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Multicollinearity in transportation models Chan , Sheung-Ling 1970

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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).  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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 . 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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  

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