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Stochastic models of travel-demand behaviour : a comparison of three disaggregate model forms using incomplete… Kahkeshan, Siavoche 1982

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STOCHASTIC MODELS OF TRAVEL-DEMAND BEHAVIOUR: A COMPARISON OF THREE DISAGGREGATE MODEL FORMS USING INCOMPLETE DATA by SIAVOCHE KAHKESHAN DiplSme d'Ingenieur C i v i l , Ecole Polytechnique Federale de Lausanne, 1980 A THESIS SUBMITTED IN PARTIAL FULFILMENT OF THE REQUIREMENTS FOR THE DEGREE OF MASTER OF APPLIED SCIENCE i n • THE FACULTY OF GRADUATE STUDIES C i v i l Engineering We accept this thesis as conforming to the required standard THE UNIVERSITY OF BRITISH COLUMBIA A p r i l 1982 © Siavoche Kahkeshan, 1982 In p r e s e n t i n g t h i s t h e s i s i n p a r t i a l f u l f i l m e n t of the requirements f o r an advanced degree at the U n i v e r s i t y of B r i t i s h Columbia, I agree that the L i b r a r y s h a l l make i t f r e e l y a v a i l a b l e f o r r e f e r e n c e and study. I f u r t h e r agree that p e r m i s s i o n f o r e x t e n s i v e copying of t h i s t h e s i s f o r s c h o l a r l y purposes may be granted by the Head of my Department or by h i s or her r e p r e s e n t a t i v e s . I t i s understood that copying or p u b l i c a t i o n of t h i s t h e s i s f o r f i n a n c i a l gain s h a l l not be allowed without my w r i t t e n p e r m i s s i o n . Department of C i v i l E n g i n e e r i n g The U n i v e r s i t y of B r i t i s h Columbia 2075 Wesbrook Place Vancouver, Canada V6T 1W5 Date: 26 A p r i l 1982 i i A b s t r a c t T h i s t h e s i s i s concerned with c a l i b r a t i n g a b e h a v i o u r a l t r a v e l demand model by adjustment of an incomplete data base and a p p l y i n g i t to three s t a t i s t i c a l methods: L o g i t , P r o b i t and D i s c r i m i n a n t A n a l y s i s , comparing the f o r e c a s t i n g a b i l i t y of each of the models and a n a l y s i n g the responsiveness of d e c i s i o n v a r i a b l e s to p o t e n t i a l changes in the urban t r a n s p o r t a t i o n system. In t h i s r e s e a r c h , the c u r r e n t s t a t e of the a r t i s i n d i c a t e d and the mathematical s t r u c t u r e of the models are d i s c u s s e d . Attempts are a l s o made to determine the value of time f o r the Vancouver P o p u l a t i o n as a means of e s t i m a t i n g the v a l i d i t y of the model c a l i b r a t i o n , and a l s o to d i s c u s s the r e s u l t s i n t r a n s p o r t a t i o n p o l i c y terms. i i i T a ble of Contents A b s t r a c t i i L i s t of Tables v L i s t of F i g u r e s v i Acknowledgement v i i I. INTRODUCTION 1 1. THE MODE CHOICE PREDICTION PROBLEM 2 I I . TRANSPORTATION MARKET AND DETERMINATION OF PRINCIPAL VARIABLES 4 1 . SURVEY DATA SAMPLE 4 2. WORK-TRIP SAMPLE 6 3. DETERMINATION OF PRINCIPLE VARIABLES 11 3.1 System V a r i a b l e s 11 3.1.1 Time V a r i a b l e s 11 3.1.2 Cost V a r i a b l e s 13 3.2 Users' V a r i a b l e s 14 3.2.1 Sex 15 3.2.2 Age 17 3.2.3 Income 19 3.2.4 Occupation 21 3.2.5 Car Ownership '. 22 4. SUMMARY 23 I I I . DATA COMPLETION AND MODEL SIMULATION 25 1. LINEAR DATA SIMULATION MODELS 26 1.1 Regression of TTT on IVTT 26 1.2 Regression of MOCOST on IVTT ....28 2. POPULATION COMPARISON 30 2.1 Comparison of P o p u l a t i o n V a r i a n c e s 32 2.2 Comparison of P o p u l a t i o n Means 33 3. SIMULATION OF WAITING TIME VARIABLE 37 4. COMPLETION OF PARKING COST 39 5. SUMMARY 44 IV. BEHAVIOURAL TRAVEL DEMAND THEORY 45 1. MATHEMATICAL THEORY OF BEHAVIOURAL MODELS 46 1.1 M u l t i v a r i a t e L o g i t Model 48 1.2 M u l t i v a r i a t e P r o b i t Model 53 2. ESTIMATING MODEL PARAMETERS ...56 2.1 Maximum L i k e l i h o o d E s t i m a t i o n 56 2.2 E s t i m a t i n g L o g i t Model Parameters 59 2.3 E s t i m a t i n g P r o b i t Model Parameters 60 2.4 Goodness of F i t 61 3. DISCRIMINANT ANALYSIS 63 3.1 C l a s s i f i c a t i o n of an Observation X 66 3.2 Goodness of F i t 68 V. ANALYSIS 69 1. DETERMINATION OF SYSTEM VARIABLES 70 1.1 S e l e c t i o n C r i t e r i a 70 1.2 Model Development 71 1.2.1 L o g i t Treatment 72 1.3 P r o b i t Treatment 72 1.4 D i s c r i m i n a n t A n a l y s i s 79 2. DETERMINATION OF SOCIO-ECONOMIC VARIABLES 81 i v 2.1 S e l e c t i o n C r i t e r i o n s 81 2.1.1 L o g i t and P r o b i t Models 84 2.1.2 D i s c r i m i n a n t A n a l y s i s 89 3. COMPARISON OF THE METHOD OF ANALYSIS 96 4. SUMMARY 100 VI. SENSITIVITY ANALYSIS 102 1. COEFFICIENT INTERPRETATION 102 1.1 P r o b i t 102 1 .2 L o g i t 103 2. SENSITIVITY ANALYSIS 104 2.1 E f f e c t of Income 105 2.2 Income-Sex I n t e r a c t i o n 108 2.3 Income-Age I n t e r a c t i o n 111 2.4 E f f e c t of T r a v e l Cost 111 2.4.1 Fare 114 2.4.2 Parking Cost 114 3. VALUE OF TIME 119 4. COMPARISON WITH OTHER STUDIES 121 V I I . GENERAL CONCLUSION 123 1. SIMULATION 123 2. MODEL STRUCTURE 124 3. VARIABLE SENSITIVITY 124 BIBLIOGRAPHY 127 V L i s t of Tables 1. Composition of Work T r i p Data 9 2. In f l u e n c e of Sex on Modal S p l i t 16 3. In f l u e n c e of Age on Modal S p l i t 18 4. Inf l u e n c e of Income on Modal S p l i t 20 5. Car Ownership D i s t r i b u t i o n 23 6. Li n e a r Models to Estimate M i s s i n g Values of T r a n s i t T r a v e l Time 27 7. L i n e a r Models to Estimate M i s s i n g Values of Monthly Operating Cost 29 8. Comparison of Two T r a n s i t T r a v e l Time P o p u l a t i o n V a r i a n c e s . 31 9. Comparison of Two Operation Cost P o p u l a t i o n V a r i a n c e s 31 10. Comparison of 'TTT' Po p u l a t i o n Means .34 11. Comparison of 'MOCOST' Po p u l a t i o n Means 35 12. S e l e c t e d Models 35 13. GAMMA Model Goodness-of-Fit 42 14. LOGNORMAL Model Goodness-of-Fit 43 15. Parameter E s t i m a t i o n of L o g i t Models ..73 16. Parameter E s t i m a t i o n of P r o b i t Models 74 17. S i g n i f i c a n c e of L o g i t Model M2 Parameters 76 18. S i g n i f i c a n c e of L o g i t Model M2.1 Parameters 76 19. S i g n i f i c a n c e of P r o b i t Model M2 Parameters 77 20. S i g n i f i c a n c e of P r o b i t Model M2.1 Parameters 77 21. C o r r e l a t i o n M a t r i c e s 78 22. Parameter E s t i m a t i o n of D i s c r i m i n a n t Models 80 23. Socio-Economic V a r i a b l e s f o r L o g i t Model M2 87 24. Socio-Economic V a r i a b l e s f o r P r o b i t Model M2 88 25. Point E s t i m a t i o n of L o g i t Model Parameters 90 26. Point E s t i m a t i o n of P r o b i t Model Parameters 91 27. D i s c r i m i n a n t A n a l y s i s : S e l e c t ion of Socio-Economic V a r i b l e s 93 28. C o e f f i c i e n t E s t i m a t i o n of D i s c r i m i n a n t and Z Functions 94 29. D i s c r i m i n a n t Models 1, 2 and 3 95 30. L o g i t O b s e r v a t i o n - P r e d i c t i o n Table 98 31. P r o b i t O b s e r v a t i o n - P r e d i c t i o n Table 99 32. D i s c r i m i n a n t O b s e r v a t i o n - P r e d i c t i o n Table 100 v i L i s t of F i g u r e s 1. T r i p Purpose D i s t r i b u t i o n 5 2. T r i p Length D i s t r i b u t i o n f o r D i f f e r e n t Purposes 7 3. T r a v e l Mode D i s t r i b u t i o n ..8 4. T r a n s i t v . s . P r i v a t e Car T r a v e l Time 36 5. Gamma Generation Model 40 6. Lognormal Generation Model 41 7. Comparison of L o g i s t i c and P r o b i t Forms 55 8. Pseudo R-Square V a r i a t i o n 85 9. Mean Square of E r r o r V a r i a t i o n 86 10. E f f e c t of Income on T r a n s i t Use P r o b a b i l i t y with Respect to ?T 106 11. E f f e c t of Income on T r a n s i t Use P r o b a b i l i t y with Respect to Walking Time 107 12. I n t e r a c t i o n E f f e c t of Income and Sex on T r a n s i t Use P r o b a b i l i t y with Respect to ?T 109 13. I n t e r a c t i o n E f f e c t of Income and Sex on T r a n s i t Use P r o b a b i l i t y with Respect to Walking Time 110 14. I n t e r a c t i o n E f f e c t of Income and Age on T r a n s i t Use P r o b a b i l i t y with Respect to ?T 112 15. I n t e r a c t i o n E f f e c t of Income and Age on T r a n s i t Use P r o b a b i l i t y with Respect to Walking Time 113 16. E f f e c t of Fare V a r i a t i o n on T r a n s i t Use P r o b a b i l i t y with Respect to ?T 115 17. I n t e r a c t i o n E f f e c t of Income and Fare V a r i a t i o n on T r a n s i t Use P r o b a b i l i t y with Respect to ?T 116 18. E f f e c t of Fare V a r i a t i o n on T r a n s i t Use P r o b a b i l i t y with Respect to Walking Time 117 19. I n t e r a c t i o n E f f e c t of Income and Parking Cost V a r i a t i o n on T r a n s i t Use P r o b a b i l i t y with Respect to ?T 118 v i i Acknowledgements I am s i n c e r e l y indebted to my s u p e r v i s o r , Dr. G.R. Brown, f o r h i s c o n s t r u c t i v e c r i t i c i s m and guidance d u r i n g the p r e p a r a t i o n of t h i s t h e s i s , and f o r h i s p a t i e n c e and d i l i g e n c e i n going over the f i r s t d r a f t . I would l i k e to express my a p p r e c i a t i o n to Dr. F.P.D. Navin and Dr. W.F. C a s e l t o n f o r t h e i r c o n t r i b u t i o n s as readers. I am g r a t e f u l to Malcolm G r e i g , s e n i o r a n a l y s t at the Computing Center, U n i v e r s i t y of B r i t i s h Columbia , f o r h i s v a l u a b l e a d v i c e s on s e v e r a l computer programming problems; and to V i c t o r i a Lyon-Lamb, c o n s u l t a n t at the Computer Center, U n i v e r s i t y of B r i t i s h Columbia , who p a t i e n t l y s o l v e d s e v e r a l t e x t p r o c e s s i n g problems for me. I would a l s o l i k e to thank the N a t i o n a l Science and E n g i n e e r i n g Research C o u n c i l of Canada f o r t h e i r f i n a n c i a l support. In a d d i t i o n , I am indebted to my parents f o r t h e i r c ontinued moral support d u r i n g the p r e p a r a t i o n of t h i s t h e s i s and over the course of the past years. F i n a l l y , I would l i k e to mention Jody B u t l e r and J e f f Smyth, two f r i e n d s whose h e l p g r e a t l y improved the s t y l e of t h i s p r e s e n t a t i o n . Having r e c e i v e d the kind h e l p and support of so many people, I must nonetheless take t o t a l r e s p o n s a b i l i t y f o r any e r r o r s and omissions which might be d i s c o v e r e d i n t h i s r e s e a r c h . 1 I. INTRODUCTION T r a n s p o r t a t i o n E n g i n e e r i n g d e c i s i o n s r e q u i r e s u b s t a n t i a l i n s i g h t i n t o the p r e d i c t i o n of t r a v e l demand, i n ge n e r a l and i n the s p e c i f i c region understudy. Methodology and t r a v e l data need to be brought together i n a meaningful way to c a l i b r a t e a mathematical model which can then be used f o r p r e d i c t i n g t r a v e l demand. T h i s r e s e a r c h i s to examine a l o c a l data base (The Vancouver Area T r a v e l Study, VATS) and to apply i t to three commonly used demand model s t r u c t u r e s (the L o g i t , P r o b i t and D i s c r i m i n a n t models) f o r the purpose o f : i . determining adjustments necessary, and f e a s i b l e f o r an incomplete data base, i i . c a l i b r a t i n g , and comparing each of the model s t r u c t u r e s f o r a p p l i c a t i o n i n p r e d i c t i n g t r a v e l demand, and i i i . to t e s t the s e n s i t i v i t y of the models on the r e l e v a n t v a r i a b l e s s e l e c t e d . 2 1. THE MODE CHOICE PREDICTION PROBLEM In order to h e l p t r a n s p o r t a t i o n a u t h o r i t i e s decide on d i f f e r e n t investment schemes, t r a n s p o r t a t i o n planners should be able to diagnose the impact on t r a f f i c flows of changes i n t r a n s p o r t a t i o n p o l i c i e s . Such changes may i n c l u d e the t r a n s i t o p e r a t i o n p o l i c y , modifying the p r i c i n g s t r u c t u r e of the t r a n s p o r t a t i o n system and/or adding a new f a c i l i t y . For t h i s purpose, f o r e c a s t i n g models which a c c u r a t e l y assess the consequences of a l t e r n a t i v e p o l i c i e s on the t r a v e l behaviour of the p o p u l a t i o n becomes necessary. The d e r i v a t i o n of the models should be based on the theory of human behaviour and should i n c l u d e the necessary v a r i a b l e s to formulate t r a v e l l e r ' s socio-economic c h a r a c t e r i s t i c s as w e l l as the a t t r i b u t e s of the t r a n s p o r t a t i o n system. S u b s t a n t i a l e f f o r t s i n the a r t of modal c h o i c e m o d e l l i n g have been expended i n the past 20 years. The e a r l i e r modal ch o i c e models, t r i p - e n d and t r i p - i n t e r c h a n g e modal s p l i t models, used z o n a l l y aggregated data c o n t a i n i n g both c a p t i v e and c h o i c e r i d e r s . For t h i s reason they are not adequately p o l i c y s e n s i t i v e . Two examples of the e a r l i e r model are the Southeastern Wisconsin and Toronto modal s p l i t model(Hutchinson, 1974) . The v e r s i o n of modal s p l i t model s t r u c t u r e d i s c u s s e d i n t h i s t h e s i s uses disaggregated data which w i l l overcome some of the problems of aggregation by p r o v i d i n g estimates of t r i p 3 d i s u t i l i t y . These models attempt to a s s o c i a t e a p r o b a b i l i t y to a t r a v e l l e r ' s d e c i s i o n to use one mode over another and hence the name s t o c h a s t i c , disaggregate modal choice model has been a p p l i e d . However, these models need i n t e n s i v e data which renders t h e i r a p p l i c a t i o n l e s s p r a c t i c a l with c u r r e n t l y a v a i l a b l e data bases. Therefore, t h i s t h e s i s i s an attempt to use an e x i s t i n g data base by e s t i m a t i n g s t a t i s t i c a l l y the inf o r m a t i o n needed to c a l i b r a t e three forms of disaggregate s t o c h a s t i c mode c h o i c e models. 4 I I . TRANSPORTATION MARKET AND DETERMINATION OF PRINCIPAL VARIABLES The c a l i b r a t i o n of the econometric modal s p l i t models f o r Vancouver i s based on a survey conducted i n the Vancouver M e t r o p o l i t a n Area i n the e a r l y s p r i n g 1972. T h i s survey , c a l l e d Vancouver A c t i v i t y T r a v e l Study ( VATS ), was c o l l e c t e d over approximately 3600 households r e p r e s e n t i n g 1% of the households i n the Region. 1. SURVEY DATA SAMPLE Relevant data f o r t h i s study are a v a i l a b l e i n four VATS data f i l e s . These f i l e s c o n t a i n household and per s o n a l i n f o r m a t i o n , t r i p records and modal choice i n f o r m a t i o n and are l i n k e d together by means of i d e n t i f i c a t i o n numbers as s i g n e d to each r e c o r d . A t o t a l of 26,652 t r i p s were r e p o r t e d . According to F i g u r e 1, 12.18% of the t r i p s were made f o r work purposes. T h i s r e l a t i v e l y low p r o p o r t i o n of work t r i p s might be due to the time of the day that the survey was conducted. The VATS survey was c a r r i e d out d u r i n g the day and the t r i p s c o l l e c t e d are those f o r the day b e f o r e . T h e r e f o r e , most of the workers were not p e r s o n a l l y i n t e r v i e w e d but the work t r i p i n f o r m a t i o n was c o l l e c t e d from at home members of the household. V IP rv 5?" 00 C7 r-o • n 1 5 o -r v VrjRK VORX VORK VDRK VORK VORK "WORK VfJRK "WORK VDRK W)RK SHOP SHDP SHOP SHOP S H O P SHOP SHOP SHOP SHOP SHOP SHDP SHOP SHDP ^ H D P S C H L S C H L S C H L S C H L S C H L S C H L sr.Hi R E C R RECR RECR RECR RECR RECR RECR RECR RECR R E C R RECR R E C R R E C R R E C R OT MR DTHR OTHR OTHR QIHR OTHR DThR OTHR DTHR OTHR OTHR OTHR OTHR OTHR OIHR OTHR OTHR OTHR OTHR DTHR OIHR OTHR DTHR OIHR DTHR OTHR DTHR OIHR OIHR DTHR OTHR OIHR DTHR DTHR OTHR DTHR DTHR O T H R n r HR " W O R K SHDP S C H L RECR OIHR F i g u r e 1 - T r i p Purpose D i s t r i b u t i o n 6 The l a r g e s t category i s that of "OTHER"(46.26%). T h i s i s mostly due to the aggregation of purposes other than those i l l u s t r a t e d in F i g u r e 1 f o r t h i s category. The "OTHER" category embodies purposes such as: e a t i n g , p e r s o n a l b u s i n e s s , r i d i n g along, s t r o l l i n g e t c . . T r i p l e n g t h d i s t r i b u t i o n f o r d i f f e r e n t purposes i s i l l u s t r a t e d i n F i g u r e 2. A l l t r i p s have a d i s t r i b u t i o n common to other surveys. Average work t r i p l e n g t h i s between 20 and 30 minutes. Compared to the o t h e r s , the work t r i p d i s t r i b u t i o n has the g e n t l e s t s l o p e , which might be due to people having l e s s f l e x i b i l i t y to choose t h e i r work l o c a t i o n . The modal s p l i t d i s t r i b u t i o n of the sample i s shown in F i g u r e 3. T r a v e l l i n g by automobile p r e s e n t s the l a r g e s t p r o p o r t i o n of t r i p s (46.64%). The second most popular t r a v e l mode i s walking which i s 23.71% of the 26419 t r i p s . However, fo r work t r i p s , the most important means of t r a v e l are auto and bus (See T a b l e l ) . 