, No. 2 (May 1959),'pp.67-72. H a r r i s , B r i t t o n , "Plan or Projection; An Examination of Use of Models in Planning", Journal of the American Institute of Planners, Volume 26 (November i 9 6 0 ) , pp.265-272. , "New Tools f o r Planning and A Gloss on Lacklustre Terms", Journal of American Institute of Planners, Volume 31, No. 2 (May 1965), PP.90-95- . : _, "The Use of Theory i n the Simulation of Urban Phenomena", Journal of the American Institute of Planners, Volume 32, No. 5 (September 1966), pp.258-272. Heathington, Kenneth V/., and Rath, Gustave J . , "Computer Simulation . for Transportation Problems", T r a f f i c Quarterly, Volume 22, No. 2 ( A p r i l 1968), pp.271-282. H i l l , D. M., "Travel Mode S p l i t i n Assignment Programs", Highway Research Board B u l l e t i n , No. 347 (1962), pp.290-301. • , and von Cube, Hans G., "Development of a Model for Forecasting Travel Mode Choice i n Urban Areas"> Highway Research Record, No. 38 (1963), pp.78-96. 113 Hurst, Michael E . E l i o t , "Land Use/Travel Movement Relationships", T r a f f i c Quarterly, (April 1 9 6 9 ) , pp.263-274. J e f f e r i e s , Wilbur C , and Carter, Everett C , for "Simplified Techniques Developing Transportation Plans. T r i p Generation i n Small Urban Areas", Highway Research Record, No. 2 4 0 ( 1 9 6 8 ) , Keefer, pp.66-87. Louis E . , "Alternate Approaches to T r i p Generation", Pittsburg Transportation Research Newsletter, (May Volume 2 - 4 I960), pp.11-21. , "Characteristics of Captive and Choice Transit T r i p s in the Pittsburg Metropolitan Area", Highway Research Board B u l l e t i n , No. 3 4 7 ( 1 9 6 2 ) , pp.24-33. u K i n d l i c k , Walter, F i s h e r , E . S., Brinkerhoff L t d . , Vance, J . A . , Parker, C . C , Parsons "Intermediate and F i n a l Quality Checks, Developing a T r a f f i c Model", Highway Research Record, No. 3 8 (1963) , pp.1-19. Kuhn, Herman A . J . , "Factors Influencing T r a f f i c Generation at Rural Highway Service Areas", Highway Research Record, No. 2 4 0 ( 1 9 6 8 ) , pp. 1 - 1 0 . Levinson, Herbert S., and Wynn, F. Houstin, "Some Aspects of Future Transportation in Urban Areas", Highway Research Board B u l l e t i n , No. 326 (1962), Loewenstein, Louis K . , pp.1-31. "On The Nature of A n a l y t i c a l Models", Urban Studies, Volume 3 , No. 2 (1966), p p . 1 1 2 - 1 1 9 . Longmaid, David D . , "Prologue: Metropolitan Land Use and Transportation P o l i c i e s " , American I n s t i t u t e of Planners Conference (1964) , Lowry, I r a , Proceedings, pp.90-91. "A Short Course in Model Design", Journal of the American I n s t i t u t e of Planners, Volume 3 1 , No. 2 (May 1 9 6 5 ) , pp.158-165. 11.4 Mertz, W. L . , "A Study of T r a f f i c Characteristics in Suburban Residential Areas", Public Roads, Volume 29, No. 7 (1957), pp.208-212. • , and Hamner, L . B . , "A Study of Factors Related to Urban T r a v e l " , Public Roads, Volume 29, No. 7 (1957), pp.170-174. Niedercorn, J . H . , and Cain, J . F., "Suburbanization of Employment & Population 1948-1975", Highway Research Record, No. 38 (1963), pp.25-39. Osofsky, Lam, "The Multiple Regression Method of Forecasting Volumes", T r a f f i c Quarterly, (July 1959), pp.423-445. Overgaard, K. Rask, "Urban Transportation Planning: T r a f f i c T r a f f i c Quarterly, Pedersen, Traffic Estimation", ( A p r i l 1967), pp.197-218. Paul Ove, "Multivariate Models of Urban Development", Socio-Economic Planning Sciences, Volume 1 (1967), pp.101-116. Rice, R. G . , "Urban Form arid the Cost of Transportation: The P o l i c y Implications of Their Interactions", Plan: Canada, Volume 10, No. 2 (1969), pp.34-45. Schocken, Thomas D . , " S p l i t t i n g Headaches", T r a f f i c Quarterly, (July 1968), pp. 389-396. Schofer, R. E . , and L e v i n , B. M . , "The Urban Transportation Planning Process", Socio-Economic Planning Sciences, Volume 1 (1967), pp.185-197. • Schwartz, A . , "Forecasting Transit Usage", Highway Research Board B u l l e t i n , No. 297 (1961), pp.l8-35. Sharpe, G. B . , Hansen, W. A . , and Hamner, L . B . , "Factors Affecting the T r i p Generation of Residential Land Use Areas", Highway Research Board B u l l e t i n , No. 203 (1950), pp.2-36. 115 S i l v e r , Jacob, "Trends in Travel to the C.B.D. by Residents of the Washington, D.C. Metropolitan Area 1948-1955", Public Roads, Volume 30 (1959), pp.153-176. Tintner, G . , -"A Note on Rank, M u l t i c o l l i n e a r i t y and Multiple Regression", Anna Is of Mathmatical S t a t i s t i c s , Volume 16 (1945), pp.304-308. Weiner, Edward, "Modal S p l i t R e v i s i t e d " , T r a f f i c Quarterly, (January 1969) pp.5-28. Wilson, A . G . , "Models in Urban Planning, A Synoptic Review of Recent Literature", Urban Studies, Volume 5, No. 3 (November 1968), pp.249-276. Wohl, Martin, Bone, A . J . , and Rose, B i l l y , "Traffic Characteristics of Massachusettes, Route 128", Highway Research Board B u l l e t i n , No. 230 (1959), pp.53-84. Voorhees, A . M.> "The Nature & Uses of Models i n C i t y Planning", Journal of American Institute of Planners, Volume 25, No. 2 (May 1959), pp.57-60. ' , "Transportation Planning and Urban Development", Plan; Canada, Volume 4, No. 3 (1963), pp.100-110. , "Application of Model Techniques in Metropolitan Planning", American Institute of Planners Conference Proceedings, (1964), pp.110-119. Wong, Shue Tuck, "A Multivariate S t a t i s t i c a l Model for Predicting Mean Annual Flood in New England", Annals, Association of American Geographers, Volume 53.(1963), pp.298-311. , "A Multivariate Analysis of Urban Travel Behavior i n / Chicago", Transportation Research, Volume 3 (1969), pp.345-363. 116 Wynn, F . Houston, "Studies of T r i p Generation i n the Nation's Capital 1956-58", Highway Research Record, No. 230 (1959), pp.1-52. Zupan, Jeffrey M . , "Mode Choice: Implications for Planning", Highway Research Record, No. 251 (1962), pp.6-25. Others B j e r r i n g , James H . , Dempster, J . R. H . , and H a l l , Ronald H . , U . B . C . T r i p (Triangular Regression Package), (The U n i v e r s i t y of B r i t i s h Columbia Computing Centre, January 1968). , U . B . C . FACTO - Factor Analysis Program, (The University of B r i t i s h Columbia Computing Centre, May 1969). Cherukupalle, N . d : , "Regression Analysis - Interpretation of Computer O u t p u t . E t c . " , ' P l a n n i n g 508 Course Notes, (School of Community & Regional Planning, University of B r i t i s h Columbia, October 1969). C u r t i s , William H . , A Study of the Relationship Between Transportation and Urban Structure, Unpublished Thesis for the Planning Institute of B r i t i s h Columbia, Vancouver, B . C . (torch 1962). P r i c k e r , Urs Josef, Regional Land Use A l l o c a t i o n Models and Their Application to Planning, Unpublished M.Sc. Thesis, School of Community and Regional Planning, University of B r i t i s h Columbia . (May 1969). Kain, J . F . , Commuting and the Residential Decisions of Chicago & Detroit Central Business D i s t r i c t Workers, Report P-2735, (Santa Monica, C a l i f o r n i a : The Rand Corporation, 1963). 117 Kain, J . F . , The Journey to Work as a Determinant of Residential Location, Report No. 29, Real Estate Research Program, Institute of Business and Economic Research, U . C. Berkely, (December 1 9 6 l ) . Taylor, M. A . , Studies of Travel in Gloucester, Northampton & Reading, Road Research Laboratory Report 141, of Transport, (Great B r i t a i n : Ministry 1968). The Rand Corporation, Transportation for Future Urban Communities: A Study Prospectus, RM - 2824-7F, (Santa Monica, C a l i f o r n i a : The Rand Corporation, August 10, 1961). U. S. Department of Transportation/Federal Highway Administration, Bureau of Public Roads,. Guideline for T r i p Generation A n a l y s i s , (June 1967). 118 APPENDIX A A LIST OF THE VARIABLES USED IN THIS STUDY The 29 independent variables 1. 2. 3. 4. 5. 6. 7. 8. 9. 10. 11. 12. 13. 14. 15. 16. 17. 18. 19. 20. 21. 22. 23. 24. 25. 26. 27. 28. 29. are: Population, T o t a l Population, Single Family Population, Multiple Family Labour Force, T o t a l Labour Force, Single Family Labour Force, Multiple Family Dwelling Units, T o t a l Single Family Dwelling Units Multiple Family Dwelling Units Dwelling Units with Car Single Family Dwelling Units With Car Multiple Family Dwelling Units With Car Cars, T o t a l Cars Per Dwelling Unit Population Per Dwelling Unit % of Dwelling Units With Car Students (4 - 6 p.m.) Time to C.B.D. in Minutes Gross Income X 1 0 ~ 5 Bus Miles Area (in Acres) Income Per -Dwelling Unit Employment, T o t a l Employment: Public U t i l i t i e s , Government and I n s t i t u t i o n a l Employment: I n d u s t r i a l , Wholesale and U n c l a s s i f i e d Employment: Service Industries Employment: Entertainment Employment Density Per Acre Population Density Per Acre The dependent variables i s : T o t a l Trips G-nerated per day trips only). Services (vehicle 119 APPENDIX 5 . STATISTICAL TEST OF AUTOCORRELATION OF MODEL 2 BY USING THE "CONTIGUITY MEASURE POR k-COLOR MAPS" TECHNIQUE NOTATION B B Joins: Joins w i t h p o s i t i v e r e s i d u a l s i n contiguous zones. W W Joins: Joins with negative r e s i d u a l s i n contiguous zones. B W Joins: Joins w i t h p o s i t i v e r e s i d u a l s i n c o n t i n g u i t y with negative residuals. L^ '••. Number of contiguous zones of a t y p i c a l zone k , k = 1 to N . Two zones are considered contiguous i f they had an edge and/or vertex i n common. K = t o t a l number of zones that are not common to a t y p i c a l zone k , k = 1 to N . Sum of B B Joins.= z Sum of B W Joins = y Sum of W W Joins = x x + y + z = L , Appendix B ( c o n t ' d ) . Page ZONES 2. 1 4 4 2 3 3 7 7 5 • .'• 6 7 4 5 6 7 8 9 10 11 12 13 4 14 15 16 17 18 19 20 21 22 23 24 25 26 • 27 28 29 30 0 42 42 20 2 6 z\- 159 6 4, 30- 5 • . 6 42 12 3 5 4 N 12 12 6 •• 5 6 2 32 3 3 2 6 4 31 1 3 7 2 3 5 6 5 8 2 5 .' 6 6 5 5 . . 7 6, 4 \ 'V* 1 V r O 42 2 1 2 4 ' • 5 4 6 20 30 20 56 7 1 4 2 20 30 30 20 20 42 30 12 30 20 30 12 2 :.12 5 5 4 4 r O 5 3" 4 5 3 1 3 : 1 N k=l . Appendix B ( c o n t ' d ) . Page 3 . B 3 JOINS B W JOINS 12-13 12- 2 13- 2 2-1 2- 4 1-4 3- 4 3 -7 4- 7 1-6 17-16 29-16 17-29 13-14 6-14 6-19 6-20 6-5 1-5 4-5 7-8 7-11 7-9 3_9 21-20 21-5 21-8 21-22 29-15 21-23 21-24 16-28 16-23 ' 16-15 16- 29 17- 18 17-14 17-26 17-27. 17-28 17-30 . 29-30 29-31 29-32 .. -'•• z = 13 L = z + y + x = y = 31 13 +31 + 31 Let p = p r o b a b i l i t y for z q - p r o b a b i l i t y for x p + q = 100$ p = 13/(X+Z) = l £ = q - 70.5$ =75 / 29.5$ : , W W JOINS 14-19 14-25 14-26 19-25 19-20 19-24 20-5 20-24 25-24 28-24 25-27 25-26 26-27 27-24 27-28 24-23 23-22 23-15 22-15 22-11 22-8 8-5 - 8-11 9-10 9-11 10-11 11-32 . 11-15 15-32 30-31 30-18 x - 31 122 4. •Appendix B ( c o n t ' d ) . Page |i(z) = p ( L ) *' 0.2952 x 75 2 •'••.=. 0.087 x 75 = 6.55 ji(x) - q ( L ) = 0.7052 x 75 = 0.497 x 75 2 = 37,3 n(y) ~ 2pq (L) = 2 x 0.705 x 0.295 x 75 z 1.51 x 0.295 x 75 = 0.445 x 75 = 33.4 5 {z) 2 = p L+ p K - p (L+K) 2 3 , 4 = 6.55 + 0.295 (712) - 0.295 (75+ 712) = 6.55 + 0.02567 x 712 - 0.007573 x 787 = 6.55 + 18.3 - 5.96 - 18.89 3 5 (x) 2 = q L + q K - g 2 3 4 (L + K) r 37.3 + 0.705^ (712) - (0.705) 4 r 37.3 + 0.35 x 712 - 0.247 x 787 (75 + 712) = ••37.3 + 249 - 194 = 92.3 5 (y) = 2pqL 2 + pqK - 4 p q 2 2 ( L +K) 33.4 +0.445 (712) - (4 x O.087 x 0.497)(75 +712) = 33.4 + 0.445 x 712 - 4 x O.087 x 0.497 x 787 = 33.4 + 317 - 136 = = 214.4 To compute t h e Z s c o r e s f o r t h e t h r e e s e t s : Zx z =• 31-37.3 - -6.3 - -O.656 9.607 A/92.3 y =31-33.4 A/214.4 z z _ _2.4 = -0.164 14.64 = 13-6.55 = 6.45 = 1.483 A/18.89 Conclusion: 4.346 S i n c e a l l o f t h e t h r e e t e s t s t a t i s t i c s a r e l e s s than I.96 a t 5$ l e v e l o f s i g n i f i c a n c e , i t i s c o n c l u d e d t h a t t h e r e s i d u a l d i s t r i b u t i o n i s random, i . e . t h e r e i s no s i g n i f i cant a u t o c o r r e l a t i o n i n t h i s s e t o f d a t a . 123 APPENDIX'C METHOD OF USING MODEL 3 FOR PREDICTION Model 3 takes the form of T o t a l t r i p s generated = 0.9904x(Factor I) +0.2675x(Factor II) - 0.2772X(Factor III) + 0.1776x(Factor I V ) . Since the independent variables i n t h i s model are in the form of factor scores;, the following steps are necessary in order to project them to some future year: 1) Project the variables that, constitute the factors in question for each t r a f f i c d i s t r i c t . Transform the variables into standard scores by using Z = x i - 6 x , x (the mean) ando"(standard deviation) have been calculated for the previous set of data, e . g . Factor VI consists of one v a r i a b l e , variable "Students". Its present mean i s 1003, and standard deviation, i s 3490. D i s t r i c t 1 now has 1,600 students. An estimate of the future number of students in D i s t r i c t 1, say five years ahead, i s , for the sake of i l l u s t r a t i o n , 2,000. Transform t h i s number into standard scores according to the formula Z = 2,000 - 1,003 =•997 3,490 37^90 ' 2) Convert the standard scores for the variables in the factors according to the formula: F - (a,,z . l i + li a ll z li z ) (iAi) ml mi i s the factor loading of variable 1 on factor 1 1 a z + 21 2 i . ...a 1. i s the standard score of i n d i v i d u a l i on variable 1. 