Incorporating Coplex Substitution Patterns and Variance Scaling in Long Distance Travel Choice Behavior Frank S. Koppelan Professor of Civil Engineering and Transportation Northwestern University Evanston, IL 60208-3109, USA Phone:, (847) 491-8794 Fax:, (847) 491-4011 E-ail: f-koppelan@northwestern.edu and Vaneet Sethi ZS Associates 180 North Stetson Avenue Chicago, IL 60601, USA Phone:, (312) 233-4873 Fax:, (312) 233-4801 E-ail: Vaneet.Sethi@ZSAssociates.Co
Vaneet Sethi and Frank S. Koppelan i ABSTRACT The assuption of independently and identically distributed (IID) error ters in the Multinoial Logit, (MNL) odel leads to its infaous IIA property. Relaxation of the IID assuption has been undertaken along a nuber of isolated diensions leading to the developent of a rich set of discrete choice odels, that are ore flexible than the MNL odel. In soe cases, these ore general odels lose the atheatically convenient closed-for structure of the MNL. In this paper, we cobine the ost flexible isolated closed-for extensions of the MNL and Nested Logit (NL) odels in an integrated odel structure to yield a behaviorally rich, yet coputationally tractable choice odel. Specifically, we cobine the Generalized Nested Logit odel that allows for non-independent errors, the Heteroscedastic MNL which allows non-constant errors across observations, and the Covariance Heterogenous NL odel which allows for non-constant correlation structure across observations. The resulting odel, called the Heterogeous GNL odel extends our ability to represent the coplex behavioral processes involved in choice decision-aking. The value and need for the additional odeling coplexity of the HGNL odel is tested in the epirical context of ode and rail service class choice behavior for long distance intercity travel. An increental odeling approach is adopted, i.e., we start fro the siple MNL odel and sequentially relax soe of its restrictive assuptions to estiate progressively ore flexible odel structures. The statistical fit and behavioral appeal of the estiated odels iprove substantially with each additional relaxation, strongly supporting the concept of integrating isolated generalizations.
2 1. INTRODUCTION Multinoial Logit (MNL) and Nested Logit (NL) odels have traditionally been used to odel the choice aong alternative odes/classes in the intercity travel deand odeling literature, (Stopher and Prashker, 1976; Grayson, 1981; Forinash and Koppelan, 1993). These odel structures ipose rigid inter-alternative substitution patterns which liits their ability to represent the coplexity of the ode/class choice decision process of intercity travelers. Advances in discrete choice odeling have led to the developent of a nuber of generalized odel structures that relax soe of the restrictive assuptions iplied by the MNL and NL odels. These odels are ore flexible in their ability to represent coplex choice behavior and in ost cases can incorporate a wider range of inter-alternative substitution patterns. The priary objective of this research is to identify the ost flexible closed-for generalizations of the MNL/NL odels that are relevant to the epirical context of long distance intercity travel and cobine the in an integrated odeling fraework. Specifically, the integrated odel cobines: 1) Heteroscedastic MNL odel, (Swait and Adaowicz, 2001) which allows for non-unifor variances across observations, 2) Generalized Nested Logit odel, (Wen and Koppelan, 2000; Papola, 2004) which allows coplete flexibility in representing inter-alternative substitution patterns, and 3) Covariance Heterogenous Nested Logit odel, (Bhat, 1997) which enables heterogeneity in substitution patterns aongst alternatives. The resulting odel structure is the ost coprehensive relaxation of the MNL that retains its atheatically convenient closed-for. This fraework is used to conduct an epirical analysis of ode and rail class choice behavior of long distance, greater than 250 iles, leisure travelers a soewhat neglected aspect of intercity travel arkets. Hensher s (1998) study of the Sydney-Canberra corridor in Australia and soe recent work for Swedish rail (Algers and Beser, 2000) are the only published research studies that address ode and rail service class choice in long distance travel.