2. WORK-TRIP SAMPLE The sample of journey to work forms the b a s i c sample for t h i s study. I t c o n s i s t s of 3172 cases. A case i s c o n s i d e r e d incomplete i f only one p i e c e of i n f o r m a t i o n about one v a r i a b l e ( i . e . T r a v e l Time, Age, Income, ... ) i s m i s s i n g . From 3172 w o r k - t r i p records c o l l e c t e d from the f i r s t three VATS data f i l e s ( household i n f o r m a t i o n , p e r s o n a l i n f o r m a t i o n 7 F i g u r e 2 - T r i p Length D i s t r i b u t i o n f o r D i f f e r e n t Purposes SHOP S H O P S H O P V O R K S H O P V O R K S H O P . V O R K S H O P j—- W O R K S H O P r~ - V O R K S H O P S C H L V O R K S H O P S C H L V O R K S H O P S C H L V O R K S H O P S C H L " W O R K S H O P S C H L S H O P S C H L CD VflRK q r m C3 W O R K S H O P S C H L R E C R R E C R R E C R RECR R E C R RECR R E C R R E C R R E C R RECK R E C R R E C R R E C R RECR REHR OTHR DTHR DTHR DTHR OIHR OIHR OIHR OIHR OIHR OIHR DTHR DTHR DTHR DTHR DTHR OIHR DTHR DTHR DTHR DTHR DTHR DTHR DTHR DTHR DTHR DTHR DTHR DTHR DTHR DTHR DTHR DTHR DTHR DTHR DTHR OIHR DTHR DTHR RECR OIHR F i g u r e 3 - T r a v e l Mode D i s t r i b u t i o n TRAVEL MODE NUMBER % Auto 538 79.23 Auto Pass. 58 8.54 Bus 83 1 2.23 Walk 0 ' 0.0 Z 679 1 00 Table 1 - Composition of Work T r i p Data 10 and t r i p r e c o r d s ) , only 841 cases have complete i n f o r m a t i o n about home-based w o r k - t r i p s made between the two morning peak hours(7-9 AM). Out of these 841 responders, 130 persons d i d not have a d r i v e r s l i c e n c e and 32 persons belonged to a household without any c a r s . T h e r e f o r e , t h i s sample has 162 c a p t i v e t r a n s i t passengers (19.26%) who were excluded from t h i s set as changes in s e r v i c e p o l i c y r e l a t e d do not i n f l u e n c e t h e i r t r a v e l behaviour. T h i s reduces the sample s i z e to 679 cases. The modal c h o i c e composition of the u n d e r l y i n g sample i s given i n Table 1. Since t h i s study i s concerned about two t r a v e l modes: p r i v a t e car and bus, the cases corresponding to auto passengers are a l s o d e l e t e d . The remaining data set of 621 records i s f u r t h e r c o n s i d e r a b l y reduced i n s i z e when an attempt i s made to l i n k the modal c h o i c e i n f o r m a t i o n , s i n c e the f i l e c o n t a i n i n g t h i s i n f o r m a t i o n i s the most incomplete f i l e . T h i s i s due to the f a c t that people can not e a s i l y estimate the s e r v i c e c h a r a c t e r i s t i c s of t h e i r t r a v e l mode. To solve t h i s problem, an attempt should be made to complete the data s e t . T h i s i s d i s c u s s e d i n d e t a i l i n the next chapter. 11 3 . DETERMINATION OF PRINCIPLE VARIABLES Before completing the data base, the s e l e c t i o n of the r e l e v a n t v a r i a b l e s and t h e i r a p p r o p r i a t e ' f o r m to be i n c l u d e d i n the modal c h o i c e a n a l y s i s should be determined. In order to d e f i n e the set of the explanatory v a r i a b l e s , we s h a l l use the r e s u l t s of p r e v i o u s s t u d i e s as w e l l as e n g i n e e r i n g judgement. We should a l s o t r y to f i n d a compromise between the number of e x p l i c a t o r y v a r i a b l e s , the s i m p l i c i t y of the v a r i a b l e form and the goodness of f i t obtained, s i n c e models with a l a r g e number of v a r i a b l e s are expensive to manipulate. V a r i a b l e s i n f l u e n c i n g modal c h o i c e d e c i s i o n s can be c l a s s i f i e d i n two c a t e g o r i e s : . i . System v a r i a b l e s , i n c l u d i n g v a r i a b l e s r e l a t e d to the f u n c t i o n i n g of the t r a n s p o r t a t i o n system, and i i . Users' v a r i a b l e s , i n c l u d i n g v a r i a b l e s d e s c r i b i n g the socio-economic c h a r a c t e r i s t i c s of the users as w e l l as t h e i r t a s t e . 3.1 System V a r i a b l e s 3 . 1 . 1 Time V a r i a b l e s There i s ample evidence that the components of o v e r a l l t r a v e l time ( w a l k i n g , w a i t i n g , i n - v e h i c l e and t r a n s f e r time) are p e r c e i v e d d i f f e r e n t l y from t r a v e l l e r s s i n c e the degree of 12 inconvenience a s s o c i a t e d with these components are not e q u a l . M e r l i n and Barbier(1965) found that P a r i s i a n s c o n s i d e r e d walking time to be 1.75 times more inconvenient than i n - v e h i c l e time. Waiting and t r a n s f e r time was valued three times and twice the value of i n - v e h i c l e time r e s p e c t i v e l y . Another study c a r r i e d out i n Manchester showed that walking and w a i t i n g time was valued 2.60 and 3.60 times the value of i n - v e h i c l e time r e s p e c t i v e l y (Rogers, Townsend and M e t c a l f , 1970). D e s p i t e t h i s evidence, i n s e v e r a l s t u d i e s the o v e r a l l t r a v e l time was i n t r o d u c e d i n t o the model s i n c e d e t a i l e d i n f o r m a t i o n r e g a r d i n g the components of the t r a v e l 'time was u n a v a i l a b l e . L i s c o ( 1 9 6 7 ) , Lave(1969), and De Donnea(l97l) c o n s i d e r e d the o v e r a l l t r a v e l time, whereas Brown(l972), N a v i n ( l 9 7 4 ) , 0' F a r r e l and Markham(1975) and S e g a l d 9 7 8 ) i n v e s t i g a t e d measuring the e f f e c t of d i f f e r e n t time a c t i v i t i e s on the t r a v e l d i s u t i l i t y . T r a v e l t i m e ( e i t h e r walking, w a i t i n g or i n - v e h i c l e ) can be expressed i n d i f f e r e n t ways. Warner(l962) and McGi11ivray(1970) have used the l o g a r i t h m of t r a v e l time r a t i o , whereas L i s c o , Quarmby(1967), Lave and De Donnea have c o n s i d e r e d the d i f f e r e n c e between t r a v e l times, arguing that commuters p e r c e i v e r e l a t i v e times i n terms of d i f f e r e n c e than i n terms of r a t i o . In h i s study, Brown has used both d i f f e r e n c e and r a t i o of t r a v e l times, and Watson(l974) formulated the r a t i o between t r a v e l times d i f f e r e n c e and the t r i p l e n g t h as being the average of a c t u a l and a l t e r n a t i v e mode t r a v e l time ((T1-T2)/[(T1+T2)/2]). He 13 argued that f i v e minutes saved on a ten-minutes journey may be more important than on a four-hour journey. Ben Akiva and Atherton(1977) used i n - v e h i c l e t r a v e l times d i f f e r e n c e and the r a t i o between excess t r a v e l time and t r a v e l e d d i s t a n c e to express the e f f e c t of time v a r i a b l e s on mode ch o i c e p r o b a b i l i t y . In the present study, we w i l l t e s t two f o r m u l a t i o n s of each time a c t i v i t i e s . The f i r s t one i s to introduce i n t o the model the d i f f e r e n c e s and the second i s to adopt the d i f f e r e n c e - r a t i o f o r m u l a t i o n , where f o r each tripmaker the r a t i o of the d i f f e r e n c e between the a c t u a l and the a l t e r n a t i v e t r a v e l time to the a c t u a l o v e r a l l t r a v e l time i s computed. For i n s t a n c e , i n the case of an i n d i v i d u a l t r a v e l l i n g by c a r , the r e l a t i v e i n -v e h i c l e time i s equal t o : [(Tb-Ta)/TTa] and s i m i l a r l y , i n the case of a person t r a v e l l i n g by bus we have: [(Tb-Ta)/TTb]. 3.1.2 Cost V a r i a b l e s Cost v a r i a b l e s r e f l e c t a l l out-of-pocket expenses. In the case of bus r i d e r s the i n d i v i d u a l c o s t of a t r i p i s s t r a i g h t f o r w a r d . I t i s the bus f a r e . Whereas f o r car d r i v e r s , the one-way t r i p c ost per c a p i t a can not be determined e a s i l y , s i n c e i t i s more convenient to gather data r e g a r d i n g the monthly o p e r a t i n g c o s t and monthly p a r k i n g charge than the d a i l y c o s t . To compute the t o t a l cost of a t r i p made by c a r , we s h a l l use the f o l l o w i n g h y p o t h e s i s : we suppose that two t r i p s are made dur i n g business days and there are 20 business days i n a month. Th e r e f o r e , the t o t a l cost per t r i p w i l l be equal t o : (Monthly Operating Cost+Monthly Parking Charges) / 20x2 14 and we assume that the above f i g u r e r e p r e s e n t s the one-way t r i p c o s t per c a p i t a s i n c e the d a i l y v e h i c l e occupancy rate i s not a v a i l a b l e from the survey. The f o r m u l a t i o n of co s t v a r i a b l e should f o l l o w the p r i n c i p l e of the t r a v e l time f o r m u l a t i o n s i n c e both r e f l e c t the co s t occured to a commuter due to t r a v e l l i n g . T h e r e f o r e , we w i l l use c o s t s d i f f e r e n c e ( C b - C a ) wherever time d i f f e r e n c e s are adopted and use c o s t s d i f f e r e n c e - r a t i o [ ( C b - C a ) / C ] wherever t r a v e l time components are formulated i n t h i s manner. 3.2 Users' V a r i a b l e s T h i s f a m i l y of v a r i a b l e s r epresent socio-economic q u a l i t i e s of t r i p m a k e r s . They are normally i n t r o d u c e d i n t o modal c h o i c e models to express the change i n the p r o b a b i l i t y of a mode use r e s u l t i n g from a change i n the socio-economic c h a r a c t e r i s t i c s of i n d i v i d u a l s . In t h i s category, one may c l a s s i f y v a r i a b l e s such as: Household Income, M a r i t a l S t a t u s , Age, Sex, number of Cars Owned by the Household, Type of Household, Household Composition, Education, O c c u p a t i o n , . . . e t c . De Donnea(l971) has s t u d i e d the e f f e c t of Income, Age, Sex and M a r i t a l " S t a t u s , number of People i n the Household and the R e l a t i o n of the Tripmaker to the Head of Household. O ' F a r r e l and Markham(1975) c o n s i d e r e d Income, Age, Sex and M a r i t a l S t a t u s , Car-Demand r a t i o , Importance of car at work and Household Composition as users' v a r i a b l e s . Lave(l969) i n c l u d e d v a r i a b l e s such as Auto Ownership, Family S i z e and Composition, 15 Income, Sex and Age of commuters i n h i s study. Quarmby(1967) has measured the impact of Income, Car-Demand r a t i o , Ownership of car by f i r m and Use of car f o r work on the p r o b a b i l i t y of t a k i n g t r a n s i t mode. In s h o r t , a v a r i e t y of socio-economic v a r i a b l e s a f f e c t s the d e c i s i o n of an i n d i v i d u a l f a c i n g s e v e r a l a l t e r n a t i v e s . However, two major reasons prevent one from using some of these v a r i a b l e s : t h e i r a v a i l a b i l i t y and t h e i r ease of f o r e c a s t i n g . For i n s t a n c e , v a r i a b l e s such as M a r i t a l S t a t u s , Household Composition and S i z e , Car-Demand r a t i o , Importance of car at work, while they might i n c r e a s e the model refinement, are very d i f f i c u l t to p r e d i c t and consequently t h e i r i n c l u s i o n l i m i t s d r a s t i c a l l y the p r e d i c t i o n boundary of the model. The present study w i l l use Income, Age, Sex, Occupation and Car Ownership to formulate the p e r s o n a l c h a r a c t e r i s t i c of t r a n s p o r t a t i o n demand. T h e i r s i g n i f i c a n c e i n improving the c a p a b i l i t y of the model w i l l be s t a t i s t i c a l l y t e s t e d . 3.2.1 Sex There i s evidence which s t r o n g l y suggests that sex of a t r i p maker may i n f l u e n c e modal c h o i c e p r o b a b i l i t y . According to M o r a l l ( 1 9 7 1 ) , the g r e a t e r p r o p o r t i o n of t r a n s i t r i d e r s are female. He used s e v e r a l surveys(1969-1976) and found that i n most Canadian c i t i e s , the.percentage of female t r a n s i t users was above 60%. T h i s might be due to the h i s t o r i c a l dominance of male owned and operated c a r s , and the d i f f e r e n c e between the 16 t h r e s h o l d l e v e l of preference and comfort. The a n a l y s i s of our data r e v e a l s that only 22.12% of auto-d r i v e r s p o p u l a t i o n are female(see Table 2). We w i l l use the i n d i c a t o r v a r i a b l e SEX to formulate the sex i n f l u e n c e i n the models. SEX = 1 i f the tripmak er i s male and 0 i f the tripmaker i s female Mode Ni imber Pei "centage T o t a l Male Female of Tot. Male Female Auto Bus 538 83 419 35 1 19 48 86.63 13.37 77.88 42. 1 6 22. 12 57.84 T o t a l 621 454 1 67 1 00 Table 2 - In f l u e n c e of Sex on Modal S p l i t 17 3.2.2 Age Table 3 d i s p l a y s the e f f e c t of age on the p r o b a b i l i t y of t r a n s i t use. Regarding modal d e c i s i o n , t r a v e l l e r s may be d i v i d e d i n t o two groups: i . The f i r s t group c o n s i s t s of young and e l d e r l y people s i n c e t h e i r behaviour toward t r a n s i t use may be formulated i n a s i m i l a r f a s h i o n . A m a j o r i t y of them use t r a n s i t mode due to the d i f f i c u l t a c c e s s i b i l i t y to c a r s : the lack of a d r i v e r ' s l i c e n c e and a budget c o n s t r a i n t i n the case of young people and the lower p s y c h o l o g i c a l p r e f e r e n c e f o r the car and d i f f e r e n t p e r c e p t i o n of the degree of comfort i n the case of e l d e r l y persons. They are l e s s s e n s i t i v e to t r a v e l time, l e s s s e n s i t i v e to l o s s of p r i v a c y but very a f f e c t e d by the out-of-pocket expenses. i i . The second group c o n s i s t s of the remaining people. We w i l l c o n s i d e r the dummy v a r i a b l e AGE to express the i n f l u e n c e of age on the t r a n s i t use p r o b a b i l i t y . I t i s formulated as below: AGE = 0 i f t r a v e l l e r ' s age belongs to the semi-closed i n t e r v a l [25,60[ and AGE = 1 otherwise 18 AgeBracket T o t a l AutoDriv. TransPass > %Auto %Trans. <18 1 1 9 2 82 18 18-25 1 29 95 34 74 26 25-30 1 02 89 1 3 87 1 3 30-35 93 86 7 92 8 35-40 87 76 1 1 87 1 3 40-45 80 75 5 94 6 45-50 60 54 6 90 10 50-55 40 38 2 95 5 55-60 1 4 1 2 2 86 14 >60 5 4 1 80 20 T o t a l 621 538 83 Table 3 - I n f l u e n c e of Age on Modal S p l i t 19 3.2.3 Income I n t u i t i v e l y , income should p l a y a great r o l e i n the modal s e l e c t i o n p r o c e s s . Table 4 d i s p l a y s d i f f e r e n t income brack e t s a v a i l a b l e i n the sample p o p u l a t i o n and t h e i r observed modal s p l i t f i g u r e s . I t a l s o r e v e a l s that the f o r m u l a t i o n of the Income v a r i a b l e i n the model i s more l i k e l y to be a combination with other v a r i a b l e s r a ther than to be a separate v a r i a b l e , s i n c e there i s no tendency of the t r a n s i t r i d e r s h i p to vary with the income v a r i a b l e i n a sigmoid(S-shape) f a s h i o n . Income i s combined with the components of the g e n e r a l i z e d t r a v e l cost' on the b a s i s of time-money t r a d e - o f f s i t u a t i o n . Two c l a s s e s of i n d i v i d u a l s may be c o n s i d e r e d : i . The c l a s s of i n d i v i d u a l s who are w i l l i n g to g i v e up time to save money. i i . The c l a s s of i n d i v i d u a l s who p r e f e r s to spend e x t r a money to save time. There always e x i s t s a t h r e s h o l d value beyond which an i n d i v i d u a l changes groups. For example, a time chooser who remains a time chooser i n a s i t u a t i o n where he should spend 4 d o l l a r s to save 1 minute, but becomes a money chooser when he i s faced with the s i t u a t i o n where spending 6 d o l l a r s i s r e q u i r e d , has a t h r e s h o l d value between 4 and 6 d o l l a r s . T h i s value i s c a l l e d the i n d i v i d u a l ' s marginal value of time. However, i n the r e a l world, the determination of t h i s value Income No. %of i n Number i n Percentage Bracket Tnt-=i 1 Auto Bus Auto Bus <3000 1 1 1 .77 10 1 90.91 9.09 3000- 5999 34 5.48 30 4 88.24 1 1 .76 6000- 8999 1 05 6.91 88 17 83.81 16.19 9000-11999 1 48 23.83 •1 28 20 86.49 13.51 12000-14999 131 21.10 113 18 86.26 1 3.74 15000-17999 85 13.69 74 1 1 87.07 1 2.94 18000-20999 54 8.70 48 6 88.89 11.11 21000-23999 1 6 2.58 1 3 3 81 .25 18.75 24000-26999 20 3.22 20 — 100. 