21 i s the factor loading of variable 2 on factor 1. i s the standard score of i n d i v i d u a l i on variable 2. Z 2i a 3 z ml i s the factor loading of variable m on factor 1. mi i s the standard score of individua1 i on variable m. i s the eigenvalue of the factor under consideration. A i 124 Appendix C ( c o n t ' d ) . Page 2 A g a i n , u s i n g F a c t o r V I and D i s t r i c t .1 as an example, 1 F*-. 6 l " (-,~ 17.6/- 17. V) ( A i1 )• a z F means t h e f a c t o r s c o r e o f F a c t o r V I f o r D i s t r i c t 1. 61 17.6 t h e f a c t o r l o a d i n g o f v a r i a b l e 17 ( s t u d e n t s ) on .* F a c t o r V I . z means t h e s t a n d a r d s c o r e o f v a r i a b l e 15 f o r D i s t r i c t 1. a m .'. F e a n = s (0.64 6 1 = 3) x 997 ) ( A . 5 3 ) 3490 1 0.119 A f t e r o b t a i n i n g the scores o f a l l the four f a c t o r s f o r D i s t r i c t 1 i n t h e model, d i r e c t s u b s t i t u t i o n o f t h e s e v a l u e s i n t o t h e e q u a t i o n w o u l d y i e l d t h e t o t a l number o f t r i p s g e n e r a t e d t h e r e f i v e y e a r s from now. / / RFS NO. 019807 UNIVERSITY OF B C COMPUTING APPENDIX MULTIPLE > REGRESSION OUTPUTS CENTRt E OF MTS I AN0!>9 ) MODEL I $SsIGNON P L A K TIME=5M PAGES=;50 C 0 P l E S = 3 6 PRIO=V * * L A S T SIGNON WAS: 21:27:25 02-24-70 USER " P L A K " S I G N E D ONI AT 1 5 : 1 1 : 3 5 ON 0 3 - 2 5 - 7 0 $RUN *WATFOR 5=*,S0URCE* 6 = * S I N K * 4=-A EXECUTION BEGINS ^COMPILE C PROGRAM TO F I N D RESIDUALS' C 1 DIMENSION X( 32 4) , F ( 32 , 2 )j, B ARX ( 4) , S T D X ( 4 ) , S T D S C 0 ( 3 2 , 4 ) , ~~ 1 Y ( 3 2 ) , R E S Y ( 3 2 ) , Y Y ( 3 2 ) » R E SYY ( 32 ) 2 R E A D ' S , ! ) ( ( X I I , J ) t J = l »4)» 1 = 1, 32) 3 1 FORMAT ( F 1 0 . ' 5 / 3 F 1 0 . 3 ) 4 WRITE ( 6 * 4 ) ' 5 W R I T E ( 6 , 3 ) ( ( X ( I ,'J ) , J = 1 , 4 ) , 1=1,32) 6_ 4 FORMAT (' TRIPGN LABFiOR DUWC AREA ) T % 6 FORMAT ( 2 F 1 0 . 7 ) 8 £ R E A D ( 5 , 3 ) (13 AR X ( J ) , J= 1, 4 ) 9 H READ(5,3) ( S T D X ( J ) , J = l , 4 ) 10 =j|;3 FORMAT ( 4 F 1 0 . 3 ) 11 WRITE (6,-7) 12 7 FORMAT ('MEANS AND STANDARD D E V I A T I O N S ' ) T3 WRITE(6,8) (BARX(J),J=2,4) 14 W R I T E ( 6 „ 8 ) ( S T D X ( J ) » J=2 ,4) 15 |jf=58 FORMAT ( 3 F 1 0 . 3 ) 16 f DO 20 J = 2,4 17 £ DO 2 5 1=1,32 18 ^ " 2 5 STDSCO( U J ) = ( X ( 1 , J ) - B A R X ( J ) )/STDX( J ) 19~ 20 CONTINUE 20 WRITE(6,9) 21 9 FORMAT ('STANDARD SCORES OF THE V A R I A B L E S ' ) 22 00 30 1=1,32 23 WRITE ( 6 , 10) X( I , 1) , ( STDSCO ( I , J ) , J=2,4 ) 24 3D WRITE(4,.I0) X( I , 1 ) , ( S T D S C O ( I , J ) , J = 2,4 ) 25 10 FORMAT ( 4 F 1 0 . 3 ) 26 STOP 27 END t 1 : blARI: I b: 11:61 0 3 - 2 b - /U (CONTINUED) — : JUB (STANDARDIZED ^ VARIABLES) — { 0 1 0 6 0 9 1 1 1 2 5 8 2 1 2 4 2 6 2 8 CONTROL CARDS J. . ...IN MS DC 2. STPREG 3. S T P R E G 4 . PARCOR 5. END MOTE: OUTDATED 3 3 0 1 • • • • • • • • • • 3 5 4 4 5 5 0 5 6 0 6 7 7 • • » • • * 0 5 • ' • 5 0 5 8 0 • • • * 4 3 3 2333 333 1 5 6 * I N V R * OR *HULREG* ROUTINES HAVE B E E N R E P L A C E D BY THE EQUIVALENT *STPREG* CONTROL CARD NO. * INMSDC FORMAT CARDS (F10.5/3F10.3 ) INPUT DATA TRIPGN LABFOR DUWC 288.0 2880. 864.0 0.1532D 05 0.11880 0 . 2 2 5 1 D 05 0 . 3 4 8 6 D 05 0.1995D 0 . 2 7 5 3 D 05 0 .2216 0 i.0.5 0 . 1 7 2 7D J0_OiL8JLD_CL5_ 0.2739D 05 0 .2118D 0 . 3 7 4 5 0 05 0 . 1 3 9 2 D 05 9743. 0 . 1 7 5 3 D 05 0 . 1602D 05 0 .11 U P Q.1471D 05 0 . 38210 0.5 0 . 2 9 6 6 0 05 0.2245D 8190. 8190. 4809. _83_95_. .__7_7_1_7_._ _J>X$XL 3364. 1553. 1551. 0 . 1 2 7 4 D 05 0 . 1 5 4 5 0 05 9464. 0 . 1 5 9 7 D 05 0 . 2 1 9 3 0 05 0.1164D 0 . 1 0 2 2 D 05 8674. 0 . 1272D 05 2862. 2225. 5406. C.J. 51.00 .03 _2_Q5J_. .0_U.2.8.6.D_Q5_ 0 .1889D 05 0 .11620 0 . 1 6 8 2 D 05 300 1 . 4388. 5081 . 6910. 0 . 1 0 6 6 D 05 0 . 1 HOD 05 1425. 1997 . 2238. 4965. 7074. 7638 . _3677. _6.1.U... 3j55_3_._ 5119. 6955 . 7816. 1139. 2038. 1634. 2530.' 5449. 3778. 23.00 46.00 69.00 1095. 706.0' 960. 0 29.76. _2.178_._ _32.5JL._ 0 . 1 8 4 4 D 05 0 . 2 1 4 5 D 05 0 .13020 6772. 5925.' 3807. 6469. 7505. 6728 . 2862. 2864. 3436. 32 OBSERVATIONS 31 _D_E.GRJE_E_S__D,F FRJEJLDOM AREA 05 05 05 05 05 05 05 05 05 1. 229 0.6820 4.895 4.818 7.420 4.050 5.480 9.810 15. 40 11.97 18.75 16. 01 9.788 21.64 2.073 4.312 27.68 27.68 4.031 2.451 2.203 5.892 3.654 3 .728 3.15 8 3.296 2.855 3.397 35.83 2 6 . 97 5 1 . 09 51 .02 NAME MEAN S.D. TRIPGN 0 . 1 0 7 9 D 05 9314. LABFOR 0 . 1 1 7 8 D 05 0 . 1 0 5 1 0 05 DUWC 7512. 6185. AREA 12.29 U.._7_Z_ CORRELATION MATRIX VARIABLE TRIPGN TRIPGN 1.0000 LABFOR 0.9211 DUWC . 0 .9700. AREA -0.0364 ARRAY WR ITT FN LABFOR 1.0000 0 ._97_5_2_ • 0 . 0835 TN AR FA DUWC JL.OQPJL 0.0117 AREA 1.0000 * CONTROL CARD DEPENDENT V A R I A B L E RSQ F PRO B. STD ERR VAR CONST . LA8F0R DUWC AREA = IS NO. * ST PREG * TRIPGN 0.9647 0.0000 1841.7498 COEFF 338.7013 -0.6510 2.5415 -79.4897 STD ERR 594.0954 0 .1 5 74 0.2666 26.5996 F-RAT 10 FPROB. 17 . 1 0 1 1 90.8980 8.9304 0.0003 0.0000 0.0057 NO. OBSERVED CALCULATED RESLOUAL 1. 2. 3. 4. 5. 6. 7. 8. 9. 10. 11. 12. 13. 14. 15. 16. 17. 18,. 19. 20. 21. 22. 2 3. 24. 25. 26. 27. 28. 29. 30. 31. 32. -1.1270 0.48700 2.5840 1 .2210 1.7820 0.33600 0.56200 2.0260 -0.27900 - 0 . 3 2 900 -0.99100 .. . 0 . 50100 1.1960 -0.60000E-01 -0.85100 0.46300 0.87000 -0.68700 -0.13000E-01 -;0 . 9 4 4 00 -0.39800 -0.50200 ^0.41100 -0.93900 -0.57300 - 1 . 1560 -1 . 0 4 0 0 -0.83800 1. 1450 -0.74900 -0.43600 - 0 . 8.5100 -1.0983 0.49451 2.4627 1.3269 2.0342 0.296C4 0.40358 2.3681 - 0 . 3 50<83 -0.19231 -0.88771 0 . 2 5862 0.65623 0.10716 -0.82867 -0.94176E-01 0.59127 -0.56298 -0.81321E-01 -0.95528 -0.37340 -0.58431 -0.45361 -1 .0132 -0.72477 -1.1762 -0.80554 -0.79665 0.93311 -0.18437 - 0 . 6 7188E-01 - 0 . 7 0268 -0.28742E-01 -0.75117E-02 0.12125 -0.10593 -0.2522 4 0.39964E-01 0.15842 -0.34206 0 . 71834E-01 -0.13669 -0;10329 0.24238 0 . 53977 -0;16716 -0.22332E-01 0.55718 0.27873 -0.12402 0.68321E-01 0.11279E-01 -0.24604E-01 0 . 8 2 3 06E-01 0.42615E-01 0.74238E-01 0 . 15177 0.20213E-01 -0.23446 -0.:41347E-01 0.21189 -0.56463 - 0 . 3 6 8 81 - 0 . 1 4 83 2 END OF, CONTROL SET * STOP 0 •XECUTI.ON TERMINATED iSlGNOFF CALCULATED OBSERVED NO. ... - - - .... - --- ._ - - •- _- . • — -— - - -- — .. ... . -- . -- - - - - - - - --- • - -- -- i • - •-- RES I DUAL -• — - .. - - ' -\ NO. i ..... . . 1. 2. 3. 4. 5. 6 . 7. 8. 9. 10. 11. 12. 13. 14. 15. 16. 17 . 1.8.. 19. 20 . 21. 22. 23. 24. . 25. 26 . 27. 28 . 29. 30_.__ 31. 32. ARRAY RESTORED FROM AREA WRITTEN 562.13 15816. 32729. 21149 . 29210. 13367. 18573 . 31731. 6005.7 10069 . 600.11 148 2 3 . 18745 . 12381. 2309.7 14640. 16731. 2458.2 10356. 2 30 8 . 7 7810.2 5535.6 '7 9 7 0 . 4 1873.5 4058.4 148.70 1281.1 3485.9 18573 . 8844.7 7833.5 3151.1 288 .00 15318. 34857. 22162. 27387. 13916. 16019. 29656. 8190.0 7717. 0 1553.0 15448. 21928. 10222. 2862.0 15102. 18891. 4388.0 10660. 1997.0 7074.0 6111.0 6955.0 2038.0 5449.0 23 .000 1095.0 2976.0 21454. 3807. 0 6728.0 2862.0 CONTROL ARRAY CALCULATED OBSERVED CARD NO. ARRAY RESTORED FROM AREA -274.13 -497.81 2127.5 1013.1 -1823 .1 548.51 -2554.3 -2075.4 2184.3 -2351 .7 952.89 624. 52 3183.0 -2158. 7 552.27 462.12 2159.5 1929.8 303.52 - 3 1 1 . 69 -736.20 575.36 -1015.4 164.52 1390.6 -125.70 -186.15 - 5 0 9 . 88 2880.7 -5037.7 -1105.5 -289 .11 CARD NO. s RESIDUAL \ 3 * STPREG * 4 * PARCOR * 6 AREA -1.000 t END OF CONTROL 0 TERMINATED CALCULATED / 6 F>ART IAL CORRELATIONS LABFOR DUWC TRIPGN VVARIABLE 1.000 1"RIPGN -1.000 t. A B F O R - 0 . 6 1 5 8 0.8744 0.9797 -1.000' [)UWC I^REA -0.4917 -0.4289 0.4223 STOP EiXECUTION OBSERVED NO. 5 IN AREA CONTROL RES I DUAL SET * ( RFS NO. 0 1 9 8 0 8 U N I V E R S I T Y OF B C COMPUTING CENTRE APPENDIX MULTIPLE R E G R E S S I O N OUTPUTS $ S I G N O N PLAK T I M E = 5 M P A G E S = 5 0 C 0 P I E S = 3 6 P R I O = V - =**.LA.ST SIGNON WAS : 1 6 : 0 4 : 2 9 03-25-70 USER " P L A K " S I G N E D ON AT 1 6 : 1 0 : 2 7 ON 0 3 - 2 5 - 7 0 :»RUN * T R I P 4 = * S 0 U R C E * EXECUTION BEGINS T R I P / 3 6 0 IMPLEMENTATION 3/18/70 MTS(AN059) JOB START: E OF MODEL I (UNSTANDARDI ZED VARIABLES) 16:10:22 03-25-70 0.7280 0.0242 0.0697 0 . 1639 0 .189 3 0.2396 0.0753 0.0769 -0. 1420 0.2309 21 22 23 24 25 26 27 28 29 30 CORRELATION V A R I ABL E 1.5 :.6 317 8 .19 2.0 ill 22 23 2 4 25 2'6 27 28 ^9 30 CORRELATION VARIABLE 23 24 25 26^ 27 23 29 30 $ S I G 0 .7302 0.0829 •0.0418 0 . 1585 0 . 1116 0.3131 0.0647 0.0573 -0 .1590 0.0323 0 .1868 -0.2781 -0.2089 0 .345 3 0.3984 0.3598 0 .2670 0.2686 0.1909 0.7750 0.6816 0.0094 -0.0449 0.2473 0.2394 0.3493 0.1522 0.1555 -0.0558 0.3928 0.7217 0.1468 0.0096 0.1406 0 .1027 0 . 2783 0.0539 0 .0459 -0._172_0 0.0436 0.2092 •0 . 2 6 3 2 -0 .1248 0.3118 0 .3671 0.2852 0.2556 0 .2792 JD.2026 0.845 2 0 . 7 046 0.0590 0.0168 0.1991 0.1977 0.3059 0.1041 0 .0996 -0.1187 0.2752 16 17 18 19 20 21 22 1 .0000 0.6271 -0.1111 0.3438 -0.2742 -0 .2037 0.1005 0 . 1042 -0.3907 •0.4302 -0 .2207 -0.4228 -0.4483 -0.4389 -0.4776 1.0000 -0.0909 0.4903 -0.1074 -0 .2169 0.2350 0.3557 -0 .7367 -0.6757 -0.6264 -0.7494 •0.7813 -0 .7998 -0.4457 1.0000 -0.2082 0.3939 0 .3579 -0.0560 0. 03 34 0 .1075 0.2180 -0.0278 0.1027 0.13'05 0.0281 -0.0009 1.0000 -0.3616 -0.5949 0.6541 0.1695 - 0 . 5900 -0.5675 -0.5813 -0 . 5 2 7 2 -0.5778 -0.4796 -0 . 5094 1 .0000 0.6075 0.1029 0.3378 0.1252 0.1973 0. 1107 0.0724 0.0894 -0.0967 0.4335 1.0000 •0.3087 0 .0202 0.4197 0 .3714 0.4279 0.3846 0 .3735 0.1840 0 .0439 26 27 28 29 30 1.0000 0.7970 0.7732 0.7001 0.1050 1 .0000 0.9690 0.9459 0.0735 1.0000 0.9590 0.1818 1.0000 0 .1650 1.0000 0 .2318 0.2312 -0.2643 -0.1814 0.3692 0.4244 0.3783 0.2906 0.2875 0.2008 0.7552 13 14 15 0.4794 -0.4124 -0.5530 -0 . 3 9 4 2 0.2170 -0.5575. 0 .6140 0.2714 -0.2840 - 0 . 1451 0.2691 iL^32_12_ 0.2619 0.2089 0.2199 0.1298 0.8057 1.0000 -0 .6 118 -0. 1300 •0.0337 0.3063 •0/2979 0.8942 0 . 7 004 0.0918 0.0798 0 . 1 82 3 0.1968 0 . 2 800 0.0905 0 .0718 -0.1448 0.1970 1.0000 0.5949 0.8222 -0.0785 0 . 5 110 -0.0655 -0. 1871 0.2618 0.3841 -0.6377 •0.5620 -0.5539 -0.6471 •0.6886 -0.69 36 -0.4750 24 25 1.0000 0.9517 J)..900_4_ 0.9726 0.9501 0.8975 0.1041 1.0000 0 . 7 89 3 0.9183 0.8940 0.8336 0.0982 L0I_-J.J)_62_ MATRIX 23 1 .0000 - 0 . 2650 -0.1336 H3_.462.1. - 0 . 1806 -0.1957 -0.1989 -0.0868 J.JL_QOOO__ ' STOP EXECUTION 0 .6846 •0.0489 -0.1229 0 .2773 0.2608 0.4116 0.1664 0.1611 -0 .0511 0.3556 0 . 7 000 0.1450 0 .0 146 0.1342 0 . 0 8 74 0.^2900 0.0420 0 . 0 356 -0_-\lZ6_ 0.029 5 MATRIX 12 1 .0000 0.2489 0.9579 0.0319 0.0191 0 .0 525 0.2838 :0..19_7 50.7880 0.7094 0.1729 0.0730 0.1293 0.108 5 0.2548 0.0424 0.0332 •0.1844 0.0183 :i2 JL3__ :.4 0.7016 0.0562 -0 .0354 0.2218 0 . 1925 0.3655 0.1177 0.1111 0 TERMINATED * END OF CONTROL SET * 1.0000 0.4107 -0.2434 -0.2064 -0.2684 -0.2158 -0.2411 -0.2379 -0.3025 32 .00 2.8 5 5 5710. 1362. 161.0 898.0 234.0 19 .00 477.0 1140. 2976. 2178 . 60.00 9102. 2127. 3.397 8902. 51.00" 5182 ." 200.0 3021. 828 .0 3254 . 1.208 255.0 3152. 3.640 243.0 102.0 0.8690 283.0 2502. 0 .0 47.00 2425. 23.60 244.0 77. 00 9 7 . 58 2682. 542 . 0 0.9220 1635 . 0 . 1 4 1 0 D 05 0. 0 290 . 0 1084. 0 . 2 0 3 4 D 05 6208. 05 0 . 1 7 9 0 D 05 3.670 2942. 0.1356D 3 8.30 173.0 05 542.0 695 .3 1442. 0.50650 271 .0' 6.30X.I 3807. 5925. 1.000 0 . 2 7 9 3 D 05 5502. 26.97 0 . 2 5 8 1 D 05 423. 0 7467 . 2115 . 7616. 3532. 6772. 1. 286 595. 0 6349. 4.710 1076. 423. 0 1.000 1620. 5925. 0.0 241.0 5502. 3 1 . 00 131.0 423 .0 322.9 1035. 672.8.._ 6469 . 1. 000 _0_._2_7J_6.Q_Q.5_ 6469. 5 1 . 09 0.0 6649. JL73_JL 9315. 2842 . 75 0 5 . 1. 091 551.0 6987. 3 . 180 621.0 518 .0 0 .7590 1369. 8538. 0 .0 301.0 8021. 42.00 56. 00 517.0 465.3 531 . 0 2864. 1.152 828. 0 2864. 2.630 203.0 0.0 0.9220 620 .0 3722. 0. 0 77.00 3722. 3 1 . 00 35.00 0.0 255 . 0 192.0 2862. 3436. ._1...0.0.0_ 05 0.18440 1.440 1341. 0 . 5 1 7 4 D 05 0 . 1275D 05 _J3_5_._83 0 .2145D 05 0.1302D 05 .._1...0.0JO_ 0.0 4294. 1796. 9789. 0.0 __3563. 9789. 3436. 3X._Q.2_ 32 OBSERVATIONS 31 DEGREES OF FREEDOM NAME 1 2 3 4 5 6 7 8 9 10 MEAN 0.1079D 0. 2939D 0.2565D 3834. 0.1178D ^94_41_.. 2 33 8 . 8877. 709 2 . 1783. XORREL.AT ION MA.TR INVARIABLE 1 1 1.0000 _2J 0 .9478 0.9147 3 I 0 .4442 4! .0_._9_2r>.9_ 0.9081 6 0.3960 7 0.9439 8 0.9285 9 0 .4278 10 J..1.0...9.7JDLCL 12 0.9356 13 0 . 4 844 0 9705 -0 ..0382 15 -0. 1457 16 ^.A39JL 17 0.4101 18 • 0.3300 19 0 .8915 _2J) S.D. NAME 05 9 3 1 4 . 05 0 . 2 4 4 9 0 05 05 0. 2226D 05 6599 . 05 0 . 1 0 5 ID 05 87 2 5 . 4438 . 7599. 6247. 3210. 1.0000 0.9642 0.4672 ..0_.J9J_Q5_ 0.9575 0.4162 0.9741 0.9687 0.4211 _P_._9_8jb7_ 0.9592 0.4725 0.9704 -0 . 1 2 3 9 -0.0702 -0.0984 0.2702 -0.3546 0.8300 11 12 13 14 15 16 17 18 19 20 1.0000 0.2170 _o_.JL&__L 0.~9 856 0.1621 0 . 8 89 1 0.9935 0.1717 _0 .9.362 0.9901 0.2319 0.9324 -0.0027 0 . 0 735 0.0292 0 .2354 -0.2120 0.7431 MEAN S .0 . NAME 7 512. 6185. 6303. 5381 . 1209. 1990. 0.1047D 05 8460. 1.220 0.2574 0 . 5529 3.40 8 0.8661 0.1253 1003. 3490 . 21.34 9.686 414.1 357.2 1. 0000 0.6191 0.2396 0.9930 0.6216 0.2528 0.9800 0 .5133 0.2316 0.9688 0.4655 -0.4447 -0.5023 -0.4532 0.2043 -0.5895 0.5782 1.0000 0 .9095 0.5800 0.9858 0.9044 0.5741 0 .9751 0 .8910 0 .6211 0.9529 -0.1882 -0 .2005 -0.1770 0.2510 -0 .4453 0.8264 21 22 23 24 25 26 27 28 29 30 1.0000 0.1889 0.8932 0 .9885 0 . 1911 0.9.379 0.9838 0.2548 0.9353 •0.0010 0.0259 0.0199 0.2273 -0 .2511 0.7227 MEAN 133.8 12.60 5885 . 9695. 2485. 3378. 3067. 749.3 3 452 . 5453. 1 .0000 0.5783 0 .1994 0.9813 0.4654 0 .1770 0.9677 0.4183 -0.4424 -0 . 5 2 4 1 -0 .4570 0. 