2. EVOLUTION OF RANDOM UTILITY CHOICE MODELS 3 The basic assuption ebodied in the rando utility approach to choice odeling is that decision akers are utility axiizers, i.e., given a set of alternatives the decision aker will choose the alternative that axiizes his/her utility, U it. The utility of an alternative i for decision aker t is assued to consist of a deterinistic portion, (that can be estiated) tered as the systeatic coponent, V it, and a rando portion called the error ter, it, as follows: U V (1) it, it, it, The systeatic coponent of the utility includes characteristics of the decision aker and the attributes of the alternative. The error ter is a coposite of errors fro any sources such as iperfect inforation and easureent errors, oission of iportant variables in the systeatic coponent of the utility, etc. and is assued to be a rando variable. Different assuptions about the distribution of these error coponents result in different choice odels. The ost restrictive set of assuptions leads to the siplest and the ost widely used discrete choice odel, the Multinoial Logit (MNL) odel, (Doencich and McFadden, 1975; McFadden, 1975). The specific assuptions regarding the distribution of the error ters in the utility specification that lead to the developent of the MNL odel are: 1. The error coponents ( I,t ) have an extree-value type I, or Gubel, distribution; 2. The error coponents are identically and independently distributed, IID, across alternatives; and 3. The error coponents are identically and independently distributed, IID, across cases. These assuptions, taken together, lead to the following atheatical structure that gives the choice probabilities of the alternatives as a function of the systeatic portion of the utility of all the alternatives, assuing the systeatic utility to be linear-in-paraeters V X : ' it, it,
' exp( X it, ) it, ' X j, t jc P n exp( ) 4 (2) where, P i,t is the probability of the decision-aker t choosing alternative i, is the scale paraeter of the error variances, typically noralized to 1, X i,t are the attributes of alternative i and the characteristics of the individual t, ' is the paraeter vector associated with the vector X i,,t In contrast to the MNL, the Multinoial Probit (MNP) is a uch less restrictive choice odel in ters of the assuptions ade with regard to the distribution of the rando error ters. The MNP assues the error ters to have a ultivariate noral distribution; thus, the error ters are neither restricted to be identical nor independent, iplying substantial relaxation of the IID assuption of the MNL, (Daganzo, 1979). The only reaining restriction of the MNP is that all error and paraeter distributions are noral. However, the choice function of the MNP cannot be written in closed for and, thus, ust be evaluated nuerically. A soewhat less known odel, the Negative Exponential Distribution, NED, odel, (Daganzo, 1979), assues that the error ters are independently distributed with an exponential distribution but are not restricted to be identical. The odel retains a closed-for solution, but has seen liited application as it restricts the range of the perceived utility of alternatives to an upper bound. The MNL odel is ost widely used for its ease of interpretation and estiation. However, it suffers fro the Independence of Irrelevant Alternatives (IIA) property which fails to accoodate different degrees of cross-alternative substitution, rendering the MNL inappropriate for choice situations where soe pairs of alternatives are ore copetitive or substitutable than others. The need to overcoe the IIA property of the MNL has been the priary otivation behind the developent of a variety of generalized logit discrete choice odel structures over the last two decades. Figure 1 presents a conceptual overview of the different rando utility based discrete choice odels, all but two of which, MNP and NED, have evolved as generalizations of the MNL.
