PART II DEVELOPMENT, TESTING, AND VALIDATION OF A WORK-TRIP MODE-CHOICE MODEL

PART II DEVELOPMENT, TESTING, AND VALIDATION OF A WORK-TRIP MODE-CHOICE MODEL 47

CHAPTER 1 PRETESTING THE SAMPLE AND INITIAL MODEL SPECIFICATIONS Introducton The development of travel demand models requres substantal data collecton and complaton efforts. It s wse to pretest not only the data collecton and data complaton methods but also the model specfcaton the analyst has n mnd. In ths way, mportant nformaton s obtaned regardng the potental value of varables and varable specfcaton n addton to the help t provdes to the data collecton actvtes themselves. Pretestng s also lkely to sharpen the analyst's ntuton and gve an ndcaton of what mportant varables may be mssng from the model. The Urban Travel Demand Forecastng Project (UTDFP) was armed wth ths type of pretest data set for about 200 households. In addton, there exsted consderable pror experence n the development of work-trp mode-choce models (McFadden, 1974; Ben-Akva, 1973; Lsco, 1968; Stopher, 1969; Talvte, 1971; and many others). Nevertheless, several ssues requred further examnaton, among them the followng. Frst, addtonal knowledge was needed n modelng multnomal choce stuatons, partcularly wth regard to modelng access mode. In the prevous studes the transt mode was always vewed to be a sngle alternatve. However, even wthn a sngle lnehaul transt mode, such as bus, there are two ways to reach the mode lne: by foot and by car. A multnomal logt (MNL) model that consders bus-wth-walk-access and bus-wth-car-access as separate modes s estmated, and two problems assocated wth the varable 48

defnton and model specfcaton are dscussed. Second, the MNL model s the only way n whch two or more access modes can be consdered. The MNL model s vald only f the odds of choosng one alternatve over another are ndependent of the exstence of a thrd alternatve (the assumpton of the ndependence from rrelevant alternatves, or the IIA assumpton). It may not be the case that the odds of choosng auto over bus-wth-walk-access are the same whether or not the possblty exsts of takng bus-wth-auto-access. A choce model that does not ncorporate the IIA assumpton--the "maxmum model"--s also estmated and dscussed. Thrd, all the earler models are based upon utlty functons n whch the attrbutes of alternatves are ntroduced drectly as arguments. However, accordng to the neoclasscal theory of consumer behavor, there exsts a tradeoff between goods and lesure tme. Models are then derved that ncorporate ths theory wth the attrbutes of work-trp alternatves enterng the constrants on utlty maxmzaton. The advantage of ths approach s that t allows the functonal form of the choce model to be related to the presumed structure of preferences between goods and lesure. We examne these ssues n the followng three sectons. All the models presented are based upon the sample of 161 ndvduals ntervewed regardng ther work-travel behavor; ths sample s often referred to as the WTS sample. 49

The Basc Three-Alternatve Model Specfcaton Table 2 gves estmates for a partcular specfcaton of the relatve mpedance of the three alternatves. The "representatve" utlty 1 functon s assumed to be of the form β x, where x s a vector of attrbutes of alternatves (or attrbutes of the ndvdual nteractng wth attrbutes of the alternatves), and β s a vector of coeffcents. The frst column n Table 2 lsts the elements of x and the second column gves the pont estmates of the elements of β. The estmaton s the method of maxmum lkelhood descrbed earler. The cost and tme varables are self-explanatory. The socoeconomc varables are alternatve-specfc varables and can be nterpreted as proxes for unmeasured attrbutes of the alternatves n whch the varable s entered; for example, "length of resdence n communty" enters the representatve utlty of the auto alternatve, but not that of the bus alternatves because ths varable assumes the value of zero for the bus alternatves. An explanaton of ts sgnfcantly postve coeffcent could be that persons who resde for a long tme n the same communty happen to value the prvacy that an automoble offers more than less sedentary persons. Or, that the length of resdence s an ndcaton of wealth not captured by the ncome varable. 1 Representatve utlty s that part of the random utlty functon that s common to all ndvduals. 50

TABLE 2 Work-Trp Mode-Choce Basc Model Mode 1: Auto Data: Work Travel Survey, East Bay Mode 2: Bus, Walk Access Model: Multnomal Logt, Ftted by Mode 3: Bus, Auto Access the Maxmum Lkelhood Method Explanatory Varables Estmated Coeffcents t-statstcs Cost dvded by post-tax wage, n cents/ (cents per mnute) e/ -.0358 2.78 On-vehcle tme, n mnutes e/ -.0185 1.38 Walk tme, n mnutes a/ e/ -.0190.972 Transfer-wat tme, n mnutes -.0534 1.54 Number of transfers a/ e/ -.0723.249 Headway of frst bus, wth a celng of 8 mnutes, n mnutes a/ e/ An ndex of dstance to parkng at home c/ Famly ncome wth celng of $7000, n $ per year b/ Famly ncome mnus $7000 wth floor of $0 and celng of $3000, n $ per year b/ Famly ncome mnus $10,000 wth floor of $0 and celng of $5000, n $ per year b/ Length of resdence n communty, n years b/ -.218 2.32 -.318.933.000434 1.51.000785 1.66 -.000617 2.47.143 2.90 51