27000-29999 5 0.81 5 — 100. >30000 12 1 .93 9 3 75. 25. T o t a l 621 100. 535 86 Table 4 - I n f l u e n c e of Income on Modal S p l i t 21 i s very d i f f i c u l t . One way of surmounting t h i s d i f f i c u l t y i s to r e l a t e the value of time to income, assuming that a saving i n t r a v e l time can be assigned to more pro d u c t i o n and hence l e a d to more employee-hours. However, there are many o b j e c t i o n s to t h i s method which are not d i s c u s s e d i n the present study ( i . e . see Quarmby and Harrison(1969) ). Lave(l969) and De Donnea(l97l) have assumed that t r a v e l time savings are p r o p o r t i o n a l to income, whereas Quarmby(1967), B.en Akiva and Atherton (1977) have r e l a t e d the income to the out-of-pocket expenses; the formers have formulated an Income-Time v a r i a b l e (I.AT) and the l a t t e r s have used an Income-Cost v a r i a b l e (AC/I). We w i l l t e s t both approaches fo r our model and keep the one which leads to a b e t t e r f i t . The value of income used f o r each bracket i s i t s mid-range value i n 1000$. 3.2.4 Occupation Even though occupation might be h i g h l y i n t e r r e l a t e d with income, we have decided to i n c o r p o r a t e i t i n t o the model, s i n c e income might not express adequately the impact of the i n d i v i d u a l ' s s o c i a l s t a t u s and p r e s t i g e on modal c h o i c e p r o b a b i l i t y . I n d i v i d u a l s are c a t e g o r i z e d i n t o 4 occupation-groups a c c o r d i n g to S t a t i s t i c Canada: Primary, P r o f e s s i o n a l and Managerial, C l e r i c a l - S a l e s and L a b o u r - S e r v i c e s . Two b i n a r y dummy v a r i a b l e s , OCC1 and OCC2, were used to formulate t h i s 22 c l a s s i f i c a t i o n as f o l l o w s : Primary 0CC1 0 0CC2 = 0 P r o f e s s i o n a l and Managerial 0CC1 0 0CC2 = 1 C l e r i c a l - S a l e s 0CC1 0CC2 = 0 Labour-Services 0CC1 0CC2 = 1 3.2.5 Car Ownership Number of c a r s in a household may a f f e c t the modal c h o i c e . However, s i m i l a r to the occupation v a r i a b l e , Car Ownership might be c o n s i d e r e d only i f i t s i n c l u s i o n a m e l i o r a t e s the f i t n e s s of the model and does not le a d to l a r g e v a r i a n c e s of the coef f i c i e n t s . C a t e g o r i c a l v a r i a b l e CO formulates t h i s f a c t o r as f o l l o w s : CO = 0 i f the household possesses one car CO = 1 i f the household possesses more than one car 23 No.Car/Hous. in Number in Percent. 1 249 40.1 0 2 266 42.83 3 67 10.79 >3 39 6.28 T o t a l 621 100. Table 5 - Car Ownership D i s t r i b u t i o n 4. SUMMARY In summary, we b e l i e v e that the modal ch o i c e model s p e c i f i e d should c o n s i s t of those v a r i a b l e s which not only a f f e c t the t r a v e l l e r s ' p r e f e r e n c e s but are easy to f o r e c a s t . R e f e r i n g to the l i t e r a t u r e a v a i l a b l e on modal c h o i c e m o d e l l i n g , we e s t a b l i s h e d the f o l l o w i n g set of v a r i a b l e s and v a r i a b l e forms: I n - V e h i c l e T r a v e l W aiting Time Walking Time Time AT AT/T I.AT I.AT/T WAIT WAIT/T I.WAIT I.WAIT/T AWALK AWALK/T I.AWALK I.AWALK/T 24 Out-of-pocket Expenses AC AC/C AC/I AC/I.C Age AGE Sex SEX Occupation OCC1 and OCC2 Car Ownership CO •' where: AT = Tb-Ta T i s the a c t u a l o v e r a l l t r a v e l time WAIT i s the w a i t i n g time at bus stops AWALK = (walk to from bus stop) -(walk to from c a r ) AC = Cb-Ca C i s the a c t u a l t o t a l t r a v e l c o s t I i s the household annual income ( i n 1000 $) We s h a l l b u i l d a p p r o p r i a t e models which enable us to deduce those v a r i a b l e s (system and users') that appear most promising from the p o i n t of view of a tripmaker. T h i s i s t r e a t e d i n the next c h a p t e r s . 25 I I I . DATA COMPLETION AND MODEL SIMULATION T h i s chapter i s concerned with the c o n s t r u c t i o n of models i n order to complete the data base. For t h i s purpose, i t was decided f i r s t to c o l l e c t those cases which are complete on the f o l l o w i n g p a i r of v a r i a b l e s : I n - V e h i c l e T r a v e l Time(lVTT) and T r a n s i t T r a v e l Time(TTT) 1, and IVTT and Monthly Operating Cost(MOCOST) 2; to f i n d by means of s t a t i s t i c a l t o o l s a r e l a t i o n s h i p between them, and then apply t h i s r e l a t i o n s h i p to those d e l e t e d cases which are incomplete on only o n e - v a r i a b l e i n order to estimate the mis s i n g value of the ot h e r . Note that t h i s approach may be a p p l i e d l o g i c a l l y only to simulate TTT and MOCOST, si n c e one may r e l a t e TTT and MOCOST to IVTT but not the ot h e r s . For i n s t a n c e , as IVTT i n c r e a s e s , f u e l consumption and hence t r a v e l c o s t i n c r e a s e s . A l s o the r i d i n g time i n a bus l o g i c a l y has some r e l a t i o n s h i p with the i n - v e h i c l e t r a v e l time. On the other hand, w a i t i n g time seems to be independant of TTT or IVTT. T h e r e f o r e , to complete m i s s i n g values on w a i t i n g time other methods should be i n v e s t i g a t e d , and t h i s i s f u r t h e r d i s c u s s e d . For par k i n g c o s t s , the average zonal parking charges are c o n s i d e r e d . 1 rprprp _ f (complete I V T T ) 2 MOCOST = f(complete I V T T ) 26 1. LINEAR DATA SIMULATION MODELS L i n e a r r e g r e s s i o n models are a p p l i e d to the p a i r s of v a r i a b l e s i n d i c a t e d above. T h e r e f o r e , the dependant v a r i a b l e s TTT and MOCOST are regressed on IVTT. Tables 6 and 7 show the r e s u l t s of t h i s attempt. In order to s e l e c t the best model, the f o l l o w i n g c r i t e r i a are used : i . R e l a t i v e magnitude and s i g n of r e g r e s s i o n c o e f f i c i e n t s i i . Squared c o e f f i c i e n t of c o r r e l a t i o n i i i . V a r i a t i o n i n sum of square due to r e g r e s s i o n (SSR) and sum of square due to e r r o r term (SSE) from one model to the other. 1.1 Regression of TTT on IVTT F i v e models are candidates to express the v a r i a t i o n of TTT as a f u n c t i o n of IVTT. Acco r d i n g to Table 6, the magnitude of the constant term i n models 1 and 3 renders the i n c l u s i o n of the constant term i n t o the models q u e s t i o n a b l e , s i n c e the constant term r e p r e s e n t s the mean of the p r o b a b i l i t y d i s t r i b u t i o n of. TTT at IVTT=0, that i s an o v e r e s t i m a t i o n of t r a n s i t t r a v e l time by 16 and 13 minutes r e s p e c t i v e l y . A f u r t h e r reason which might support the d e l e t i o n of the constant term from the models i s the v a r i a t i o n i n SSR i n r e l a t i o n to SST(the t o t a l sum of square). No Y X Models SSR SSE SST Overal1 S i g n i f . Std. Er •ror ol • Coeff . c >1gnif Cnst . X X ! ,LogX Cnst. X X' ,LogX 1 TTT IVTT TTT=17.30+1 .5 1 (IVTT ) .27709 97400 .25 +6 .35 +6 .0000 3 .90 . 15 --- .0000 .0000 ---2 TTT IVTT TTT= 2. 1(IVTT ) .27709 .86 +6 .27 +6 .11+7 0. --- .071 --- --- 0. ---3 TTT IVTT, IVTT' TTT=15.43+1.G8(IVTT) -.003(IVTT)> .27734 97489 .25 +6 .35 +6 .0000 7 . 20 .55 .009 .0330 .0026 .7570 4 TTT IVTT , IVTT' TTT=2.79(IVTT)-.02(IVTT)' .27079 .86 +6 .26 +6 .11+7 0. --- . 193 .005 --- .0000 .0001 5 TTT LoglVTT TTT=18.04Log(IVTT) .25688 .85 +6 .28 +6 .11+7 0. --- --- .62 --- --- 0. Table 6 - Linear Models to Estimate Missing Values of Transi t Travel Time 28 For i n s t a n c e , by d e l e t i n g the constant term from Model 1, the r a t i o SSR/SST i n c r e a s e s from 0.278 to 0.782. T h i s i n c r e a s e i n d i c a t e s that the t o t a l v a r i a b i l i t y i n TTT which i s accounted f o r by Model 2, i s g r e a t e r than that accounted f o r by the f i r s t model, s i n c e SSR may be c o n s i d e r e d as a measure of the v a r i a b i l i t y of TTT a s s o c i a t e d with the r e g r e s s i o n l i n e , and the l a r g e r SSR i s i n r e l a t i o n to SST, the g r e a t e r i s the e f f e c t of the r e l a t i o n s h i p . T h e r e f o r e , Models 2, 4, 5 are c a n d i d a t e s f o r f u r t h e r a n a l y s i s . 1.2 Regression of MOCOST on IVTT Table 7 shows three models r e l a t i n g the monthly o p e r a t i n g c o s t of an auto d r i v e r to h i s d r i v i n g time to work. Since the d r i v i n g time to work i s not the only v a r i a b l e which determines the monthly o p e r a t i n g c o s t , the presence of a constant term i n the models may be j u s t i f i e d by assuming that i t would take account of other r e l e v a n t v a r i a b l e s which are not i n c l u d e d i n the model due to the l a c k of data. T h i s constant term can be c o n s i d e r e d to be the average f i x e d c o s t of owning a c a r . The low squared c o e f f i c i e n t of c o r r e l a t i o n and the l a r g e standard e r r o r of the r e g r e s s i o n c o e f f i c i e n t s are a matter of s e r i o u s concern. These are mainly due to the nonconstancy of e r r o r v a r i a n c e ( H e t e r o s k e d a s t i c i t y ) . In other words: tf2(e) = k(lVTT) To a v o i d the problem of h e t e r o s k e d a s t i c i t y , the c l a s s i c a l t r a n s f o r m a t i o n : No Y X Models R' SSR SSE SST Overa11 Signi f . Model Expression C o e f . S t d . E r r S1gn1f . CNST . X, 1/X CNST . X. 1/X 1 MOCOST IVTT MOCOST=35.30+.0541VTT .00052 77.175 .15+6 .15+6 .7661 M0C0ST=35.30-.054(IVTT) 4 .98 . 18 .0000 .7661 2 MOCOST IVTT 1 IVTT MOCOST 27.70 = .42 + IVTT IVTT .27090 199.44 536.79 736.23 .0000 M0C0ST=27.70+.41(IVTT) .24 3.48 .0797 .0000 3 MOCOST IVTT 1 IVTT MOCOST 5 = .03 + IVTT IVTT .72964 6.5517 2.4277 8.9794 .0000 M0C0ST=25+.3(IVTT) +.001(IVTT)' .016 .234 .0310 .0000 Table 7 - Linear Models to Estimate Missing Values of Monthly Operating Cost 30 (MOCOST)' = MOCOST/IVTT , (IVTT)' = 1/IVTT and (MOCOST)' = SQRT(MOCOST)/IVTT , (IVTT)' = 1/IVTT are used. Note the inc r e a s e i n the squared c o e f f i c i e n t of c o r r e l a t i o n and the r e d u c t i o n i n the c o e f f i c i e n t s ' standard e r r o r i n r e l a t i o n to t h e i r magnitude. These changes are s p e c i a l l y c o n s i d e r a b l e f o r Model 3. Th e r e f o r e , the two f o l l o w i n g r e g r e s s i o n models are chosen: MOCOST = 27.90 + .40(IVTT) MOCOST = 2 5 + .30(IVTT) + ( I V T T ) 2 2. POPULATION COMPARISON In order to s e l e c t the f i n a l models, i t i s u s e f u l to compare f o r each v a r i a b l e the mean of survey p o p u l a t i o n ( P o p u l a t i o n 1:with missing data) and the mean of p o p u l a t i o n r e s u l t e d from completing the mi s s i n g data by means of r e g r e s s i o n models ( P o p u l a t i o n 2). To i l l u s t r a t e the problem, assume that each p o p u l a t i o n i s normally d i s t r i b u t e d , even though the d i s t r i b u t i o n of some v a r i a b l e s are l a r g e l y skewed (see Tables 8, 9). Let », and J I 2 represent the two p o p u l a t i o n means, <r, and c2 the two p o p u l a t i o n v a r i a n c e s and m,, m2, s , 2 and s 2 2 t h e i r e s t i m a t o r s , r e s p e c t i v e l y . No Models Popi j l a t ion 2 F Stat . Sign i f P O p U 1 c t I on 1 N Mean Var . Skew. N Mean Var . Skew. 2 TTT = 2.1(IVTT) 679 51 . 27 916.37 1 .056 1 .594 .0000 4G3 51 .61 1461 .5 2 . 76 4 TTT=2.79(IVTT)-.02(IVTT)' 679 52 . 24 511.24 . 107 2.858 .0000 5 TTT=18.04 Log(IVTT) 679 54 . 74 131.81 - .476 11.088 0. Table 8 - Comparison of Two Transi t Travel Time Populat ion Variances PopU1c it i on 1 No Models Popu at lon 2 F Stat . Sign If N Mean Var . Skew. N Mean Var . Skew. 275 35 . 36 644 . 10 3 . 77 1 M0C0ST=27.70+.42(IVTT) 679 37 . 95 36.42 1 . 27 17.68 0. 2 M0C0ST=25+.30(IVTT) +.001(IVTT) ! 679 33 . 24 28.40 1 . 35 22 .68 0. Table 9 - Comparison of Two Operation Cost Populat ion Variances 32 If the u n d e r l y i n g p o p u l a t i o n s have the same v a r i a n c e s , then the sampling d i s t r i b u t i o n of (n^-nia) w i l l have the t -d i s t r i b u t i o n with (N!+N2-2) degrees of freedom. On the other hand, i f the v a r i a n c e s d i f f e r , then the sampling d i s t r i b u t i o n of (m1-m2) w i l l have a t - d i s t r i b u t i o n with DF degrees of freedom, where DF i s equal to(see A f f i f i and Azen(l972), Dixon(1969)): ( A, + A 2 ) DF = - 2 A, 2/(N, +1) + A 2 2 / ( N 2 +1) where Ai '= s i 2 / N i . 2.1 Comparison of Po p u l a t i o n V a r i a n c e s For t h i s purpose, we s h a l l use the s t a t i s t i c s , 2 / s 2 2 and t e s t the n u l l hypothesis H 0 : 2 = <j2 2 v i s - a - v i s H , : <r 1 2 ^ c r 2 2 . Under the assumption of no r m a l i t y , t h i s s t a t i s t i c has an F-d i s t r i b u t i o n with (N,,-1, N 2-1) degrees of freedom. The a n a l y s i s of v a r i a n c e s of Po p u l a t i o n s 1 and 2 are shown i n Tables 8, 9. These t a b l e s c l e a r l y i n d i c a t e that the v a r i a n c e of Pop u l a t i o n 1 (Survey Population) d i f f e r s from that of the Simulated P o p u l a t i o n , s i n c e the n u l l hypothesis H 0 can be r e j e c t e d even at 1% l e v e l of s i g n i f i c a n c e . In other words, the p r o b a b i l i t y of having <f-i2^a22 i s more than 99%. T h i s d i f f e r e n c e i n p o p u l a t i o n v a r i a n c e s i s due t o : i . the s i z e of p o p u l a t i o n s . One expects l a r g e r d e v i a t i o n 33 from the mean as the sample s i z e decreases. i i . the property of r e g r e s s i o n models. Model parameters estimated by l e a s t square method are unbiased and have minimum v a r i a n c e among a l l unbiased l i n e a r e s t i m a t o r s . T h i s property i m p l i e s t h a t Y' (the value of the estimated r e g r e s s i o n f u n c t i o n ) i s an unbiased estimator of E(Y), with minimum v a r i a n c e i n the c l a s s of unbiased l i n e a r e s t i m a t o r s . T h e r e f o r e , the simulated p o p u l a t i o n s TTT and MOCOST w i l l have the s m a l l e s t v a r i a n c e . 2.2 Comparison of P o p u l a t i o n Means In order to compare the p o p u l a t i o n means, we form the s t a t i s t i c (m, -m2) - (\> 1 -u 2 ) S Q R T [ ( S 1 2 / N , ) + ( s 2 2 / N 2 )] and t e s t the n u l l hypothesis H 0 : ( i 1 - ( i 2 = 0 . Since the v a r i a n c e s are not equal, the t - d i s t r i b u t i o n has DF degrees of freedom, where DF i s d e f i n e d i n S e c t i o n ( 2 ) . Table 10 shows the comparison of T r a n s i t T r a v e l Time p o p u l a t i o n means f o r Models 2, 4, and 5. By t e s t i n g at 5% l e v e l of s i g n i f i c a n c e , Model 5 can be d e l e t e d . Model 4 y i e l d s to the s m a l l e s t 95% c o n f i d e n c e i n t e r v a l l e n g t h . However, NO MODELS T - S t a t 1 s t i c D . F . S i g n i f Conf i dene ;e Interv /. ( 95) L.L imi t U . L i m i t Range 2 TTT = 2. 1(IVTT ) O. 160 838.71 0.4365 -3.825 4 .496 8 . 321 4 TTT = 2.79UVTT) - .020(IVTT) ! -0.319 683 .17 0.3749 -4.499 3 . 251 7.750 5 TTT = 18.08 Log(IVTT) -1 .710 519.48 0.0439 * * --- ---** Not s i g n i f i c a n t at 5% Tab l e 10 - Comparison of 'TTT' P o p u l a t i o n Means No Models T-Stat 1 st 1c D.F. S igni f Conf idenc :e Inter\ ( .95) L .L imit U.Limit Range 1 TTT = 27.70 + .42(IVTT) -1.673 286.72 0.0477 * * --- ---2 TTT=25+2.79(IVTT)-.020(IVTT)* 1 . 373 283.90 0.0855 -0.903 5 . 149 6.052 * * Not s i g n i f i c a n t at 5% Table 11 - Comparison of 'MOCOST' Populat ion Means Dep.Var. Indep.Var Model Expression R-Square TTT IVTT TTT = 2.1(IVTT) .27709 MOCOST IVTT MOCOST = 25 + .30(IVTT) + .OOI(IVTT)' .72964 Table 12 - Selected Models 36 F i g u r e 4 - T r a n s i t v.s. P r i v a t e Car T r a v e l Time 37 by i n s p e c t i n g F i g u r e 4, the output of Models 2 and 4 d i f f e r s s l i g h t l y f o r i n - v e h i c l e t r a v e l time l e s s than 40 minutes. Note that the range of o b s e r v a t i o n s i s l i m i t e d to about 60 minutes. Model 2, t h e r e f o r e , was chosen as the f i n a l model e x p l a i n i n g the r e l a t i o n s h i p of TTT. and IVTT due to the s i m p l i c i t y of i t s e q u a t i o n a l form. For MOCOST, the i n s p e c t i o n of Table 11 i n d i c a t e s that Model 2 can be s e l e c t e d f o r the f i n a l model e x p r e s s i n g the v a r i a t i o n of MOCOST i n terms of IVTT, s i n c e the n u l l h y p o thesis H 0 can be r e j e c t e d at 5% l e v e l of s i g n i f i c a n c e . S e l e c t e d models are represented i n Table 12. The low squared c o e f f i c i e n t of c o r r e l a t i o n i s due to the nature of data and i s c o n s i d e r e d a c c e p t a b l e . One should expect more v a r i a b i l i t y i n disaggregate data than i n z o n a l l y aggregated data, s i n c e the l a t t e r tends to submerge the observed v a r i a b i l i t y at an i n d i v i d u a l l e v e l by g i v i n g only one f i g u r e r e p r e s e n t i n g the zonal r e s i d e n t c h a r a c t e r i s t i c s . Moreover, we need to make sure that our p r e d i c t i o n models are used only f o r v a l u e s f a l l i n g w i t h i n the range of o b s e r v a t i o n s , s i n c e beyond these l i m i t s the r e s u l t s may not longer apply. 