Figure 1 illustrates the three priary diensions along which the MNL can be generalized, and 5 identifies the specific odels that have evolved fro each of these generalizations. Bhat and Koppelan (1999) provide a conceptual review of Type II and Type III generalizations and briefly describe ost of the specific odels that have evolved fro these two generalizations. This structure is adapted and extended here to include Type I generalization and soe ore recently developed choice odels. [Place Figure 1 about here] 3. MODEL DEVELOPMENT AND ESTIMATION APPROACH Substantial progress has been ade in discrete choice odeling, priarily through the relaxation of one or ore diensions of the IID assuption of the MNL, resulting in ore flexible odel structures. In general, the additional flexibility of these advanced odels coes at the cost of increased coputational burden, and in soe cases losing the atheatically aenable closed-for structure. This research cobines the ost flexible closed-for generalizations of the MNL/NL odels in an integrated fraework yielding a behaviorally richer, yet coputationally tractable, choice odel. The integrated odel structure cobines the priary eleents of the Generalized Nested Logit (Wen and Koppelan, 2000), Covariance Heterogeneous Nested Logit (Bhat, 1997) and the Heteroscedastic Multinoial Logit odel (Swait and Adaowicz, 2001). The GNL odel 1 and the siilar GenMNL odel, Swait (2000) are the ost recent ebers of the faily of Generalized Extree Value odels. Wen and Koppelan (2001) show that all GEV odels proposed to date, can be derived as restricted versions of the GNL odel, with the exception of three or higher level nested odels, which can be closely approxiated by the GNL. The GNL odel provides a higher degree of flexibility in the estiation of substitution or cross-elasticity between pairs of 1 The GNL odel has been proposed by others (Papola, 2004) but was referred to as the Cross Nested Logit (CNL) odel. However, since (Vovsha, 1997) proposed a different CNL odel, we retain the GNL designation to avoid confusion. Further, it is interesting that the Cross-Correlated Logit odel proposed by Willias and Ortuzar, (1982) was an approxiate precursor of these odels. Willias and Ortuzar attepted to accoplish the objectives of the CNL/GNL but proposed a odel for that could not be estiated directly at that tie.
6 alternatives than previously developed GEV odels. It does so by allocating alternatives proportionally to ultiple nests. The choice probabilities for the GNL odel are given by: 1 Vi e i jn / 1 V j j P P P i i jn e j e jn j e V j V 1 j 1 (3) where V i is the observable portion of the utility for alternative i, N is the set of all alternatives included in nest, is the siilarity paraeter for nest, 0 < 1, i is the allocation paraeter that characterizes the portion of alternative i assigned to nest, and ust satisfy the conditions that i 1, i and i i, where is assigned an arbitrarily sall, positive value. The only prior epirical study of the GNL odel (Wen and Koppelan, 2001) shows that the GNL odel substantially outperfored all tested GEV odels 2 in ters of statistical fit and behavioral interpretation. Bhat (1997) proposed a odification to the nested logit odel that allows heterogeneity across individuals in the covariance of nested alternatives, tered as Covariance Heterogenous Nested Logit (COVNL) odel. He argues that the restriction of equal correlation in the NL odel, which iplies equal substitution between pairs of nested alternatives across individuals, and/or choice contexts ay be untenable in any choice contexts. For exaple, in an intercity ode choice situation, a stronger substitution effect can be expected between high-speed rail and air for longer distance trips, relative to short trips. Covariance heterogeneity is incorporated by paraeterizing the logsu, dis-siilarity, paraeter(s) in the NL odel as a function of individual and trip related characteristics as follows: 2 The PD and OGEV odels were not tested as they are structured to represent different and highly specific relationships aong alternatives.
t, F( ' Zt) (4) where, is the siilarity paraeter for nest for observation t, t, Z t is a vector of individual and trip related characteristics for observation t,, are paraeters to be estiated, and F is a transforation function that ensures t, is bounded between 0 and 1 7 If 0 in the above equation, covariance heterogeneity is absent and the COVNL reduces to a NL odel. Any proper continuous cuulative distribution function will ensure that t, is bounded by zero and one. The resultant odel is ore coplex than the siple NL odel but retains a closed for structure. In its only epirical application (Bhat, 1997), the COVNL odel was found to be statistically and behaviorally superior to the corresponding NL and MNL odels 3. Swait and Adaowicz (2001) forulated a Heteroscedastic Multinoial Logit (HMNL) odel that allows the variance of the rando coponent of the utilities to vary across individuals/observations. The odel is otivated by the hypothesis that individuals with the sae systeatic utility for an alternative ay have different abilities to discriinate between the utilities of different alternatives. These differences can result fro task coplexity and/or choice environent and ay be particularly iportant when using stated preference data since the coplexity of experients ay influence the choice process itself. Further, different data collection procedures can influence choice variability differentially. These potential differences in error variances are incorporated in the HMNL as a paraeterization of the variance, or scale, of the rando error ter of the utility function. This iplies that the error ters are independent and identically distributed across alternatives, but not identically distributed across observations. The choice probabilities for the HMNL odel are given as follows: 3 The iproveent in goodness of fit is due to both the introduction of additional variables and the incorporation of the COVNL structure.