Table 2, contnued Explanatory Varables An ndex of populaton densty n neghborhood b/ Estmated Coeffcents -.741 t-statstcs 2.58 Dummy f respondent s over 44 years of age -.781 1.22 Dummy f there s chld n household b/ -1.63 2.54 Number of persons n household who can drve c/ 1.10 2.68 Auto alternatve dummy d/ -5.49 2.27 Bus-wth-auto-access dummy d/ -2.76 3.45 Lkelhood rato ndex:.5983 Log lkelhood at zero: -176.9 Log lkelhood at convergence: -71.05 Values of tme saved as a percent of wage: On-vehcle tme: 52 Walk tme: 53 Transfer-wat tme: 149 One transfer wth no watng or walkng s valued the same as 3.9 mnutes of on-vehcle tme. 52

Table 2, contnued All cost and tme varables are calculated round-trp. Dependent varable s alternatve choce (one for chosen alternatve, zero otherwse). Sample sze: 161. a/ The varable s zero for the auto alternatve, and takes the descrbed value for the other alternatves. b/ The varable s zero for the bus alternatves, and takes the descrbed value for the auto alternatve. c/ The varable s zero for the bus-wth-walk-access alternatve, and takes the descrbed value for the remanng alternatves. d/ The varable s one for the bus-wth-auto-access alternatve and zero otherwse. e/ Sum of home-to-work and work-to-home. 53

The alternatve-specfc dummy varables also reflect the mpacts of the alternatve s unmeasured level-of-servce attrbutes not captured n the ncluded varables; ther omsson could bas the coeffcents estmated for the observed and ncluded varables. Consequently, alternatve-specfc dummy varables appear n most contemporary dsaggregate models. Three ncome varables were ncluded to allow for a non-lnear relaton between ncome and representatve utlty of the auto alternatve. These varables can be understood most readly by reference to Fgure 4. The postve coeffcent for the frst ncome varable ndcates that the representatve utlty of the auto alternatve ncreases wth ncome up to an ncome level of $7000 per year. The second ncome varable also has a postve coeffcent, ndcatng that representatve utlty of auto ncreases wth ncome for ncremental ncome above $7000, up to a total ncrement of $3000 (total ncome of $10,000). 54

The slope of ths segment s greater than that of the frst segment because the coeffcent of the second ncome varable exceeds that of the frst. The negatve coeffcent of the thrd ncome varable ndcates that representatve utlty of auto decreases wth ncome for ncremental ncome above $10,000, up to a total ncrement of $5000 (total ncome of $15,000). No fourth ncome varable beng ncluded ndcates that representatve utlty does not vary wth ncome for ncremental ncome over $15,000. (A fourth ncome varable was ncluded orgnally and found to have a very small coeffcent and a t-statstc below 0.1.) The values of on-vehcle and walk tmes conform to prevous estmates (Quarmby, 1967; Thomas and Thompson, 1971). Because walk tme was calculated at the rate of two mnutes per block, the walk and on-vehcle tme coeffcents beng smlar ndcates that the representatve ndvdual s practcally ndfferent between walkng an extra block and rdng for two extra mnutes. The value of transfer-wat tme, whch s much larger than the values of walk and on-vehcle tmes, conforms to the authors expectatons. The estmated coeffcent for "headway of frst bus" s dffcult to explan. Headway s the mnutes between buses for the partcular bus route at the tme the respondent would travel. If the average amount of tme a person wats for a bus s half of ts headway wth a maxmum wat of four mnutes, the estmated value of tme spent watng for the frst bus s 1,190 percent of wage. 1 Apparently the headway varable s capturng somethng more than tme spent watng. It s possble that ndvduals dslke long headways not so much for the watng tme nvolved as the rgdty that nfrequent servce places upon ther schedules. Ths does not seem to be the case, however, unless ndvduals thnk that headways of seven or eght mnutes mpose sgnfcant schedulng rgdty. Three addtonal headway varables were ncluded n a model n a manner analogous to the ncome varable explaned above. Ths specfcaton allowed for a non-lnear relaton between headway and representatve utlty of the bus alternatves. The addtonal headway varables affected the coeffcents of many varables. Naturally, most affected were the the frst headway varable and the transfer-wat tme varables whose coeffcents and t-values were cut by ffty percent or more. An example of the model estmated wth two headway varables s n Table 3. (Note that the model s estmated wth only 142 data ponts.) 1 Ths large value of frst wat tme s partally confrmed by Algers, et al. (1974) on Stockholm data. They found a rato of twelve between the values of wat and on-vehcle tme. However, Algers, et al. were usng a wat tme formula from Brtsh studes (O'Flaherty and Mangan, 1970) that gves approxmately half our value of wat tme (measured as one-half headway) at the mean of the headway values for our sample. 55