3. SIMULATION OF WAITING TIME VARIABLE As p r e v i o u s l y mentioned, w a i t i n g time seems l i k e l y to be u n c o r r e l a t e d with the in-bus t r a v e l time. T h e r e f o r e , the r e g r e s s i o n approach was not used to simulate t h i s v a r i a b l e . 38 The approach taken i n the present study i s o f t e n used i n p r a c t i c e , when the a - p r i o r i p r o b a b i l i t y d i s t r i b u t i o n of the v a r i a b l e i s known to a n a l y s t s , the s o - c a l l e d random generating model c o n s i s t s of ge n e r a t i n g random w a i t i n g times u t i l i z i n g t h e i r estimated p r o b a b i l i t y d i s t r i b u t i o n . T h i s e s t i m a t i o n i s made by the method of moments on the sample of the complete v a l u e s of the w a i t i n g time v a r i a b l e . The method of moments assumes that the estimator u of » should be the sample mean and the estimator s 2 of the v a r i a n c e e2 i s the sample v a r i a n c e . Since the observed d i s t r i b u t i o n i s skewed to the l e f t , two types of d i s t r i b u t i o n are co n s i d e r e d : i . GAMMA d i s t r i b u t i o n 1 : X.x .exp(-Xx) f ( x ) = r(k ) where x>0 and X and k are the model parameters a = k/X ; c = SQRT(k)/X The use of the method of moments i m p l i e s that : u = k/X. = 5.59 and s = SQRT(k)/X = 4.18 Theref o r e : k = 1.79 and X = 0.32 1- A c c o r d i n g to the comment made by Dr.Navin, 2-step d e n s i t y f u n c t i o n s can a l s o be a p p l i e d ( i . e . l i n e a r + n e g a t i v e exponent i a l ) . 39 i i . LOGNORMAL d i s t r i b u t i o n : f ( x ) = l/[SQRT(2n )< r '.x] exp{-1/2[ ( l n x - „ ' ) A ' ] 2 } where u' and *' are the mean and standard d e v i a t i o n of the n a t u r a l l o g a r i t h m of x. u' = 1.48 and s' = 0.70 In order to make a f i n a l s e l e c t i o n between these d i s t r i b u t i o n s , the X 2 c l o s e n e s s - o f - f i t s t a t i s t i c with (k-r-1) degrees of freedom i s used. The l e t t e r k r e p r e s e n t s the number of c a t e g o r i e s c o n s i d e r e d and r i s the number of parameters estimated from the data. The r e s u l t s of t h i s i n v e s t i g a t i o n are shown i n F i g . 5 and 6, and Tables 13 and 14. Due to the lower magnitude of the t o t a l normalized squared d i f f e r e n c e and the b e t t e r f i t obtained i n the range of 4-7 minutes, one might conclude that the use of a lognormal d i s t r i b u t i o n i s more a p p r o p r i a t e to the present data s e t . 4. COMPLETION OF PARKING COST The Aggregation method i s used to complete missing i n f o r m a t i o n on pa r k i n g c o s t . T h i s method was a p p l i e d as f o l l o w s : Land use c h a r a c t e r i s t i c s of the work t r i p d e s t i n a t i o n s are a v a i l a b l e from the survey. For each type of d e s t i n a t i o n land use, an average p a r k i n g c o s t i s " c a l c u l a t e d from 40 co CD CD 3-D 5.D 1D.D 15.D 20.11 25.1) 3D 0 "WAITING TIME (MINJ F i g u r e 5 - Gamma Generation Model 41 oo rv o CM a a 0.0 5.0 10.0 15.0 20.0 25.0 30.0 WAITING TIME (MIN) F i g u r e 6 - Lognormal Generation Model 42 Xi Ni f (Xi) n . f ( X i ) ( N i - n . f i ) V n . f i 1 21 .1017 25.83 .90 2 22 . 1277 32.43 .36 3 25 .1278 32.44 1.71 4 37 | . 1 1 64 29.57 | 1 .86 I 5 64 | 125 1 .1008 2 5.61 | 94 .28 57.52 I 10.0 6 22 | .0846 21.48 | .013 I 7 2 1 .0694 17.62 1 1 3.85 1 8 6 1 .0560 14.22 | 4.75 1 9 3 1 .0446 1 1 .33 | 6.13 1 1 0 29 | 39 .0352 8.94 | 46 .93 45.00 1 1 .34 1 1 o 1 .0276 7.00 | 1 1 2 1 1 .0214 5.44 1 3.63 1 1 3-20 1 5 .0868 22.05 2.25 254 254 16. 56 1- Aggregated Value X 2-Stat.(.01,4) = 13.28 X 2-Stat.(.005,4)= 14.86 Table 13 - GAMMA Model Goodness-of-Fit 43 Xi Ni f (Xi) n . f ( X i ) (Ni-n. f i ) V n . f i 1 21 .0612 15.48 1 .96 2 22 .1515 38.48 2.33 3 25 .1638 41 .59 6.62 4 37 | .1412 35.86 | .035 I 5 64 | 125 1 .1121 28.46 | 102.74 44.37 I 4.82 6 22 | .0861 21.85 | .001 I ' 7 2 1 .0652 16.57 J_ 12.81 1 .8 6 1 .0494 12.54 | 3.41 1 9 3 1 .0375 9 . 51 | 4.46 1 10 29 | 39 .0286 7.25 | 39.17 65. 1 4 1 0.001 1 1 o 1 .021 9 5.57 | 1 1 2 1 1 .0169 4.30 1 2.53 1 1 3-20 1 5 .0646 16.45 . 13 254 254 15. 85 1- Aggregated Value X 2 - S t a t . ( .01,4) = 13. 28 X 2 - S t a t . ( .005,4)= 14. 86 Table 14 - LOGNORMAL Model Goodness-of-Fit 44 the complete i n f o r m a t i o n . These zonal average f i g u r e s are then used to complete the m i s s i n g values of p a r k i n g charges a c c o r d i n g to t h e i r corresponding land use i n f o r m a t i o n . 5 . SUMMARY Since the c a l i b r a t i o n of a l o g i t or p r o b i t model r e q u i r e s a r e l a t i v e l y l a r g e sample s i z e , the completion of the missing data was a necessary p r e - r e q u i s i t f o r c a l i b r a t i o n . For t h i s purpose, three data s i m u l a t i o n methods were used: Regression A n a l y s i s to simulate TTT and MOCOST; Random Generation Model to generate w a i t i n g time and Aggregation method to estimate the m i s s i n g value of parking c o s t . S t a t i s t i c a l i n f e r e n c e s were made to s e l e c t the more a p p r o p r i a t e model. However, the use of t h i s s imulated data may have some i m p l i c a t i o n s i n the c a l i b r a t i o n of the modal c h o i c e models. For i n s t a n c e , the true w a i t i n g time d i s t r i b u t i o n might not be lognormal which w i l l b i a s the impact of t h i s v a r i a b l e on the c h o i c e p r o b a b i l i t y . T h e r e f o r e , the comparison of the c a l i b r a t e d model and r e s u l t s found i n previous s t u d i e s may be a r e l e v a n t way of e v a l u a t i n g t h i s attempt. 45 IV. BEHAVIOURAL TRAVEL DEMAND THEORY In t h i s s e c t i o n , we are i n t e r e s t e d i n e s t i m a t i n g modal ch o i c e p r o b a b i l i t y models r e l a t i n g the p r o b a b i l i t y of using a t r a v e l mode to a s e r i e s of exogenous v a r i a b l e s . S i m i l a r to other demand models, m u l t i v a r i a t e r e g r e s s i o n models can be used. But, s i n c e the dependant v a r i a b l e has a b i n a r y form (the user w i l l or w i l l not choose the t r a v e l mode i ) , the e s t i m a t i o n of the model parameters becomes complicated. Sources of d i f f i c u l t i e s are as f o l l o w s : i . The e r r o r v a r i a n c e i s not constant i f we l e t E[Y=1|X] = p then Var[Y] = E [ Y 2 ] - { E [ Y ] } 2 = p-p 2 = p ( l - p ) t h e r e f o r e , the v a r i a n c e around Y i s f u n c t i o n of the estimated v a l u e . i i . The r e g r e s s i o n f u n c t i o n i s c o n s t r a i n e d 0 < E[Y|X] < 1 T h i s l a t t e r i s the most troublesome, s i n c e f o r some value of the independant v a r i a b l e s , the estimated value of the dependant v a r i a b l e exceeds 1 or or i s l e s s than 0. Due to the f a c t that we can not make a t r a n s f o r m a t i o n of the dependant v a r i a b l e to l i n e a r i z e i t s expected value, we are f o r c e d to f i t a model which i s n o n - l i n e a r i n the parameters. Two models are 46 commonly used f o r t h i s purpose : the l o g i t and p r o b i t model. 1. MATHEMATICAL THEORY OF BEHAVIOURAL MODELS Accord i n g to the approach taken by economists to diagnose consumer p r e f e r e n c e s , a b e h a v i o u r a l model should i n v e s t i g a t e the manner that an i n d i v i d u a l uses t o decide among s e v e r a l c h a r a c t e r i s t i c s . A p p l y i n g t h i s approach t o t r a n s p o r t a t i o n demand, i t i s hypothesized that the t r a v e l l e r reaches a d e c i s i o n by c o n s i d e r i n g the p e r c e i v e d a l t e r n a t i v e t r a v e l modes by v a l u i n g the a t t r i b u t e s of each mode. His behaviour f o l l o w s the r e l a t i v e v a l u e s he a s s o c i a t e s to d i f f e r e n t a t t r i b u t e s and may be d e s c r i b e d by h i s i n d i f f e r e n c e curves s i n c e these l a t t e r s represent a l l combination of c h o i c e s among which the t r a v e l l e r i s i n d i f f e r e n t . The f a m i l y of i n d i f f e r e n c e curves i s c a l l e d the u t i l i t y f u n c t i o n and can be expressed by the f o l l o w i n g equat i o n : Umi = V( Xm,Si ) 4.1 where Xm i s the set of s e r v i c e a t t r i b u t e s of a l t e r n a t i v e m and S i the set of the a t t r i b u t e s of i n d i v i d u a l i . However, t h i s f o r m u l a t i o n i m p l i c i t l y assumes that the consumer i s aware of a l l p o s s i b l e combination of a t t r i b u t e s and makes a d e c i s i o n with p e r f e c t i n f o r m a t i o n . In the r e a l world, t h i s assumption i s r a r e l y s a t i s f i e d and t h e r e f o r e a s t o c h a s t i c term should be i n t r o d u c e d i n t o (4.1) to express the 47 p r o b a b i l i s t i c e r r o r made each time the u t i l i t y of a given mode i s e v a l u a t e d . The f u n c t i o n a l form of U w i l l then be as : Umi = V( Xm,Si ) + emi 4.2 where V i s the n o n - s t o c h a s t i c term and re p r e s e n t s a common element shared by a subset of p o p u l a t i o n p r e d e f i n e d a c c o r d i n g to t h e i r socio-economic c h a r a c t e r i s t i c s , and emi i s the i t h t r a v e l l e r ' s t a s t e not shared by o t h e r s , and s i n c e i t can not be assessed , the assignment of a p r o b a b i l i t y d i s t r i b u t i o n to any i n d i v i d u a l t a s t e becomes necessary. T h e r e f o r e emi forms the s t o c h a s t i c component of the u t i l i t y f u n c t i o n . Note that i n t r a n s p o r t a t i o n demand, s i n c e the s e r v i c e a t t r i b u t e s ( t r a v e l time, t r a v e l c o s t , . . . ) are n e g a t i v e l y valued, that i s , the u t i l i t y of a mode i n c r e a s e s as t r a v e l c o s t or time decreases, one should use the term d i s u t i l i t y f o r the f u n c t i o n U. However, in the present study, we w i l l use the term u t i l i t y and a s s i g n a p p r o p r i a t e s i g n s to the parameters of the s e r v i c e a t t r i b u t e s . With t h i s i n mind, a consumer i s assumed to be a u t i l i t y maximizer and p r e f e r s mode m to remaining modes, only i f : Umi > Uki f o r k=1,...,M ; k*m 4.3 T h i s i s the e q u i v a l e n t of s t a t i n g that a consumer w i l l choose that mode f o r which the g r e a t e s t u t i l i t y ( d i s u t i l i t y ) i s p e r c e i v e d . 48 In order to analyse the behaviour of commuter i when he i s faced with M a l t e r n a t i v e s , we should a s s i g n a p r o b a b i l i t y t o h i s d e c i s i o n , t h a t i s : Pmi = P( Umi > Uki) ' f o r k=1, ,M ; k*m 4.4 where Pmi i s the p r o b a b i l i t y t h a t i n d i v i d u a l i takes mode m to work. Pmi = P(V(Xm,Si)+emi > V(Xk,Si)+eki) f o r k=1,...,M ; k*m or to f a c i l i t a t e the n o t a t i o n Pmi = P(Vmi + €ini > Vki + eki ) f o r k=1, ,M ; k#m 4.5 In the present study, i n order to determine a s t r u c t u r a l s o l u t i o n to Equation 4.5, we w i l l c o n s i d e r the s p e c i a l case of bina r y c h o i c e (M=2). T h e r e f o r e , 4.5 becomes : Pmi = P( V 1 i + 6 , i > V 2 i + e 2 i ) 4.6 C l e a r l y , to develop a model, i t i s necessary to assume a s p e c i f i c d i s t r i b u t i o n f o r the p r o b a b i l i s t i c components e . i and € 2 i • 1.1 M u l t i v a r i a t e L o g i t Model 49 Under the assumption of independancy of e r r o r terms e , i and € 2 i , t h e i r j o i n t p r o b a b i l i t y d e n s i t y i n the C a r t e s i a n space i s given by : f ( x , y ) = f , ( x ) . f 2 ( y ) 4.7 where x and y are the values taken on by random v a r i a b l e s e , i and e 2 i r e s p e c t i v e l y . T h e r e f o r e , p r o b a b i l i t i e s i n the ( e , i , € 2 i ) plane are assigned i n accordance with Equation 4.8 : P(E) = ^ f,(x) . f 2 ( y ) dx dy 4.8 E By a r r a n g i n g (4.6) we have for mode 1: P1i = ( e z i - d i < V 1 i - V 2 i ) 4.9 Consider a new random v a r i a b l e e such that € = t i i - e 2 i . Thus the event E = {e<V1i-V2i } i s d e f i n e d by the r e g i o n of ( € , i , € 2 i ) plane such that : y - x < V I i - V2i y < V1i - V2i + x 4.10 T h e r e f o r e , P l i = P U 2 i - e 2 i < V 1 i - V 2 i ) 50 f . ( x ) . f 2 ( y ) dx dy f ,(x) f 2 ( y ) dx dy 4.11 Let F 2 ( y ) denote the p r o b a b i l i t y d i s t r i b u t i o n of e 2; t h e r e f o r e : Equation 4.12 i s c a l l e d the c o n v o l u t i o n of two d e n s i t y f u n c t i o n s f, and f 2 . The l o g i t model r e s u l t s i f € has a W e i b u l l d i s t r i b u t i o n with 0 mean v a l u e . One of the most important c h a r a c t e r i s t i c s of the W e i b u l l d i s t r i b u t i o n i s that the d i f f e r e n c e of W e i b u l l d i s t r i b u t e d v a r i a b l e s has a l o g i s t i c d i s t r i b u t i o n which leads to the d e s i r e d sigmoid shape. The W e i b u l l d i s t r i b u t i o n i s given by: W(x) = ProbU<x) = exp[-exp(-x) ] 4.13 Th e r e f o r e , the a s s o c i a t e d frequency d i s t r i b u t i o n i s : w(x) = dW(x)/dx = exp(-x) . exp[-exp(-x)] 4.14 By s u b s t i t u t i n g Eqs. 4.13 and 4.14 i n t o 4.12, we o b t a i n : P1 i f,(x) . F 2(V1i-V2i+x) dx 4.12 — oo P1i= I (exp(-x)exp[-exp(-x)]}.{exp[-exp(-(V1i-V2i+x))]} dx 51 3 exp(-x).exp[-exp(-x).(1+exp(V2i-V1i))] dx Using the t r a n s f o r m a t i o n t=exp(-x) r e s u l t s i n : + oo P I i = J' exp[-t(1+exp(V2i-V1i))] dt o = -{exp[-t(1+exp(V2i-V1i))] / (1+exp(V2i-V1i))} And hence : P I i = 1 / (1+exp(V2i-V1i)) = exp(V1i-V2i) / 1+exp(V1i-V2i) 4.15 Equation 4.15 s t a t e s that the p r o b a b i l i t y of t a k i n g mode 1 i s a f u n c t i o n of the d i f f e r e n c e between the n o n - s t o c h a s t i c terms V1i and V2i of the u t i l i t y f u n c t i o n s . Note that the n o n - s t o c h a s t i c term of u t i l i t y f u n c t i o n s has the form: Vmi = Po + 0 ,xA ! + . . .+0n x*n +. . .+fik x*k = jj.Xmi and the only r e s t r i c t i o n imposed on i t , i s that t h i s r e l a t i o n must be l i n e a r i n parameters (pi), whereas, X may be a complex t r a n s f o r m a t i o n of the raw data. I t should a l s o be n o t i c e d that the parameters of the u t i l i t y f u n c t i o n s (0i) c o n s i s t of two components : d e t e r m i n i s t i c and random. 52 8 = B + uim where uim repr e s e n t s the unobserved random t a s t e of the i n d i v i d u a l i f o r mode m. However, to enable the form u l a t i o n to be used f o r consumer behaviour, i t i s u s u a l l y assumed that uim=0. By l e t t i n g V I i - V2i = V( Xi,£ ) the p r o b a b i l i t y of choosing mode 2 w i l l be e q u i v a l e n t to : P2i = 1-PIi = 1-[ exp(V(Xi,£)) / 1+exp(V(Xi ,e ) ) ] P2i = 1 / [ 1+exp(V(Xi,£)) ] 4.16 By t a k i n g the lo g a r i t h m of the p r o b a b i l i t i e s r a t i o : L = l o g ( P 1 i / 1-P1i) = l o g ( P 1 i ) - l o g ( 1 - P 1 i ) and s u b s t i t u t i n g P1i by i t s e x p r e s s i o n , we form the l o g i t L : L = V(Xi,£) 4.17 The major p r o p e r t i e s of t h i s t r a n s f o r m a t i o n are as f o l l o w s : i . The l o g i t i s a l i n e a r f u n c t i o n of the d i f f e r e n c e between the n o n - s t o c h a s t i c components of u t i l i t y f u n c t i o n s (see Eq. 4.17), whereas the p r o b a b i l i t i e s themselves are not (see Eqs. 4.15,-4.16) 53 i i . While the p r o b a b i l i t i e s are bounded, the l o g i t i s unbounded with respect to the value of V(Xi,£). T h i s i s i n accordance with the c o n s t r a i n t imposed on the r e g r e s s i o n f u n c t i o n (see p.45). 1.2 M u l t i v a r i a t e P r o b i t Model The s p e c i f i c a t i o n of the p r o b i t model i s s t r a i g h t f o r w a r d and f o l l o w s the d e f i n i t i o n of the normal p r o b a b i l i t y f u n c t i o n . However, s e v e r a l assumptions are needed. I t i s assumed that each person possesses a d i f f e r e n t c r i t i c a l value or a t h r e s h o l d Ui which determines whether the u n d e r l y i n g person w i l l take car or bus. T h i s c r i t i c a l value i s unobserved and hence u n a v a i l a b l e to a n a l y s t s . In order to make a d e c i s i o n , i n d i v i d u a l i compares h i s own c r i t i c a l value to choice index G(Xmi,Si,£). T h i s index i s produced by the l i n e a r combination of the exogenous v a r i a b l e s or "s t i m u l u s " . Exogenous v a r i a b l e s , themselves, can be a complex t r a n s f o r m a t i o n of the raw data. G(Xmi,Si,£) = G(Xi,£) = p0 + p , x^ +. . . +*n x^n +. . . +pk x*k 4.18 where Xmi and S i are the s e t s of the s e r v i c e a t t r i b u t e s of the mode m and socio-economic c h a r a c t e r i s t i c s of commuter i , r e s p e c t i v e l y . 