P it, e st, it, s, tv j, t (5) jc n e V 8 where, s, t is the scale paraeter for data source, s, observation t and V i,t is the systeatic coponent of the utility for alternative i, observation t. The heterogeneous logit odel is based on paraeterization of the scale of the ultinoial logit odel as follows: E( ' Z ) (6) s, t s, t where, s, t is the scale paraeter for observation s of individual t, Z s,t is a vector of individual and trip related characteristics for observation t,, are paraeters to be estiated and E is a transforation function that ensures s, t is bounded between 0 and 1. The epirical findings of Swait and Adaowicz (2001) support the idea that the choice context and level of choice coplexity influences the variance of the rando error ter. We integrate the above three generalizations of the MNL/NL odels into a single odel, tered the Heterogeneous Generalized Nested Logit (HGNL) odel. The HGNL odel can be viewed as an extension of the GNL odel the extensions being the paraeterization of the siilarity coefficients, s, and the variance scale,. The choice probability equations of this odel are
P P P ist,, ist,, / st,, 1 s, tvi, t s, t, i, e 1 V j, e jn jn jn j, s, t j, t s, t, e j, V 1 s, t j, t s, t, e V 1 s, t j, t s, t, s, t, s, t, where V i,t is the systeatic coponent of the utility for alternative i, observation t N is the set of all alternatives included in nest, s,, t is the siilarity paraeter for nest for observation t that is a function of individual and trip related characteristics and satisfies the condition, 0 st,, 1, This siilarity paraeter can be paraeterized in ters of attributes of the individual and/or the observation for each nest. i, is the allocation paraeter which characterizes the portion of alternative i assigned to nest and ust satisfy the conditions i) 1, ii) i, i, i (7) 9 s, t is the scale paraeter for individuals and observation t, and can be paraeterized in ters of attributes of the individual and/or observation. The GEV generating function for the above odel is given by G GNL 1 s,, t V j, (8) jn st, jt, st,, e Integration of the generalizations described above is based on direct incorporation of the paraeterized functions, equations 4 and 6, into 7 as follows:
10 G HGNL 1 F( ' Zs,, t) E( ' Z ) V F( ' Z ) j, e (9) jn st, jt, st,, This structure is closed for, requires no integration; however, the estiation proble is coplex as the cobined odel includes significantly ore paraeters than the underlying GNL, utility, siilarity, allocation and scale paraeters and is subject to a larger nuber of constraints. A full inforation axiu likelihood ethodology is applied to siultaneously estiate all these paraeters, taking account of appropriate restrictions as iplied in the odel forulation and/or to ensure consistency with rando utility axiization principles. The axiization of the constrained log-likelihood function can be atheatically represented as follows: Maxiize LL I ist,, log Pist,, t s i1 (10) s.t. 0 1, t, t,, t i, st, i, 1, it,, 0 1, i, 0, st, where, ist,, is 1 if individual t fro data source s chooses alternative i, and 0 otherwise, and P i,s,t is the estiated probability that individual t fro data source s chooses alternative i The odel estiation is ipleented in the GAUSS statistical software package using its constrained axiu likelihood odule (Aptech, 1995). 4. THE DATA SET This above odeling fraework is used to conduct an epirical analysis of ode and rail service class choice of intercity travelers. The data for this study is drawn fro Stated Preference, SP, surveys
of both existing rail users and travelers using other intercity travel odes; air, autoobile, and bus; 11 referred to as non-users. Rail users were recruited by a self-adinistered survey conducted in Fall 1998 on-board long distance intercity trains serving long distance travel arkets, ore than 250 iles. The non-user saple, selected to provide a coprehensive geographical coverage across the U.S., was recruited fro a rando saple of households in which at least one eber had ade a long distance intercity trip in the recent past. The stated choice experients, developed for the user saple, required each respondent to choose fro a set of three train service classes, each with and without auto train option, an alternative ode of travel (bus, auto, air), and the option to not travel at all a total of eight alternatives. The three train alternatives included existing service classes, coach and sleeper, and one of two proposed service classes, preiu coach or econoy sleeper. The user survey saple consisted of over 1,000 respondents, each of who was presented with up to 8 choice scenarios. The non-user survey required each respondent to choose fro two train classes, each with and without auto train option, their existing ode of travel, and the option to not travel at all a total of six alternatives. The two train alternatives presented to a respondent consisted of one of the following three cobinations of coach and sleeper classes: Existing Coach and Existing Sleeper Existing Coach and Econoy Sleeper Preiu Coach and Existing Sleeper The non-user survey saple consisted of over 400 respondents, each of who were presented with up to 9 choice scenarios. 