1479 -0.5598 0.53 5 5 S.D. 129 .9 13.71 1806. 0 . 1 4 0 2 D 05 3533. 4313. 5611. 1369. 0 . 1 0 7 0 D 05 0 . 1 0 3 7 0 05 1.0000 0.9108 0.5952 0.9860 0.8966 0.6400 0.9593 -0.2120 -0.2406 -0. 1947 0.3191 -0.4237 0.8829 1.0000 0 .2103 0.9517 0.9931 0.2722 0.9474 -0.0235 -0.0034 0.0005 0.2843 -0.2220 0.7678 10 11 1 .0000 .0.482 5 0.1900 0.9857 0.4276 -0 .4561 •0.5628 •0.4619 0 .2024 -0.5712 0.5963 1.0 000. 0 . 9 502 0.5384 0.9877 -0 . 1 0 4 9 -0.1613 lO.OSl 1 0 .3168 -0.3512 0 . 8 832 0 . 1 0 7 5 0 05 16.01 348. 0 5121 . 624.0 1302. 2763. 432. 0 320.0 0.21930 05 0 . 1 1 6 4 0 05 294.0 0 . 4 9 7 9 0 05 0 . 1110D 05 9.788 0.4817'D 05 542.0' 7743 v 1623. 0 . 1 7 8 6 D 05 9806. 0.1597D 1.347 1996. 05 0 . 1 4 8 8 D 05 3.760 3229. 1082 . 0.8780 3770 . 0 .1326D 05 0.0 811.0 0 . 1 0 2 2 D 05 8674. 168.0 0 . 3 2 3 9 D 05 . 7903. 21.64 0 . 3 0 4 6 0 05 771. 0 6307." 1930. 0 . 1214D 05 4794. 0 . 1 2 7 2 D 05 1.370 1424. 0 . 1 1 7 6 0 05 3.560 2398. 964. 0 0.9780 782 . 0 8867. 0.0 190.0 8096. 18.00 221.0 771 . 0 437. 1 1492. 2862. 3.000 3512. 2544 . 0.7010 380.0 3178. 0.0 91.00 2225. 28.50 2022. 953.0 49.26 4610. 7625. 2.730 4741. 5240. 0.9500 5120 . 9532. 0. 0 1103. 57 2 0 . 2 7 . 00 3735. 3 813. 371 .7 6030. 895.0 0 .9560 1283. 0 . 12 1 70 05 0.0 287.0 5081. 3.540 465 .0 0 .0 0.8670 586.0 3463. 0.0 150. 0 3463. 38 . 5 0 8 5 . 00 0.0 161 .9 441.0 9743. 3.500 739 .0 1354. 0.8480 1309. 8152. 0.0 172 . 0 7113. 1 7 . 00 729 .0 1040. 343.2 7070. 7314.' 953.0 1891 . 9540. 1272. 2 . 0 73 2862. 2225 . 62.00 2226. 349 7 . 4187 . 5406. 1. 002 806.0 1083. 815 .6 5090. 0 . 1 5 1 0 0 05 9057. 219.0 0 . 2 6 0 2 D 05 5720. 4.312 0 . 1 95 3D 05 3337 . 48 23.'" 6484. 0 . 1 1 9 1 0 05 0 . 1 6 0 9 D 05 0.1286D 1.250 4526. 0 . 1 8 8 9 D 05 0 . 1 1 6 2 D 05 37.00 0 . 4 4 2 0 0 05 0 . 1 0 7 1 D 05 27.68 0 . 4 4 1 5 D 05 908.0 57 77." 3045 . 0 . 1 6 1 0 D 05 8757. 0 . 1 6 8 2 D 05 1.323 3682. 0.12240 3001 . 2 7 . 68 0.1224D 0.0 6003. 05 0.0 5541. 2361 . 0 .2582D 05 843.0 5888T 2724. 9662 . 2940. 4742. 177.0 5018 . 404.0 2013. 7480. 2238 . 1.310 2206. 1951 . 3.350 1862. 287.0 0.9280 1990. 1536. 0 .0 1421 .__ 1359. 16.00 3050. 177.0 6 5 . 89 2097. 4388. 3001 . 1. GOO 0 . 1 0 6 6 D 05 6910. 22 6 . 0 05 0 . 2 8 5 4 D 05 6067. 4 . 0 31 5146. 1248. 2 .451 1997. 1425. 66.00 1 x 05 0.1218D 0 5 17.00 1003. 2 02 5 . 5081. 1. 599 1160 . 0.1110D 05 1 .183 725. 0 0 . 1592D 05 3.640 3505. 0 . 1 1 0 8 D 05 28.60 316.0 1090. 530.5 1595. 7074. 4965. 97.00 0 . 1 7 8 4 D 05 3103 . 2.203 0 . 12 2 5D 05 1862 . 5156. 5588. 6732. 4371 . 7638. 1.153 340.0 4815. 3.060 1958. 2823. 0.8510 1836. 5840. 3500. 236.0 3414. 19.70 1986. 2427. 227 .8 8110. 6111 . 3677. 114.0 0.J.445D_ 05 3506. 5.892 0 . 1 4 0 I'D 05 1 7 1 . 0" 4 7 54." 439.0 5616. 2111. 5653. 1.364 202.0 5454. 3.510 1536. 199. 0 0 .8920 310.0 4119, 0.0 63 .00 3900. 2 5 . 60 358.0 219.0 146. 2 2455. 6955. 5119. 115.0 0 . 1 8 8 5 D 05 4270. 3.654 0 . 1 6 6 8 D 05 849. 0 5005 ." 2167. 7019. 6997 . 7816 . 1. 153 1644. 6657. 3 . 100 3301. 1159. 0.8420 1816. 6087. 0.0 236.0 5093. 22.90 1919. 944.0 237.4 5160. 142.0 1678 . 3351 . 1634. 1.300 2293. 1615. 3.160 659 .0 19.00 0.8840 252.0 1289 . 0. 0 147.0 1256. 17.40 898.0 33.00 51.69 1236. 475 .0 3678. 700.0 3778. 1.370 59.00 3586. 4.040 3 51.0 192. 0 0.9500 261 . 0 2664. 0.0 29.00 2530. 2 2 . 20 222.0 134.0 124.5 3405. 0. 0 46.00 1554. 69.00 1. 000 0 .0 69.00' 3 . 500 242.0 0 .0 1.000 1300. 46 .00 0.0 1 2 . 00 4 6 . 00 26 . 4 0 472.0 0. 0 1.794 49.00 0.0 1133 . 960.0 1.504 960.0 4.320 0.0 0.9480 753.0 5000. 753.0 25.20 0.0 35.20 460 7 . 1120. 3.728 2038 . 1139. 40 .00 0 . 1 0 7 5 0 05 2396. 3.158 5449. 2530. 58.00 23.00 46 .00 15.00 1095. 706. 0 f 4465. 19.00 5502." 0 .1028D 05 134. 0" 62 95 . 161.0 46.00 3.296 161.0 0.0 5050." 3254. 706. 0 32 5 4 . ' 0.0 ' ) <\ J CONTROL CARD N O . 1 * INMSDC * CRM AT CARDS 1F10.5/8F9.3/8F9.3/8F9.3/5F9.3) INPUT DATA I II 21 2 12 22 2.8.8..J0 864.0 245.0 547-2.. 288.0 1.229 3 13 21 4 1.13-2 576. 0 4555."' 5 15 25 14 24 4320. 864.0 0 . 7 4 6 9 D 05 2880. 0 . 3 750 0.17610 05 6 1.6 26 7 17 27 576.0 2 .370 0.1812D 05 2304. 0 .3750 0.3132D 05 8 18 28 9 19 29 2304. 1600. 7634. 10 20 30_ 288.0 2. 600 0 . 6 0 7 5 D 05 2016. 5 8 . 18 4550. 0 . 1 7 8 2 0 05 0.6220 247 1 . 0 . 1 9 0 6 D 05 0.0 1470. 4376. 3.600 0.1032D 05 0.2348D 05 3.060 1058. 4041. 0.8410 3828. 0 . 2 3 7 4 D 05 0.1900D 05 1313. 0 .1944D 05 13.10 810.0 0.1858D 05 0 .7570 0.1011D 05 0.2281D 3000. 1768. 05 05 0 . 1 6 2 9 D 05 2.840 0 . 1 1 6 8 0 05 0.1140D 05 6 . 300 7130. 9779. 0 . 2 5 9 9 D 05 0 . 2 3 0 7 D 05 0 . 3 7 4 5 D 05 0.9700 3274 . 0.3186D 05 3 . 430 0 . 1 0 5 9 0 05 5588. 0.7890 7877. 0 . 2 6 8 3 D 05 0. 0 1330. 0.2316D 05 10 .40 3110. 0 . 4 3 9 7 0 05 835. 0 3751. 5562. 0 . 1 1 9 7 D 05 0.2252D 05 0.1753D 0.9750 4085. 05 0.1419D 05 4.050 0.1388D 05 3338. 0 .7950 3248. 0 . 1 2 2 5 D 05 0.0 1306 . 0 . 1 0 5 8 D 05 8.500 5555. . 0 , 3.9880 03 9153. 5.480 0 . 3 4 6 5 D 05 1961. 9294 ." 5232. 0 . 1 7 3 3 0 05 0 . 1 1 3 4 D 05 0 .1471D 05 1.433 4257. 0.1177D 05 3 . 300 3269. 2943. 0 .9100 3397. 0.1209D 05 0 .0 618.0 9480. 12.80 2070. 0 . 8 8 7 1 D 05 0 . 2 1 6 4 D 05 9.810 0.8684D 802.0 4738. 1869. 0 . 3 0 9 9 D 05 0 . 1291D 05 0.3821D 05 1. 170 1628. 0.3714D 05 3. 350 6458. 1069. 0.8750 3422. 0 .2565D 05 0.0 1404. 0.24580 15.20 1318. 8190. 4809. 106.0 0 . 2 0 8 4 D 05 4809. 15.40 0 .20 840 05 0.0 848 2 v 0 .0 6590. 7742. 8190 . 1.273 4148. 8190. 4.030 1164. 0.0 0 .9300 1885. 5165. 0.0 545.0 5165. 22.90 503.0 0.0 302.7 1354. 7717. 6 353. 76.00 0..24.05D. 05 6126. 11.97 0 . 2 360D 05 227.6 7224 454.0 8849. 2346. 8395. 1. 298 495.0 8168. 3 . 520 1276. 227 .0 0.9340 476.0 6 807. 0.0 9 9 . 00 6580. 28 . 10 196.0 227.0 349.9 2 001 . 3364. 1.668 76.00 3364. 4 . 170 4138. 0.0 1.000 389.0 1551. 0. 0 135.0 1551. 2 4 . 10 263.0 0.0 48.24 345.0 579.0 0 .9410 0.10040 0^0 8884. 19.50 1159. 876.3 0 . 153 2D 05 0 . 4 0 3 1 D 05 3126. 0 . 1 1 8 8 D 05 _ 0_. 6.82Q 45...0LC 0.14370 8750 . 5900.' 05 0 . 2 5 9 4 D 05 0 . 1 2 50D 05 7044. 0 . 2 2 6 1 D 05 0.6560 1742 . 0 . 3 4 8 6 D 05 0.1995D 05 414.0 0 . 7 2 4 8 D 05 0.1692D 05 14.89 0 .6 415D 0 5 303 0 . 6735. 8337. 0 . 2 6 7 7 D 05 0.1208D 05 0.27530 1.12 8 5287. 05 0..64 85D _Q5__ 0 . 10 430 05 4.818 0.3682D 05 6843. 4536." 0 . 2 8 0 3 D 05 0.2574D 05 0 . 3 4 3 8 D 05 0.34870 1. 128 0.10820 05 0 .2739D 05 0 . 2 1 1 8 0 05 4_Q_8_. 0 0.9194D 05 0 . 1 7 7 9 D 05 7.420 0 . 8 216D 05 3390 . 2910.' 0 . 1 3 9 2 D 05 9743. 189.0 0 . 4 9 5 4 D 05 8908. 4.050 _.0.. .1.6.0.2 D._0_5_ 0 . 1111D 05 210. 0 0 .2966D 05 0 . 2 2 4 5 D 05 416.0 _0...2.2i.6D_-0.5 0 . 1 7 2 7 0 05 190.0 1553. 1551. . 2 8- .-0.0 0 . 1 5 4 5 D 05 9464. 646 8 . 1551. 18.75 0.3244D 8498. 05 05 6468. 0.0 4374.' 0.0 2 587 . 4938. 0 . 3 070D 05 966.0 1737. 0 . 1564D 05 • 0 . 1 2 7 4 D 05 1. 55.6 4690. 2 . 120 1361. 0 . 1 2 1 6 D 05 3.230 05 0.1469D 0 5 1039. 0 . 5 9 1 0 D 05 4 293. 1288. 4860. 0 . 1 1 4 0 D J35 988.9 0 . 1345D 0 5 3672. 626.5 2400. 1670. 362. 5 0 . 1222D 05 2613.. 908.3 7275 . 05 1069. 965 . 8 9040. ) • • • • • 0 1 • • • • • • • • • 6 0 9 * • • • 0 :ONTROL :AR.DS 1. JNJi.SJ3_C__3_P_ 2 . END • • • 1 2 • * • • • • 1 5 • • • • • • 1 8 • • • • • • 2 1 • * • 2 4 2 6 2 8 3 3 0 1 3 5 4 0 4 5 5 0 5 5 6 0 6 5 7 0 7 5 8 0 r R F S /NO. 019809 UNIVERSITY OF B C COMPUTING CENTRE MTS( AN059) JOB START: 16:03:13 APPENDIX D INPUT DATA AND THE CORRELATION MATRIX S S I G N C N P L 0 8 TIME=5M PAGES=50 C 0 P I E S = 3 6 PRIO=V THE. ISK SPACE A L L O T T E D T H I S USER I D HAS BEEN EXCEEDED. * * L A S T SIGNON WAS: 1 4 : 5 3 : 5 2 03-25-70 USER " P L 0 8 " S I G N E D ON AT 1 6 : 0 3 : 2 2 ON 0 3 - 2 5 - 7 0 $RUN * T R I P 4 = * S 0 U R C E * "_ EXECUTION BEGINS T R I P / 3 6 0 IMPLEMENTATION 3/18/70 VARIAR1.F NAMFS 1. Total T r i p Generated _2-.-_Popu-1-a-t-i-on ---T-o-ta-l 3. Population, Single Family 4. Population, M u l t i p l e Family JL. Labour F o r c e , Tot a 1 Labour Force, S i n g l e Family 6. Labour Force, M u l t i p l e Family 7. - 8 — -D-we-l-l-i-n-g—U-n-i-t-s-/—-Total 9. Single Family Dwelling Units 10. M u l t i p l e F a m i l y D w e l l i n g U n i t s 11. D w e l l i n g U n i t s P e r C a r 12. S i n g l e D w e l 1 i n g U n i t s W i t h C a r 13. M u l t i p i e F a m i 1 y D w e l 1 i n g Uni t s W i t h C a r -1-4-.—Car-s-,—Total 15. C a r s , P e r D w e l l i n g Uni t 16. P o p u l a t i o n P e r D w e 1 1 i n g U n i t 17. % o f D w e l l i n g U n i t s W i t h C a r 1 8 . S t u d e n t s ( 4 - 6 p.m.) 1 9 . T i m e t o CBD Xn M i n u t e s -2-0^—G ross—l-neome—(-1-GvO—E—-5-)21. Bus M i l e s 22. A r e a i n A c r e -2-3-.—Income P e r D w e l l i n g U n i t Emp&oyment, T o t a l 24 25. E m p l o y m e n t : P u b l i c U t i l i t i e s , G o v e r n m e n t and I n s t i t u t i o n a l S e r v i c e s -2-6-.- -Employment-: -l-ndus-t-r-i-a-1-,—V>/ho-l-esa-l-e-and—Unc-l-ass-i-f-i-ed — 27. E m p l o y m e n t : S e r v i c e I n d u s t r i e s 28. E m p l o y m e n t E n t e r t a i nment -13- E m p l o y m e n t D e n s i t y P e r A c r e 30. P o p u l a t i o n D e n s i t y ^ P e r Acre / ; 03-25-70 $DATA LABFOR AREA TRIPGN OUWC 2880.000 1.229 -1.127 864.000 0.682 0 . 4 8 7 225G8.COO 1 1 8 7 6 . 0 0 0 4.895 2 . 5 8 4 2 7 5 2 6 . 0 0 0 19948.CCO 4.818 1.221 3 4 8 7 0 . 0 0 0 1 7 2 7 0 . 0 0 0 1.782 3 7 4 4 6 . 0 0 0 21183.OQO 7.420 9743.000 0 . 336 1 7 5 3 0 . 0 0 0 4.050 0 . 5 6 2 1 4 7 1 1 . 0 0 0 11114.OCO 5.480 2.026 38212.000 22446.000 9.810 8190.000 4809.000 -0.279 15.396 6353.OCO - 0 . 329 8395.COO 11.970 18.753 -0.991 3364.COO 1551.000 16.006 0.501 12744.000 9464.000 9.788 1.196 1 5 9 6 7 . 0 0 0 1 1 6 3 8 . 0 0 0 21.642 - 0 . 0 6 0 1 2 7 2 4 . 000 8674.OCO 2.073 -0.851 5406.000 2225.000 4,312 0.463 12865.000 9057.000 0.870 16817.000 11623.000 27.678 -0.687 5.081. 000 3 0 0 1 . 0 0 0 27.678 - 0 . 0 1 3 11097.000 6910.OCO 4.031 -0.944 2238.000 1425.000 2.451 - 0 . 398 7638.000 4965.000 2.203 -0.502 5653.COO 3677.OQO 5.892 -0.411 7816.000 5119.000 3. 654 -0.939 1634.000 1139.000 3.728 -0.573 3778.000 2530.000 3.158 -1.156 69.COO 46.000 3.296 -1.040 960.COO 706.OCO 2.855 -0.838 32 5 4 . 0 0 0 2178.OCO 3.397 35.827 1. 145 1 8 4 3 9 . 0 0 0 1 3 0 1 8 . 0 0 0 5925.000 26.974 -0.749 6772.COO 6469.OCO 51.086 -0.436 7505.000 3436.000 51.024 -0.851 2864.000 MEANS AND STANDARD DEVIATIONS 12.289 11779.780 7511.938 10509.600 6184.680 13.767 STANDARD SCORES OF THE VARIABLES 1. 127 - 0 . 847 -1.075 -0.803 0.487 1.021 0.706 -0.843 1.498 2.584 2.011 -0.537 2 . 197 1.221 1.578 -0.543 -0.354 I 1.782 2.210 2.442 -0.598 0.336 0,361 0.547 -0.495 0.562 0.582 0.279 -0.180 2.026 2.415 2.515 0.226 -0.279 -0,437 -0.342 -0,023 -0.329 -0,187 -0.322 -0. 801 0.470 -0.991 -0.964 0.092 0.270 0 . 501 0.316 0.398 -0.182 1.196 0.667 0.090 0.679 ,-0.060 0.188 -0.606 -0.742 -0.851 -0.855 0.103 -0.579 ! 0.463 0.250 1.118 0.665 0.870 0.479 -0.729 1.118 -0.687 -0.637 -0.097 -0.600 -0.013 •0.065 -0.984 -0.715 -0.944 -0.908 -0.412 -0. 733 -0.398 -0.394 -0.620 -0.465 -0.502 -0.583 -0.411 -0.939 -0.573 -1.156 -1.040 -0.838 1.145 -0.749 '-0.436 -0.851 COMPILE -0.377 -0.965 -0.761 - 1 . 114 -1.030 -0.811 0.634 -0.476 -0.:407 -0.848 T I ME= - 0 . 3 87 -1.030 -0.806 -1.207 -1.100 -0.862 0.890 -0.257 -0.169 -0.659 -0.627 -0.622 - 0 . 663 -0.653 -0.685 -0.646 1.710 1.067 2.818 2.814 0 . 2 1 S E C E X E C U T ION T IME= 0.50 SEC,OBJECT COOE = 1824 BYTES,ARRAY AREA • 1824 BYTES,UNUSED : 98752 BYTES m O TI m • $STOP EXECUTION TERMINATED $RUN -LQAD#+;*TRI P 4--A 5-.*SOURCE* EXECUT I.ON B E G I N S T R I P / 3 6 0 IMPLEMENTATION 3 / L 8 / 7 0 6=*SINK* 0 1 0 6 0 9 •*• ••• • * NOTE: OUTDATED 1 5 1 8 2 1 2 4 2 6 2 8 3 3 C 1 * • • • -• * • • * • • • * • » • • • * • • * • CONTROL CARDS " L. INMSDC 2. S T P R E G 3. END 1 2 4 3 1 1 1 1 1 1 3 4 • • 5 0 4 5 5 0 5 5 • 6 6 * • 0 5 7 0 2333 *INVIR* OR *MULREG* R O U T I N E S HAVE BEEN R E P L A C E D BY THE EQU1V A L E N I * S « P R b G * 7 5 >• 8 0 INMSDC CONTROL CARD NO, FORMAT CARDS (4F10.3) INPUT DATA TRIPGN LABFOR DUWC AREA -1.075 - 1 . 127 - 0 . 8470 . 7060 0.4870 1.021 2.011 2.584 1.498 1.578 1.221 2 . 197 2 .210 1.782 2 . 4 42 . 3 610 0.3360 0.5470 . 5 8 20 0.5620 0.2790 2.515 2.415 2.026 -0.3420 -0.4370 •0.2790 -0.3220 -0.1870 • 0 . 32-90 0 . 8 0 1 0 C.9640 •0.9910 0 . 9 2 0 0 D 0 1 0.3160 0.5010 0 . 3 9 8 0 0 .6670 1.196 •0.6:0000-01 0.9000D-01 0.1880 •0.8510 -0.6060 -0.8550 0.4630 0.1030 0.2500 0.8700 0,4790 0.6650 •0.6870 -0.6370 -0.7290 •0.1300D-01 -0.65000-01 -0.97C0D-01 0 . 9 0 8 0 •0.9440 -0 .9840 -0.3940 ^0. 3980 -0 .4120 -0.5830 '0.5020 -0 .6200 • 0 . 3770 •0.4110 -0 .3870 -0.9650 •0.9390 1.030 -0 .8060 -0.7610 -0.5730 1.207 -1.114 -1.156 1.100 1 . 0 3 0 -1.040 . 8620 0 . 8 1 1 0 -:0 -0.8380 . 89C0 0 . 6 3 4 0 0 1. 145 . 2 570 0 . 4 7 6 0 0 -0.7490 . 1690 0 . 4 0 7 0 -:0 -0.4360 -0.8510 -0.8480 -0.6590 32 OBSERVATIONS 31 DEGREES OF FREEDOM -0.8030 -0.8430 -0.5370 -0.5430 -0.3540 -0.5980 -0.4950 -0.1800 0.2260 •0.23C0D-01 0.4700 0.2700 -0.1820 0.6790 -0.7420 -0.5790 I.118 1.118 -0.6C00 -0.7150 -0.7330 -0,4650 -0.6270 -0.6220 -0.6630 •0,6530 -0.6850 -0,6460 1.710 1. 067 2.818 2.814 NAME MEAN S.D. TRIPGN-0.3125D-04 0.9999 LABFOR 0 . 3 1 2 5 0 - 0 4 0 . 9 9 9 9 DUWC 0.6250D-04 1.000 AREA 0.6250D-04 1,000 CORRELATION VARIABLE TRIPGN LABFOR DUWC AREA MATRIX TRIPGN 1,0000 0,9211 0.9700 -0.0366 LABFOR 1.0000 0.9752 -0.0836 DUWC AREA 1.0000 0.0115 l.COOO CONTROL CARD NO. 2 * STPREG * DEPENDENT VARIABLE RSQ = FPROB.; = STD ERR Y = VAR CONST. LABFOR DUWC AREA IS TRIPGN 0*9648 0.0000 0.1974 COEFF -0.1064D-03 -0.7354 1.68 83 -0.1175 STD ERR 0.0349, 0.1773 0 . 