54 Therefore, t r a v e l l e r i w i l l choose : i . mode 1 i f G(Xi,£) > Ui i i . mode 2 i f G(Xi,£) < Ui .4.19 Since a l l i n d i v i d u a l s w i l l not possess the same t h r e s h o l d l e v e l , a p r o b a b i l i t y d i s t r i b u t i o n i s a s s i g n e d to c r i t i c a l v a l u e s . I f we assume t h a t t h i s d i s t r i b u t i o n i s normal N(0,1), the p r o b a b i l i t y that commuter i possesses a c r i t i c a l value g r e a t e r than the c h o i c e index, and consequently chooses mode 1, i s given by : G(Xj/£) P I i = l/SQRT(2n) ^ e x p ( - t 2 / 2 ) dt 4.20 — oo Since we are l i m i t i n g the present study to the case of b i n a r y c h o i c e , the p r o b a b i l i t y of choosing mode 2 w i l l be e q u i v a l e n t t o : G(Xj^) P2i = 1-PIi = 1-[ l/SQRT(2n) I e x p ( - t 2 / 2 ) dt ] + C© _oo P2i = 1/SQRT(2n) j ( e x p ( - t 2 / 2 ) ) dt 4.21 G(Xi,£) i s d e f i n e d as the p r o b i t of Pmi(m=2) One of the above assumptions which needs j u s t i f i c a t i o n i s the n o r m a l i t y of the c r i t i c a l v a l u e s . These values are a complex combination of a l a r g e number of p s y c h o l o g i c a l , p h y s i o l o g i c a l , s o c i a l and c u l t u r a l f a c t o r s which can be assumed F i g u r e 7 - Comparison of L o g i s t i c and P r o b i t Forms 56 independant.Therefore, a c c o r d i n g to the C e n t r a l L i m i t theorem which s t a t e s that sums of independant random v a r i a b l e s are a s y m p t o t i c a l l y normally d i s t r i b u t e d , we can accept the f a c t that c r i t i c a l v alues are normal. S i m i l a r to the l o g i t t r a n s f o r m a t i o n , p r o b i t t r a n s f o r m a t i o n of the cho i c e index produces a sigmoid curve. I t i s worthwhile to remark that the d i f f e r e n c e between a l o g i s t i c d i s t r i b u t i o n (obtained from l o g i t t r a n s f o r m a t i o n ) , and a cumulative normal d i s t r i b u t i o n (obtained from p r o b i t t r a n s f o r m a t i o n ) i s not very l a r g e ( s e e F i g . 7 ) . 2. ESTIMATING MODEL PARAMETERS The model parameters can be estimated by means, of weighted l e a s t square or maximum l i k e l i h o o d method. 2.1 Maximum L i k e l i h o o d E s t i m a t i o n The method of maximum l i k e l i h o o d chooses as es t i m a t o r s of the parameters p i , a set of s t a t i s t i c s which maximize the l i k e l i h o o d f u n c t i o n f o r the given v a l u e s . The l i k e l i h o o d f u n c t i o n i s d e f i n e d as being the p r o b a b i l i t y of the occurence of random sample (W1,W2,...,Wn) as a f u n c t i o n of the unknown parameters ( p 0 , p 0 n ) with the c o n d i t i o n that the unknown parameters (^) belongs to the parameter space n which has to be s p e c i f i e d f o r the case study n f ( X i , * ) JB£ n g<!> = 57 where N i s the number of o b s e r v a t i o n s . Since our problem c o n s i s t s of determining whether or not a person s e l e c t e d randomly chooses a p a r t i c u l a r t r a v e l mode, random v a r i a b l e s Wi are d e f i n e d as Wi=1 i f commuter i s e l e c t s mode 1 and Wi=2 i f he does not: P[Wi=l] = P1 and P[Wi=2] = P2 = 1-P1 4.22 Assuming Wi's are mutually independant, the l i k e l i h o o d f u n c t i o n of the parameters i s given by: N, N g(£)=P(W1,W2,...,WN1, , WN )=n(P[Wi=1] ) .n(P[Wi=2]) 4.23 where N1 i s the number of o b s e r v a t i o n s that s e l e c t mode 1. S u b s t i t u i n g 4.22 i n t o 4.23 y i e l d s to : Ni N g(£) = n ( P 1 i ) . n(1-P1i) 4.24 and the parameter space w i l l be d e f i n e d as : n : { R k } But, s i n c e P1i i s a f u n c t i o n of s e r v i c e a t t r i b u t e s and s o c i o -economic c h a r a c t e r i s t i c s of the t r i p makers, 4.23 can be w r i t t e n as : gU) = x(x,£) 58 4.25 The aim of the maximum l i k e l i h o o d e s t i m a t i o n i s to choose the ve c t o r of unknown c o e f f i c i e n t s £ i n a manner which make the p r o b a b i l i t y d e f i n e d i n 4.23 as l a r g e as p o s s i b l e . For t h i s , the set of js maximizing x ( X , £ ) and t h e r e f o r e s a t i s f y i n g the f o l l o w i n g equation has to be determined. 6 ( X ( X , £ ) ) / 6 ( 0 3 ) = 0 f o r j=1,...,k 4.26 To f a c i l i t a t e the d i f f e r e n t i a t i o n task, l o g [ x ( X , £ ) ] i s co n s i d e r e d r a t h e r than x ( X , £ ) In g e n e r a l , maximum l i k e l i h o o d e s t i m a t o r s f o r l a r g e sample s i z e s possess some d e s i r a b l e p r o p e r t i e s . They are as f o l l o w s : i . Maximum l i k e l i h o o d e s t i m a t o r s are c o n s i s t e n t . That i s , l i m Prob( I b.-0.| <e) = 1 where e i s an a r b i t r a r y p o s i t i v e v a l u e . i i . Maximum l i k e l i h o o d e s t i m a t o r s are unbiased minimum v a r i a n c e . That i s , bj has among a l l unbiased estimators(E[bj] = 0j) , the s m a l l e s t v a r i a n c e . i i i . Maximum l i k e l i h o o d e s t i m a t o r s are approximately normally d i s t r i b u t e d . Since the data a v a i l a b l e f o r t h i s study c o n s i s t s of 621 o b s e r v a t i o n s , these p r o p e r t i e s apply. 5 9 2.2 E s t i m a t i n g L o g i t Model Parameters According to 4.24 and 4.25, the l i k e l i h o o d f u n c t i o n f o r the l o g i t model i s : X(X,£) = X. = n ( P l i ) . S d - P 1 i ) and the l o g a r i t h m of t h i s f u n c t i o n w i l l be : Hi N A = l o g U ) = E [ l o g ( P 1 i ) ] + E [ l o g ( l - P 1 i ) ] 4.27 A»l A=N,*1 S u b s t i t u t i o n of 4.16 and 4.17 i n t o 4.27 y i e l d s to : A= El {log[exp(V(Xi , B) ) / 1 +exp(V(X, B)) ]} +E { l o g [ l / 1+exp(V(X, *))]} = .E ( l o g [ e x p ( V ( X i ,£) ) ] }-E {log[1+exp(V(Xi , p ) ) ] } -E { l o g [ l + e x p ( V ( X i , p ) ) ] } T h e r e f o r e : A = E' {V(Xi,/,)}-? {log[ 1+exp(V(X,£) ) ]} 4.28 A ' 1 " ~ _ A.I — In order to f i n d the v e c t o r of the estimated value of £ which maximizes A, we d i f f e r e n t i a t e p a r t i a l l y A with r e s p e c t to aj : 6A/6pj=E ( x i j ) - E [ x i j / ( 1 + e x p ( V ( X , f l ) ) ) ] f o r j=1,...,K 4.29 A=l A=l 6 0 By equating 4.29 to zero, we o b t a i n a system of K n o n - l i n e a r equations that can be s o l v e d i t e r a t i v e l y (K=number of exogenous v a r i a b l e s ) . 2.3 E s t i m a t i n g P r o b i t Model Parameters According to 4.27, the l o g l i k e l i h o o d f u n c t i o n f o r the p r o b i t model i s : A = l ' {log[P1 (Xi,?) ]} + E ( l o g [ P 2 ( X i ,B ) ]} 4.30 A-1 ~ ~ /i=N.-»1 — where G(Xj,/3) P1(X,JS) = l/SQRT(2n) f e x p ( - t 2 / 2 ) dt and r P2(X,<?) = l/SQRT(2n) e x p ( - t 2 / 2 ) dt J G(Xj^) And by d i f f e r e n t i a t i n g 4 . 3 0 with r e s p e c t to pj we have : 6A/60J = ? '{[6(Pl(Xi,£))/6*j]/Pl(Xi,£)} +E { [ 6 ( P 2 ( X i , 0 ) ) / 6 p j ] / P 2 ( X i , ^ ) } 4.31 ;=N,.I — — — — In order to c a l c u l a t e 6 ( P 1 ( X i , £ ) ) / 6 B j , we apply L e i b n i t z ' r u l e and t h e r e f o r e : 6(P1(Xi,£))/6*j = !/SQRT (2n) exp[- 1 / 2(G(Xi,£)) 2]xij 4 . 3 2 61 and 6(P2(Xi,£))/6*j = l/SQRT(2n) exp[-1/2(G(Xi,£)) 2]xij And f i n a l l y , by s u b s t i t u t i n g 4.32 and 4.33'irtto 4.31 we w i l l have a system of K equations which are non l i n e a r i n 0 and which must be s o l v e d by an i t e r a t i v e p r o c e s s . 2.4 Goodness of F i t Since the parameters of l o g i t and p r o b i t models are estimated by the maximum l i k e l i h o o d method, hypothesis about the o v e r a l l s i g n i f i c a n c e of the r e l a t i o n s h i p may be t e s t e d by the l i k e l i h o o d r a t i o method. T h i s method t e s t s the n u l l hypothesis that the p r o b a b i l i t y of s e l e c t i n g a p a r t i c u l a r mode i s independant of the values of the explanatory v a r i a b l e s of the model. T h i s can be formulated as t e s t i n g H 0 : a l l 0=0 versus H, : Not a l l 0=0 Let assume that b 0 i s the estimated value o f 0 o f o r which A ( b 0 , 0 , 0 , . . . , 0) i s maximum and ( b 0 ,b, , .. . ,b„) are those which maximize A ( b 0 , b 1 , . . . , b ) . The l i k e l i h o o d r a t i o w i l l then be : A = A(b o,0,0,...,0) / A(b 0,b,,..., bfl) 4.35 62 with the f o l l o w i n g c r i t i c a l r e g i on : Reject H 0 i f 0<X<c Accept H 0 i f c<X<1 where c i s chosen i n a f a s h i o n that the c r i t i c a l r e g i on has the d e s i r e d s i z e . I t i s found that -21ogX. i s a more convenient s t a t i s t i c , s i n c e f o r l a r g e samples, t h i s s t a t i s t i c i s d i s t r i b u t e d as a c h i -square random v a r i a b l e , with a degree of freedom equal to the d i f f e r e n c e between the number of parameters i n the model. An a l t e r n a t i v e s t a t i s t i c i n v o l v i n g the l i k e l i h o o d r a t i o X. i s the Pseudo R-Square c o e f f i c i e n t . T h i s c o e f f i c i e n t , a l s o c a l l e d l i k e l i h o o d r a t i o index, i s denoted by p2 and i s analogous with the well-known R 2 index of l e a s t - s q u a r e r e g r e s s i o n a n a l y s i s . p2 = A(b 0,0,0,...,0) / A(b 0,b,,...,b) = 1-X However, the p2 index values are c o n s i d e r a b l y lower than those of R 2 index. For i n s t a n c e , i f values of 0.8 to 0.9 f o r R 2 represent an e x c e l l e n t f i t , f o r p2 index these values correspond to 0.2 to 0.4 (McFaddenl976). 63 3. DISCRIMINANT ANALYSIS The t h i r d s t a t i s t i c a l t o o l s e l e c t e d to c a l i b r a t e the r e l a t i o n s h i p between a b i n a r y c h o i c e v a r i a b l e and a set of exogenous v a r i a b l e s i s D i s c r i m i n a n t A n a l y s i s . In t h i s approach, two separate p o p u l a t i o n s are co n s i d e r e d : p r i v a t e car and t r a n s i t u s e r s . The aim of t h i s method i s to a s s i g n p r o b a b i l i t y to an o b s e r v a t i o n as coming from one of these p o p u l a t i o n s such that the c o s t of m i s c l a s s i f i c a t i o n i s minimum. For t h i s purpose, explanatory v a r i a b l e s are l i n e a r l y combined to form a s c a l a r c a l l e d the d i s c r i m i n a n t s c o r e . The l i n e a r d i s c r i m i n a n t f u n c t i o n i s of the type : K ymi = I (a .x\ ) m=1, 2 4.36 where k=1,...,K=number of exogenous v a r i a b l e i=1,...,N=number of t r i p makers(=observations) m=1 ,2 The m a t r i c i a l p r e s e n t a t i o n of 4.36 i s : I = l ' 4 » 3 7 where £' and X' are the transposed m a t r i c e s of £ and X r e s p e c t i v e l y . 64 Let f T ( X ) and f 2 ( X ) represent the d e n s i t y f u n c t i o n s of these two p o p u l a t i o n s , q and 1-q denote the a - p r i o r i p r o b a b i l i t y t h a t an o b s e r v a t i o n comes from p o p u l a t i o n l and 2 r e s p e c t i v e l y , and and c 2 the r e s p e c t i v e m i s c l a s s i f i c a t i o n c o s t s . The c o n d i t i o n a l p r o b a b i l i t y t h at an observed set of exogenous v a r i a b l e s (X), comes from p o p u l a t i o n 1 i s : P1(X) = ( q.f,(X) )/( q . f , ( X ) + ( 1 - q ) f 2 ( X ) ) 4.38 and i t s expected c o s t of m i s c l a s s i f i c a t i o n i s : E(C,) = c,( q.f,(X) )/( q . f , ( X ) + ( l - q ) f 2 ( X ) ) 4.39 S i m i l a r i l y we have: P2(X) = ( q . f 2 ( X ) )/( q . f , ( X ) + ( 1 - q ) f 2 ( X ) ) 4.40 and E ( C 2 ) = c 2 ( q . f 2 ( X ) )/( q . f , ( X ) + ( l - q ) f 2 ( X ) ) 4.41 To r e l a t e P(X) to the the set of explanatory v a r i a b l e s , we form the l o g r a t i o of p r o b a b i l i t i e s : log(P1 (X)/1-P1 (X)) = l o g ( q . f , ( X ) / d - q ) f 2 ( X ) ) = l o g ( f , ( X ) / f 2 ( X ) ) + l o g ( q / l - q ) 4.42 65 Now i n order to f i n d a mathematical e x p r e s s i o n f o r the c o n d i t i o n a l p r o b a b i l i t i e s P i ( X ) , some assumptions about the c h a r a c t e r i s t i c s of p o p u l a t i o n s should be made. The most 'convenient' and w e l l known hypothesis i s that the explanatory v a r i a b l e s of each p o p u l a t i o n are j o i n t normal with mean v e c t o r s , £ 2 and a common covari a n c e matrix E. T h e r e f o r e , a c c o r d i n g to the e x p r e s s i o n of a m u l t i v a r i a t e normal d e n s i t y f u n c t i o n we have: K 1/2 f,(X) = ( l/SQRT(2n)). E . exp ( - l/2A' 1 E" 1 A,) and f 2 ( X ) = ( 1/SQRT(2II) K I1. exp ( - l/2A' 2 E" 1 A 2) where A, = X-£, , A 2 = X~iL2 and k = number of f a c t o r s . Hence: f 1 ( X ) / f 2 ( X ) = exp [ - l/2 ( A , ' . I " 1 . A , - A 2 ' . E - 1 . A 2 ) ] 4.44 and consequently: l o g ( f , ( X ) / f 2 ( X ) ) = - l / 2 [ ( A , ' . E " 1 . A , - A 2'.E- 1.A 2) = Y(X) 4.45 Note that y(X) i s l i n e a r i n parameters and has the form of Eq. 4.37 s i n c e one can w r i t e : Y ( X ) = - l / 2 [ ( X - £ 1 ) ' E" 1 (X-*-! ) - ( X - M 2 ) ' I' 1 ( X - ^ 2 ) 1 66 = - I / 2 [ X ' E - 1 X - x ' i - % ! - £ ' i E ~ 1 x + i ' , ? : - 1 ^ , - X ' E ' 1 X + X ' Z - % 2 + 2 E~ 1 X - 2 Z " V 2 Using the p r o p e r t y that E" 1 i s symmetric and thus X 1 E" 1jf . i=V 1 E " 1 £ w e have: Y(X)= X' E" 1 (j.,-£2 ) " l / 2 U i ' E - V i " JL2 " ^ 2 ) = X'£ - K 4.46 where X' i s the row v e c t o r v a r i a b l e s and £ i s the column v e c t o r of parameters. By s u b s t i t u t i n g 4.45 i n t o 4.46 we o b t a i n : l o g ( P 1 ( X ) / l - P 1 ( X ) ) = Y(X)+log(q/1-q) = G(X) 4.47 which i s e q u i v a l e n t t o : P1(X)/1-P1(X) = exp(G(X)) and t h e r e f o r e P1(X) = exp(G(X)) / [ 1+exp(G(X))] 4.48 Equation 4.48 has the f a m i l i a r s t r u c t u r a l form of the modal s p l i t models a l r e a d y developed i n the case of l o g i t model (Eq. 4.15) 3.1 C l a s s i f i c a t i o n of an Observation X 67 R e c a l l from 4.46 and 4.47 t h a t : G(X) = X' E" 1 (^- L 2) - K + l o g ( q / l - q ) 4.49 The f i r s t term of 4.49 rep r e s e n t s the l i n e a r c l a s s i f i c a t i o n f u n c t i o n which w i l l be denoted by Z(X). Z i s normally d i s t r i b u t e d with parameters: The v a r i a n c e a2 i s known as the Mahalanobis d i s t a n c e D 2 of the two m u l t i v a r i a t e normal p o p u l a t i o n s . Now, assume that the v e c t o r o b s e r v a t i o n x i s taken on X. According to Bayes, we w i l l minimize the c o s t of m i s c l a s s i f i c a t i o n by c l a s s i f y i n g x as coming from p o p u l a t i o n 1 i f : E ( Z ) i = (j/ , - ^ 2 ) • E~ 1 .Jii i = 1 ,2 and Var(Z) = U . - j . 2 ) ' . I " 1 . ( P , - ^ ) = <y2 4.50 E(C,) > E ( C 2 ) That i s c,.q f,(X) > c 2 ( 1 - q ) f 2 ( X ) 68 f 1 ( X ) / f 2 ( X ) > c 2 d - q ) / c,q 4.51 Accord i n g to Eqs. 4.45, 4.46 and 4.49, 4.51 becomes: Y(X) > l o g ( c 2 ( 1 - q ) / c , . q ) or Z(X) > l o g ( c 2 ( 1 - q ) / c,.q) + K and i f we assume that c, and c 2 are equal, the c l a s s i f i c a t i o n r u l e w i l l be: The v e c t o r o b s e r v a t i o n x i s drawn from: p o p u l a t i o n 1 i f Z(X)>log((1-q)/c,q)+K p o p u l a t i o n 2 otherwise 3.2 Goodness of F i t Te s t s concerning parameters B can be set up using the t -d i s t r i b u t i o n . The general l i n e a r t e s t which t e s t s the o v e r a l l s i g n i f i c a n c e of the f u n c t i o n can be set up using the F-d i s t r i b u t i o n . An a l t e r n a t i v e o v e r a l l t e s t of s i g n i f i c a n c e i s the Mahalanobis s t a t i s t i c D 2. T h i s s t a t i s t i c , d e f i n e d by Equation 4.50, re p r e s e n t s the s t a n d a r d i s e d d i s t a n c e between groups. I t i s assumed that the l a r g e r D 2 i s , the b e t t e r i s the f i t , s i n c e the region of m i s c l a s s i f i c a t i o n i s s m a l l e r . 