TABLE 3 Work-Trp Mode-Choce Maxmum Model Mode 1: Auto Data: Work Travel Survey, East Bay Mode 2: Bus, Walk Access Model: Multnomal Logt, Ftted by Mode 3: Bus, Auto Access the Maxmum Lkelhood Method Explanatory Varables Estmated Coeffcents t-statstcs Cost dvded by post-tax wage, n cents/(cents per mnute) e/ -.0448 3.1 On-vehcle tme, n mnutes e/ -.0303 1.8 Walk tme, n mnutes a/ e/ -.0185 0.90 Transfer-wat tme, n mnutes a/ e/ -.00202 0.04 Number of transfers a/ e/ -.312 0.90 Headway of frst bus, wth a celng of 8 mnutes, n mnutes a/ e/ Headway of frst bus exceedng 8 mnutes a/ e/ An ndex of dstance to parkng at home c/ Famly ncome wth celng of $7000, n $ per year b/ Famly ncome mnus $7000 wth floor of $0 and celng of $3000 n $ per year b/ Famly ncome mnus $10,000 wth floor of $0 and celng of $5000, n $ per year b/ -.138 1.1 -.0639 1.7 -.382 1.0.000343 1.1.00107 1.9 -.000748 (2.6) 56

Table 3, contnued Explanatory Varables Length of resdence n communty, n years b/ Estmated Coeffcents.176 t-statstcs (2.8) An ndex of populaton densty n neghborhood b/ Dummy f respondent s over 44 years of age b/ -.713 2.1 -.714 1.0 Dummy f there s chld n household b/ -1.679 2.4 Number of persons n household who can drve c/ 1.251 2.6 Auto alternatve dummy b/ -5.190 1.8 Bus-wth-auto-access dummy d/ -2.730 3.0 Lkelhood rato ndex:.613 Log lkelhood at zero: -156.0 Log lkelhood at convergence: -60.4 Values of tme saved as a percent of wage: On-vehcle tme: 68 Transfer-wat tme: 5 One transfer wth no watng or walkng s valued the same as 10.3 mnutes of on-vehcle tme. 57

Table 3, contnued All cost and tme varables are calculated round-trp. Dependent varable s alternatve choce (one for chosen alternatve, zero otherwse). Sample sze: 161. a/ The varable s for the auto alternatve, and takes the descrbed value for the other alternatves. b/ The varable s zero for the bus alternatves, and takes the descrbed value for the auto alternatve. c/ The varable s zero for the bus-wth-walk-access alternatve, and takes the descrbed value for the remanng alternatves. d/ The varable s one for the bus-wth-auto-access alternatve, and takes the descrbed value for the remanng alternatves. e/ Sum of home-to-work and work-to-home. 58

Thus, t appears that the headway coeffcents are capturng many effects. The most plausble explanaton for the headway coeffcents s that they capture not only the watng tme effects but also some other unobserved effects correlated wth headways. Clearly, the headway varable needs more attenton and study. We wll return to the problems assocated wth the headway varable on several occasons n ths volume. 59

Examnaton of an Emprcal Issue wth the Basc Model In the basc model (Table 2), varables were calculated for the bus-wth-auto-access alternatve under the assumpton that ndvduals would drve ther auto to the bus stop, park the auto, and then take the bus. If, nstead, a member of the ndvdual s famly were to drve the ndvdual to the bus stop and return home n the auto, then the actual cost and tme varables would be dfferent from those calculated. To determne the extent to whch ths problem could affect the estmates n the basc model, the followng steps were taken. Frst, respondents whose households had more drvers than autos were dentfed. Second, the cost and tme varables for the bus-wth-auto-access alternatves were recalculated for these respondents under the assumpton that these respondents would be drven to the bus stop. Thus, new estmates were made ncorporatng these varables. Ths method provdes an upper lmt to the actual dfference between the estmate n Table 2 and the estmates based on exact knowledge of whether a person would drve to the bus stop and park or be drven. The actual number of respondents who would be drven s probably less than the number of households n whch there are fewer autos than drvers. (Respondents whose households have more autos than drvers have no ncentve to be drven and thus were always assumed to drve-and-park.) The results of ths recalculaton are presented n Table 4. The estmates are close to those presented n Table 2. The ndcaton, then, s that the choce of access mode can be ncluded n an MNL model wthout materally affectng the coeffcent estmates. Whether the IIA assumpton s volated s not known; nevertheless, the stablty of the coeffcent estmates s a good omen. We wll return to both the dagnostc tests for the volaton of the IIA property and the choce of access mode later n greater detal. 60