1767 0.0392 F-RATIO FPROB. 17.1945 91.2683 8.9625 0.0003 0.0000 0.0056 r~ V i > ' i 1 NO. 08SERVED CALCULATED RESIDUAL 1. 2. 3. 4. 5. 6. 7. 8. 9. 10. 11. 12. 13. 14. 15. 16. 17. 18. 19. 20. 21. 2 2. 2 3. 2 4. 25. 26. 27. 28. 29. 30. 31. 32. -1.1270 0.48700 2.5840 1.2210 1. 7820 0.33600 0.56200 2.0260 -0.27900 -0.32900 -0.9910,0 0.50100 1.1960 -0.60000€-01 -0.85100 0.46300 0.87000 -0.68700 -0.13000E-01 -0.94400 -0.39800 -0.50200 -0.41100 -:0.93900 -0.57300 -1.1560 -1 .0400 -0.83800 1.1450 -0.74900 -0.43600 -0.85100 -1.0979 0.54008 2.3566 1.1123 1 .'9769 0 . 2 7739 0 . 8 3549 2.2489 -0.51296 -0.76331E-01 -1.0938 0.43403 0.85472 0.17134 -0.91082 0.41426 0 , 6 3 904 -0.89382 -0.45581E-01 -0.9C970 -0.31984 -0.56352 -0.30259 -0.95637 -0.72339 -1.142 0 -1.0194 - 0 . 7 8316 0.83537 - 0 . 2 0933 -0.31722 -0.81974 -0.29144E-01 -0;53082E-01 0.22737 0 . 10874 -0.19491 0.58610E-01 -0.27349 -0.22289 0.23396 - 0 . 2 5267 0.10284 0.66970E-01 0.34128 - 0 . 2 3134 0.59815E-D1 0.48740E-01 0.23096 0.20682 0.32581E-01 -0.34301E-01 -0.78159E-O1 0.615156-01 -0.10841 0.17372E-01 0.15039 -0.14006E-01 -0.20646E-01 -0.54840E-01 0.30963 -0.53967 -0.11878 -0.31263E-01 I STOP 0 EXECUTION TERMINATED $ SIGNOFF * END OF CONTROL SET * NO. OBSERVED CALCULATED \ RESIDUAL ) - RFS UNIVERSITY NO. 0 1 9 8 1 0 OF B C COMPUTING CENTRE APPENDIX FACTOR A N A L Y S I S : MTS<AN059) JOB START: 14:53:37 F OUTPUTS FOR THE T R I P G E N E R A T I O N DATA < • SSIGNON P L 0 8 TIME= 5M P A G E S = 5 0 C 0 P I E S = 3 6 P R I O = V THE DISK SPAC.F A L L O T T E D T H I S USER ID HAS BEEN E X C E E D E D . * * L A S T SIGNON WAS: 1 1 : 5 5 : 1 4 03-16-70 USER " P L 0 8 " S I G N E D ON AT 1 4 : 5 3 : 5 2 ON 0 3 - 2 5 - 7 0 $RUN * F A C T 0 + * S S P 1=-A 2=-B 4 = * S 0 U R C E » ' EXECUTION BEGINS VARIABLE ~ C 03-25-70 . . NAMES 1. Population, Total 2. Population. Single Family 3. Population, M u l t i p l e Family 4. Labour Force, Total 5. Labour Force, S i n g l e Family 6. Labour Force, M u l t i p l e Family 7. Dwelling Units, Total 8. Single Family Dwelling Units 9. M u l t i p l e Family Dwelling Units 10. D w e l l i n g U n i t s W i t h C a r 11. S i n g l e F a m i l y D w e l l i n g U n i t s W i t h C a r 12. M u l t i p l e F a m i l y D w e l l i n g U n i t s W i t h C a r 13. C a r s , T o _ a l 14. C a r s P e r D w e l l i n g U n i t 15. P o p u l a t i o n P e r D w e l l i n g U n i t 16. % o f D w e l l i n g U n i t s W i t h C a r 1 7 . S t u d e n t s ( 4 - 6 p.m.) 1 8 . T i m e t o CBD i n M i n u t e s 19*. G r o s s I n c o m e ( 1 0 . 0 E - 5 ) 20. Bus M i l e s 21. A r e a i n A c r e 22. Income P e r D w e l l i n g U n i t 23. Employment. T o t a l 24. E m p l o y m e n t : P u b l i c U t i l i t i e s , Government! a n d I n s t i t u t i o n a l 25. E m p l o y m e n t : I n d u s t r i a l , W h o l e s a l e a n d U n c l a s s i f i e d .2 6 _ . _ E m p l o y m e n t : S e r v i c e I n d u s t r i e s 27. E m p l o y m e n t : E n t e r a i n m e n t 28. E m p l o y m e n t D e n s i t y P e r A c r e 29. P o p u l a t i o n D e n s i t y P e r A c r e Services FACTOR A N A L Y S I S . . . . . T R I P G E N 2 NO. OF CASES NO. OF VARIABLES MEANS 29389.28125 32 29 25649.21875 7 5 1 1 . 9 3 7 50 21.34370 3067.12500 3833.81250 6 302.84375 414.07495 749.28125 11782.90625 1209.09375 133.84375 3452.40625 9441.43750 10470.53125 12.60173 5452.56250 2338.34375 1.21984 5885.34375 8876.59375 3.40843 9695.43750" 7092.12500 _ 0» 866 06 '2485.15625 STANDARD DEVIATIONS 24488.33984 22260.67187 3209.49121 6184.67969 3490.11816 9.68599 4312.88281 5610.63672 6598.98828 '5381 . 2 9 2 9 7 357.14966 1369.17456 10512.91797 1990.07080 129.91887 10704.78125 87 24. 8 5156 8460.46875 13.70711 10368.73047 4438.44922 0.25744 1806.46021 7599.47656 0. 55 295 14019.75391 6247.10156 0.12532 3532.57666 IJ3X. 03125^ 1 0 0 3 . 1 2 50 0 3378.18750 CORRELATION COEFFICIENTS ROW I 1.00000 0.95925 0.05621 0.96421 0.47248 -0.03540 0.46717 0.97036 0.22177 0 .97052 -0. 12388 0.19247 0 .95753 -0.07017 0.36548 0.41619 -0 . 0 9 8 4 4 0 . 11773 0.97406 0 .27016 0.111 14 0.96871 -0 . 3 5 4 5 5 -0 .10617 0.42106 0.82999 0.23183 0.98667 0.70158 _0_. 96421 0.99015 0.14496 1.00000 0.23192 0.01458 0.21696 0.93236 0.13422 0.88620 -0.00271 0.08740 0.98556 0.07354 0.29002 0.16207 0.02918 0 .04199 0.88914 0.-23537 0 .03557 0.99349 -0.21202 •0.17758 0 . 17174 0 . 7 4 3 07 0.02955 0.93615 0.70002 3 0.46717 0.23162 -0.26425 0.21696 0.96876 -0.18142 1.00000 0.46551 0.36923 0 .61909 -0.44471 0.42439 0 .23959 -0.50232 0.37834 0.99295 -0.45322 0.29059 0.62164 0.20434 0.28751 0.25279 -0.58950 0.20075 0.98001 0.57817 0 . 7 5 5 17 0.51326 0.23116 4 0.97052 0.89098 . rJ3„.0A8_8_9_ 0.88620 0.62114 -Q, 12287 0.61909 0.95289 0.27727 1.00000 - 0 . 18825 0.26078 0.90949 -0.20048 0.41159 0.58002 -0.17698 0 .166 39 0 .98582 0.25101 0.16112 0 .90438 -0. 44528 -0.05111 0.57409 0.82642 0.35555 0.97511 0 .68464 ROW ROW fkOW f^OW 5 0.95753 0.98376 0.08289 0.98556 0.25476 -0.04176 0.23959 0.93531 0.15853 0.90949 -0.00096 0.11165 1.00000 0.02592 0.31306 0 .18888 0 .01994 0.06473 0.89319 0 .227 33 0.05732 0.98845 -0.25107 -0.15895 0.19108 0.72274 0.03227 0.93794 0.73019 HOW 6 0.41619 0.17698 -0.27810 0.16207 0.96770 -0.20894 0.99295 0 .4182'7 0 . 3 4 5 26 0.58002 -0.44240 0.39837 0.18888 -0.52412 0.35983 1.00000 •0.45699 0.26696 0.57827 0.14788 0.26859 0 .199 39 -0. 55983 0.19093 0.98126 0.53547 0.77497 0.46537 0.18677 RO.W... 7 0.97406 0.89656 0.00941 0 .88914 0.64002 -0.04492 0.62164 0.95931 0.24726 0.98582 -0.21200 0.23939 0.89319 -0.24062 0.34926 0.57827 •0.19470 0.15223 1.00000 0.31910 0.15553 0.91080 -0.42371 -0.05584 0.59516 0.88286 0.39276 0.98604 0.68157 JO.JL9349_ 0.27223 0.00960 0.25279 0.94738 0.14064 0 .90438 -0. 02353 0.10268 0.98845 -0.00345 0.27831 0 . 19939 0.00048 0.05393 0.91080 0.28429 0.04586 1.00000 -0 . 2 2 2 0 3 -0. 17201 0.21032 0.76776 0.04360 0.95172 0.72169 ROW . . 8 0.96871 0 .99313 0.14682 ROW 9 0.42106 0.18999 -0 . 2 6 3 1 7 0 . 17174 0.98567 -0.12481 0.98001 0.42765 0.31179 0 .57409 -0.45607 0.36706 0.19108 -0 . 5 6 2 7 8 0.28524 0.98126 -0 . 4 6 1 8 6 0.25559 0.59516 0.20235 0 .27919 0.21032 -0.57118 0 .20265 1 .00000 0. 59630 0.84516 0.48247 0.20918 0.93615 0 .53836 JCL.J11A.8A. 0.51326 0.98773 0.19910 0.97511 - 0 . 10492 0.19773 0.93794 • 0 . 16131 0 .30595 0.46537 -0.08114 0 . 10411 0 .98604 0.31679 0.09963 0.95172 -0.35121 -0.11871 0.48247 0 .88319 0 . 2 7 5 18 1.000 00 0.70461 .ROW 11 0.95925 1.00000 0.17289 0.99015 0.24892 0.07303 0 .23162 0 . 9 5 788 0.12930 0.89098 0.03192 0.10847 0 . 9 83 76 0.01911 0.25478 0. 17698 0.05252 0.04241 0. 89656 0.28383 0.03319 0.99313 • 0 . 19745 -0.18443 0.18999 0.78797 0.01832 0.95020 0.70943 ROW 12 0.47248 i 0.24892 -0.28399 0.23192 1.00000 -0.14514 0.96876 0.47945 0.26910 0.62114 •0.41239 0 .32118 0.25476 • 0 . 55300 0.26189 0.96770 -0.39419 0.20889 0.64002 0.21701 0.21989 0.27223 -0.55755 0.12978 0.98567 0 .61404 0.80566 0.53836 0.27140 -ROW 130.97036 0.95788 0 .09183 0.93236 0.47945 0.07984 0.46551 1.00000 0.18233 0.95289 -0.01176 0 . 19680 0.93531 •0.13004 0.27999 0.41827 -0 .03365 0.09050 0.95931 0.30629 0.07184 0 . 9 4 7 38 -0. 29792 - 0 . 14484 0.42765 0.89425 0 .19696 0.98773 0.70037 •JL.JW21X.- 0 . 4 4 4 7 1 -0.41239 0.38410 -0.01176 -0.63 767 -0.18825 1.00000 -0 .56198 -0.00096 0.59437 -0 . 5 5 3 9 2 -0.44240 0.82225 -0.64713 -0.21200 -0.07846 -0.68859 -0.02353 0.51103 -0.69365 •0.45607 -0 . 0 6 5 4 7 -0.47498 l p . 10492 - 0 . 18714 -0.50232 -0.13004 -0.39073 -0.20048 0.59487 -0.43024 0.02592 1 .00000 -0.22067 -0.52412 0.62714 -0.42280 -0.24062 -0. I l l 14 -0.44825 -0.00345 0.34379 -0.43887 -0.56278 • 0 . 27420 -0 . 4 7 7 5 8 -0.16131 -0.20370 0.10051 0.07354 -0. 55300 0.10420 feOW 16 -0.09844 0.05252 0.2349 8 0.02918 -0.39419 0.35569 -0.45322 -0.03365 -0.73668 - 0 . 17698 0.82225 -0 . 6 7 5 6 9 0.01994 0.62714 -0 .62640 -0.45699 1.00000 -0 . 7 4 9 3 5 -0. 19470 -0.09087 -0.78134 0.00048 0.49030 -0.79977 •0.46186 -0 . 10739 -0.44571 -0.08114 -0 . 2 1 6 9 3 fcOW 17 0.27016 0.28383 - 0 .05604 0.23537 0.21701 0.03345 0 .20434 0.30629 0 . 1 0 74*8 0.25101 -0.07846 0.21802 0.22733 -0.11114 -0.02778 0.14788 -0 . 0 9 0 8 7 0.10268 0.31910 1.00000 0.13052 0.28429 •0.20824 0.02806 0.20235 0.39389 -0 . 0 0 0 8 7 0.31679 0.35792 f^OW 18 -0.35455 -0 .19745 0.65409 -0.21202 -0.55755 0.16952 -0.58950 - 0 . 2 9 79'2 -0.58996 -0.44528 0.51103 -0 . 5 6 7 5 2 -0.25107 0.34379 -0.58126 -0.55983 0.49030 -0. 52716 -0.423 71 -0.20824 -0.57782 •0.22203 1 .00000 -0.47958 -0.57118 -0.36158 -0.50942 • 0 . 35121 -0 . 5 9 4 9 0 ftOW 19 0.82999 0.78797 0.10289 0.74307 0.61404 0.33779 0 .57817 0.89425 0.12524 0.82642 -0.06547 0.19734 0.72274 -0 . 2 7 4 2 0 0.11071 0.53547 -0 .107 39 0.07242 0. 88286 0 . 3 9 3 89 0.08936 0.76776 •0.36158 -0.09667 0.59630 1.00000 0.43346 0 .88319 0.60755 HOW 20 - . 0.7CL58 0.70943 -0.30866 _0.7000L2_ 0.27140 0.02019 0.23116 Q.7003'7 0.41969 0.68464 •0.18714 0.37144 0.73019 •0.20370 0 .42789 0 . 18677 0.21693 0.38458 0.68157 0.35792 0 . 3 7 3 54 0.72169 -0.59490 0.18402 0.20918 0 .60755 0.04389 _0. 70461 1.00000 ROW 10 0.98667 0 .95020 JX.D-5-9..0.5- ROW 14 -JL.J..23_8.8_ 0 .03192 0.26180 *0W 15 -0.07017 0.._OL15L1.1_ ROW 21 0.05621 0.17289 1.00000 0.14496 •0.28399 0.41068 -0.2642 5 0.09183 -0.24342 -0 .04889 0.26180 -0.20642 0 .08289 0.10051 -0.26838 -0.27810 0 .23498 -0.21582 0.00941 -0 .05604 -0.24105 0.14682 0 .65409 -0.23787 -0.26317 0.10289 -0.30253 0.05905 -0.30866 ROW 22 -0.03540 0.07303 0.41068 0.01458 -0.14514 1.00000 -0.18142 0.0798™4 -0. 26495 -0.12287 0.38410 -0.13360 -0.041 76 0.10420 -0.46210 -0.20894 0.35569 -0 .18059 -0 .04492 0.03345 -0.19 5 73 0.00960 0.16952 -0. 19888 -0. 12481 0.33779 •0.08684 0.01684 0.02019 ROW 23 0.22177 0.12930 -0.24342 0.13422 0.26910 -0.26495 0.36923 0. 18233 1 .00000 0 .27727 -0.63767 0 .95173 0 .15853 -0.39073 0.90045 0.34526 -0.73668 0.97259 0.24726 0 .10748 0.95007 0.14064 -0 .58996 0.89748 0.31179 0.12524 0.10408 0 . 19910 0.41969 ROW 24 0 .19247 0.10847 •0.20642 0.08740 0.32118 -0.13360 0.42439 0 .196 8'0 0.95173 0.26078 -0.56198 1.00000 0. 11165 -0.43024 0.78928 0.39837 -0.67569 0.91831 0 .239 39 0.21802 0.89397 0 .10268 -0. 56752 0.83357 0.36706 0.19734 0.09824 0.19773 0.37144 ROW 25 0.36548 0.25478 -0.26838 0.29002 0.26189 -0 .46210 0.37834 0.2 7999 0.90045 0 .41159 -0.55392 0.78928 0 .31306 -0.22067 1.00000 0 .35983 -0 .62640 0.79698 0.34926 -0 .02778 0.77321 0.27831 -0 .58126 0.70013 0.28524 0.11071 0.10495 0.30595 0.42789 ROW 1 26 P_._l_l_7_73_ 0.04241 -0.21582 J3.04199_ 0.20389 -0.18059 0.29059 0 .09050 0 .97259 0.16639 -0.64713 0.91831 0.06473 -0.42280 0.79698 0.26696 •0.74935 1.00000 0 .15223 0.10268 0.96904 0.05393 -0. 52716 0.94589 0.25559 0.07242 0.07346 J > » 10411 0.38458 ROW 27 0.11114 _ 0_.„033_19L -0.24105 0 .03557 0_t2198_9_ -0. 19 573 0.28751 0.07184 0.95007 0.16112 -0 .68859 0.89397 0.05732 •0.44825 0.77321 0.26859 0.78134 0.96904 0.15553 0.13052 1.00000 0.04586 -0.57782 0 . 9 5903 0 .27919 0.08936 0.18176 0 .09963 0.37354 ROW 28 - 0 . 10617 -0.18443 -0.23787 -0. 17758 0.12978 -0_«.1_9_8_8_8_ 0.20075 -0.14484 JL.8974'8 -0 .05111 -0.69365 0 .83357 -0.15895 -0.43887 0.70013 0.19093 -0.79977 0.94589 -0.0 55 84 0.02806 0.95903 -0.17201 -0 .47958 1.00000 0.20265 -0.09667 0.16498 -0.11871 0.18402 ROW 29 0.23183 0.01832 •0.30253 0.02955 0.80566 -0.08684 0.75517 0.19696 0.10408 0.35555 -0 .47498 0.09824 0.03227 -0.47758 0.10495 0.77497 -0.44571 0.07346 0.39276 •0.00087 0.18176 0.04360 -0. 50942 0.16498 0 .84516 0.43346 1.00000 0.27518 0.04389 EIGENVALUES 12.38673 0 .13926 0.00156 7.31579 0.10893 0.00086 3.63327 0.06794 0.00031 1.52941 0.03440 0 .00011 1.23195 0.03048 0 .00008 0.88700 0.01884 0.00000 0 .77393 0.01307 0.00000 0.45769 0.00457 0.20327 0.00350 0.15438 0.00203 C U M U L A T I V E PROPORTION OF E I G E N V A L U E S 0.42714 0.67942 0.80470 0.99012 0.99388 0.99622 0.99999 1.00000 0 .99996 0.85744 0.99741 1.00000 0.89993 0.99846 1.00000 0.93051 0.99911 1.00000 0.95720 0.99956 1.00000 0.97298 0.99971 0.97999 0.99983 0.98532 0.99990 I EIGENVECTORS 0.24842 0 .21422 -0.04884 VECTOR 2 -0.16741 -0.21844 -0.14789 VECTOR 3 0.05551 0.14271 0.08019 -v-ECXOR- -0.04501 0.01192 0.56068 VECTOR 5 VECTOR 6 _j ! i a...0.8.5.2.8_ 0 . 0 4591 0.42497 -0.01275 JX._0.20_9_5_ 0 .07163 VFf*. TOR 7 0.00478 -0.04223 0.15361 VECTOR 8 0.09751 0.01449 0 .10578 VECTOR 9 \ ECTOR 10. VECTOR 11 -0 .041 18 0 .01382 -0.24297 -0.02965 -0.03895 0.43811 0.05143 -0.09355 -0.05986 VECTOR 12 VFf. TDR 13 -0.01479 -0_.1.1_7_9_9__ 0 .31871 0.21196 0.20454 -0 . 0 3 8 9 4 0.20757 0.24249 0 . 15754 0.26292 •0.12817 0.15377 0.21695 -0. 11785 0 . 16811 0 .19594 -0.13284 0 . 13256 0.26418 0.09391 0.13454 0.21924 -0.18417 0.08027 0 . 19885 0.23050 0.13472 0.25220 0.20831 -0 . 2 1 3 6 1 0.06978 -0.12219 0.09524 -0. 18263 0.25185 -0.12432 -0.24919 0.24101 •0.20314 -0.17669 0.18858 0.10450 -0.27251 0.26738 -0.12996 -0 .03198 0.27403 -0.20867 - 0 . 15641 0.31497 0 .09840 -0.14039 0 .10091 - 0 . 16761 -0.05301 0.15144 -0.33599 -0.01267 -0.30194 0 .03773 0.22941 •0.01338 0.00480 0.18155 0.14891 0.14629 0.22180 •0.32285 0.02893 0.23565 -0.03086 0.01420 0.21353 0 . 1 3 6 50 0.03283 0.18983 -0.33874 -0.09247 -0.37520 0.01606 0.18845 -0 . 0 4 9 8 7 0.01033 0.57454 0.00597 0.04890 0.04319 -0.08045 0.00365 0.15173 -0.09265 -0.26187 •0.18703 -0.00877 -0.07425 0.12219 0.00246 0.14369 0 . 