69 V. ANALYSIS The present chapter i n v e s t i g a t e s the s e l e c t i o n of r e l e v a n t v a r i a b l e s and v a r i a b l e forms which appear to a f f e c t the i n d i v i d u a l t r a v e l p r e f e r e n c e s . For t h i s purpose, we f i r s t f i n d that combination of system v a r i a b l e s which best express the v a r i a t i o n i n i n d i v i d u a l u t i l i t y f u n c t i o n , and then attempt to ob t a i n the best set of socio-economic v a r i a b l e s which seem l i k e l y to succeed i n e x p l a i n i n g the behaviour of t r a v e l l e r s . Before e n t e r i n g i n t o the a n a l y s i s , i t may be noteworthy to o u t l i n e two problems u s u a l l y encountered i n these s o r t . of s t u d i e s : a. The f i r s t one i s the problem of m u l t i c o l l i n e a r i t y . . T h i s occurs when the independant v a r i a b l e s are h i g h l y or even p e r f e c t l y c o r r e l a t e d among themselves. b. The second type of problem i s the use of z o n a l l y aggregated data. Horowitz (1981) has shown that the use of z o n a l l y average v a r i a b l e s i n maximum l i k e l i h o o d e s t i m a t i o n normally w i l l produce i n c o n s i s t e n t estimates of disaggregate c h o i c e p r o b a b i l i t i e s unless the z o n a l l y averaged explanatory v a r i a b l e s have the same j o i n t d i s t r i b u t i o n f u n c t i o n i n each zone and are not c o r r e l a t e d with any di s a g g r e g a t e v a r i a b l e s . 70 T a l v i t i e ( 1 9 7 6 ) suggests that zonal average v a r i a b l e s are adequate when p o l i c y q u e s t i o n s a f f e c t zones in a r e l a t i v e l y homogeneous way, such as parking c o s t i n the case of t h i s study(see C h . I I I . 4 ) . 1. DETERMINATION OF SYSTEM VARIABLES 1.1 S e l e c t i o n C r i t e r i a Below, we w i l l l i s t c r i t e r i a and set l i m i t s of s i g n i f i c a n c e which are used to r e j e c t h y p o t h e s i s : i . Sign Test We expect that the p a r t i a l model c o e f f i c i e n t s have the c o r r e c t s i g n . V a r i a b l e s i n v o l v i n g i n - v e h i c l e time d i f f e r e n c e ( A T ) , walking time d i f f e r e n c e and w a i t i n g time should have a negative sig n s i n c e an i n c r e a s e i n r i d i n g time, walking time to bus stop and w a i t i n g time r e s u l t s i n a r e d u c t i o n of the p r o b a b i l i t y of t r a n s i t use. On the other hand, the c o s t d i f f e r e n c e v a r i a b l e (AC) should have a p o s i t i v e s i g n . i i . L i k e l i h o o d R a t i o Test We should at l e a s t be a b l e to r e j e c t at f i v e percent l e v e l of s i g n i f i c a n c e the n u l l h y p o t h e s i s that the s e l e c t e d set of v a r i a b l e s does not e x p l a i n the v a r i a t i o n i n the u t i l i t y f u n c t i o n . As mentioned before, the use of the l i k e l i h o o d r a t i o t e s t i s a way to measure the e x i s t e n c e of t h i s r e l a t i o n s h i p . T h i s 71 r a t i o i s d i s t r i b u t e d as a ch i - s q u a r e with K degrees of freedom, where K r e p r e s e n t s the number of independant v a r i a b l e s i n c l u d e d i n the model. i i i . T Test T h i s t e s t i s used to v e r i f y whether c o e f f i c i e n t ak i s s i g n i f i c a n t l y d i f f e r e n t from zero. As i n d i c a t e d i n Chapter IV, we can make t h i s i n f e r e n c e about ak only i n the case where we hol d a l a r g e data s e t , s i n c e f o r . small sample ak i s not assumed to be normally d i s t r i b u t e d ( s e e I V . 2 . 1 . i i i ) i v . C o r r e l a t i o n between V a r i a b l e s Since one of the major sources of e r r o r i s due to the c o r r e l a t i o n that might e x i s t s between v a r i a b l e s , we must check the c o r r e l a t i o n c o e f f i c i e n t matrix f o r each model. In the presence of the c o r r e l a t i o n , we should take the a p p r o p r i a t e remedial measure. 1.2 Model Development The f o l l o w i n g models, i n v o l v i n g d i f f e r e n t forms of the four system v a r i a b l e s : AT, AWALK, WAIT and AC, are c o n s i d e r e d and t r e a t e d with L o g i t , P r o b i t and D i s c r i m i n a n t A n a l y s i s : M1 : L = 0 o + 0iAT+02AWALK+0 31WAIT+0 j,(AC/l) M2 : L = 0 Q + 0.I.AT+0 21.AWALK+0 3I.WAIT+0„AC M3 : L = 0o + 0i (AT/T) + 02(AWALK/T)+0 3(WAIT/T)+0 4(AC/I.C) M4 : L = 0o+01(lAT/T)+0 2(IAWALK/T)+0 339I.WAIT/T) + 0i, (AC/C) 72 and M1 . 1 M2. 1 M3. 1 M4. 1 L = 0 o + 01AT+0 2AEXC+0 3(AC/I) L = 0 0 + 0,(I.AT)+ p 2(I.AEXC)+ 0 3(AC) L = 0o +0i(AT/TT)+0 2(AEXC/TT)+0 3(AC/IC) L = 0o + 0i(I.AT/TT) + 0 2(I.AEXC/TT)+0 3(AC/C) where AEXC = AWALK + WAIT 1.2.1 L o g i t Treatment R e s u l t s of t h i s treatment are shown in Table 15. By a p p l y i n g the s i g n t e s t , we e l i m i n a t e those models which have a p o s i t i v e AT, AWALK, WAIT c o e f f i c i e n t ; or a negative AC c o e f f i c i e n t . Models 2 and 2.1 s a t i s f y t h i s c r i t e r i o n : M2 : L = 1.93 - .0015(IAT) - .00175(IAWALK) - .00199(1.WAIT) + .2245AC and M2.1 : L = 1.94 - .0015(IAT) - .00196(I.AEXC) + .2289.AC where I i s expressed i n 1000$ They l e a d almost to the same l i k e l i h o o d r a t i o , root mean square error(RMSE) and sum of absolute e r r o r ( S A E ) . However, i n the case of M2, the ch i - s q u a r e d i s t r i b u t i o n of the l i k e l i h o o d r a t i o has 4 degree of freedom whereas, that of M2.1 has 3. 1.3 P r o b i t Treatment Table 16 d i s p l a y s the estimated c o e f f i c i e n t s f o r models M1,...,M4.1. F o l l o w i n g the same reasoning made i n the case Models Constant In-Veh i c l e Wa1k i ng Wa i 11ng Out-of-Pocket Excess L1keli hood RMSE 1 SAE ! T i me T1 me T i me Expenses T1 me Rat i o M1 1 .697 -0.0173 -0.0586 -0.0093 -0.1944 6. 153 . 3392 142 .65 M1 . 1 1 .573 -0.018 -0.1845 -0.0158 5. 176 . 3394 142.85 M2 1 .931 -0.0015 -0.00175 -0.0020 0. 2245 8. 136 . 3380 141.90 M2 . 1 1 .942 -0.0015 0.2289 -0.0020 8. 130 .3381 141.92 M3 8.G61 7 .847 1 . 1955 0.1517 0. 1823 106 . 1 .3012 110.78 M3 . 1 8 . 731 7 .908 0.1447 0.4203 105 . 1 . 3003 110.54 M4 2 . 107 0.06 17 0.7042 -0.0206 -0.3252 13.69 . 3362 140.72 M4 . 1 2.569 0.0628 -0.0454 -0.0017 1 1 .63 .3372 140.97 1- Root Mean Square of E r r o r 2- Sum of A b s o l u t e E r r o r T able 15 - Parameter E s t i m a t i o n of L o g i t Models Models Constant In-Vehicl e T i me Wa1k i ng T i me Wa i t i ng T i me Out-of-Pocket Expenses Excess T i me L i ke1i hood Ratio RMSE 1 SAE' M1 0 972 -O 01054 -0.0296 -0.0044 -0 . 0829 6 63 . 3393 142.59 M1 . 1 0 938 -0 01012 -0 . 0797 -0.0089 5 54 . 3394 142.75 M2 1 125 -0 00084 -0.0011 -0.0010 0. 1 126 8 28 . 3382 141.98 M2 . 1 1 122 -0 .00084 0 1 109 -0.0011 8 28 .3382 141.97 M3 4 415 3 . 780 0.4847 0.0542 0 0758 103 28 . 3027 111.47 M3 . 1 4 462 3 .825 0 0581 0.1613 102 58 . 3020 111.37 M4 1 231 0 .03690 0.0431 -0.0103 -0 1943 14 22 . 3363 139.91 M4 . 1 1 495 0 .03636 -0 0317 -0.00094 1 1 45 .3371 140.86 1- Root Mean Square of Error 2- Sum of Absolute Error Table 16 - Parameter Estimation of Probit Models 75 of the l o g i t models, models M2 and M2.1 are s e l e c t e d . M2 : L = 1.13 - .00084(1.AT) - .00113(I.AWALK) - .00104(1.WAIT) + .1126AC and M2.1 : L = 1.22 - .00084(1.AT) - .00105(I.AEXC) + .1109AC T h e i r l i k e l i h o o d r a t i o , obtained l e v e l of s i g n i f i c a n c e , RMSE and SAE are given i n Table 16. Tables 17, 18, 19 and 20 show the r e s u l t of t e s t i n g the n u l l hypothesis that ak i s equal to zero. T h i s hypothesis can be r e j e c t e d at 5% l e v e l of s i g n i f i c a n c e f o r a l l parameters except a2 and 0„ f o r M2 and a3 f o r M2.1 . The c o r r e l a t i o n matrix i s shown i n Table 21. I t i n d i c a t e s that v a r i a b l e s i n c l u d e d i n models M2 and M2.1 have an accept a b l e l e v e l of c o r r e l a t i o n . The maximum c o r r e l a t i o n ( 0 . 32) i s found to be between (I.AT) and (AC). 76 Parameters T-Ratio S i g n i f . 0o 3.008 0.001 01 -2.3957 0.008 02 -0.5904 0.227 03 -1.9771 0.024 04 0.4148 0.339 (d.f.=616) Table 17 - S i g n i f i c a n c e of L o g i t Model M2 Parameters Parameters T-Ratio S i g n i f. 0 0 3.0893 0.001 0 i -2.4351 0.007 02 -2.1136 0.017 03 0.4254 0.335 (d.f.=617) Table 18 - S i g n i f i c a n c e of .Logit Model M2.1 Parameters 77 Parameters T-Ratio S i g n i f . 0 0 3.2751 0.000 01 -2.4757 0.006 0 2 -0.7484 0.227 0 3 -1.8147 0.035 0 u 0.3869 0.349 (d.f.=616) Table 19 - S i g n i f i c a n c e of P r o b i t Model M2 Parameters Parameters T-Ratio S i g n i f . 0 0 3.3194 0.000 01 -2.5128 0.006 0 2 -2.0079 0.002 0 3 0.3834 0.350 (d.f.=617) Table 20 - S i g n i f i c a n c e of P r o b i t Model M2.1 Parameters 78 Models V a r i a b l e s 1 2 3 4 1 1 = AT 2=AWALK 3= WAIT 4= AC/I 1 .00 .08 1.00 .04 -.08 1.00 .01 -.01 .02 1.00 2 1 =IAT 2=1AWALK 3=IAWAIT 4=AC 1 .00 -.27 1.00 -.31 .19 1.00 .32 .03 .001 1.00 3 1= AT/T 2=AWALK/T 3= WAIT/T 4= AC/IC 1 .00 -.22 1.00 -.28 .54 1.00 -.02 -.02 -.03 1.00 4 1=1AT/T 2=IAWALK/T 3=IAWAIT/T 4=AC/C 1 .00 -.37 1.00 -.49 .48 1.00 -.09 .21 .14 1.00 Table 21 - C o r r e l a t i o n M a t r i c e s 79 1.4 D i s c r i m i n a n t A n a l y s i s R e c a l l from Chapter IV, that the e x p r e s s i o n of modal choice p r o b a b i l i t y obtained by means of d i s c r i m i n a n t a n a l y s i s i s s i m i l a r to that r e s u l t e d from the l o g i t treatment(see Eqs. 4.15 and 4.48). T h e r e f o r e , time v a r i a b l e s should have a negative s i g n , whereas the cost v a r i a b l e should c o n t r i b u t e n e g a t i v e l y to the c l a s s i f i c a t i o n f u n c t i o n ( E q . 4.49). Table 22 d i s p l a y s the r e s u l t s of t h i s a n a l y s i s 1 . Only model 4 leads to c o r r e c t s i g n s . I t a l s o leads to the second l a r g e s t D 2 (Mahalanobis d i s t a n c e ) . The o v e r a l l F - s t a t i s t i c found for t h i s model i s 4.27 with an a t t a i n e d s i g n i f i c a n c e of 0.002. Th e r e f o r e , a c c o r d i n g to 2.49, f u n c t i o n G w i l l be: G = -.45 - .07(IAT/T) - .10(1AWALK/T) - .004(I.WAIT/T) + .31(AC/C) - K + log(83/538) where 83 and 538 are the t r a n s i t and auto sample s i z e r e s p e c t i v e l y . K i s c a l c u l a t e d i n t e r n a l l y by the computer programme ac c o r d i n g to Eq. 4.46. Since i n the case of l o g i t and p r o b i t a n a l y s i s , no s i g n i f i c a n t changes has been det e c t e d between the magnitude of c o e f f i c i e n t s of models 1, 2, 3 and 4 and that of models 1.1, 1.2,1.3 and 1.4 c o e f f i c i e n t s , these l a t t e r s were not pursued f u r t h e r 80 Models V a r i a b l e s F-Stat S i g n i f Discrim.E 'unction Z-Func. Auto T r a n s i t 1 Constant AT AWALK WAIT AC/I 3.5916 1.4566 .2143 .0279 .0585 .2279 .6436 .8674 -4.933 -.1391 .6644 .2963 -7.786 -4.802 -.1237 .7179 .3060 -7.608 .1310 .01 54 .0553 .0097 .1786 D 2 = 0.0807 F-ST/ ^T. = 1 .i 1436 « 5IGNIF. = .2180 2 Constant I AT IAWALK I.WAIT AC 6.0254 0.2646 3.7322 .2037 .0144 .6072 .0538 .6519 -13.499 -.0036 .0304 .0104 -22.064 -13.670 -.0049 .0318 .0127 -22.305 -.1710 .001 3 .001 4 .0023 -.2410 D 2 = 0.1099 F-ST; . = 1 . < 3653 i SIGNIF. = .0982 3 Constant AT/T AWALK/T WAIT/T AC/I.C 175.72 2.044 1 .140 .01 90 .0000 . 1 533 .2883 .8909 -19.907 -47.45 .3830 -2.450 -10.468 -30.882 -59.605 -1.617 -2.864 -10.617 -10.975 -12.152 -1.234 -.415 -. 1 49 D 2 = 2.5012 F-ST; ^T. = 44 .745 < 3IGNIF. = .0000 4 Constant I.AT/T IAWALK/T IWAIT/T AC/C 8.903 3.465 2.252 .0527 .0030 .0631 . 1 340 .8185 -129.25 -.3663 1.2737 .0617 -181 .39 -129.70 -.4410 1 . 172 -.0574 -181.08 -.45 -.07 -.10 -.004 .31 D 2 = 0.2388 F-STAT. = 4.2721 SIGNIF. = .0020 Table 22 - Parameter E s t i m a t i o n of D i s c r i m i n a n t Models 81 2. DETERMINATION OF SOCIO-ECONOMIC VARIABLES The set of socio-economic v a r i a b l e s which appear to have a p o t e n t i a l r o l e i n e x p l a i n i n g i n d i v i d u a l behaviour towards t r a v e l l i n g i s l i s t e d Chapter I I . They are as f o l l o w s : Sex, Age, Occupation and Car Ownership. Note that the e f f e c t of income has a l r e a d y been c o n s i d e r e d by the combination with system v a r i a b l e s . 2.1 S e l e c t i o n C r i t e r i o n s i . P a r t i a l C o e f f i c i e n t Sign According to the sex v a r i a b l e f o r m u l a t i o n , Female=0 and Male=1, and, s i n c e the p r o b a b i l i t y that a male take the car i s h i g h e r , we should expect a p o s i t i v e s i g n f o r Sex c o e f f i c i e n t . Age v a r i a b l e must have a p o s i t i v e s i g n s i n c e i t takes on the value 1 when t r i p maker i s younger than 25 or o l d e r than 60 years o l d , and we know that people belonging to these age brackets are mostly t r a n s i t u s e r s . As f o r Car Ownership v a r i a b l e , a p o s i t i v e s i g n i s expected. CO i s equal to 1 when the household possesses more than one car and t h e r e f o r e , the p r o b a b i l i t y of t a k i n g car to work f o r people belonging to t h i s category of household i s h i g h e r . Occupation v a r i a b l e s needs more r e f l e x i o n s i n c e two 82 i n d i c a t o r v a r i a b l e s 0CC1 and 0CC2 e x p l a i n simultaneously i t s e f f e c t — Primary category i s formulated by OCC1=0 and OCC2=0. Therefore, the corresponding u t i l i t y f u n c t i o n i s : Lp = 0 o + 0,(lAT)+0 2(lAWALK)+p 3(I.WAIT)+0 a.AC +0 5.SEX+0 6AGE+0 7.O+0 8.O+0 9.CO — I n d i v i d u a l s belonging to Managerial and P r o f e s s i o n a l category have an u t i l i t y f u n c t i o n as: Lm = 0o+0 i(lAT)+0 2(lAWALK)+0 3(I.WAIT)+0„.AC +0 5,SEX+0 6AGE+0 7.O+0 8.1+0g.CO — C l e r i c a l and Salemen's u t i l i t y are expressed by: Lc = 0o+01(lAT)+0 2(lAWALK)+0 3(I.WAIT)+0„.AC +0 5.SEX+0 6AGE+0 7.1+0 8.O+0 9.CO — And f i n a l l y , we can formulate the u t i l i t y of Labours by: L l = 0 0 + 0,(IAT)+ 0 2(IAWALK)+ 0 3(I.WAIT)+ 0 ft.AC +0 5.SEX+0 6AGE+0 7.1+0 8.1+0g.CO In order to f i n d the a p p r o p r i a t e sig n f o r 0 7, the marginal e f f e c t of OCC1, we should compare Lm and L l . The d i f f e r e n c e between these two u t i l i t y f u n c t i o n s are given as: 83 L l - Lm = 0 7 and s i n c e we expect that i n d i v i d u a l s belonging to managerial and p r o f e s s i o n a l category are mostly p r i v a t e car u s e r s , i n other words they a t t a c h more d i s u t i l i t y to the t r a v e l a c t i v i t y ( L m > L l ) , t h e r e f o r e 0 7 should be n e g a t i v e . S i m i l a r i l y , when we compare Lm and Lp, we f i n d that the c o r r e c t s i g n f o r BS i s a p o s i t i v e s i g n s i n c e : Lm - Lp = 0 8 and Lm > Lp i i . Pseudo R-square C r i t e r i o n As mentioned i n Chapter IV, />* index i s a measure of goodness of f i t . One should be forewarned that a good f i t i s expressed by value of .2 to .4, and t h i s c r i t e r i o n i s used to s e l e c t the 'best' set of v a r i a b l e s . Since as one adds v a r i a b l e s to the model p2 index i n c r e a s e s r e g a r d l e s s of the explanatory power of the v a r i a b l e 1 , the i n t e n t i o n w i l l then be not to maximize p2 index but to f i n d whether the i n c l u s i o n of the new v a r i a b l e worth the i n c r e a s e i n p2 index. i i i . Mean-Square of E r r o r C r i t e r i o n T h i s i s a l s o used to measure the goodness of f i t of the model. I t i s expressed as: MSE = SSE / n-k where k i s the number of parameters i n the model. The Since by i n c r e a s i n g the number of v a r i a b l e s , the model approaches the s a t u r a t e d model f o r which the f i t i s p e r f e c t . 84 advantage of MSE c r i t e r i o n i s that i t takes account of the number of parameters present i n the model. We should seek to minimize MSE s i n c e Min(MSE) can inc r e a s e as k i n c r e a s e s i f the r e d u c t i o n i n SSE becomes so small that a l o s s of an a d d i t i o n a l degree of freedom can not compensate i t . 2.1.1 L o g i t and P r o b i t Models F i g u r e s 7 and 8, and Tables 23 and 24 show the e f f e c t of each new users' v a r i a b l e on i n c r e a s i n g or reducing the p2 and MSE. The p2 index and MSE value f o r each e n t e r i n g v a r i a b l e are p l o t t e d i n these f i g u r e s . P o i n t s are connected by s t r a i g h t l i n e s to express the e f f e c t of adding a d d i t i o n a l independant v a r i a b l e s . Both f i g u r e s c l e a r l y d i s p l a y the important e f f e c t of SEX and CO, shown by the slope of connecting l i n e s , i n improving the model c a p a b i l i t y . On the other hand, small gains are achieved when age and occupation v a r i a b l e s are added to the model as i l l u s t r a t e d i n F i g . 7. Note a l s o that the T - r a t i o of 0CC1 and 0CC2 are low(-.73 and 1.68), whereas that of age v a r i a b l e v a r i e s between -3.66 and -3.37. T h e r e f o r e , only the occupation v a r i a b l e s 0CC1 and 0CC2 were excluded. The f i n a l modal c h o i c e r e s u l t e d by means of l o g i t and p r o b i t approach are r e s p e c t i v e l y : 85 F i g u r e 8 - Pseudo R-Square V a r i a t i o n 8 6 F i g u r e 9 - Mean Square of E r r o r V a r i a t i o n Entering Var iable Pseudo R ! MSE Est imates T-Rat io Degree of Freedom S i gn i f i cance System Variables .0239 . 