TABLE 4 Work-Trp Mode-Choce Model Mode 1: Auto Data: Work Travel Survey, East Bay Mode 2: Bus, Walk Access Model: Multnomal Logt, Ftted Mode 3: Bus, Auto Access by the Maxmum Lkelhood Method Explanatory Varables Estmated Coeffcents t-statstcs Cost dvded by post-tax wage, n cents/(cents per mnute) e/ -.0364 2.77 On-vehcle tme, n mnutes e/ -.0166 1.31 Walk tme, n mnutes a/ e/ -.0240 1.22 Transfer-wat tme, n mnutes a/ e/ -.0539 1.55 Number of transfers a/ e/ -.0957.326 Headway of frst bus, wth a celng of 8 mnutes, n mnutes a/ e/ An ndex of dstance to parkng at home c/ Famly ncome wth celng of $7000, n $ per year b/ Famly ncome mnus $7000 wth floor of $0 and celng of $3000, n $ per year b/ Famly ncome mnus $10,000 wth floor of $0 and celng of $5000, n $ per year b/ Length of resdence n communty, n years b/ -.235 2.54 -.314 0.910.000426 1.48.000790 1.66 -.000635 2.51.146 2.94 61

Table 4, contnued Explanatory Varables An ndex of populaton densty n neghborhood b/ Estmated Coeffcents -.736 t-statstcs 2.56 Dummy f respondent s over 44 years of age b/ -.845 1.31 Dummy f there s chld n household b/ -1.67 2.56 Number of persons n household who can drve c/ 1.11 2.68 Auto alternatve dummy b/ -5.74 2.36 Bus-wth-auto-access dummy d/ -2.93 3.66 Lkelhood rato ndex:.5976 Log lkelhood at zero: -176.9 Log lkelhood at convergence: - 71.18 Values of tme saved as a percent of wage: On-vehcle tme: 46 Walk tme: 66 Transfer-wat tme: 148 One transfer wth no watng or walkng s valued the same as 5.8 mnutes of on-vehcle tme. 62

Table 4, contnued All cost and tme varables are calculated round-trp. Dependent varable s alternatve choce (one for chosen alternatve, zero otherwse). Sample sze: 161. a/ The varable s zero for the auto alternatve, and takes the descrbed value for the other alternatves. b/ The varable s zero for the bus alternatves, and takes the descrbed value for the auto alternatve. c/ The varable s zero for the bus-wth-walk-access alternatve, and takes the descrbed value for the remanng alternatves. d/ The varable s one for the bus-wth-auto-access alternatve, and zero otherwse. e/ Sum of home-to-work and work-to-home. 63

An Examnaton of a Specfcaton Issue wth the Basc Model Estmaton of the basc model was performed on a sample that contaned persons who do not have an auto or do not have a lcense. How these persons should be treated depends upon how the choce to have an auto and lcense or not s made vs-à-vs the choce of work-trp alternatve. The decson could be made recursvely, as represented n Fgure 5. The person chooses to obtan an auto and lcense or not and then chooses a work-trp alternatve. On the other hand, the decsons could be made jontly, as represented n Fgure 6. 64

If the decsons are made jontly (and the four possble choces satsfy the assumpton of the ndependence of rrelevant alternatves property), then the representatve utlty functon can be estmated n ether of two ways: (l) a four-alternatve model could be estmated, wth alternatves as the nodes n Fgure 6 and attrbutes of auto and lcense ownershp ncluded as arguments of the representatve utlty functon, or (2) a model smlar to the basc model wth the three work-trp alternatves can be estmated usng the subsample of persons who have an auto or lcense. Nether of these approaches s feasble for the present study. Attrbutes of auto and lcense ownershp were not obtaned n the ntervews n the frst place, and, furthermore, the sample s too small to permt the estmaton of coeffcents. If the decsons of auto and lcense ownershp and work-trp alternatve are made recursvely, and the two decsons are ndependent, then the representatve utlty functon can be estmated by addng to the basc model a dummy varable that assumes the value of one n the auto and bus-wth-auto-access alternatves for all persons who have both an auto and a lcense, and zero otherwse. Table 5 presents estmates of such a model. The outstandng tem of the model of Table 5 s that the values of tme are about ffty percent hgher than those of the basc model, and the estmated coeffcent of "number of transfers," though nsgnfcant, has the wrong sgn. Whether or not specfcaton error s nvolved, however, s unclear. 65