1 1 3 31 -0.02450 0.23254 0.12691 0.05374 0.25619 -0 . 0 0 6 6 7 0.01369 •0.10614 J0_.JX8J3.3_ •0.00652 • 0 . 35275 J0_._Q3_653_ 0.01485 0.02307 0.09158 • 0 . 18192 -0.07939 0.07549 -0.06248 0.18825 0.06820 -0.16107 -0 . 0 1 7 4 1 0 .06285 -0.50514 -0 . 0 3 7 5 0 0 .07203 0.36864 0.00133 0.00855 -0 . 1 5 1 0 4 0.04336 0.03785 -0.35251 0 .00743 :_0.._Q.2.65_5_. -0 . 4 2 7 6 7 -0.07567 -0.. 03.565 -0.11188 -0.02543 -0. 28562 -0.10089 0.01180 -0.29147 - 0 . 17099 -0.08320 -0 . 2 2 1 4 1 -0.08262 0.03506 0 . 6 7 5 00 -0.04060 0.06910 0.20297 -0.04394 •0.05171 •0.06869 -0.08748 0 .00969 0.00316 -0.06623 0.11805 iO__lJit8.L 0.25487 0.03874 0.16163 0.03894 0.32171 0.28303 -0.06944 0.39220 0.20510 0.23002 0.21446 0 .06577 -0 .00643 0.3 7 342 -0 .02577 -0 . 0 7 2 7 4 0 . 14833 -0.05580 0.12669 -0.05566 -0.27937 0.00124 -0.27621 0.11979 • 0 . 10544 0.12852 -0.04017 • 0 . 10838 -0.04874 -0.04630 -0. 32049 • 0 . 14588 -0.01021 0.66177 -0.00886 -0.09151 - 0 . 16813 -0.04431 0.03057 0 . 2 0 9 13 0.10413 0.03528 -0.12624 0.13569 0.00423 0.06380 0 . 4 0 5 38 -0.02132 -0.24183 -0.01155 -0.01169 -0.15953 • 0 . 11955 0 .12538 -0.01312 0.00190 0.59574 -0.07756 0 . 0 3 2 87 0 . 0 6773 -0 .13295 -0.06142 -0.37861 0 .08426 0 .00036 -0.00322 0.17183 -0 . 0 1 9 4 5 0.38689 0.22902 0.03875 0.18281 0.27817 0.00826 -0.01495 -0.03009 -0.14153 -0.11713 - 0 . 0 25 58 -0.07646 -0.01737 -0.01979 0.41240 •0.10402 -0.00920 -0.15853 0.36436 - 0 . 0 2 9 13 -0.22659 -0.23757 -0.02568 0.10684 -0.03777 -0.00583 -0 . 5 0 1 4 6 -0.07471 •0.04931 0.01689 0.17308 -0.07943 0.08186 -.0_._0.0_9.03_. -0.04098 0.20093 0.16491 0 .00037 -0.02 5 8 7 -0.02921 •0 . 0 8 4 8 4 0.00273 -0.12333 0.22311 0.09538 0.17536 -0.64810 -0.02457 -0.00617 •0.06961 -0 .32775 0 .00429 0.13334 -0.22216 10 .JL2356. 0.06211 -0.32464 IO.J3.9459 - 0 .04926 _J3_._27_623_ -0.08035 0 .04198 -0.16046 0.00559 -0.00190 0.03563 -0.43365 -0.02680 0.18718 -0.04987 0.04708 0.16183 0.26037 0.04476 -0.00004 0.13845 -0.01117 0.13065 0.11517 0.12903 -0.20500 0 .00996 -0 .0 1379 0.58065 0.28852 I -0.02864 0.02488 -0.12159 0.00422 0.02186 0.07504 -0.13455 0 .05431 0 . 11649 •0.17749 - 0 . 14569 -0.06095 • 0 . 19238 -0 .0 70 48 0.62821 -0.04181 0.23173 -0 . 1 5 4 0 4 -0.07283 0.11184 -0.10949 -0.07642 0.39626 -0.08257 -0 . 0 2 5 1 5 0 . 2 4937 0.27386 0.02866 0.18062 VECTOR 14 0.05895 0.02692 -0.14794 -0.00093 -0.04792 0.10991 0.15610 0.14812 0 . 0 0 155 -0 . 0 7 1 4 7 0.03550 -0.64046 -0 . 0 7 5 7 6 -0. 13838 0.20736 -0.01919 0.01067 0.32384 0 . 0 3 3 21 0 .07894 -0.14255 0.04560 -0 . 13512 0.24812 -0.01120 0.11454 -0.27767 0.00802 -0.35132 VECTOR 15 0.05515 -0.03263 -0 .14510 0.0610 1 0.18293 0.40484 -0.00097 -0.22423 0.05009 0.17107 0 .13504 0.01180 0.25262 -0 .13288 0.19148 -0. 09206 •0.10419 -0.02302 0.08623 0.16786 - 0 . 1 5060 0.11843 0.14677 0 .03208 -0.02825 -0.63028 0.16355 0 . 0 30 57 -0. 12109 J/ECX0.RJ.6_ -0.20314 0.02095 -0.02082 -0. 31819 -0.53179 0.09674 0.15334 0 . 1086'4 -0.02711 0.40970 -0.09445 - 0 . 15125 0 .32551 -0.00305 0.05565 0.33177 0.07880 •0.03859 -0.07930 0 .07341 0. 16884 -0.08624 0 .12861 -0. 01910 -0.01943 -0.02290 0.07886 -0. 15289 0.08280 VECTOR 17 -0.14785 0.06089 0.11995 -0.10772 0.03582 -0.10325 0.02817 0.15925 -0.02444 0.10142 -0.05742 0.17308 0.19074 0.07145 -0.05461 -0. 13496 0.04139 0.07592 - 0 . 1 5 9 94 0.00276 -0 . 6 7 6 6 7 -0.18 568_ -0. 05075 0.48903 -0 .01944_ 0.03519 0.12245 0 .06452 0.12477 VECTOR 18 -0.25005 -0.27987 -0.01530 -0.17699 0.32368 -0.01276 -0.28862 •0.29415 -0.01579 0 .25035 -0.05756 -0 .01833 0 .37932 0.05964 0.13441 - 0 . 15712 -0.03206 0.00732 0.13307 -0 .02274 -0.00004 0.11715 -0.01775 -0.06843 0.08365 0.42268 -0.192 25 -0.13895 0.12679 VECTOR 19 -0 .06490 0.11649 0.00255 -0.07539 0.38483 0.07617 0.14635 0.19214 -0.19389 0.04671 -0 . 0 6 4 8 0 -0.09097 0.08972 0.06241 0.10331 -0.06127 • 0 . 11258 -0.45959 -0.2 86 75 -0.01617 0.34425 -0.22341 -0. 00450 0.22201 -0 . 2 3 6 1 8 -0.04264 -0.23070 0.22516 -0.00298 VECTOR 20 -0.12220 0.18327 -0.00951 -0.13614 0.02490 0.04002 -0.54989 -0.05577 -0 . 0 6 1 4 1 0 .02316 0.00643 -0.00656 -0.25791 0.04545 0.00801 0.55864 -0.00684 -0.15182 0.17287 -0.00082 -0.06975 0.21446 -0 . 0 6 0 0 0 0.29018 0.00086 -0 . 0 9 3 9 5 -0.10598 0.16649 -0.02418 VECTOR 21 0.01116 -0.05852 0.06822 -0 . 0 2 1 4 1 0.25059 0.01481 -0.38823 0.41743 0.08089 0.10852 -0 . 0 3 9 7 9 -0.06188 0.05461 0 .03196 -0.02327 0. 15866 •0.08840 0.37789 -0.2 73 61 0.06298 -0.02068 -0.21523 -0. 11222 -0.40859 -0 . 2 2 3 1 4 -0.10040 0 . 15553 0 .02923 -0.09592 VECLOR„22„ -0.40007 0.32929 0.01096 -0.51954 0.04104 -0.01390 0.17195 0.08720 0.11322 -0 .16710 -0.02168 -0.00780 -0 . 0 6 4 6 8 0.09946 -0.00068 -0.26141 -0.08371 0 . 10846 0.20810 -0 .04299 0.00175 0.25566 -0 . 0 3 7 5 1 -0.24411 -0.04710 •0.08587 0.09241 0.29996 •0.01749 VECTOR 23 „-0.1A59it 0.29184 -0 .00914 0.19123 -0.14052 0.04004 -0.21427 -0 . 0 6 5 2 6 0.06602 0.00911 -0 .01855 -0.08622 0.07377 0 .01375 0.03914 •0. 13489 •0.10029 -0.01923 -0.07768 •0.00065 0.02297 -0.41726 •0.03085 • 0 . 12262 0 .66972 -0.09136 -0.16261 0 .20914 -0.01794 VECTOR 24 I 0.25035 -0.28831 0.01937 -0.30252 •0.01380 0.03349 -0.17321 0.55697 0.07201 -0 .08715 -0.05942 0.00708 0 .00268 0.04085 0.01089 -0.21327 0.00558 -0.24506 0.21757 -0.00895 0.03718 0.09230 -0.01798 0.09131 0.35289 -0.18793 - 0 . 0 7 2 08 -0.25584 0.02307 VFCTOR 25 r"0.6 8 4 6 7 0.19930 -0 . 0 1 2 4 3 -0.5249 8 -0.01957 -0.02250 -0 . 0 9 2 1 4 -0.32455 0.0792*8 0 .03504 0 .02532 -0.01715 0.05425 -0 .0 2 3 4 6 -0.03298 -0.03210 -0.01482 0.02970 -0.12374 -0.00725 -0 . 0 3 6 4 8 -0.18302 0.01371 -0.0 3 0 6 7 -0 . 0 0 1 4 0 0.10408 -0.01389 0.15279 -0.01233 0.00683 -0.14829 0.00016 -0.00054 0.00133 0.04459 0.50544 -0.00021 -0 . 0 1 2 4 4 -0.41647 -0.00009 -0.01387 -0.20797 -0. 0 0 0 7 7 -0.01814 0.24600 0.00025 •0.00389 -0.20785 -0.00071 0.00055 -0.11104 0.00159 -0.00133 0.46785 -0 . 0 0 0 7 7 -0.02160 •0.07559 -0.0012 5 0 .00429 -0.0072 8 -0.02924 0.12127 0.00080 0.00946 -0.10351 -0.00008 0.00899 •0.05728 0.00047 0.01667 -0.6 8 2 7 3 •0.00053 0.00075 0.56739 0.00103 -0.00358 0.28325 0.00182 0.0037 5 0 .23802 0.00022 0.0000938 0.0000218 0.00002 94 0.0001085 0.0000615 0.0000060 0.0000592 0 . 0 0 0 0 5 29 0.0000124 0.0000181 0.0000111 0.0000136 0.0000182 0.0000172 0.0000108 0.0006304 0.0007294 0.00 10477 0.0042888 0.0009358 0.003 8 5 1 5 0.0322166 2.1083441 0.4274132 26.2166748 19.6783905 9.3527594 0.0293792 20.3988953 0.10580 -0 . 0 0 7 4 6 -0.00326 •0.05566 0.01093 0.00261 0.09466 0.00963 0.00615 -0.01201 -0.02788 -0.00001 0.06597 -0.01690 -0.01857 -0 . 0 0 3 8 4 -0.01165 -0 . 0 0 5 5 1 •0.57777 -0.01626 -0.01524 0.28866 0. 0 0 1 1 0 -0.06167 -0. 0 0 0 1 7 -0.00315 0.09693 0.01065 •0.00471 0.00700 -0.04367 0.00001 -0.05826 -0.01231 -0.00002 0.08104 -0.01196 -0.00521 -0.00446 -0 . 0 1 1 8 2 -0.00613 0.25760 0.00738 0.02895 -0.00180 0.04055 •0.00017 -0 . 0 0 0 8 3 -0.07127 0.02105 -0.00000 0.22421 0.00322 0.00000 -0 . 0 2 7 1 8 -0.01951 -0 . 0 5 3 9 0 0 .00866 •0.01005 -0 . 0 2 4 7 8 0.00505 •0.57553 -0 . 0 3 5 0 7 -0.00376 -0.33626 -0.00063 -0 . 0 0 4 9 0 -0.02550 -0. 0 4 6 2 6 0.00016 -0.09949 -0.01325 -0.00091 0.10165 0.02986 -0.02395 0.05623 0.03426 0.01159 -0.03132 0.01692 0.00086 0.00276 -0 . 0 0 7 7 7 0.00104 _a..5J_St£L6_ J1_2JEL3J_3_ -0.05014 0.083 79 0.04410 0.00049 0.01111 0.04468 -0.00067- -0.06109 -0 . 0 0 6 9 1 0.02564 0.01482^ 0.00531 _0..J)J3_36.2. -0.01162 0.00129 -0. 11458 •0.01405 0.00003 -0.61540 ._Q_.Si.l5J5±. - 0 . 0 1 0 9 1 3 -0.00237 -0.0076 7 -0.0 1 0 8 4 -0. 0 0 3 5 6 -0.00222 0.07570 -0.07836 0.20236 -0 . 0 1 5 4 3 -0.00006 -0.06191 -0.01062 VECTOR 2 6 -0.00324 -0.41005 0 . 0 0 0 15 VECTOR 2 7 0.01354 -0.19637 -0.00047 '. .6 RR.OR—B.O.U N D-S-F-0 R_EXGEflVALUES^ 0 .0001435 0.0000361 0.0000170 0.0001326 0.0000314 0 . 0 0 0 0 9 7"8 0 .0000381 0.0000151 0.0000583 0.0000237 0.0000157 0.0000235 0.0000409 0.0000121 0.0009697 0.0054044 0.0512877 E.RR-0.R^B0UNOS_E0R^EJ_G.E,NV.E.CXQR,S.. 0.0000566 0.0047718 0.0721269 FACTOR MATRIX VARTAB1 F _0_..OJO.O_4_t_ 2 0.74598 -0.00337 -0.00085 VARIABLE 0.0 00093"0 0.0022746 0.5858514 3 0.73054 0.06154 -0.01532 -0.45280 -0.00488 •JL.JLU.2A. 0.01385 0 . 0 0 3 35 —VARIABLE 0.92533 -0.01090 0.00428 I VARIABLE .. .__]. I 5 0.76356 -0.04602 0.00216 VARIABLE -0.00885 -0.00190 6 0.68961 0.06544 0.00626 _ AR T AR1 F 0.0111186 _ _ 0 . 0 3 1 3 6 56 0.0001429 ( 2 7 FACTORS) L 0.87431 0.01919 VARIABLE 0.0000720 0.0020713 0.0544068 0.28264 0.00031 -0.01607 -0.00029 0.00081 0.04554 -0.00033 0 .00420 -0.01690 0 .00001 0.00013 0.02181 -0 . 0 0 0 1 1 •0.02769 •0.00362 -0.01144 0.02517 0.92979 -0.00230 -0.01080 VARIABLE 8 0.77160 0.00160 -0.00850 VARIABLE 9 0 .69984 -0.00879 -0.00881 VART ABI F 10 0.88763 -0.03530 0.00115 VARIABLE 11 VARIABLE 12 VARIABLE 13 VARIABLE 14 0.75393 -0.03491 -0 .00231 0.71986 -0.01529 0.00989 0.85345 0.00014 0.01648 -0.45109 -0 .03166 -0.00157 VARIABLE 15 -0.41478 0.08326 0.00126 . VARIABLE.16 -0.46752 -0.24186 -0.00349 VARIABLE 17 VARIABLE 18 0.33052 -0.02598 0.00249 -0.64819 0.04976 -0.00443 V A R I A B t F 19 -0.35151 0.01477 0.00611 -0.05882 -0.01898 -0.00136 0.00304 0 .00616 0.00227 0.06975 0 .01505 -0.00111 0.03302 -0.01088 0.00040 -0.00565 -0.01828 -0 .00075 0 .02068 0.00899 0.00016 -0.01696 -0.01009 0.00779 -0.56441 -0.00369 0.00750 0.26019 -0.01992 -0 . 0 0 7 32 -0.03030 0.00846 0.00096 0.07995 0.02068 -0.00164 0.06508 -0.01184 -0.00033 -0.06399 -0 .02123 0.00062 0.02387 0 .00792 -0.00877 -0.01321 -0.00229 0.00966 0.26615 0.04258 -0.00138 -0.64568 -0.0065*6 0.01174 0.06646 -0 .00208 0.00368 0.00949 -0 .00493 -0.00001 -0.04870 -0.00267 -0.00018 0.11145 -0.00222 0.00031 0.00286 0.00565 0 .01747 - 0 . 0 1 3 97 - 0 . 0 1937 0.00004 .... -0.4533 5 -0.00455 0.00880 0.03061 0.00747 0.00367 0.01694 0.00149 -0.00267 0.04201 0.00534 0 .00137 0.00912 -0.02098 0.00075 0.00109 0.00738 0.00026 -0.01442 -0.00939 0.00373 0 .01332 ~-6T03121 0.00750 -0.59083 -0.03894 0.00966 0.27201 0.00649 0.00512 0.01474 0 .00499 -0.00300 0.05096 -0 .00570 0.00179 0.01973 0.00288 -0.00066 -0.03715 0.00696 -0 .00022 0.00981 -0.01892 0 . 0 0 6 23 0.00689 -0.01530 0.00826 0.18874 0.09117 0.00120 -0.64043 0 . 0 0 5 70 -0 . 0 0 2 4 6 0.01278 -0.00889 -0.00014 -0.00724 0.03194 -0 .00018 -0.02500 - 0 . 0 7 299 -0.00024 0 .10385 0.00409 -0.00008 - 0 . 0 7134 0 .02188 -0.00527 0 .02276 -0.05561 0.00112 - -0.49398 -0.05296 0.00256 0 .07192 0.0141"6 -0.00114 0.06047 0 .02747 0.00580 0.01648 -0 .03915 -0.00291 -0.03358 0.01491 0.00000 0.03408 0 .01821 -0 .00001 -0.07332 -0.01988 0.05653 0.01137 -0.03004 -0.00251 -0.67401 0 .01176 -0.00064 0.00916 -0.03797 -0.00033 0.00451 0.00658 -0.00062 -0.20192 0.02357 0 .00023 -0.26900 -0.01296 -0.00000 0 .28302 -0.00656 0.00000 -0.21682 -0.00389 0.26859 -0 . 0 0 3 8 3 0.16204 0.00029 -0.47790 0.06178 0.00292 0.2788 5 -0.01837 0.00024 -0.32386 -0.02566 0.00043 -0.06935 -0.02320 -0.00021 -0.27450 -0.00042 -0.00000 0.34503 0.00817 -0.00000 0.44770 0.00403 0 .03054 0.00369 -0.06229 0.002 05 -0.73709 0.05341 -0.00246 -0.05514 0.06040 -0.00176 -0.09182 0.00198 0 .00006 -0 .17878 -0.01819 -0.00013 -0.20853 0.01082 -0.00000 0.18866 0.00473 0.00000 -0.11375 -0.00217 -0.17069 -0.00666 -0TO8963""- -0.08649 -0.00001 -0.00126 0 .02707 0.02915 -0.00001 0.17770 0.01464 -0.00009 -0.56067 0.02930 -0.00007 0.63572 0.01008 0.00000 0.32851 0.00032 -0.00000 0.14148 -0.00154 -0.00145 -0.00096 0 .04198 -0.00004 -0.42305 0.04312 -0.00110 0.06258 0.10329 -0.00054 0.28758 -0.02506 -0.00019 0.40917 0.02562 0 .00012 0 . 19115 0.01765 -0.00000 0.13049 -0.00580 0.00000 -0.03540 -0.00120 0 . 17443 -0 . 0 0 0 2 7 -0. 19703 -0.00270 -0.00031 \ J \ 0.81124 0.0 2318 -0.00396 -0.37972 -0.06766 -0.00252 -0.176 26 0.06500 -0.00160 0.31683 0.02124 -0.00196 -0 .16765 -0.11003 0.00093 -0.06469 -0.00314 0.00000 -0.04897 0 .00402 0.00000 0.04316 0.02857 0.08242 •0.00252 0.00664 -0.00423 VARIABLE 20 0.73313 0.10767 -0.00379 -0.14338 0.19164 •0.00051 0.35921 0.04708 -0.00031 -0.13126 •0.06516 0.00024 -0.39126 -0.02114 -0.00011 0.00298 0.01137 -0 .00000 -0.24299 0.01426 0 .00000 - 0 . 1 6 361 -0.00857 -0.00674 -0.00018 0.03216 -0.00109 VARIABLE 21 -0.17188 -0.02234 0.00269 -0.40000 0. 10519 0.00032 0 .15285 -0.03169 -0.00016 0.69339 -0. 02744 0.00020 0.47169 -0.02533 -0.00011 0.06747 -0.00286 0.00000 0. 13513 0.01371 -0.00000 0.07157 •0.00103 -0.10954 0.00015 0.17214 -0.00043 -VAR LA B-L-E—22— -0.13705 0.07498 0.00058 -0.33049 -0 .02652 -0.00041 -0.02415 0.019 56 0.00070 0.71053 0.02038 0.00035 -0.39152 0.07068 -0.00020 -0.40279 0.01328 0 .00000 -0 .153 78 •0.01180 -0.00000 0 .08695 -0.00086 -0.07192 0.00451 •0.04602 0.00180 VARIABLE 23 Jl._5_5A4.5_ -0.00965 0.00319 -_Q_._6_8_120_ 0.00185 0.00332 JX«t4_3728_ 0.03036 0.00116 0.05341 0.00029 0.00075 0.02561 0.00874 0.00071 - 0 . 10537 -0.00372 0.00007 0. 14219 -0 .00279 -0.00003 -0.03297 -0.00107 -0.00592 0.01147 -0 .00682 -0.00277 0.65188 :0.UA3JL2_ 0.18764 -0. 11879 0.00007 -0.08811 0.00206 -0 .00015 -0.09502 •0. 02076 -0 .00002 0.24899 0.01979 0.00001 -0 .09869 -0.00124 - 0 . 0 349 7 -0 .00538 •0.04087 -0.00030 -0.00023 0.34605 _0_.JDX58'9_. -0.00151 0.51008 -0.01646 i0.000.0 2. 0.42278 0. 16374 0.00069 -0 .23130 0.03846 0.00011 0.20895 0.03343 -0.00030 -0. 16104 0.00764 -0.00002 0.18043 -0 .00624 0.00001 -0.00600 0.00908 -0.05994 0.00611 0 .14316 0.00036 VARIABLE 26 0.46655 -0.00917 0.01492 0.72319 0.08593 0.00318 0.44918 -0.04015 -0.00034 0.15111 0.06006 -0.00255 -0.019 32 -0.00402 0.00027 -0.07781 -0.00530 -0 .00003 0.05786 0.00868 0.00002 -0.02998 0.00049 0.03799 -0 .02719 -0.09335 -0.00684 VARIABLE 27 0.47350 -0.12231 -0.00082 0.74118 0.04570 0.00005 0.40702 -0.02 854 0.00040 0.14013 -0. 