1 143 Sex . 1400 . 1053 1 .591 6.83 615 .0000 Age . 1748 . 1003 - .956 -3.66 614 .0001 0CC1 i - . 259 -0 . 73 . 2330 | . 0CC2 . 1959 •.1974 -. 575 1 .66 .0480 CO . 2831 .0889 1 .054 5.52 611 .0000 Table 23 - Socio-Economic Var iables for Logit Model M2 Entering Var iable Pseudo R! MSE Est 1 mates T-Rat io Degree of Freedom S1gn i f1cance System Variables Sex Age 0CC1 0CC2 CO .0243 1396 1695 1868 . 2730 1 144 1054 1009 .0980 .0899 .869 - .494 .096 .319 .841 6 . 37 -3 . 37 - .54 1 .68 5.60 615 614 612 61 1 .0000 .0OO1 . 2940 .0460 .OOOO Table 24 - Sodo-Economic Var iables for Probi t Model M2 89 L = 1.10 - .0012(IAT) - .0053(IAWALK) - .0026(I.WAIT) + .490AC + 1.70SEX - 1.03AGE + 1.56CO L i k e l i h o o d R a t i o A t t a i n e d s i g n i f i c a n c e l e v e l Pseudo R-Square Mean Square of E r r o r 95.74 .0000 .2623 .0916 L = 0.63 - .0006(IAT) - .0025(IAWALK) - .0013(I.WAIT) + .256AC + 0.90SEX - 0.54AGE + 0.82CO L i k e l i h o o d R a t i o A t t a i n e d s i g n i f i c a n c e l e v e l Pseudo R-Square Mean Square of E r r o r 92.52 .0000 .2542 .0924 Tables 25 and 26 i l l u s t r a t e the standard d e v i a t i o n s and the t - r a t i o s t a t i s t i c s obtained f o r the parameters of the above models. 2.1.2 D i s c r i m i n a n t A n a l y s i s In the present stepwise r o u t i n e , we f i r s t examine the sign of the new v a r i a b l e s a c c o r d i n g to the s i g n t e s t c r i t e r i o n , then choose those v a r i a b l e s which l e a d to the l a r g e s t i n c r e a s e i n D 2 and F s t a t i s t i c s . In a d d i t i o n , we should examine that the s i g n i f i c a n c e l e v e l a t t a i n e d by these F-values f a l l s below our predetermined l i m i t of s i g n i f i c a n c e ( 5 % ) . Param. Est imates Std.Dev. T-Ratio 0 o 1.10 .72 1 .52 0 1 -.0012 .0007 -1 .75 0 2 . -.0053 - .0032 -1 .67 0 3 -.0026 .001 2 -2.26 0 « .490 .6007 .82 0 5 1 .70 .2681 6.35 0 6 -1 .30 .2745 -3.74 0 7 1 .56 .2835 5.15 Table 25 - Point E s t i m a t i o n of L o g i t Model Parameters 91 Param. Est imates Std.Dev. T-Rat i o 0o .63 .3901 1.61 0 1 -.0006 .0004 -1.71 02 -.0025 .0017 -1 .52 03 -.0013 .0006 -2.01 0U .256 .3208 .80 05 .90 . 1 460 6.20 06 -.54 . 1 528 -3.56 07 .82 . 1483 5.53 Table 26 - Point E s t i m a t i o n of P r o b i t Model Parameters 92 Table 27 shows the r e s u l t s of the a n a l y s i s . I t i s n o t i c e d that SEX and CO v a r i a b l e s c o n t r i b u t e to a s i g n i f i c a n t i n c r e a s e i n F and D 2 s t a t i s t i c s , whereas 0CC1 and 0CC2, e x p r e s s i n g the e f f e c t of o c c u p a t i o n , reduce the F-value and s l i g h t l y augment the D 2. Table 28 d i s p l a y s the estimated c o e f f i c i e n t s when System, Sex, Age and Car Ownership v a r i a b l e s are present in the model. According to these r e s u l t s , the f o l l o w i n g d i s c r i m i n a n t modal cho i c e model i s formulated: Z = -6.91 - .124(IAT/T) - .066(IAWALK/T) - .0065(I.WAIT/T) - 1.25(AC/C) + 2.08SEX - 1.30AGE + 1.94CO Mahalanobis Distan c e : 1.955 F - S t a t i s t i c : 19.89 A t t a i n e d S i g n i f i c a n c e L e v e l : .0000 Note that by i n c l u d i n g socio-economic v a r i a b l e s , the sign of the cost v a r i a b l e becomes negative and hence v i o l a t e s s i g n c r i t e r i a . T h e r e f o r e , we decided to take the reverse 'cheminement' which i s : having the present set of socio-economic v a r i a b l e s (SEX, AGE and CO), look f o r the 'best' model(M1, M2 or M3) which r e s p e c t s the sign c r i t e r i o n and leads to an accep t a b l e F-value. Table 29 compares these three models and i n d i c a t e s that only M3 s a t i s f i e s the s i g n c r i t e r i o n . E n t e r i n g V a r i a b l e S i g n F - S t a t . S i g n i f . D- S q u a r e A D ' O v e r a 1 1 F - S t a t . A F S i g n i f . S y s t e m V a r i a b l e s 2 3 8 8 4 . 2721 . 0 0 2 0 S e x + 47 . 6 5 3 .OOOO 9 2 3 0 . 6 8 4 2 13 . 189 8 9 1 6 9 . 0 0 0 0 A g e - 1 5 . 8 9 3 . 0 0 0 1 1 1697 . 2 4 6 7 13 . 9 0 5 0 7 160 . 0 0 0 0 0 C C 1 i - 1 . 7 8 9 . 1818 1 11 3 4 8 3 1768 — 1 1 9 8 2 —0 1923 • — . 0 0 0 0 — 1 0 C C 2 3 . 3 4 3 . 0 6 8 2 CO + 4 9 . 8 3 7 .OOOO 2 1604 . 8 1 2 1 17 0 3 8 5 0 5 6 1 . 0 0 0 0 T a b l e 27 - D i s c r i m i n a n t A n a l y s 1 s : S e l e c t i o n o f S o c i o - E c o n o m i c V a r i a b l e s V a r i a b l e s D i s c r i m . Funct ion Z-Funct ion 1Auto T r a n s i t Constant -154.24 -161.75 -6.910 I AT/T -0.6087 -0.7332 • -0.124 IAWALK/T 1.4999 1.4340 -0.066 IWAIT/T 0.1054 0.0402 -0.065 AC/C -185.24 -186.49 -1.250 SEX 10.679 -12.759 2.08 AGE 8.110 6.810 -1 .30 CO 10.368 12.312 1 .944 1- See Eq. 4.49 Table 28 - C o e f f i c i e n t E s t i m a t i o n of D i s c r i m i n a n t and Fun c t i o n s Z-Funct ion V a r i a b l e s Model 1 Model 2 Model 3 Constatnt -3.167 -4.277 -13.042 InVehTime 0.01 1 0.001 -11.36 WalkTime 0.089 0.005 -1.272 WaitTime 0.014 0.003 -0.354 O.P.E. 1 -2.187 -0.586 1 .584 SEX 2.047 2.083 1 .908 AGE -1.278 -1.236 -1.069 CO 1 .682 1.647 1 .430 1- Out-of-Pocket Expenses Table 29 - D i s c r i m i n a n t Models 1, 2 and 3 96 T h e r e f o r e , the f i n a l model i s formulated as below: Z = -13.04 - 11.36(AT/T) - 1.27(AWALK/T) - .35(WAIT/T) + 1.58(AC/I.C) + 1.91 SEX - 1.07AGE + 1.43CO Mahalanobis Distan c e 3.69 F - S t a t i s t i c 37.58 A t t a i n e d S i g n i f i c a n c e L e v e l .0000 3. COMPARISON OF THE METHOD OF ANALYSIS Although the s t r a t e g y employed to develop the modal c h o i c e models was the same for the three approaches, the form of System v a r i a b l e s obtained by means of D i s c r i m i n a n t a n a l y s i s d i f f e r s from that obtained by L o g i t and P r o b i t methods. However, the use of a l l three methods suggests that Occupation v a r i a b l e does not s i g n i f i c a n t l y a f f e c t the i n d i v i d u a l u t i l i t y f u n c t i o n . I t i s very d i f f i c u l t to base the comparison and the assessment of the three approaches, on the one hand, to the q u a l i t y of estimates s i n c e they a l l produce s i g n i f i c a n t o v e r a l l s t a t i s t i c , and on the other hand, to the magnitude of estimates s i n c e t h e i r d e f i n i t i o n s are not the same(see C h a p t e r l V ) . T h e r e f o r e , we decided to evaluate the e f f i c i e n c y of each method by i t s p r e d i c t i o n c a p a b i l i t y . For t h i s purpose, we apply the models obtained from these three methods to the present data and measure t h e i r a b i l i t y to reproduce the a c t u a l s i t u a t i o n . 97 O b s e r v a t i o n - p r e d i c t i o n 1 Tables 30, 31 and 32 show the r e s u l t s of t h i s comparison . Note that e n t r i e s t i j represent the mode i o b s e r v a t i o n followed by mode j p r e d i c t i o n . T h e r e f o r e , a l l o f f - d i a g o n a l cases represent a p r e d i c t i o n e r r o r . The o v e r a l l frequency of c o r r e c t p r e d i c t i o n i s then c a l c u l a t e d by E ( t i j ) / t . In some cases t h i s score might be m i s l e a d i n g . For i n s t a n c e , the o v e r a l l frequency score obtained by means of D i s c r i m i n a n t a n a l y s i s i s 78% which r e p r e s e n t s the highest among the two others (73% f o r L o g i t and 69% f o r P r o b i t ) . But the percent of c o r r e c t p r e d i c t i o n f o r t r a n s i t mode obtained through D i s c r i m i n a n t approach i s only 2% whereas, L o g i t and P r o b i t a n a l y s i s l e a d to b e t t e r s c o r e s ( l 6 . 9 % and 15%)). In order to take account of t h i s f a c t , we s h a l l use the success index which i s the normalized p r e d i c t i o n success p r o p o r t i o n : ( t i j / t . , ) / ( t , J t ) f o r i=j According to the above three t a b l e s , one may choose the L o g i t approach s i n c e i t leads to a gr e a t e r success index. Note a l s o the c l o s e n e s s of P r o b i t success index to that of L o g i t model. See f o r example T h e i l d 9 6 6 ) and McFadden(1 976) 98 P R E D I C T I O N Auto T r a n s i t T o t a l Observed % Auto 435 103 538 87 T r a n s i t 62 21 83 1 3 T o t a l 497 124 621 P r e d i c t e d % 80 20 100 C o r r e c t % 87.5 16.9 73 Success Index 1.01 1 .30 Table 30 - L o g i t O b s e r v a t i o n - P r e d i c t i o n Table 99 P R E D I C T I O N Auto T r a n s i t T o t a l Observed % Auto 407 131 538 87 T r a n s i t 59 24 83 13 T o t a l 466 1 55 621 P r e d i c t e d % 75 25 100 C o r r e c t % 87.3 15.5 69 • Success Index 1 .003 1.19 Table 31 - P r o b i t O b s e r v a t i o n - P r e d i c t i o n Table 100 P R E ! ) I C T I 0 N Auto T r a n s i t T o t a l Observed % Auto 486 52 538 87 T r a n s i t 82 1 83 13 T o t a l 568 53 621 P r e d i c t e d % 91 .5 8.5 100 C o r r e c t % 85.5 1.9 78 Success Index .98 .15 Table 32 - D i s c r i m i n a n t O b s e r v a t i o n - P r e d i c t i o n Table 4. SUMMARY In t h i s chapter, we a p p l i e d three c l a s s i c a l approaches: L o g i t , P r o b i t and D i s c r i m i n a n t a n a l y s i s to our data s e t , i n order to determine the set of s e r v i c e a t t r i b u t e s and t r a v e l l e r s ' socio-economic c h a r a c t e r i s t i c s which 'best' a f f e c t the t r a n s p o r t a t i o n consumers' behaviour. T h i s a n a l y s i s attempted to o b t a i n those unco'rrelated v a r i a b l e s with the a p p r o p r i a t e form 101 which l e a d to a s i g n i f i c a n t o v e r a l l s t a t i s t i c and to s i g n i f i c a n t c o e f f i c i e n t estimates with the c o r r e c t s i g n . We found t h a t , while the models r e s u l t e d from L o g i t and P r o b i t treatment i n c l u d e the same set of v a r i a b l e s with s i m i l a r forms, the D i s c r i m i n a n t approach leads to d i f f e r e n t v a r i a b l e forms. Although not very d i f f e r e n t from the f o r e c a s t i n g a b i l i t y of P r o b i t model, the L o g i t model has the g r e a t e s t p r e d i c t i o n a b i l i t y . 102 VI. SENSITIVITY ANALYSIS T h i s chapter i n v e s t i g a t e s a s s e s s i n g the impact of a change in s e r v i c e a t t r i b u t e s and users' socio-economic c h a r a c t e r i s t i c s on the mode c h o i c e p r o b a b i l i t y . The l o g i t model s i n c e , as i t was shown, the d i s c r i m i n a n t a n a l y s i s performs badly on p r e d i c t i o n , and as i t w i l l be seen, the v a r i a t i o n of p r o b i t model v a r i a b l e s f o l l o w s the same trend as those of l o g i t model. F i r s t , the i n t e r p r e t a t i o n of c o e f f i c i e n t s i s given, then a s e n s i t i v i t y a n a l y s i s i s c a r r i e d out and f i n a l l y , the value of time i s d e r i v e d . 1. COEFFICIENT INTERPRETATION I t should be r e c a l l e d that i n the l o g i t and p r o b i t models, the dependant v a r i a b l e i s not the index L, the l i n e a r combination of independant v a r i a b l e s , but the p r o b a b i l i t y of choosing a mode given a set of independant v a r i a b l e s which i s found by means of l o g i t or p r o b i t t r a n s f o r m a t i o n of index L. Due to t h i s f a c t , the i n t e r p r e t a t i o n of c o e f f i c i e n t s needs to be e x p l a i n e d . 1.1 P r o b i t R e c a l l from Eq. 4.20: 103 L(X) P(X) = l/SQRT(2n) e x p ( - t / 2 ) dt where L(X) = e' .X A one u n i t change i n Xk w i l l l e a d to a change of p i n the index L which changes the area under the standard normal curve. In other words, the marginal e f f e c t of Xk i s e q u i v a l e n t to the B standard d e v i a t i o n u n i t s . Since t h i s o r d i n a t e i s l a r g e r near the c e n t e r of the d i s t r i b u t i o n , t h e r e f o r e , the l a r g e s t v a r i a t i o n i n the p r o b a b i l i t y i s obtained i n t h i s a r e a . Moreover, one can e a s i l y d e r i v e the d e f i n i t i o n of the constant term 0 O ; i t i s the p r o b a b i l i t y corresponding to B0 standard d e v i a t i o n of N (0 ,1) of t a k i n g the bus when a l l independant v a r i a b l e s are zero. In the present case, i t represents the p r o b a b i l i t y that a female t r a v e l l e r belonging to age group 25-60 and to a household owing one c a r , takes bus when a l l the system c h a r a c t e r i s t i c s ( t r a v e l time, t r a v e l c o s t , . . . ) of a l t e r n a t i v e s are s i m i l a r ; i t i s equal to 74%. 1.2 L o g i t According to the l o g i t t r a n s f o r m a t i o n of the index L, the p r o b a b i l i t y of s e l e c t i n g the t r a n s i t mode i s : P(X) = exp(L) / 1+exp(L) = 1 / (l+exp(-L)) The marginal e f f e c t of one of the v a r i a b l e s can be seen by ta k i n g the p a r t i a l d e r i v a t i v e of P with respect to the 104 u n d e r l y i n g v a r i a b l e , f o r example Xk: 6P/6Xk = 6[l/(1+exp(-£'.X))]/6Xk = 0k exp(-0_'.X) l / ( 1+exp(-£'.X)) 2 and by s u b s t i t u t i n g the e x p r e s s i o n of P(X) we have: 6P/6Xk = B . P . ( 1 - P ) k As one can n o t i c e , the marginal e f f e c t of v a r i a b l e Xk depends on where the p r o b a b i l i t y P i s e v a l u a t e d . S i m i l a r to the p r o b i t t r a n s f o r m a t i o n , the hi g h e s t e f f e c t occurs at the mid-point of the d i s t r i b u t i o n ( P = 5 0 % ) . The s t r a i g h t f o r w a r d n e s s and the s i m p l i c i t y of the l o g i s t i c f u n c t i o n parameters render the use of t h i s approach more convenient. The i n t e r p r e t a t i o n of B0 f o l l o w s the p r i n c i p l e of the l o g i t t r a n s f o r m a t i o n ; i t corresponds to the l o g a r i t h m of the r a t i o of bus ch o i c e and car choice p r o b a b i l i t i e s when a l l Xk are zero. In the a c t u a l case, the t r a n s i t use p r o b a b i l i t y a s s o c i a t e d to B0 f o r the- type of t r a v e l l e r mentioned before i s 75%. 2. SENSITIVITY ANALYSIS Before e v a l u a t i n g the e f f e c t of each v a r i a b l e , i t i s i n t e r e s t i n g to note a p o i n t deduced from the f a c t that the hi g h e s t marginal e f f e c t occured at. the cente r of the d i s t r i b u t i o n . In the t r a n s p o r t a t i o n context i t means that disaggregate modal c h o i c e models are p r i m a r i l y u s e f u l only i n 105 c i t i e s where the whole t r a n s p o r t a t i o n system i s w e l l developed and p u b l i c t r a n s i t i s h i g h l y c o m p e t i t i v e with the p r i v a t e mode in a manner that makes the p r o b a b i l i t y of t a k i n g a bus or a car to work equal. Below, the e f f e c t of each v a r i a b l e on the t r a n s i t use p r o b a b i l i t y i s s t u d i e d . 2 .1 Ef f e c t of Income I t i s expected that an in c r e a s e i n income reduces the p r o b a b i l i t y of t r a n s i t use. T h i s i s due mostly to the f a c t that the d i s u t i l i t y a s s o c i a t e d with t r a v e l a c t i v i t y v a r i e s d i r e c t l y with the l e v e l of income: higher income people a t t a c h a grea t e r value to t h e i r p r i v a c y and l e v e l of comfort which makes the use of t r a n s i t l e s s probable. F i g u r e 10 compares the p r o b a b i l i t y of t r a n s i t use obtained by l o g i t and p r o b i t model f o r three annual income l e v e l s : 10,000$, 20,000$ and 30,000$. I t can be concluded that higher income persons are more s e n s i t i v e to changes occured i n t r a v e l time d i f f e r e n c e . For i n s t a n c e , a d i f f e r e n c e of ±15 minutes between car and bus t r a v e l times only m a r g i n a l l y a f f e c t s mode ch o i c e p r o b a b i l i t y of low-income people(72%<Pt<75%) whereas, i t has a l a r g e e f f e c t on the high-income people's mode ch o i c e p r o b a b i l i t y ( 6 4 % < P t < 8 4 % ) . Note a l s o that p r o b a b i l i t i e s obtained by p r o b i t model are s l i g h t l y lower and favor the p r i v a t e mode r e l a t i v e to the t r a n s i t mode up to a c e r t a i n p o i n t v a r y i n g with the income l e v e l . For i n s t a n c e , f o r a 30,000$ l e v e l of income t h i s p o i n t 106 <X3 CO DELTA JN-VEH. TJME F i g u r e 10 - E f f e c t of Income on T r a n s i t Use P r o b a b i l i t y with Respect to AT 107 F i g u r e 11 - E f f e c t of Income on T r a n s i t Use P r o b a b i l i t y with Respect t o Walking Time 108 i s about AT=1Ominutes. From t h i s comparison, i t can be seen that the v a r i a t i o n s of the l o g i t and p r o b i t models v a r i a b l e s f o l l o w s the same trend, and f o r t h i s reason, only the l o g i t model w i l l be co n s i d e r e d f u r t h e r . F i g u r e 11 d i s p l a y s the e f f e c t of income on the v a r i a t i o n of t r a n s i t use p r o b a b i l i t y i n terms of the r e l a t i v e walking time. I t c l e a r l y i l l u s t r a t e s the f a c t that the d i s u t i l i t y of walking i s higher than that of i n - v e h i c l e time. A change of ±1.5minutes i n walking time to or from the bus stop m o d i f i e s the u n d e r l y i n g p r o b a b i l i t y from 22% to 97% f o r high-income people and from 55% to 83% f o r low-income people. These r e s u l t s are used i n the next chapter f o r t r a n s p o r t a t i o n p o l i c y p r o p o s a l s . 