TABLE 5 Work-Trp Mode-Choce Model Mode 1: Auto Data: Work Travel Survey, East Bay Mode 2: Bus, Walk Access Model: Multnomal Logt, Ftted Mode 3: Bus, Auto Access by the Maxmum Lkelhood Method Explanatory Varables Estmated Coeffcents t-statstcs Cost dvded by post-tax wage, n cents/(cents per mnute) e/ -.0308 2.29 On-vehcle tme, n mnutes e/ -.0226 1.61 Walk tme, n mnutes a/ e/ -.0231 1.16 Transfer-wat tme, n mnutes a/ e/ -.0689 1.74 Number of transfers a/ e/.0330 0.106 Headway of frst bus, wth a celng of 8 mnutes, n mnutes a/ e/ An ndex of dstance to parkng at home c/ Famly ncome wth celng of $7000, n $ per year b/ Famly ncome mnus $7000 wth floor of $0 and celng of $3000, n $ per year b/ Famly ncome mnus $10,000 wth floor of $0 and celng of $5000, n $ per year b/ -.249 2.55 -.222 0.461.000470 1.62.000529 1.11 -.000481 1.88 66

Table 5, contnued Explanatory Varables Length of resdence n communty, n years b/ Estmated Coeffcents.136 t-statstcs 2.70 An ndex of populaton densty n neghborhood b/ Dummy f respondent s over 44 years of age b/ Dummy f there s chld n household b/ Number of persons n household who can drve c/ -.716 2.44 -.663 0.990-1.55 2.26.272 0.578 Auto alternatve dummy b/ -6.92 2.72 Bus-wth-auto-access dummy d/ -3.56 4.05 Dummy f respondent owns an auto and a lcense c/ 2.59 3.14 Lkelhood rato ndex:.6315 Log lkelhood at zero: -176.9 Log lkelhood at convergence: - 65.17 Values of tme saved as a percent of wage: On-vehcle tme: 73 Walk tme: 75 Transfer-wat tme: 224 67

Table 5, contnued All cost and tme varables are calculated round-trp. Dependent varable s alternatve choce (one for chosen alternatve, zero otherwse). Sample sze: 161. a/ The varable s zero for the auto alternatve, and takes the descrbed value for the other alternatves. b/ The varable s zero for the bus alternatves, and takes the descrbed value for the auto alternatve. c/ The varable s zero for the bus-wth-walk-access alternatve, and takes the descrbed value for the remanng alternatves. d/ The varable s one for the bus-wth-auto-access alternatve, and zero otherwse. e/ Sum of home-to-work and work-to-home. 68

The ncluson of a dummy varable for ownng a car and lcense s only one way to deal wth alternatve avalablty. Another would have been to exclude auto and bus-wth-auto-access choces from those ndvduals who do not own a car or a lcense to drve. If ths course of acton was chosen then another alternatve nvolvng auto, such as shared-rde and drve-drop to the bus stop, should have been ncluded n the choce set. However, the WTS sample of 161 workers was not rch enough to permt the nvestgaton of such ssues. The fact that the dummy varable for ownng a car and a lcense obtans a sgnfcance coeffcent of substantal magntude ponts toward the mportance of ncludng the consderatons of auto ownershp and drver-lcense status n both the model buldng and travel forecastng work. Later n the sequel, models wll nclude such alternatves as shared-rde and specfc models wll be developed for auto ownershp, though no attempt wll be made to model the choce of obtanng or not obtanng a drver's lcense. Perhaps n ths day and age all elgble persons wll know how to drve. 69

Testng of Hypotheses n the MNL Model McFadden (1973) shows that the followng statstc s dstrbuted as ch-square wth degrees of freedom equal to the number of restrctons mpled by the null hypothess: (1) 2[L(ˆθ H ) L(ˆθ)] where L ( ( ) s the log lkelhood functon, ˆθ s the unconstraned maxmum lkelhood estmate of the vector of coeffcents, and ˆθ H s the maxmum lkelhood estmate of the vector coeffcents under the null hypothess. Usng ths statstc we obtan the followng results. (1) We accept at the fve percent level the hypothess that auto and bus on-vehcle tmes are weghted the same. The test-statstc s 0.70, whch s below the crtcal value (wth one restrcton) of 3.84. The pont estmates mply that bus on-vehcle tme s weghted 2.3 tmes as much as auto on-vehcle tme. (2) We accept at the fve percent level the hypothess that auto and bus costs are weghted the same. The test-statstc s 0.24, whch s below the crtcal value (wth one restrcton) of 3.84. (3) We accept at the fve percent level the hypothess that auto mleage costs, tolls, auto parkng costs, and bus costs are all weghted the same. The teststatstc s 6.5, whch s below the crtcal value (wth three restrctons) of 7.81. (4) We accept at the fve percent level the hypothess that walk, on-vehcle, and transfer-wat tmes are weghted the same. The test-statstc s 1.02, whch s below the crtcal value (wth two restrctons) of 5.99. 70