02644 0.00039 -0.04162 -0.02629 -0.00033 -0.03824 0.02317 -0.00001 -0.02267 -0 .07736 0.00000 0.07045 -0.00000 0.07747 0.02036 -0.01484 -0.00314 VARIABLE .28. 0.28252 -0.08290 -0.01613 0.85191 0.03801 -0.00716 0.36183 -6.0215"2 -0. 00215 0. 156 94 0 .04602 0.00095 0.00147 0.00560 -0.00028 -0.04138 -0.00262 0 .00000 -0 .04909 0.055 91 -0 .00000 0 .09180 -0.0046 3 0 .10326 0.01313 •0.02936 0.01307 -0.00824 -0.05150 •0.00075 0.04813 0.02855 -0.00012 -0.08239 0 .01082 -0.00000 -0.245 77 0 .01400 0.00000 0.27425 -0.01299 0.12541 •0.01365 0_.0_68 01 -0. 00478 VARIABLE 24 0 .54119 -0-..0.0-L0.2„ -0.00244 V A R T ARI F 25 0.59167 0.03560 _n0„.JXO„O92_ VARIABLE 29 0.474.16 -0.12115 0 .00614 ITERATION CYCLE 0 1 2 .0..27294_ 0.00329 0.00271 VARIANCES 0.146480 0.460208 0.498346 -A.. 71518_ 6. 07138 -0.00285 3 4 5 6 7 8 0.501713 0.502589 0.502614 0.502614 0.502614 0.502614 0.502614 ROTATED FACTOR MATRIX { 27 FACTORS) VARIABLE 1 0.96682 0.01495 -0.00200 0.06288 -0.00104 -0.01467 -0.22594 0.02656 -0 .00 164 -0 .02900 0.00359 0.00097 0.00165 0.00820 0.01116 0.02715 -0.01283 -0.00000 0.05941 -0.006 36 0.00001 0.03944 -0.03118 -0. 01299 -0.00758 0.02688 -0.00450 VARIABLE 2 0.99344 0.0101.4 -0.00650 0.00682 -0.00096 -0.02221 0.02852 0.01791 0.00166 -0.01585 -0.00219 -0.00117 0.02433 0.00734 -0.00155 0.01164 •0.02923 -0.00002 0.07473 -0. 00331 0.00004 0.04100 -0.03505 0 . 0 1 9 29 -0.00604 0 .0 1827 -0.00346 VARIABLE 3 0.24727 0 .00577 0.01113 0.20952 -0.02490 -0.00024 -0.93085 0.03023 -0.00340 -0 .05894 -0.00395 -0.00002 -0.05973 0.00165 0.00043 0.05619 0.03144 •0.00015 -0.02784 0 .00613 0.00011 0.00446 0.00340 -0. 10943 0.00403 0.03783 -0.03460 _V„ARJAB.L.E_ 0.90640 -0.01042 0.00154 0.09730 0.00144 -0.00063 -0.38689 0.00260 -0.00249 -0.09119 -0.01406 0.00002 -0.05006 0.0 38 10 0.00046 0.00367 0.03523 0.00086 -0.00938 •0.00336 -0.00011 -0 .01503 0.05721 -0 .03126 0.00152 0 .02681 0.00659 VARIABLE 5 0.99234 -0.01873 -0.00159 0.02181 0.00505 -0.00210 0.01690 -0.01498 0 .00162 -0 .06575 -0.00417 -0.00007 -0.02861 0.05246 0.00028 -0.00055 0.01563 •0.00062 0.01800 0 .00394 0.00011 -0.01585 0.06436 0.00906 0.00124 0.02813 •0.00536 VARIABLE 6 0.19655 jO_.J0UJ.5_ 0.00737 0.18782 L0..0_0_642_ 0.00282 -0.94657 0.0 3627 -0.00931 -0.08709 -0.02537 0.00025 -0.06226 -0.01294 0.00052 0.01018 0.05292 -0.00032 -0.05724 •0.01574 0.00024 -0 .00538 0.00894 -0 .09382 0.00102 0 .00802 0.02601 VARIABLE 0.90526 0.01812 -0.02804 0.07792 0.00984 0.00524 -0.40389 -0.00 263 0 .00194 -0.01732 0.01191 0.00540 -0.00163 0.01071 -0.00094 0.07326 -0.00760 0.00001 -0.042 73 -0 .00928 -0.00100 0.03964 -0 .00714 0.00614 -0 .00964 0.01654 0.00485 VARIABLE 8 0.99564 0.01727 -0.02617 0.00738 0.01308 0.00705 •0.00194 0 .00668 -0.00404 •0.02060 0.01240 0.00716 0.03264 0 .02041 -0.00080 0.05740 •0.00953 -0.00017 -0.00110 •0. 01247 0 .00064 0.03250 -0.00833 -0 .00732 -0.00894 0 .01878 0.00530 VARIABLE 9 0.20568 0.00912 -0.01496 0.17023 -0.00193 -0.00245 -0.95272 •0.01971 0.0 130 7 -0.00069 0.00415 -0.00097 -0 .06706 -0.01440 -0.00102 0.06184 0.00066 -0.00012 -0.09857 0 .00185 0.00008 0.0 3047 -0 .00080 0.02888 -0 .00487 0.00287 0.00192 0 .03798 0.00152 0.01023 -0.28686 0 . 0 0 248 0.00147 0.02204 -0 .00698 -0.00644 0.00369 -0 .00458 -0.003 27 0.07188 •0.01936 0.00090 -0.03463 0.00575 0.00011 -0.00617 -0. 01441 -0 .00206 0.00955 -0 .00674 0.00081 JV:AR.LAB.L.E_LO._ 0.95230 -0.03642 0.00971 V A R I A B L E 11 0.99530 -0.02782 0.01152 0.00308 -0.01446 0.01170 0.01510 0.01045 0.002 39 0.03716 •0.01166 -0.00755 0.04342 -0.01411 -0 .00371 0 .05832 0.01045 -0.00066 0 .00778 0.00336 0.00016 -0.01325 -0.01200 0.00338 0.01045 0.00090 -0 .00088 0.10967 0.04385 JL.0002_7_ -0.93233 -0.02 054 L0_.JL0J_?J. -0.03201 0 .00984 0.00018 -0.10595 0.02391 0.00024 0.06568 -0.08840 -0.00002 -0.12865 0.00880 0.00007 0.01665 •0.01241 -0.01555 0. 00143 -0.02340 0 . 005 03 V A R I A B L E 13 0.95505 -0 .01621 0.03949 0.02941 -0.02306 0.00539 -0.23467 0.01264 -0.000 81 0.07456 •0.01191 0 .00323 0.01308 •0.06418 -0 .00065 0.06427 0.02434 -0.00003 -0.04002 0.01534 •0.00017 -0.09603 -0.00355 0.04470 0.00309 -0.03892 -0.00201 V A R I A B L E 14 0.00746 -0.06727 0.00035 -0.60275 0.00301 0.00014 0.32194 0.002 7"4 0.00002 0.2018 5 -0.00371 -0.00012 0.05985 -0.00181 -0.00004 •0.01273 0.00045 0.00001 0.21239 0.00032 -0.00001 -0.65986 0.00026 -0.05251 -0. 00008 -0.02351 0.00004 -VAR.I-ABL.E_L5— 0.00794 -0 .03766 -0.00003 -0.34124 -0.01477 -0.00014 0.44154 0 . 0 0 723 0.00004 -0. 00431 0.00091 -0 .00003 -0.02608 0.00194 -0 .00004 -0 .03747 0.00005 -0.00001 0 .82019 0.00013 -0.00000 -0.10823 -0.00015 -0.01394 -0 .00011 -0.00022 -0.00002 V A R I A B L E 16 JX..0.263SL. -0.46447 0.00017 -0..J-253-.0-. •0.01079 0.00018 JL..3J.8.2.L 0.00574 0.00001 0.17644 -0.00557 -0.00004 0.01599 0.00364 0.00001 -0.02447 -0.00042 0.00000 0.23642 •0.000 10 0.00000 -0.24091 0.00017 -0.09824 - 0 . 00024 0.00987 -0.00002 V A R I A B L E 17 0.23090 0.0025.9 -0.00001 0.06195 J0_.OJDJ94 5-_ 0.00003 -0. 08327 -0.00501 -0.00011 0.03194 •0.00161 0.00003 -0.04473 •0.00145 -0.00006 0.96537 -0.00034 0.00000 -0 .023 56 -0.00022 0.00000 0.00417 -0.00016 •0.01048 -0.00006 0.00660 0.00001 V A R I A B L E 18 -0.22401 0.01114 _..0..0.0.0.2.2_ -0.47492 -0.04429 -0.00001 0.41593 -0.03092 -0.00002 -0 .04223 0.00167 0.00003 0.58644 -0 .00172 0.00001 •0.06405 -0.00048 -0.00001 0.003 83 0.00080 -0.00000 -0.07939 -0.00034 -0.0501 1 0. 00006 -0 .43530 0.00009 V A R I A B L E 19 0.77195 0 .05341 -0.00026 0.00552 -0.00836 -0.00009 - 0 . 4 3 504 0.00212 0.00011 0.36336 •0.01467 -0.00019 0.02536 •0.18036 0.00002 0 . 17298 0.00291 0.00000 -0.09599 -0.001.34 0.00000 -0.04037 -0.00227 0.07342 0.00012 -0.01411 0.00007 V A R I A B L E 20 0.72345 0.01985 -0 .00022 0.30221 0.37013 0.00003 0 .04364 0.01263 -0.00006 0.11550 0.01262 0.00001 -0.42231 0 .00141 0.00005 0.15043 -0.00098 -0.00000 •0.15470 •0.00050 0.00000 -0.01832 •0.00003 -0 .04136 -0.00026 0.07267 0.00012 -VAR.LABLE_2-1_ 0.12376 -0.00443 -0.00001 -0.15449 0.00866 -0.00012 0.21036 0 . 0 0 3 73 0.00010 0.21676 0.00098 0 .00002 0.93064 -0.00036 -0.00017 -0 .02470 0.00014 -0.00000 -0 .02808 -0.00014 -0.00001 -0.00967 -0.00056 •0.00875 0 .00021 0.04107 0.00001 V A R I A B L E 22 0 ..02662 -0.02368 0.00031 -IL..17-2 50_ 0.00854 0.00030 0.09221 -0. 003 95 0.00019 0.96364 -0.00038 0.00020 0.16344 0 .00354 -0.00014 0.02336 -0.00013 -0.00000 0.00714 0.00030 -0.00001 -0.06560 -0.00036 0 .00070 0.00007 _0 ,0_0 7 2 3^ -0.00016 VAR TABI F 1? 0.26817 -0.03796 _n0_..0-0_0_93_ L VARIABLE 23 0.13351 -0.02947 -0.00271 0.97154 0.01561 0.00125 -0.12164 0 . 0 66 39 0.00163 -0.08175 -0.02105 0.00027 -0.07387 0.00359 0.00059 0.00490 0.00158 0 .00004 -0 .01423 0.00427 - 0 .00011 -0 . 0 0 7 3 6 -0.00545 -0.06630 -0.01230 0.01967 -0.00042 VARIABLE 24 0.09701 -0.05385 0 .00054 0.92269 -0.06967 0.00004 -0.19166 0.02481 -0,00015 0.04211 -0.20732 -0.00006 -0.04850 -0.00961 -0.00001 0 . 12116 0.00137 -0.00002 -0.05042 6 .00481 0.00002 -0.07538 0.00080 -0.15420 -0.00036 0.01091 0.00006 VARIABLE 25 0.27525 -0.01944 0.00021 0.81557 " 0.02380 ' -0.00006 -0.11678 0.29936 -0.00008 -0.31218 -0.00972 -0.00000 -0.09761 -0.00093 0 .00003 -0.14828 0.00122 0.00000 0 .11228 0.00132 0.00002 -0.02079 -0.00031 -0.05278 -0 . 0 0 0 0 0 0.09938 -0.00013 VARIABLE 26 0.04550 -0.00702 -0.00221 0.98168 0.05041 -0 . 0 0 1 1 7 -0.07000 -0.05911 -0.00192 0.00568 0.08098 0.00008 -0.06095 0.00877 0.00005 0.02059 -0.00249 -0.00000 -0.06671 0.03098 0.00009 0.03109 -0.00297 -0.06232 -0.02792 -0 . 0 3 5 9 5 0.00077 VARIABLE 27 0.03525 -0.00137 -0.00111 0.96825 0.02383 0.00001 -0.08680 -0.07981 -0.00027 -0.00518 0.03068 0.00007 -0.08263 -0.00489 0.00002 0.05160 0.00813 -0.00001 -0 .08534 -0.09304 -0.00000 0 .06859 0.00065 0.12138 0.00188 ) 1 0.02734 0 .00064 , VARIABLE 28 -0.18063 0.05268 0.00604 „__0.. 95293 " -0.00542 0.00128 -0.05891 -0.06867 -0.00072 -0.00705 0.07401 -0.00017 -0.05804 0.00118 -0.00018 0.00212 -0.00849 -0.00000 -0.08895 0.05301 0.00001 0.09751 0.00972 0.13039 0.04284 -0.00562 -0.00174 VARIABLE 29 0.04659 0.11873 -0.00037 0.02304 -0.02854 -0.00014 -0.83722 -0.01632 0.00002 0.03761 0.00965 -0.00001 -0.14708 -0.00595 -0.00005 -0.09260 - 0 . 00006 -0.00002 -0.09046 -0.00096 0.00001 0.21245 -0.00018 0.43992 -0.00026 0.04986 -0.00011 CHECK ON COMMUNALITIES VARI A8LE 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 13 19 20 21 IQB-IGJNAL 0.99994 0.99993 0.99993 0.99994 0.99993 0 .99993 0.99994 0.99994 0.99993 0.99994 0.99994 0.99993 0.99995 0.99994 0.99993 0.99993 0.99993 J3.99993 0.99994 0.99994 0.99992 , FINAL 0".99991 6.99991 0 . 99990 0.99991 0.99991 O". 99990 6". 9999 1 0.99991 0.99990 0.99992 0.99992 0.99990 0". 99992 0.99989 0'. 99989 0.99989 6.99990 0.99988 0.99990 0.99990 0.99988 DIFFERENCE 0.00003 0.00002 0.00003 0.00003 0.00002 0 . 0 0 003 0 . 0 0 00 3 0.00002 0.00003 0.00003 0.00002 0.00003 0.00003 0.00004 0.00004 0.00004 0.00003 0.00004 0.00003 0.00004 0.00004 . . . , . „ . ; , - ; J 22 23 24 25 26 27 28 29 0 .99992 0.99994 0.99993 0.99993 0.99994 0.99994 0 .99994 0.99994 0.99989 0.99991 0". 99990 0.99989 0.99991 6.99990 0.99991 0.99990 0. 00003 0.00003 0.00003 0.00004 0.00003 0.00003 0.00003 0.00004 / THE PERCENTAGE OF THE TOTAL VARIANCE OF THE ORIGINAL STANDARDIZED VARIABLES WHICH IS ACCOUNTED FOR IN THE FACTOR SPACE IS 100.0 REGRESSION EQUATIONS FOR COMPUTING FACTOR SCORES _ 0.07219 0. 15848 -0.03349 0.17069 0.0 -0.00681 -0.02652 0.10647" 0 . 2 4 0 4 5" 0.07970 -0.00187 -0.12154 0.17242 -0.01834 - 0 . 16050 0. 0 -0 .0 1392 -0.16493 -0.20511 -0 .04961 -0.03642 0.36826 0.00603 0.01447 0.0 0.06006 -0 .01966 0.07372 -0.00386 0.02025 0.06033 0.04616 -0.01503 0.0 0.03225 rJ3_Jj3_105_ 0.01572 0.29179 - 0 . 05409 0.03269 0 . 13872 0.06013 0.05494 -0.0082 2 0.0 -0.03043 0.05037 -0.11079 - 0 . 01256 0.21568 -0 . 0 2 9 6 3 0.02238 0.20395 0.0 -0.01308 -0.00218 -0.07170 -0.03526 0.07991 0.84662 -0.08414 -0.04028 0.0 -0.01363 - 0 . 2 848 8 0.05431 -0.55664" -0.78101 -0.03116 0.20638 0.64307 - 0 . 10455 0.22806 0.0 -0.00995 0. 27488 -0.43188 0 .02166 0.07788 0.45117 -0.02367 0.01042 0. 0 -0.02177 -0.12364 -0.87964 0.00604 0.09328 0.20608 ..._-0.._98LOLl_ - 0 . 11034 0.0 1.12481 -0.02130 0.02307 - 1. 33105' -0.19333 -0.10016 0.37016 0.07823 0.02217 0.44963 0. 0 -0.03235 0.54715 0.47359 -0.03302 0.15401 -0.33003 0.01478 -0.00334 0 .0 - 0 . 0 4 8 51 -0.02327 - 0 . 16884 0. 00028 0.G0620 -0.12645 1.09472 -0.06622 0.0 -0.22589 0.05339 0.00220 -0.42519' 0.04562 0.01102 0.17365 -0.09991 0.07936 0.16879 0 .0 0 . 01570 0.24411 0.31030 0.04489 0.08470 -0.25706 0.11492 0.02290 0. 0 -0 .04308 0.07993 0. 11455 0.00505 -0.01212 0.04399 0.03662 -0.03719 0.0 -0.03471 -0.0127 1 -0.02174 -0.17839 -0.07245 0.00874 0.03415 0.00972 0.01242 0.07495 0.0 0.01508 0.05930 -0.00105 1.06727 0.00956 -0.03841 0.02012 -0.00924 0.0 -0.02592 0.02849 -0 . 0 9 6 6 6 -0.02145 -0.04706 -0.45932 0. 12001 -0.00124 0.0 0.03849 0.14020 -0.02098 0.39445 0.36888 -0.16946 -0.03437 -0.32385 1.38310 -0.11432 0.0 -0.07488 -0 .07157 0.19328 0.02011 0 .02282 -0.19946 0.03649 0.05626 _ -..0_. 0.1343 0.0 0.30533 -0.02679 -O.OOUO 0.22244' -0.01070 -1.76923 - 0 . 19792. 0 .04578 ' 0.33701 -0.17181 0.0 0.18130 -0.30557 - 0 . 18872 -0.01923 - 0 . 19101 0.13599 0.06467 -0.14577 0. 0 C'05563 -0.16934 . - 0 , 0 3 674 0.03909 0.02863 0.0 -0.36977 -0.42448 -0.01054' 0.63940' -0.98486 0.56915 0.13022 0.84869 -0.12368 0.08499 0. 0 0.54657 -0.02654 -1.28345 0.34506 0.08539 1 .04907 0.35671 0 . 16385 0 .0 -0.05872 2.44150 -1.25620 0.23341 - 0..-0.1A22_ 0.03429 -0.05022 -0.01303 1. 11394 0.24023 • — oTo 0.02332 0.07852 0.43932 0.04855 . _ 0.02828 0.23486 1.26761 -0.10487 0.0 -0.48918 -0.10400 0.00584 -0.58669' -0.35498 0.25749 0.00831 0.21394 -0.09853 -0.16367 0.0 -0.14877 0.06459 -0.12573 -0.10077 -0 . 0 8 5 0 3 0.08643 - 2 . 38535 -0.12123 0.0 -0.01879 -0.36517 -0.31487 -0.30617 0.04572 -0.02798 -0.13870 -0.02252 0.0 0.32770 0.01071 0.01562 - 0 . 4 1 1 8 7^ 0.00021 0.60637 -0.14163 0.03636 0.38500 -0.0726 5 0.0 -2.40857 - 0 . 21359 0.05162 -0.07780 -0.23720 0.00981 -0.19403 -0.26451 0. 0 -0.02202 -0 . 4 8 5 6 6 0.10396 - 0 . 08900 -0.28907 - 0 . 9 3 064 1.20183 -0.22332 0.0 -0.61234 0.23942 -0.24496" 0.65952 0.45096 -0.16342 - 0 . 03003 -0.76489 0.69624 -0.32574 0.0 0.25767 -0.37925 0 . 12724 -0.24026 - 0 . 15213 -0.59070 0.60383 -0.07917 0.0 -0.05602 0.57092 0.43788 2.88005 - 0 . 1 5 842 -0.18970 -0.51666 0.02742 0.0 1.19035 -0.05449 -0.18793 1.02413 0 .00717 - 0 . 18030 -0.96177 0.01503 - 0 . 64891 2.84328 0.0 0.01432 -1.14523 -1.01799 0.70028 - 0 . 62505 0.61995 0.72080 -0.48724 0. 0 0.10091 0.31840 0.01584 -0.18240 0.14230 - 0 . 33685 -0.52203 -0.03024 0.0 0.68515 0.23445' 0 . 16750 - 1.00744 0.55887 0.27442 - 3 . 83030 -0.44688 -0.43454 1.27257 0.0 0 .0 2464 1.82009 1.69815 0.41281 0 .90925 -1.26285 -0.38802 0.86334 0.0 0. 20676 -1.67235 0.57310 -1.24400 0.60412 0.70544 -0.27724 0.4563 6 0.0 2 . 10050 0.22086 0.35230 -1.08292 0.98704 0.75290 0.57175 -0.25617 - 0 . 52123 0.31780 0.0 - 0 . 53521 0.27359 1.44470 0.80442 0.06326 -0.36914 0.43513 -0.02834 0.0 -5.58346 1. 18888 0.41384 0.04335 -0.02536 24.82422 -0.24989 1.34770 0.0 0.00582 2.27674 -0.14471 1.91804 10.51530 0.31948 -0.49861 -10.27223 -0.85211 -1.66717 0.0 1.50692 -0.02443 5. 