2.2 Income-Sex I n t e r a c t i o n F i g u r e 12 d i s p l a y s the i n t e r a c t i o n of income l e v e l and the sex of the tripmaker on the bus ch o i c e p r o b a b i l i t y . I t i s i n t e r e s t i n g to n o t i c e that income seems to have a small i n f l u e n c e on mode choice p r o b a b i l i t y of women whereas, f o r men, i t seems to more d r a s t i c a l l y a f f e c t t h e i r d e c i s i o n s . In a d d i t i o n , i t might be concluded that women are l e s s s e n s i t i v e and t o l e r a t e more v a r i a t i o n of i n - v e h i c l e time than men. The r e f o r e , women may be l e s s a f f e c t e d by t r a n s p o r t a t i o n p o l i c i e s modifying the i n - v e h i c l e t r a v e l time. However, the e f f e c t of an in c r e a s e i n walking time to/from bus stop on the t r a n s i t use p r o b a b i l i t y of high income women i s important. T h i s i s shown i n Fi g u r e 13. 109 03 cn -30.0 -20.0 -]0.0 0.0 10.0 20.0 30 0 DELTR I N - V E H . TIME F i g u r e 12 - I n t e r a c t i o n E f f e c t of Income and Sex on T r a n s i t Use P r o b a b i l i t y with Respect to AT 110 F i g u r e 13 Use - I n t e r a c t i o n E f f e c t of Income and Sex on P r o b a b i l i t y with Respect to Walking Time T r a n s i t 111 2.3 Income-Age I n t e r a c t i o n The a n a l y s i s of F i g u r e 14 f o l l o w s the same l i n e as f o r the above case. People belonging to age group 25-60 are more s e n s i t i v e to v a r i a t i o n i n t r a v e l time. I t should a l s o be noted that the l e v e l of income p l a y s a s i m i l a r r o l e i n mode c h o i c e d e c i s i o n making for both age groups. I f the p r o b a b i l i t y curves of two age groups are completely separated when the i n - v e h i c l e time i s c o n s i d e r e d , they become c l o s e r when the p r o b a b i l i t i e s are c a l c u l a t e d i n terms of walking time (see F i g u r e 15). T h i s i m p l i e s that the d i s u t i l i t y occured to people under 25 or above 60 years o l d towards walking i s c l o s e to that of the people belonging to the other age group. Note a l s o that f o r low-income people of 25-60 age group the p r o b a b i l i t y of t r a n s i t i s p r a c t i c a l l y zero when the walking time to/from bus stop i n c r e a s e s by over 15 minutes. 2.4 E f f e c t of T r a v e l Cost R e c a l l t h a t i n our model the f o r m u l a t i o n of t r a v e l cost i s : AC = Fare - ((MOCOST+PKCOST)/20x2) where MOCOST i s the monthly o p e r a t i n g cost PKCOST i s the monthly p a r k i n g c o s t and 20 working days per month i s assumed The e f f e c t of t r a v e l cost can be assessed by changing on one hand the t r a n s i t f a r e and on the other hand the c o s t of o p e r a t i n g a c a r . For the l a t t e r the i n t e r v e n t i o n of p o l i c y makers can be e i t h e r on MOCOST component, by i n c r e a s i n g the p r i c e of f u e l or c r e a t i n g a t o l l system, on PKCOST component 112 F i g u r e 14 - I n t e r a c t i o n E f f e c t of Income Use P r o b a b i l i t y with Respect and Age on to AT T r a n s i t 113 cn I 1 1 1 1 1 1 -]5.0 -]0.0 -5.0 0.0 5.0 10.D 15.C DELTA WALKING TIME F i g u r e 15 - I n t e r a c t i o n E f f e c t of Income and Age on T r a n s i t Use P r o b a b i l i t y with Respect to Walking Time 1 1 4 or on both. In t h i s study, the e f f e c t of changes i n fa r e and par k i n g c o s t s are only c o n s i d e r e d , s i n c e a v a r i a t i o n i n MOCOST w i l l have a s i m i l a r e f f e c t on the c h o i c e p r o b a b i l i t y as that of PKCOST. 2.4.1 Fare F i g u r e 16 i l l u s t r a t e s the expected r e s u l t s that the p r o b a b i l i t y of t r a n s i t s e l e c t i o n decreases as the t r a n s i t f a r e augments. F i g u r e 17 d e p i c t s the i n t e r a c t i o n e f f e c t of f a r e i n c r e a s e and income. For high l e v e l income c l a s s , the downward s h i f t of the curve i n the r e s u l t of a f a r e i n c r e a s e i s the s m a l l e s t , s i n c e they are l e s s concerned about the co s t of t r a v e l . T h i s s h i f t becomes more important as the income l e v e l decreases. The above curves correspond to the three income l e v e l s mentioned before and f o r f a r e s of .50 and .75 d o l l a r s . F i g u r e 18 d i s p l a y s the e f f e c t of a fa r e change i n terms of r e l a t i v e walking time. Since the range of the v a r i a t i o n of the t r a n s i t use p r o b a b i l i t y i s almost the same as when the i n -v e h i c l e time was taken i n t o account, the c o n s i d e r a t i o n of r e l a t i v e walking time i s not f u r t h e r pursued. 2.4.2 Parking Cost F i g u r e 19 presents the i n t e r a c t i o n e f f e c t of parking charge i n c r e a s e and income. Three l e v e l s of income 10,000 $, 20,000 $ and 30,000 $; and two monthly p a r k i n g c o s t s 10$ and 40$ were used. They produce s i m i l a r curves as i n the above case. In these f i g u r e s the e f f e c t of income l e v e l on the curves s h i f t 115 CO -30.0 -20.0 ~]0.0 0.0 10.0 20.3 DELTR J N - V E H . TIME F i g u r e 16 - E f f e c t of Fare V a r i a t i o n on T r a n s i t Use P r o b a b i l i t y with Respect to AT 1 16 .0 DELTA JN-VEH. TIME F i g u r e 17 - I n t e r a c t i o n E f f e c t of Income and Fare V a r i a t i o n on T r a n s i t Use P r o b a b i l i t y with Respect to AT 1 17 CO i 1 r -]5.0 -30.0 -5.0 0.0 5.0 10.0 15. DELTA WALKING TIME F i g u r e 18 - E f f e c t of Fare V a r i a t i o n on T r a n s i t Use P r o b a b i l i t y with Respect to Walking Time 118 F i g u r e 19 - I n t e r a c t i o n E f f e c t of Income and P a r k i n g Cost V a r i a t i o n on T r a n s i t Use P r o b a b i l i t y with Respect to AT 119 i s more accentuated. E v e r y t h i n g being equal i n terms of system a t t r i b u t e s , an i n c r e a s e of 30$ i n monthly parking c o s t r i s e s the p r o b a b i l i t y of t r a n s i t use from 70% to 74%. However, one might keep i n mind that t h i s study c o n s i d e r s only work t r i p s which are i n e l a s t i c towards parking c o s t v a r i a t i o n . The e f f e c t of parking c o s t on the t r a n s i t use p r o b a b i l i t y of other purposes i s important. 3. VALUE OF TIME The value of time i s d e f i n e d as to be the amount of money that a t r a v e l l e r i s ready to pay i n order to save 1 minute of h i s / h e r t r a v e l time. For t h i s purpose, f i r s t , the l i n e a r f u n c t i o n L i s d i f f e r e n t i a t e d with respect to i n - v e h i c l e t r a v e l time(6L/6(AT)) and then with respect to t r a v e l c o s t ( 6 L / 6 ( A C ) ) ; the value of i n -v e h i c l e t r a v e l i s d e r i v e d by the 6L/6(AT) to 6L/6(AC) r a t i o when 6(AT) i s equal to u n i t y . A c c o r d i n g to the c a l i b r a t e d l o g i t model, the f u n c t i o n L i s equal t o : L = 1.10-.0012(IAT)+...+.490AC By d i f f e r e n t i a t i n g the above f u n c t i o n with respect to AT and AC, we have: 6L/6(AT)=-.00121 , 6L/6(AC)=.49 Thus: 1 20 [6L/6(AT)]/[6L/6(AC)] = 6(AC)/6(AT) = -(.0012/.49) I And by equating 6(AT) to 1 minute, the value of i n - v e h i c l e time i s found: Vt = |6(AC)| = 2.45 10- 3 I [$/min] = 0.147 I [$/hr] where I i s the household annual income i n 1000 $. U s u a l l y , the value of time i s expressed i n terms of the h o u r l y wage rate(w) . If 2000 hours per year i s assumed, the value of the i n - v e h i c l e time w i l l be: Vt = (.147 x 2000/1000)W = .29 w [$/min] The d e t e r m i n a t i o n of walking and w a i t i n g time f o l l o w s the same l i n e of r e a s o n i n g : Vwalk = (.0053/.490)1 = 1.08 10" 2 I [$/min] = 0.65 I [$/hr] = 1.30 w [$/min] and Vwait = (.0026/.490)1 = 5.31 10" 2 I [$/min] = 0.32 I [$/hr] = 0.64 w [$/min] The use of the p r o b i t model leads to the f o l l o w i n g value of time: Vt Vwalk = Vwait = 0.28 w [$/hr] 1.18 w [$/hr] 0.60 w [$/hr] 121 One might take the average and determine the value of d i f f e r e n t components of t r a v e l time i n $/hour f o r commuters i n the year of the survey(1972): Vt = 0.285 w Vwalk = 1.240 w Vwait = 0.620 w These values i n terms of i n - v e h i c l e t r a v e l time v a l u e ( V t ) a r e : Vt = 1 Vwalk = 3.50 Vwait =2.17 4. COMPARISON WITH OTHER STUDIES From the above s e c t i o n , i t can be concluded that commuters c o n s i d e r walking and w a i t i n g time 3.50 and 2.17 times more onerous than i n - v e h i c l e time. T h i s d i f f e r e n c e of walking time being valued 1.6 times as h i g h l y as w a i t i n g time i s not i n agreement with the r e s u l t s obtained by p r e v i o u s s t u d i e s (see C h a p . i l ) . Two p o s s i b l e e x p l a n a t i o n s f o r t h i s d i s c r e p e n c y a r e : i . The w a i t i n g time g e n e r a t i o n model d i d not simulate c o r r e c t l y the survey s i t u a t i o n . T h i s needs more d e t a i l e d a n a l y s i s and can be the su b j e c t of f u r t h e r r e s e a r c h . 1 22 i i . The second e x p l a n a t i o n assumes that the w a i t i n g time was c o r r e c t l y simulated and commuters indeed c o n s i d e r walking time twice as inconvenient as w a i t i n g time. T h i s may be true when environmental f a c t o r s such as weather are c o n s i d e r e d ; f o r example on r a i n y days, the d i s u t i l i t y of walking i s gr e a t e r than that of w a i t i n g under a s h e l t e r . And, s i n c e the data set used f o r t h i s a n a l y s i s were obtained from a survey c a r r i e d out du r i n g s p r i n g , the obtained r e s u l t i s reasonable. The value of i n - v e h i c l e time i s estimated to be about 28.5% of the ho u r l y wage r a t e . The t r a v e l time value found i n the l i t e r a t u r e have a high v a r i a n c e . I t v a r i e s between 25% and 67% of the h o u r l y earning r a t e . However, a value of 30% of the hou r l y wage r a t e i s u s u a l l y c o n s i d e r e d (Foster and Beesley, 1963). 123 V I I . GENERAL CONCLUSION T h i s t h e s i s determines the components of the journey to work demand i n the Vancouver M e t r o p o l i t a n Area. 1. SIMULATION The major problem encountered i n e s t i m a t i n g the c o e f f i c i e n t s of the u n d e r l y i n g demand f u n c t i o n was the amount of m i s s i n g i n f o r m a t i o n on the responders' a l t e r n a t i v e t r a v e l mode a t t r i b u t e s . I t was decided to simulate the m i s s i n g data rather to attempt o b t a i n t h e i r obsrved value s i n c e i t i s b e l i e v e d that the important element in the b e h a v i o u r a l model development i s the t r a v e l l e r s ' p e r c e p t i o n of the system a t t r i b u t e s , and to supplement the e x i s t i n g data would be too c o s t l y . Three types of s i m u l a t i o n model were used: r e g r e s s i o n model to estimate the m i s s i n g values of t r a n s i t t r a v e l time, the random ge n e r a t i o n model to simulate the m i s s i n g values on w a i t i n g time from an estimated a - p r i o r i frequency d i s t r i b u t i o n and an aggregation method to complete p a r k i n g c o s t data. The aptness of the above models was e v a l u a t e d by comparing the obtained value of d i f f e r e n t components of t r a v e l time with those a v a i l a b l e from p r e v i o u s s t u d i e s (see S e c t i o n 3 : V a r i a b l e S e n s i t i v i t y ) . 124 2. MODEL STRUCTURE From the three s t a t i s t i c a l approaches used to c a l i b r a t e the demand model, only l o g i t and p r o b i t e s t i m a t i o n methods succeeded i n reproducing the a c t u a l s i t u a t i o n s a t i s f a c t o r i l y . Models c a l i b r a t e d with these two methods embody e x a c t l y the same v a r i a b l e s and v a r i a b l e s form, whereas d i s c r i m i n a n t a n a l y s i s l e d to a d i f f e r e n t form of the v a r i a b l e s . The determinant of t r a n s p o r t a t i o n modal c h o i c e which were s t a t i s c a l l y s i g n i f i c a n t f o r the a v a i l a b l e data set are as f o l l o w s : I n - v e h i c l e t r a v e l time, walking time to and from the modal i n t e r f a c e , w a i t i n g time at the modal i n t e r f a c e , the t r a v e l out-of-pocket expenses, the sex and age of the tripmaker, the household income and the number of c a r s a v a i l a b l e to the t r a v e l l e r ' s household. 3. VARIABLE SENSITIVITY The value of i n - v e h i c l e time obtained i n t h i s study (28.5% of w, the hourly earning rate) corresponds approximately to the value used i n p r a c t i c e (30% of w). But the walking time was valued 1.6 times more onerous than w a i t i n g which i s not i n agreement with other s t u d i e s . I t i s , however, b e l i e v e d that environmental f a c t o r s had h i g h l y i n f l u e n c e d the a t t i t u d e of responders and m o d i f i e d the shape of the p e r c e i v e d w a i t i n g time d i s t r i b u t i o n . 125 Since on an i n t u i t i v e ground, income p l a y s an important r o l e i n modal d e c i s i o n , the e f f e c t of other v a r i a b l e s on the modal s p l i t p r o b a b i l i t y was combined with three income l e v e l s : 10,000$, 20,000$ and 30,000$. I t i s found that a d i f f e r e n c e of ±15 minutes i n i n - v e h i c l e t r a v e l time changes the p r o b a b i l i t y of low-income people from 72% to 75% whereas, f o r high-income people t h i s change i s between 64% and 84%. However, with respect to walking time, t h i s range of v a r i a t i o n i s g r e a t e r ( 5 5 % to 83% f o r low-income people and 22% to 97% f o r wealthy p e o p l e ) . Although female t r a v e l l e r s seems to be l e s s s e n s i t i v e to changes in i n - v e h i c l e time, they r e a c t s t r o n g l y to p o l i c i e s r e l a t e d to walking time. T h i s a l s o a p p l i e s to a l l t r a v e l l e r s below 25 and above 60 years o l d . These r e s u l t s might be used as p r o p o s a l s i n order to i n c r e a s e the t r a n s i t r i d e r s h i p : i . To capture the higher s o c i a l s t r a t a , the performance of t r a n s i t system in terms of i n - v e h i c l e time should be h i g h l y c o m p e t i t i v e with that of p r i v a t e t r a n s p o r t a t i o n . For t h i s , modern technology, o f f e r i n g a b e t t e r a c c e l e r a t i o n and d e c e l e r a t i o n r a t e should be i n c o r p o r a t e d i n t o the system, stop spacing should be o p t i m i z e d and boarding and a l i g h t i n g time should be reduced to i t s minimum. i i . Since people r e g a r d l e s s t h e i r age and sex are s t r o n g l y 126 a f f e c t e d by walking time i n c r e a s e s , a r e s t r i c t i o n on p a r k i n g p l a c e s or on p a r k i n g development i n the downtown core w i l l h e l p i n c r e a s e the t r a n s i t r i d e r s h i p . Park and r i d e c o n f i g u r a t i o n s combined, with modern t r a n s i t technology such as Automated L i g h t R a i l T r a n s i t ( A L R T ) , Rapid R a i l Transit(RRT) and so f o r t h , would provide d e s i r a b l e r e s u l t s s i n c e the walking time can be reduced to i t s minimum and the i n - v e h i c l e time to a p o i n t which can compete with the p r i v a t e c a r . Although parking c o s t can have a great importance i n the d e t e r m i n a t i o n of the magnitude of the t r a n s i t use, i n t h i s study an increase i n parking c o s t had no c o n s i d e r a b l e e f f e c t on the p r o b a b i l i t y of t r a n s i t use. T h i s i s p o s s i b l y due to the f a c t t h a t in the present r e s e a r c h , only work t r i p s were c o n s i d e r e d and the p a r k i n g demand of these t r i p s i s h i g h l y i n e l a s t i c with r e s p e c t to p a r k i n g c o s t . 127 BIBLIOGRAPHY 1. A f i f i , A.A. and Azen, S.P. 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