A Non-Logt Choce Model: The Maxmum Model In the logt model, the probablty that an ndvdual wll choose the frst of three alternatves s: P 1 e U 1 e U 1 e U 2 e U 3, where U 1, U 2, U 3 denote the representatve utltes of the three alternatves. The odds of choosng the frst alternatve over the second alternatve are: P 1 P 2 e U1 e U 2 e U 1 U 2. Thus, the odds of choosng the frst alternatve over the second are ndependent of the exstence of the thrd alternatve (.e., the magntude of U 3 ). For choce stuatons n whch ths ndependence does not exst, the logt model s not the approprate model. A plausble model that does not ental the assumpton of the ndependence of rrelevant alternatves s the maxmum model. Wth ths model a person s assumed to choose between auto and bus and then, f bus s chosen, to choose between walk- and auto-access. The auto-bus decson s based on attrbutes of the partcular bus alternatve (ether bus-wth-walk-access or bus-wth-autoaccess) that gves the hgher representatve utlty. Thus the probablty of choosng the auto alternatve s: P 1 e U 1 e U 1 e U 4, where U 4 = max [U 2,U 3 ] and 1 denotes auto, 2 denotes bus-wth walk-access, and 3 denotes bus-wth-auto-access. The maxmum model can be estmated n a two-step procedure. Frst, the sample s lmted to those respondents who chose one of the two bus alternatves. A logt analyss of the choce between walk access and auto access s performed on these respondents. Second, the logt functon estmated n step one s used to 71

determne whether waik access or auto access gves hgher representatve utlty for each person n the sample. A logt analyss s performed on the choce between auto and the bus alternatve whch gves hgher representatve utlty. The estmates obtaned n Step 2 of ths method are presented n Table 6. The estmated value of on-vehcle tme s somewhat hgher than n the basc model of Table 2, whle that of transfer-wat tme s smaller. What renders ths model mplausble, however, s the postve estmate for the value of walk tme. The t-statstc, on the other hand, s relatvely small. A tentatve concluson can be drawn, nevertheless, that t s preferable to nclude the choce of access mode wthn the MNL structure rather than to use a non-logt model to accomplsh the same purpose. Agan, a caveat must be added to note that the volatons of the IIA property have not been thoroughly nvestgated so far, and that some way must be found to deal wth the many alternatves that can exst even wthn a mode-choce model. An example of ths would be BART; four access modes (walk, bus, drve, and drven) and two egress modes (walk, bus) translates nto eght BART modes. It would be hghly presumptuous to beleve that no volaton of the IIA property would take place n usng such a choce set. 72

TABLE 6 Work-Trp Mode-Choce Maxmum Model Mode 1: Auto Data:.. Work Travel Survey, East Bay Mode 2: Bus, Walk Access Model: Multnomal Logt, Ftted by Mode 3: Bus, Auto Access the Maxmum Lkelhood Method Explanatory Varables Estmated Coeffcents t-statstcs Cost dvded by post-tax wage, n cents/(cents per mnute) e/ -.0319 2.42 On-vehcle tme, n mnutes e/ -.0233 1.65 Walk tme, n mnutes a/ e/.0330 1.23 Transfer-wat tme, n mnutes a/ e/ -.0395 1.23 Number of transfers a/ e/ -.0551.189 Headway of frst bus, wth a celng of 8 mnutes, n mnutes a/ e/ -.156 1.68 An ndex of dstance to parkng at home c/ -.243.682 Famly ncome wth celng of $7000, n $ per year b/ Famly ncome mnus $7000 wth floor of $0 and celng of $3000, n $ per year b/ Famly ncome mnus $10,000 wth floor of $0 and celng of $5000, n $ per year b/ Length of resdence n communty, n years b/.000360 1.36.000768 1.63 -.000540 2.15.138 2.80 73

Table 6, contnued Explanatory Varables An ndex of populaton densty n neghborhood b/ Estmated Coeffcents -.803 t-statstcs 2.86 Dummy f respondent s over 44 years of age b/ -.646 1.02 Dummy f there s chld n household b/ -1.26 2.02 Number of persons n household who can drve c/.448 1.34 Auto alternatve dummy b/ -2.65 1.24 Lkelhood rato ndex:.5170 Log lkelhood at zero: -111.6 Log lkelhood at convergence: - 53.90 Values of tme saved as a percent of wage: On-vehcle tme: 73 Transfer-wat tme: 124 One transfer wth no watng or walkng s valued the same as 2.4 mnutes of on-vehcle tme. 74

Table 6, contnued All cost and tme varables are calculated round-trp. Dependent varable s alternatve choce (one for chosen alternatve, zero otherwse). Sample sze: 161. a/ The varable s zero for the auto alternatve, and takes the descrbed value for the other alternatves. b/ The varable s zero for the bus alternatves, and takes the descrbed value for the auto alternatve. c/ The varable s zero for the bus-wth-walk-access alternatve, and takes the descrbed value for the remanng alternatves. d/ The varable s one for the bus-wth-auto-access alternatve and zero otherwse. e/ Sum of home-to-work and work-to-home. 75