12517 0.35356 -0.95509 -3.09556 0.92740 1.16922 0.0 -1.85664 1.74444 -27.80121 1. 52595 -0.00556 3.90921 0.68951 0.24859 0.0 -1.05270 -0.43675. -0.11521 2 . 30533" -0.00717 0.25320 1.08576 - 0 . 15646 -0.09346 -1.45187 0.90007 3.11372 -0.2175 5 0.22420 ' -7.9940 3 0.45263 - 0 . 52481 3.24147 0.0 0.15404 2.29206 -4.42096 0.60571 - 2 . 38597 -12.01273 0.21711 -3.44337 0.0 0.29124 -1.16840 - 1.59580 -1.53169 -0.18010 - 1 . 21335 -0.40945 12.17865 0.74341 1.88266 0.0 -0.64052 0.68227 -3.72487 0.24750 0 . 51811 -0.77833 0 . 14597 - 1 . 18462 0. 0 4.25171 -1 .34780 9.08600 -1.00413 1.05171 1.45645 -0.80698 1.55669 0.0 1.00258 1.72724 -1.66828 -6.00656" 6.48324 -0.16651 0. 72807 - 7 . 50906 0 . 06522 1. 74429 0. 0 0.94988 -12.85294 6.38090 -0.75642 2.55054 -2.93779 1. 28469 14.14331 0.0 - 0 . 66195 -5.87979 -0.94007 1.89542 0.27973 -5.60386 0.40069 1.84016 0.0 1.21242 - 1 8 . 72362 3.25682 4.09678' 33.47461 -0.46437 -2.18706 -27.41634 0.57868 -1.02162 0.0 -0.82449 -1 . 5 0 5 1 8 8.69985 0.94973 -2.02501 -9.20554 -2.23624 1.74904 0.0 -2.09246 -2.35797 5.99319 -1 . 6 7 0 2 3 -0.31155 -14.82066 1.70799 -0.23501 0.0 1.22496 -3 . 6 7 0 3 8 -1.99484 -1.50892 3.51379 1.35329 0.32026 0.0 - 2 . 2 5749 0.13366 -7.87424 0.96409 2.90469 -5.08703 -1.98754 -3.80095 0. 0 -6.81715 1.02761 14.67192 -0.27040 -3.18059 1 8 . 86780" -0.45810 . 0. 0 J J 3.64147 5. 4 4 1 7 7 0.26876 -32.24710 0. 0 -0.47861 -7.07 571 0.0 3.56608 4. 1 9 1 7 1 23.85272 0.51176 1.32217 0.0 2.49503 -12.80019 -47.83936 2.21488 -20.34119 -0.74603 -0.41356 10.09042 3.37977 -0. 85400 0.0 -2.3 8472 -0.44498 1.62492 -1 . 6 3 2 6 3 -0.66093 -17.96086 25.27940 -13.84796 -52.20502 -3.86004 1.98320 42.24194 2.75322 8.85872 0.0 -4. 01271 136 .03285 -0. 6323 7 5.27845 -32 .18195 -5.00917 5.30277 15.16525 3.64468 0.0 4.16964 -16. 22427 46.04 953 -0 . 2 0 9 1 5 4.31281 10.35435 -1.05087 11.49642 -2.40107' 58. 22420 -6.39455 3.23445 -5.98815 0.0 -2. 25328 1 .81474 13.68811 -1.22291 -4.05860 -0.09271 1.67473 5.62040 - 1 1 8 .8 1 0 7 3 -1.94524 -0.95804 0.0 -14.83328 -11.53463 - 0. 0 16.41324 -0.16918 16.67035 -0.57170 -31.3302 5 0.96982 -23.21570 2.84299 -3. 17453 0.1 3 8 1 9 80.76552 4.31166 -0.38718 -60. 1439 7 0.0 - 1.43669 -12.84580 -10.05247 3.38940 -17.43996 0.71019 1.30542 20.08327 -1.77684 -1.78994 0. 0 0.93163 -4.04492 4.90721 -0.63485 -1 . 8 8 2 6 9 -17.85847 1.457 39 3.24173 0 .0 4.75599 -1.91963 5.. 4 3 8 2 4 16.03757 0. 1 1 4 8 7 -7.34354 0 .0 0.38015 -2. 12888 1.87 3 0 4 -107.37512^ -13.30058 0. 1 7 8 6 1 28.70490 4.36546 0.03177 33.2499 1 0 .0 1.40572 42.83047 28.5163 1 0.37009 10.93032 -18.78526 0.92817 -1. 26232 0.0 -5 . 7 1 6 1 6 1.08614 -12.25775 1.87447 4. 7 8 6 3 4 12.-22 5 0 6 -0.08483 -6.22551 0.0 -0.42936 2 .20323 -4. 39195 -84.34357' -5.36386 0.90510 22.85645 -0.57241 -0.48855 2 5.41504 0.0 1.61445 36.15402 1. 9 3 8 8 9 0 .28609 7.9040 5 2. 1 2 7 2 3 1.21148 -1.60452 0.0 -1.78743 2 .83226 -5.40 22 6 1.38861 FACTOR SCORES ON SUB J ECT SUBJECT 0.32435 0.18707 -0.12899 -0.01469 1.28519 2.70557 0.14767 0.26096 0.26290 -0.39595 0.52146 0 .00598 0.50086 0.36707 0 .10408 -0.08356 0.10359 0.05127 -0 .545 3 1 -0.46229 0.00600 -0.49937 0.01154 -0.04634 -0 . 0 3 4 2 4 -0.3367'6 -0.36248 0.36849 -0.59181 0.06754 0.01530 -0.52207 0.53134 4.90682 -0.3548 2 -0.07489 -0.47860 -0 . 1 4 2 4 1 0 .01100 0.41194 -0.20248 0.52150 -0.64254 0.32265 -2.82124 0.38119 -0.09206 -0.41607 -0.53443 -0 . 2 7 5 7 9 0.32570 -0.40492 -0.14348 0.09488 2. 1 9 1 6 1 -0.01125 0.54153 -0.15491 0.02666 -1. 57583 0.54664 -3.02919 0 .05217 0.15360 -0.146 54 -0.09168 0.89408 -0.69719 0 .00180 -0.62 371 -0.04345 -1 .54206 1.64400 0.72832 -0.27070 1 .99182 0.04696 -0.90410 -2.14897 0.05734 0.2 92 8 0 -0. 10033 -0 . 0 3 7 2 9 0.73394 -0.28266 -1 . 6 6 7 1 7 -0.48472 0 .66673 0.44344 0.32122 0.23149 0.15838 3.98642 -0.97357 -1 .13874 -0.39081 -0.14435 -0 . 3 8 3 2 7 1 .71641 -0.70169 1.04535 -0.76478 1.26350 0 .39143 -0.46306 -0 . 1 2 9 3 8 -0.37202 -4.20216 -0.90011 0.14656 -0.26562 0.30442 0.03829 0.21736 0.45010 -0.06076 0.14902 0 .32877 0. 1 8 8 3 9 0.18239 -0.16519 5 2.24131 0.29153 -1.49993 SUBJECT 0.70171 0.32787 0.13083 -0.0078 2 0.00332 1 .19204 -0.70894 -0.00215 4 0.40769 -0.21319 0.38737 SUBJECT 0.43787 0.11744 0. 1 2 5 2 6 0.00770 -0.01497 4.23197 -0.02780 0.01814 3 1.66895 -0.093.15 -0 . 5 9 3 2 0 SIIR.lFf.T — 2 ___.____.0..A3.0-9J4_ 0.55258 -0.67112 SUBJECT ROTATED FACTORS 1 -1.94836 0.29055 0.04704 ) 6 0.38463 1.17680 -0.65350 J -0.36429 0.11268 0.70490 0 .46557 -0.31236 -0 .23635 -0.01115 -1.49080 1.68790 - 0 . 10351 0. 8704"6 -1.43527 2.14975 -0. 12910 0.45087 -1 . 1 0 8 6 9 -1 . 2 2 9 2 4 0.20076 -0.55057 -2 . 19880 0.47100 -0.34604 0.39643 0.58597 -0.92540 -0.07223 •0.38190 •0.77100 0 .26049 0 . 22914 ___0. 2779 5 -0.22560 1. 11527 1.00236 -0 . 8 0 6 54 -0.34445 •0. 65831 0.03756 -1.33400 -0.66914 0.19991 -0.136 32 -0.91881 1.94108 0 .11239 -0.70686 •0.61322 0.21596 0.20707 1.99757 1.8586 2 -0 . 1 7 5 2 0 • 0 . 0 5_5_85 6. 16066 SUBJECT 9 -0.24890 0 .06660 0 .24403 0.15305 0.0365 6 -0 . 3 7 5 4 8 0.38103 -0. 42935 1.35347 1.52688 :2._911.43. 0.46934 -0.14012 1 .68948 0.69541 -0.234 60 -0 .32461 0.20815 1.26433 -0 .52676 -0.12368 0.76086 1.98098 •0.37990 1.22112 -0 . 2 4 6 2 4 1.23819 10 SUBJECT 0.02783 -0.13477 -1.66663 •0.28798 •1 . 2 1 2 6 2 0.04488 0.37201 0 .90077 -0.36837 0.75073 1.00502 -1.54560 -0. 36485 0.90733 0 .08738 -0.23466 0.67938 0 .20997 -0 .18138 -0.52530 0.21177 0.35071 -0.48972 -0.25006 2 .61440 - 1 . 16958 0 .58330 SUBJECT 11 -0.84985 -0.72383 -1 .11328 -0.08012 I . 19566 0.80818 0.12764 0.11179 0 . 0 3620 -1.24906 0.89626 •0.83104 0.73031 -1 .09151 0.07707 0.05727 -0 . 5 8 6 6 3 - 0 . 02010 1.21426 0 . 8 2 1 14 0.03708 -2.32670 1.28628 1 .10049 -1.26601 1 .53547 0.23971 SUBJECT 12 0 .48180 0 .54197 -0.16130 •0.20987 1.98697 -1.12031 0.26604 0 .80980 0.87820 2.75519 1.08214 0.61842 -0.61183 •0.63811 -0.27102 -0.50718 -0.21301 -0.49105 -0.77665 -0.37105 -0.51384 •1 . 0 7 3 7 3 1 .72274 •0.39392 1 .179 83 0.25077 •1 . 1 6 8 9 9 SUBJECT 13 0.90554 0.84448 0.05270 -0.08030 0.75696 -2.14677 0 .45135 -0. 3135"0 0.13949 1.1480 3 0.16935 •1.50050 -0.70157 -1 . 6 7 1 0 3 •0.83334 •0.54309 1 .09004 0.25738 0.52775 0 .11165 0.14422 -0. 16138 •2.49011 0 . 2 056 8 -1.98446 -0 .02155 1. 2 7861 SUBJECT 14 0.29475 -0.96721 1.26358 •0.25820 0.81079 0.00002 0.22335 -0.77564 1 .10957 0.00586 -0.23064 0.39638 0.43534 0.62782 1.39561 -0.30938 •0.29788 0.09120 -0 .02134 1.28090 -0 .21288 -0.36526 -0.14283 -0 .06668 2.35322 i_9689.1_ 1 .05161 SUBJECT 15 -0.71481 2.05491 1 .30675 -0.32854 0.38482 -1.04175 0.08856 0.44063 -1.40823 -2.21362 -0.77072 -0.48340 •0.32399 -1.72409 -0.32024 •0.04894 -1 . 2 3 6 0 9 0.44326 -1.01208 -0 .15276 0.38021 0.63524 1.46460 •0.70686 1.12028 -0 . 8 9 4 2 7 1.51810 SUBJECT 16 -0.21071 -2.70556 0.46112 0.10029 1.99276 0.77298 -0.55230 0.47797 0.36787 • 0 . 56536 •1.26476 -0.19116 -0.28598 0.46118 -0.09021 -0.35127 -0.93594 -0 .14870 -1 .32320 •0.41243 -0 .231 19 -0 . 9 1 0 1 5 -1 . 5 9 7 0 5 0.24014 -0 .1 7242 •2.55721 0.94773 SUBJECT 17 0.74309 -1.05683 -0.24113 -0.04745 1.65326 •1.42831 0.25145 • 0 . 54433 0.62849 -0.32405 - 1 . 67929 •0.52885 1.10572 -0.27724 -3.14928 •0.35832 -1.32045 -0.00587 0.25592 1 .26844 0.01277 -0.24526 0.04834 0.26732 0.81265 -0.02124 - 2 . 01817 SUBJECT 18 -0.49184 1.30130 -0.05688 0 .25131 0.20369 -0 . 3 4 2 6 0 -0. 32538 -0.60590 1.23934 1.59102 -0.02945 0.41472 -0 .11326 -0.33734 -2 . 0 3 5 1 3 0.26595 0.67642 -0.50454 1.17038 •1.31755 -0.00175 SUBJECT 7 0.48085 0.33132 -0.09988 SUBJECT 2 . 9 0 5 88. -0.43369 0.61940 -1.61686 -1.78216 -0.03251 1.45805 -0.05928 0.29041 -0.38918 1.30978 0.84304 0.14312 - 1 . 2 5220 -1.77519 0.07721 -0.79408 1.83305 -0.87073 0.52307 -0.56200 -0.32604 1.13491 -0.14027 0.00757 1 .48302 0.19076 0.61947 -1.15763 0.13847 -0.75782 -0.14513 0.38862 -1.13053 -0.32339 -0.47148 0.12073 1.02322 -0.76116 -0.48710 -0.54286 -0.07629 -0.29476 0.07749 -0.51148 -4.29731 0 .09397 -0.66534 -0.51605 0 .08073 - 0 . 0 3 010 1.45330 -0.73397 -0.36708 -0.51993 J3.J_0J-.2J. . ... - 0 . 6 4 0 7 3 3.05837 -0.72824 -0.31847 0/8999% 1.74634 -0 . 3 4 6 1 6 1.91278 -0.89823 -0.68738 1.28605 0.29664 0.86278 -0.50140 0.00450 -0.78642 0 .37457 -0.15003 0.24472 0.09793 -0.22137 -1.28181 -0.07389 -1.46871 -0 . 3 2 3 1 0 0.57509 , O . J 7369 0 . 33321 -0.24772 1. 25582 -0.71828 0.00450 0.65908 -0.49477 0 .03459 0.54509 - 0 . 11715 1.05925 0.57711 -0.26200 0 . 5 3 2 36 0.36304 -0.59870 -0.92942 0.21095 1.15139 -0.33914 -1.05089 -0.21554 -0.43028 1 .94314 0.25096 0.96670 - 0 . 2 29 33 -0 . 4 1 4 3 3 -0.22821 -0.35459 -0 . 6 8 6 6 8 0.12932 0.3TB3 80 -0.20811 -0.31536 -0.01197 - 0 . 9 4 2 79 0.94188 0.21826 0.30045 0 .13534 -0.07404 - 1 . 13386 - 0 . 542 75 -1.38236 -0 . 1 7 7 4 6 -1 . 4 7 6 3 3 -0.66561 0.42511 -0 . 5 7 0 23 0 . 6 5677 -0.21110 -1.11327 -1.00022 -0.82728 0 .50630 1 .61808 -0.06917 0.01175 -0.40728 -1.06454 0.50398 - 0 .39760 -0.40146 -1.03001 -0.57889 -0.07828 1.59448 0.49349 -0.27759 -0.33159 0.98344 0.16362 -0.56106 -0.35793 0.25432 0.51588 1.14546 - 0 . 8 449 3 0.63694 - 1 . 14475 -0.09198 0.46594 -0.21136 1.03493 0.18043 -0.19195 0.01929 -0.28817 -0.09015 0.58380 -0.07884 0.57768 -0.31767 -0.2.3273 -1.41567 0.30760 - 0 . 1 4 9 27 -0.40343 -0.39827 0.78873 1.28933 -0.66524 -1.63185 0.58221 0.01575 1.12444 -0.06582 - 0 .08874 0.68644 0.01081 2.10721 1.31369 -0.89604 -1 . 1 2 3 2 8 - 0 . 7 7 9 78 -0.37844 -0.19326 -0.. 12681 0.16105 0.14481 0/41847 0.20138 -0.18151 1.02248 0.55832 - 0 . 6 1 9 65 1.27792 -2.13812 1.535 5 8 0 .38198 0.28829 1.67767 0.19158 0.04571 -0.84119 0.33333 0.41900 0.06949 -0.56311 1.74987 -0.66936 19 SUBJECT 0.12080 0.91628 1.23834 SUBJECT 20 SUBJECT 21 -0 .88155 -0.89406 0.47777 _ _ SUR.) EOT _' ?? -0.39019 0.39311 -0...71.78.0 .. SUBJECT 23 SUBJECT 24 SUBJECT _ 25 SUBJECT 26 SUBJECT 27 SUBJECT 28 SUBJECT 29 SUBJECT 30 -0.29355 0.25539 -0.82299 -0 . 8 4 5 9 9 0 . 3 7053 -1.17045 -0.59536 0 . 0 7033 0.05747 --L.jD.2al4 -2.40190 -1.43370 - -0.99673 _ 0 . . 3_7_6.8A. -0 . 1 2 6 9 4 -0.62132 0.74354 0.024.70 1.16062 -0.04879 1.08558 -0 .17354 -0.78634 -0.16091 -0.00669 0.50671 -1.24123 ) , .... -0.31619 -0.30915 1.38329 0.30300 -0.74261 1.07703 -0.36541 -0.06345 -1.45248 -0.74167 -0.22022 -0.30761 -0.08048 0.20271 -0 . 8 0 0 0 2 0 .16523 0.03456 -0.72422 0.59325 0.00260 - 0 . 5 2 843 -0.14032 -0.05155 0 . 1 2 8 86 -0.11056 -2.43765 -0.27747 0.43903 0 . 0 0 5 09 0 . 3 5960 -0.15417 1.00679 2.29142 1.65128 -0.64405 1.27859 -0.37988 0.13856 -0.35600 0 . 1 6 2 44 0 .47795 -0.49522 -0.81762 -0.51427 0.81133 -0.24526 -1.41117 1.19496 -0.04042 0.39149 -0.08539 -0 .950 06 0.73369 0.62168 1.06481 -0.33191 -0.13989 -0.93247 2.97536 -0.53960 1 .16998 -0.45681 -0 . 4 4 0 1 0 0.61859 -0.71751 -1.10347 0.46121 -1.35493 -1.84003 1.41563 0.15996 0 .42890 •0.11768 0.69122 1.66016 0.23303 -0. 11333 2.91168 •1.07610 -0 . 5 5 2 7 6 -0 . 14347 0 . 16 593 -0.14897 -0 . 4 2 8 9 2 - 1 . 16468 -0.32318 1.12365 -0.47819 -0.61530 -0 .55170 -0. 54517 -0.34734 1 .87290 -0.00553 0.03039 0 .47758 -0.16196 0.31208 -0.02392 0 .26544 1 .22912 -1.66498 1.04528 0.38037 -0.48928 2.54606 1.37436 -0.05251 0.74474 0.29832 •1.89633 0.89663 -0.04994 -0.38885 -0.44000 1 .85882 1.09172 1.125 51 SUBJECT 0.05366 1 .78112 -0.87873 SUBJECT -0.58342 •1.01744 0.47574 31 32 MEANS OF THE FACTORS -0.00000 G.00001 0.00002 0.00004 STANDARD DEVIATIONS 0.99226 0.99827 0.99980 0.99710 -0.00000 -0.00001 0.00000 0.00002 OF T H E FACTORS 0.80052 0.99994 0 .99993 0.28552 CORRELATIONS OF T H E FACTORS R0W._._1 1.00000 0.00714 0.00027 _ -0.32513 -0.00456 0.00016 ROW 2 -0_.J3__5_._3_ -0.00391 0 .00913 ROW _ 3 0.02183 -0.00314. -0.00081 •0.92118 0.21694 0 .41139 -0. 00000 •0.00003 0.00001 C.00003 -0.00000 0.00001 •0 . 0 0 0 0 3 0.00000 -0. 00000 -0.00008 0.00000 0.00000 -0.00005 0.98878 1.00013 1.00010 0.30033 0.99850 0.99979 0.99804 0.99744 0.99882 1.00021 1.00011 0.99988 1.00676 0.02183 -0.02453 -0.00062 0.02534 0.00475 0.00015 _______ 0.01714 -0.00234 -0.00030 -0.00031 -0.00109 0.00211 -0.00031 -0.00148 0.00061 0.00244 0.00163 .ooooo 0.18259 -0.03593 -0 . 0 0 1 3 4 0.03740 -0. 022 72 •0.00146 0.06644 -0.00111 0.00108 -0.01019 0.00377 - 0 . 00153 0.01750 -0.008 10 0.00009 -0.00997 __.0_0_2_L0_ 0.18259 0.00018 1.OOOOO p.. 0.1155 0.00049 •0.01203 •0.00261 0.00041 -0.01357 •0.00057 0.00010 0.00242 -0.00073 -0 .00686 J -0.01987 -0.00127 0.00001 __P_L°OOO_L -0.00(317 0.99955 __A0_QP _ 0.98600 6 -0.00001 o •ooooi -6.00004 0.99982 J. .00082 1 .00034 0.00301 0.02165 -0.00596 0.00036 0.00341 0.03936 0.01263 -0.00649 -0.00185 -0 .00266 0.00108 0 .00465 -0.00010 -0.00106 -0.01212 -0 . 0 0 1 9 4 0.00268 0 .00009 ROW 4 0.02534 0.00009 -0.00054 0.03740 0.00035 0.00006 -0.01203 0.00348 -0.0000*9 1.00000 -0.00056 -0.00003 -0.00360 -0.00017 -0.00025 0.00118 -0.00020 0.00312 -0.00077 0.00012 0.00217 -0.00030 -0.00033 -0.00184 -0.00041 0 .00084 0.00002 ROW 5 0.01714 -0.00088 -0.00043 0.06644 0.00098 0.00007 -0.01357 0 .00352 -0.00018 -0.00360 -0.00086 0.00006 1.00000 -0.00033 -0.00026 0.00139 - 0 . 00021 0 .00310 -0 .00159 0.00067 0.00006 -0 . 0 0 0 2 1 -0.00029 -0 .00396 -0.00092 0.00087 -0 .00009 ROW 6 0.00211 0.00025 0.00005 -0.01019 -0.00043 -0.00023 0 .00242 -0.00118 -0.00017 0.00118 -0.00026 0.00008 0.00139 -0.00001 -0.00019 1.00000 0.00009 0.00010 0.00014 - 0 . 0 0 0 15 0.00052 -0.00010 -0.00004 0.00162 0.00023 -0.00024 0.00003 0.01750 0.00058 -0.00266 0.00079 -0.00026 -0.00077 0.00075 0.00021 -0.00159 -0.00005 0.00002 0.00014 -0.00005 0.00421 1 .00000 0.00008 -0 . 