The Tradeoff between Goods and Lesure In the above analyss, cost s dvded by wage, and the tme varables do not nteract wth wage. Elsewhere (McFadden, 1976), the opposte procedure was used: tmes were multpled by wage, and cost dd not nteract wth wage. Because an ndvdual s wage conveys nformaton about hs tradeoff between goods and lesure n the neoclasscal model of consumer behavor, the manner n whch wage s treated n the logt analyss has mplcatons for the form of the goods/lesure tradeoff. In order to determne the extent of these mplcatons and choose the most satsfactory way of treatng wage, a utlty maxmzaton model s employed n whch utlty s a functon of goods and lesure, and n whch attrbutes of work-trp alternatves enter nto the constrants on maxmzaton. The general procedure s as follows. Choose some specfc functonal form for the utlty functon, U(G,L), where G s goods and L s lesure. Assumng the prce ndex to be constant and normalzed to one, the followng denttes must hold: where V = unearned ncome (gven) ; w = wage rate (gven) ; G = V + w ( W - c, L = T - W - t, T = total amount of tme (gven) ; W = number of hours worked (a contnuous varable, non-negatve) ; c = t = n = cost of transportaton to and from work (a dscrete varable that can assume values c 1,...,c n ) ; tme of transportaton to and from work (a dscrete varable that can assume values t 1,...,t n ) ; number of work-trp alternatves. The ndvdual chooses the work-trp alternatve (and hence c and t ) and the number of hours worked so as to maxmze U, subject to the denttes. 76

Defne: G = V + w ( W - c ; L = T - W - t ; U = U(G, L ). Thus U s a functon only of the varable W. Because W s contnuous, U can be maxmzed n the normal way, settng (1) U W U 1 w U 2 0 ; or, as usual: (2) w U 2 U 1. Solve ths for W and call the soluton. Substtute nto U and call the resultng value U. W Alternatve s chosen by the ndvdual f and only f for all j = l,...,n ; j U. Let U* be the functon of c and t, whch assumes values W U >U j U, = l,...,n. Ths U* s the functon used n logt analyss. Thus, one can determne what functonal forms of the goods/lesure tradeoff (that s, U(G,L) ) produce partcular forms of U*. An equvalent approach, whch s less heurstc but easer computatonally, s the followng. Choose a specfc U(G,L) and derve the correspondng expendture functon: (3) E = E(U,w), where E s expendture(s) and prces are normalzed to one. The followng dentty holds for expendtures when utlty s maxmzed: 77

(4) E = V - c + w(t - t). Substtute (4) nto (3) and solve for U. The soluton s U*. The U* obtaned n ths manner s the same or a monotonc transformaton of the U* obtaned n the frst approach. Some specfc examples follow. Example A. Let U = α 1 log G + a 2 L. Wth ths functon the dervatve of the utlty-maxmzng G wth respect to ncome s zero: all extra ncome s absorbed n lesure. The resultng U* s: U α 1 log α 1 α 2 w α 2 T α 1 α 2 V w α 2 c w t. When comparng and ( U j), all terms that do not contan ether c or U U j t drop out. Thus, operatonally, U* s (5) U α 2 c w t. In ths case, cost s dvded by wage rather than tme beng multpled by wage. There s only one parameter snce c/w s n unts of tme, and the ndvdual values work tme and transportaton tme the same on the margn. The two varables can have dfferent coeffcents by specfyng a model analogous to that n Example D below. Example B. Let U = α 1 G + α 2 log L. Wth ths functon the dervatve of the utlty maxmzng G wth respect to ncome s zero and all extra ncome s absorbed n goods. The resultng U* s: U α 1 V Tw tw α 2 α 1 c α 2 log α 2 α 1 w. Operatonally, 78

(6) U* = -α 1 (tw + c). In ths case, tme s multpled by wage rather than cost beng dvded by wage. Dfferent parameters can be gven to the two terms n a manner analogous to Example D below. Example C. Let U be a Cobb-Douglas utlty functon: U = AG 1-β L β, 0 < β < 1. The expendture functon (wth prces normalzed to one) s: E = Uk -1 w β, where k s a constant. Recall that Thus, E = V - c + w (T - t). And, Uk -1 w β = V - c + w(t - t). U* = k(w -β V - w -β c + w 1-β T - w 1-β t). Operatonally, (7) U* = -k(w -β c + w 1-β t). 79

When β approaches 0, (7) becomes (6); and when β approaches 1, (7) becomes (5). For values of β between zero and one, the dervatves of the utlty-maxmzng L and G wth respect to ncome are greater than zero. The choce of β s an emprcal ssue. Example D. The analyss n Example C can be extended so that the terms n (7) have dfferent coeffcents and components of tme and cost enter each wth dfferent coeffcents. Let U be the same as n Example C. The defntons of goods and lesure are re-specfed: (8) effectve lesure L θ 0 T W M effectve goods G γ 0 V ww N θ j t j ; γ j c j ; where t j c j M N θ j θ 0 γ j γ 0 s the j-th component of travel tme (say, on-vehcle tme); s the j-th component of travel cost; s the number of tme components; s the number of cost components; s the psychometrc weght attached to a mnute n travel component j, n work tme unts; s the psychometrc weght attached to the total tme budget, n work tme unts, when mode s used; s the psychometrc weght attached to a travel cost component, n wage ncome unts; s the psychometrc weght attached to non-wage ncome, n wage ncome unts, when mode s used. The θ j and γ j reflect the relatve onerousness or burden of dfferent tme or expendture actvtes (assocated, for example, wth the exerton, fatgue, or bother nvolved). The parameters θ 0 and γ 0 reflect the value of added unts of 80

tme or ncome (agan evaluated n workng unts). They may dffer wth mode f choce of mode tself affects the types of consumpton and lesure actvtes avalable to the consumer. "Effectve" ncome s: E (γ 0 V wθ 0 T) N γ j c j M θ j wt j. Substtutng ths expresson nto the expendture functon for Example C and solvng for U obtans: U k γ 0 Vw β θ 0 w 1β T N γ j c j w β M θ j t j w 1β. Ths formula was used assumng the γ 0 and θ 0 constant n, so that operatonally: (9) U k N γ j c j w β k M θ j t j w 1β. Example E. The analyss n Example D can be extended so that U* depends on socoeconomc varables and alternatve dummy varables. Redefne "effectve" goods and lesure as: effectve lesure L θ 0 T W M θ j t j P θ jm u j ; effectve goods G γ 0 V ww N γ j c j Q γ jn v j ; where u j v j s the j-th unmeasured tme component of travel by mode ; s the unmeasured consumpton of "good" j n travelng by mode. 81

An example of a u j s the length of tme that one would usually arrve early to work so as not to be late for work f one's travel mode were delayed. If ths varable were measured, t would assume a dfferent value for each alternatve, because the probablty of beng delayed a certan length of tme vares across modes. An example of a v j s the prvacy that one "consumes" n a partcular travel mode. Because the varables are unmeasured, they are approxmated by other, measured varables, such as socoeconomc varables: (10) where x j y j δ j ε P θ jm u j Q γ jn v j R S δ j x j C, η j y j µ, s the j-th measured varable used to approxmate the unmeasured tme components of travel by mode ; s the j-th measured varable used to approxmate unmeasured consumpton from travel by mode ; and η j are parameters; and and µ are errors that allow equatons (10) to hold exactly rather than approxmately. Substtutng (10) nto the defntons of "effectve" lesure and goods and solvng for U* as n Example D obtans: (11) U k N γ j c j w β k M k S η j y j w β k R θ j t j w 1β, δ j x j w 1β k(c µ). 82

For logt analyss, t s assumed that -k(c + µ) s dstrbuted Webull n the populaton and s ndependent across alternatves. To estmate β and the coeffcents, logt estmaton can be performed for varous values of β between zero and one, wth utlty defned as n (11). The value of β that results n the largest value of the lkelhood functon s chosen as the estmate of β. Unfortunately, equaton (11) does not help to dentfy whch socoeconomc varables are to be consdered y s and whch ones x s. The choce of whch way to treat a socoeconomc varable affects the lkelhood functon and hence the estmate of β. Several dvsons of the socoeconomc varables were consdered. When all the socoeconomc varables and alternatve dummy varables were treated as y s, the estmated β was approxmately 0.7. For ths model, wth β equal to 0.7, the estmated values of tme were generally about half as large as those estmated for the basc model. For other dvsons of the socoeconomc varables, the estmated β was generally between 0.7 and 1.0, wth the estmated values of tme rsng wth the estmated β. Thus, the results here suggest that dvdng travel cost by wage s better than multplyng travel tme by wage, and that the tme varables do not, n fact, need to be nteracted wth wage. In some ways ths result s very convenent. Frst, the specfcaton of the model s smple and the practtoner has one less varable to worry about. Second, the assumpton of neoclasscal economc theory that ndvduals are tradng off goods and lesure tme, wth the number of hours worked beng a contnuous varable, s not a fully satsfactory bass for model specfcaton, for a number of reasons. It s unlkely that most people have an unrestrcted choce of how many hours they want to work. The dvson of tme nto workng tme and lesure tme s somewhat arbtrary; there are many other consderatons whch, n real lfe, enter the dvson of tme (pad and unpad) nto varous actvtes. The emprcal conclusons that β = 1 s consstent wth a neoclasscal theory of behavor. On the other hand, t mples a smple model structure whch should appeal to pragmatsts who do not accept or consder rrelevant the neoclasscal formulaton of the choce problem. In concluson, the ssues warrantng closer examnaton wth the full sample can be dentfed as beng the avalablty of alternatves and the choce set, especally wth regard to the handlng of access modes; the ndependence from rrelevant alternatves property of the MNL model; and, naturally, further testng of the specfcaton tself. 83