0 0 1 6 4 0 .00050 0.00005 -0.00107 -0.00038 0.00015 -0.00016 _RO.W_ 7 -0.00031 -0.00015 -0.00018 0.00016 J ROW ROW -0.00997 -0.00013 -0.00010 0.00016 0 .00060 -0 .00030 -0.00035 0.00022 -0 .00021 -0.00003 0.00003 -0.00010 -0.00004 -0.00399 0.00050 1.00000 0 .00003 -0.00197 9 0 .02165 -0.00041 0.03936 0.00055 .0.00022 -0.01212 0 .00277 0.00056 -0.00184 -0.00059 0.00025 -0.00396 -0.00019 -0.00002 0.00162 -0.00013 -0.01280 10 -0.00596 -0.00008 0.00008 -0.00649 ' -0.00004 -0.00008 0.00268 -0.00039 -0 .000.56 0.00084 -0.00002 -0.00019 0.00087 -0.00006 0.00006 -0.00024 11 0.00714 1.00000 -0.00023 -0.0 0391 0.00014 0.00008 -0.00314 0 .00001 0.00008 0.00009 -0.00088 0.00053 -0.00014 -0.01987 1.00000 -0.00008 0.00230 -0_._0QO44_ ROW ROW ^.ROW- -12 . - _ -0.00456 0.00014 0.00039 ROW ROW ROW 13 -0 .J3.2453 0.00001 0.00052 14 0.00475 0.00053 -0.00009 1 5 -0.00234 0.00009 - o . - . - a o . Q . o 4 ROW ROW 16 -0.00031 -0.00011 -0.00005 17 -0.00148 0.00019 0.00018 ^ROW- ROW N 8 0.00244 0.00012 0.00011 18 . 0.00023 -0.0OO51 0.00009 _ -0 ..03593 0.01155 1 .'00000 -0.00051 -0.00017 0.00038 -0 .00006 -0.00006 -0.00004 0.00013 -0.00107 0. 0 0 0 5 0 0.00624 -0 .00006 -0.00051 1 .00000 -0.00046 0.00065 0.00060 0.00010 -0.00356 0.00015 -0.00008 0.00077 0.00014 0.00006 0.00065 0.00024 1.00000 0.00013 0.00009 -0.00003 0.00025 -0.00011 0.00451 -0.00015 0.00019 0 .00318 0.00012 -0.00005 -0 . 0 0 0 4 1 -0.00018 -0.00008 -0.00005 0.00035 -0.00033 0.00063 0.00098 -0.00008 -0.00014 -0.00043 -0.00010 0.00405 0.00058 -0.00005 - 0 . 0 0 4 97 -0.00013 0.00006 0.00055 0.00017 -0.00004 0.00007 0.00348 0.00040 0.00009 0.00352 -0.00023 0.00079 0.00016 0.00041 0 .00277 0.00046 -0.00039 0.00002 -0.00002 -0.00118 0.00026 -0.00108 -0.00086 -0.00015 -0.00013 -0.00026 -0.00001 0.00457 -0.000.35 -0 . 0 0 0 2 6 -0.00059 0.00013 -0.00002 -0.00030 -0.00001 0.00016 -0.00003 -0.00010 -0.00019 0.00030 -0.00006 0.00034 0.00075 -0.02272 -0.0003.3 -0.00019 -0.00035 -0.00056 1.00000 -0.00015 -0.00111 -0.00008 -0.00112 -0.00057 -0.00023 0.00102 -0.00017 -0.00015 -0.00001 -0.00033 0.00104 0.00932 -0.00005 0.00016 -0.00114 0.00.377 '-0.00010 -0.00131 -0.00073 0.00026 -0 .00 300 -0.00020 -0.00001 -0.00039 -0 .00021 0.00016 -0.00005 0.00018 -0.03318 -0.00004 -0.00096 -0.00013 0.00050 0.00010 0.00383 -0.00706 0.00009 1.00000 -0.09505 -0.00810 -0.00005 -0.00008 0.00108 -0 . 0 0 0 6 0 0.00032 0.00012 0 .00016 -0.00050 0.00067 0.00016 -0.00108 -0.00015 0.00018 -0.00924 0.00008 0.00003 0.00004 0.00050 -0.00055 -0.00008 0.00053 0 .01221 -0.00261 0.00040 1.00000 - 0.00163 -0.00060 0 .00282 0.00014 0.00016 -0.00912 1.00000 0.00006 0.00341 -0.00005 -0.00028 0.00006 -0.00015 -0.00106 0.00041 -0.00030 -0 . 0 0 0 3 3 -0.00026 -0.00118 -0.00029 -0.00010 0.0C462 -0.00004 -0.00096 0.03582 0.00005 0 .00004 0.01812 -0.00006 1.00000 -0.00051 -0.00036 -0.00216 19 0 .00301 -0.00018 0.00007 0 .01263 0.00017 -0.00225 -0.00194 0 .00046 -0.00041 0.00013 0.00068 -0.00092 0.00030 -0.00207 0.00023 0.00050 0.04171 - 0 . 0 0 0 38 -0.00055 - 0 .02330 -0.00004 -0.00036 -0.00046 1.00000 0.00024 -0.00056 - o . o o o s i / \ j ROW 20 0.00036 - 0 .00005 -0.00167 -0.00185 0.00007 -0.00385 0.00009 0.00002 -0.00641 0.00002 •0.00030 -0 .00359 -0.00009 0.00034 0 .00146 0 .00003 0.00383 0.20000 -0.00016 0.00053 -0.08801 0.00013 -0.00216 0.00060 -0.00056 0.00013 1.00000 ROW 21 0.00027 •0.00023 1.00000 0 .00913 0.00039 -0.00050 -0.00081 0.00052 0.00118 -0.00054 -0.00009 0.00061 -0.00043 -0 . 0 0 0 0 4 0.00497 0.00005 -0.00005 0.05234 -0.00018 0 . 0 0 0 18 -0 .00463 0.00011 •0.00028 -0.00044 0.00007 0 .00008 -0.00167 ROW 22 0.00016 0.00008 -0.00050 -0.00127 •0.00008 1.00000 0.00018 -0.00017 - 0 . 0 0 5 83 0.00006 -0.00019 0.00005 0.00007 -0.00112 0 .00246 -0.00023 -0.00131 -0.06912 0.00016 -0.00008 0.02735 0.00023 -0.00015 0.00022 -0.00225 -0.00008 -0.00385 ROW 23 -0.00062 0.00008 0 .00118 -0.00134 0.00009 -0.00583 0.00049 0.00038 1.00000 •0.00009 -0.00035 •0.00316 -0.00018 0 .00102 -0.02011 -0.00017 -0.00300 -0.28885 •0.00026 0.00032 -0.67946 0.00060 -0.00030 0.00056 -0. 00081 -0.00056 -0.00641 J30>LJ_A. 0.00015 -0.00014 0.00061 -0.00146 0.00063 0.00005 0.00041 0.0000'9 -0 . 0 0 3 1 6 -0.00003 • 0 . 00015 1 .00000 0.00006 •0.00001 0.00733 0 .00008 -0.00039 -0.09976 0.00021 -0.00050 -0.18123 0.00022 -0.00118 0.00025 0 .00068 -0.00019 •0.00359 ROW 25 -0_.j30jD3.P_ -0.00003 0.00497 0.00108 •0.00014 0.00246 0.00010 •0.00002 •0.02011 -0.00025 -0.00013 0.00733 -0.00026 0.00104 1.00000 -0.00019 -0.00706 -0.11209 0.00002 -0.00108 -0.05645 0.00003 0.00462 -0.00002 -0.00207 0.00006 0.00146 ROW 26 -0.00109 0.00451 0.05234 -0.00153 _0.0 0j40._5_ -0.06912 -0. 00686 -0.00108 -0 . 2 8 8 8 5 0.00312 0.00457 -0 . 0 9 9 7 6 0.00310 0.00932 -0 . 1 1 2 0 9 0 .00010 -0.09505 1.00000 0.00421 -0.00924 0.84306 -0.00399 0 .03582 -0.01280 0 .04171 •0.00356 0.20000 ROW 27 0.00061 0.00318 rP._Q.0463 0.00009 -0.00497 O.02735_ 0.00465 0.00282 -0.67946 0 .00217 -0 . 0 0 9 1 2 -0.18123 0.00006 -0 . 0 0 1 1 4 -0.05645 0.00052 -0 .03318 0.84306 •0.00164 0.01221 1 .00000 -0.00197 0.01812 0.00624 -0.02330 0.00077 -0.08801 STOP 0 EXECUTION TERMINATED $SIG RFS NO. 019805 UNIVERSITY OF 8 C COMPUTING CENTRE MTSIAN0 59 ) APPENDIX G M U L T I P L E REGRESSION $SIGNON PL AK T I M E= 5M PAGES=50 C 0 P I E S = 7 **J_A.SI SIGNON WAS : 1 6 : 1 2 : 3 8 03-25-70 USER " P L A K " SIGNED ON AT 1 6 : 1 4 : 2 3 ' ON $RUN * T R I P 4=*SOURCE* EXECUTION BEGINS T R I P / 3 6 0 IMPLEMENTATION 3/18/70 PRIO = V 03-25-70 OUTPUTS OF MODEL I I JOB START: 16: 14: 18 03-25-70 • 0 1 0 6 0 9 • • • • • • • • • • • • • • • • _ • * « • 1 2 1 5 1 8 2 1 2 4 2 6 2 8 3 0 3 1 • 5 3 • • 4 0 4 5 • • 5 0 • 5 5 • 6 0 • 6 5 • 7 0 • 7 5 * * • • • • • « • 8 0 CONTROL CARDS 1. 2. INMSDC STPREG 3. STPREG 4 . PARCOR 5 . END NOTE : 3 2 2 1 5 5 6 6 1 1 1 1 1 233 33 OUTDATED *INVR# OR #MULREG* ROUTINES HAVE BEEN REPLACED BY THE EQUIVALENT *S TPREG* CONTROL CARD NO. I * INMSDC * FORMAT CARDS ( F 1 0 . 5 / F 1 0 . 7 , 3 0 X ,F1Q .7 ) INPUT DATA TRIPGN FACTOl FACT05 -1.948 0.1253 -1 .127 -0.8356D-01 0 . 4 8 70 - 0 . 4 3 09 2 . 584 1.669 0.1530D-01 1 .221 0 .4077 0 .3257 -0.2707 2.241 1.782 0 . 3 360 0.3243 -0.3908 0.5620 0.4808 - 1 . 109" 2 . 906 -0.6691 2.026 -0 . 2 7 9 0 -0.2489 -0.1401 -0.3290 0.2783D-01 -0.3648 -0.8498 0.7303 -0.9910 0 .5010 0.4818 -0.6118 1. 196 0.9055 -0.7016 -0.6000D-01 0.2947 0.4353 -0.8510 -0.7148 -0.3240" 0_._4.63„0 - 0 . 2107 -0.2860' 1. 106 0.7431 0 .8700 -0.6870 -0.4918 1.239 -0.1300D-01 0.1208 -0.8707 -0.7612 -0.8815 -0 . 9 4 4 0 -0.3980 -0.5199 -0.6874 -0.5020 -0.3902 -0.4948 -0.2935 -0.6867 -0.4110 -0.9390 -0.8460 - 0 . 8 2 73 -0.5730 -0.5954 -0.8449 -1 .020 -0.6652 -1.156 - 1 . 040 - 0 . 9967 -0.6196 -0 , 8 3 8 0 -0.6213 -0.7417 1.651 1.145 1.161 -0.7490 -0.1735 1.065 -0.4360 0.5366D-01 2.912 -0.8510 -0.5834 2.546 32 OBSERVATIONS 31 DEGREES OF FREEDOM NAME MEAN S.D. TRIPGN-0.3125D-04 0.9999 FACT01-0.3125D-06 0.9923 FAC TO5 0 . 2188D-05 0 . 9 9 7 4 CORRELATION MATRIX VARIABLE TRIPGN TRIPGN 1 .0000 FACTOl 0.9078 FACT05 -0.0431 ARRAY WRITTEN FACTOl FACT05 1.0000 0.0171 IN AREA 1.0000 5 CONTROL CARD N O . ? * STPREG » J DEPENDENT VARIABLE IS RSQ = FPROB. = STD ERR Y = VAR CONST. FACT01 FACTO 5 TRIPGN 0.8275 0.0000 0.4294 . _ .CO EF F -0.3084D-04 0.9157 -0.0588 < ' STD. .ERR 0.0759 0.0777 0.0773 , F-R.A.T.LO FPROB. 138.7819 0.5786 0.0000 0.4590 NO. OBSERVED CALCULATED RES I DUAL 1. 2. 3. 4 . 5. 6 . 7. 8. 9. 10. 11 . ]2 . 13. 14. 15. 16. 17 . 1 8. 19. 20. 21.' 22. 23 . 24. 25. 26 . 27. 28. 29. 30. 31. 32. -1 . 1 2 7 0 0.48700 2.5840 1 .2210 1.7820 0.33600 0.56200 2.0260 - 0 . 2 7 9 00 -0.32900 -0.99100 0.50100 1.1960 -0.60000E-01 -0.85100 0.46300 0.87000 -0.68700 -0.13000E-01 -0.94400 -0.39800 -0.50200 -0.41100 - 0 . 9 3 900 - 0 . 5 7 300 -1.1560 -1.0400 -0.83800 1.1450 -0.74900 -0.43600 -0.85100 - 1 .7916 -0.38974 1.5274 0.35415 2.0683 0.31998 0.50552 2.7004 -0.21972 0.46916E-01 - 0 . 82123 0.47716 0.8704 7 0.2442 7 -0.63555 -0.17616 0.61540 -0.5233 3 . . 0 , 16181 -0.7625 3 -0*14357 2 - 0 . 32 824 -0.22845 -0". 7260 7 - 0 . 4955 2 - 0 . 8 9 508 -0.87632 -0.52537 0'. 96566 - 0 . 22158 - 0 . 12216 -0.68406 0.66459 0.87674 1.0566 0.86685 -0.28634 0.16023E-01 0.56483E-01 -0.67435 -0.59284E-01 -0 . 3 7 5 9 2 -0.16977 0.23839E-01 0.32553 -0.30427 -0.21545 0,63916 0.25460 -0.16367 -0.17481 -0.18147 0 .37717E-Q1 -0.17376 -0.18255 -0 . 2 1 2 9 3 - 0 . 77477E-01 -0.26092 -0.16368 - 0 . 31263 0 . 179 34 -0.52742 -0.31384 - 0 . 16694 CONTROL CARD NO. ARRAY RESTORED FROM AREA 3 * STPREG * 4 * PARCOR 6 CONTROL CARD NO. * 6 PARTIAL CORRELATIONS VARIABLE TRIPGN FACTOl FACTO 5 .XR.XPI..N i . . . a a o _ _ _ ______________ FACTOl 0 . 9 0 9 5 -1.000 FACT05-0.1399 0.1714D-01 - 1 . 0 0 0 * STOP EXECUTION 0 TERMINATED END OF CONTROL OBSERVED CALCULATED RESIDUAL ) 5 ARRAY WR TTT FN IN AREA ARRAY RESTORED FROM AREA NO. SET * RFS NO. 019808 ,$SIGNON PLAK UNIVERSITY OF B C C O M P U T I N G T I ME= 5M PAGES=50 C 0 P I E S = 3 6 PR10=V !**.LA.S_T. .5 IGNON WAS: 1 6 : 0 4 : 29 03-25-70 ! U S E R " PL A K " S I G N E D ON AT 1 6 : 1 0 : 2 7 ' ON 0 3 - 2 5 - 7 0 |$RUN * T R I P 4=*S0URCE* E X E C U T I O N B E G I NS T R I P / 3 6 0 IMPLEMENTATION 3/18/70 CENTRE MTSI AN059 ) JOB START: 16: 10:22 03-25-70 RFS NO. 019804 UNIVERSITY OF B C COMPUTING CENTRE MTS.AN059) JOB START: 16:12:31 03-25-70 APPENDIX H MULTIPLE . $SLGNON PLAK TlME=5M PAGES-50 C0PIE8=7 PRIO=V * * L A S T SIGNCN WAS: 1 6 : 1 0 : 2 7 03-25-70 USER " P L A K ' ' SIGNED ON AT 1 6 : 1 2 : 3 8 ON 03-25-70 $RUN STRIP 4=*S0URCE* EXECUTION BEGINS T R I P / 3 6 0 IMPLEMENTATION 3 / 1 8 / 7 0 REGRESSION OUTPUTS OF MODEL I I I _ J 0 6 0 1 » • • _ • 0 9, •• • • 1 ... 1 2 5 • • * CONTROL 1 • . .* 8 2 2 1 4 • * * * • 2 6 . * 6 3 5 4 0 4 C 3 I 5 5 0 5 5 6 0 5 • • • _ • * • • • • 2 8 3 •_ 7 0 7 8 5 0 • • • CARDS 1. 2. 3. NOTE i INMSDC STPREG END OUTDATED 5 4 1 1 *INVR* OR *MULREG* 1 1 1 1 ROUTINES 23333 HAVE BEEN REPLACED B!Y THE EQUIVALENT *STPREG* CONTROL CARD NO. 1 * INMSDC * FORMAT CARDS ( F I D . 5 / 3 F 1 0 . 7 , 2OX, F 10. 7 ); INPUT DATA EMPLOY DENS I -1.948 4.232 -1.127 0.4870 -0.4309 0.4631 2 . 5 84 1.669 0.2656 1.221 0.4077 0.5215 1.782 2..2 41 C.9168D-01 0.3360 0.3243 0.3212 0.562 0 0.4808 0.11150-01 2.026 2.906 0.2779 - 0 . 2 790 -0.2489 0.1530 -0.3290 0.2783D-01 -0 . 2 8 80 -0.9910 -0.8498 0.8012D-01 0.5010 0.4818 0.2099 1 . 196 0.9055 0.8030D-01 -0.60000-01 0.2947 0.2582 -0.8510 -0.7148 0.3285 0.4630 -0.2107 0.1003 0.8700 0.7431 0.47450-01 -0.6870 -0.4918 0 . 5 6 88D-01 -0.1300D-01 0.1208 0.3892 -0.94-40 -0.8815 0.1385 -0.3980 -0.5199 0.3671 -0.502 0 -0.3902 0.3231 -0.411-0 -0.2935 0.2155 -0.9390 -0.8460 0.1775 -0.57,30 -0.5954 0.2776 -1.02 0 0.3177 -1.156 -1.040 -0.9967 0.1933 -0.8380 -0.6213 - 0. 31 62 1.145 1^161 0.1106 -0.7490 -0.1735 0.4042D-01 -0.4360 0.5366D-01 -0.1177 -0.8510 -0.5834 0 . 1620 32 OBSERVATIONS 31 DEGREES OF FREEDOM L.T192 -4.1202 -0.3424D-01 -2.821 0.:1800D-02 0 . 1 5 84 -0.1035 l.:002 0.3810 0 . 3 7 20 0 . 1 2 76 0. 2660 0.4513 0 . 2 233 0..8 8 56D-O1 -0.5523 0.2514 0.2037 0.1431 0.3886 -0.3185 0. 3332 6.2510 0.4251 0. 1636 0.3076 0.1448 0.3030 0.4390 -0.8539D-01 0.2330 0. 2654 TRIPGN SI.ZE MEAN NAME TRIPGN-0 . 3 1 2 5 D - 0 4 SIZE - 0 . 3 1 2 5 D - 0 6 EMPLOY 0 . 3 1 2 5 0 - 0 6 DENS 1—0 . 1 8 7 5 0 - 0 5 STUD - 0 . 3 1 2 5 0 - 0 6 CORRELATION MATRIX TRLPGN VARIABLE TRIPGN 1. 0000 SIZE 0 . 9078 EMPLOY -0.1562 DBNSI -0.2078 STUD 0.1769 - - - - - STUD 0.1308 -0.5453 4 . 9.07 0.9488D-01 -0.9041 -0.3833 -0.5506 -0.9188 -0.2346 -0.2347 0.5727D-O1 -0.5072 -0.5431 -0.3094 -0.4894D-01 - C . 3513 -0.3583 - 0 . 29.45D-01 -0.3260 -0.7629D-01 0.8628 -0.1171 -0.208 1 -0.6917D-01 -0.9198D-01 ,0. 1575D-01 1 . 536 -0.8048D-01 -0.3799 - 0 . 1399 -0.1435 -0.52510-01 - - - • © S.'D. 0.9999 0.9923 0.8005 0.9888 1.000 SIZE l.OCOO -0.3251 0.0218 0.0021 EMPLOY 1.0000 0 . 1 8 26 -0.0102 DENS I 1.0000 0.0024 STUD 1.0000 J CONTROL CARD NO, DEPENDENT .VARIABLE IS RSQ FPROB. = STD ERR Y = VAR CONST. SIZE EMPLOY DENSI STUD * S T P R E G * TRIPGN 0.9460 0.0000 0.2489 COBFF -0. 3 1 4 8 0 - 0 4 0.9904 0.2675 -0.2718 0.1776 STD ERR 0.0440 0.0478 0.0603 0.046 2 0.0447 F-RATIO FPROB. 429.0756 19.7063 3 4 . 6 8 09 15.7943 0.0000 0.0002 0.0000 0.0005
Featured Collection
UBC Theses and Dissertations
Multicollinearity in transportation models Chan , Sheung-Ling 1970
Item Metadata
Download
- Media
- 831-UBC_1970_A8 C43.pdf [ 12.06MB ]
- Metadata
- JSON: 831-1.0093346.json
- JSON-LD: 831-1.0093346-ld.json
- RDF/XML (Pretty): 831-1.0093346-rdf.xml
- RDF/JSON: 831-1.0093346-rdf.json
- Turtle: 831-1.0093346-turtle.txt
- N-Triples: 831-1.0093346-rdf-ntriples.txt
- Original Record: 831-1.0093346-source.json
- Full Text
- 831-1.0093346-fulltext.txt
- Citation
- 831-1.0093346.ris
Full Text
Cite
Citation Scheme:
Usage Statistics
Share
Embed
Customize your widget with the following options, then copy and paste the code below into the HTML
of your page to embed this item in your website.
<div id="ubcOpenCollectionsWidgetDisplay">
<script id="ubcOpenCollectionsWidget"
src="{[{embed.src}]}"
data-item="{[{embed.item}]}"
data-collection="{[{embed.collection}]}"
data-metadata="{[{embed.showMetadata}]}"
data-width="{[{embed.width}]}"
async >
</script>
</div>
Our image viewer uses the IIIF 2.0 standard.
To load this item in other compatible viewers, use this url: http://iiif.library.ubc.ca/presentation/dsp.831.1-0093346/manifest