5. EMPIRICAL ANALYSIS AND RESULTS The odel developent and interpretation process is illustrated by an increental approach, as shown in Figure 2 and described below. The first step in odel developent is to estiate source specific reference MNL odels, followed by pooled estiation of joint user/non-user odels, accounting for variance differences between data sources, based on first and second choice preference data. The first stage
12 extension allows for different error variances within data sources. The second stage extension involves relaxation of the assuption of error independence across alternatives through the estiation of Nested Logit and the Generalized Nested Logit odels. Finally, the third stage extension relaxes the constant covariance assuption with the expectation that the substitution pattern aongst the alternatives ay vary across observations. It is iportant to note that even though the odel developent was perfored in stages to assist in evaluation of increental benefits and insights fro each specific odel extension, the paraeters at each stage are estiated using Full Inforation Maxiu Likelihood, FIML, ethod. The following sub-sections discuss the epirical results for the reference MNL odel(s) and for odels at each subsequent stage of odel extension. [Insert Figure 2 around here] 5.1 Reference MNL Models The paraeter estiates, t-statistics (in parenthesis) for the reference MNL odel for users, non-users and the pooled data set are presented in Table 1 4. The variables in the utility specification of the user odel (Model 1) are grouped into the following categories and the interpretation of the paraeter estiates for each group of variables is provided below. Schedule convenience represented by bad departure hour and bad arrival hour duy variables are specific to all rail and the not travel alternatives. The variables take a value of one if the train departs/arrives between idnight and 4:00 A.M., and zero otherwise. The paraeter results confir the undesirability of such a schedule for rail travel Overnight rail trip duy variable specific to sleeper alternatives has a positive paraeter associated with it, iplying an increased preference for sleepers for overnight train travel. Quality of service perception variables include an unreasonable delay duy variable indicating that the delay experienced by rail user was unreasonable in the view of the rail user and a quality rating
score on a scale fro 1 to 10. The paraeters confir that avoidance of excessive delays and 13 perceived quality of rail service have a positive influence on rail choice. Group size duy variable paraeters, for group size 1 and 2 with 3+ as the base specific to sleeper and autoobile suggest, as expected, that the larger travel parties are ore likely to travel by sleeper and auto relative to other alternatives. Household incoe variables specific to sleeper, air and autoobile alternatives indicate that increasing incoe ost favors air, followed by sleeper and autoobile, relative to the other three rail and not travel alternatives. Trip distance variables suggest as expected that increasing trip distance favors sleeper and air, and discourages travel by autoobile. Inertia variables represent the resistance to change fro their chosen alternative applied only to the stated preference alternatives chosen in the previously revealed preference response obtain positive signs, as expected. The specification in the non-user odel (Model 2) differs fro the preferred user specification, Model 1) in the exclusion of variables that were not available in the non-user SP choice context, i.e., schedule convenience, overnight duy, and delay duy; these variables were observed only for respondents who chose rail. In addition, the inertia variables for non-user alternatives, air and autoobile, were highly co-linear with their corresponding bias constant, thus discouraging their inclusion in the odel. Finally, the relative value of scaled rail transportation cost and low/high incoe scaled upgrade cost are restricted to those derived fro the user only odel. [Place Table 1 around here] The hypothesis of equal taste paraeters but unequal variances is tested by specifying a coon taste vector for the two data sources but distinct error variances, Model 3. The null hypothesis can be forally stated as follows: 4 The specification is selected fro aong a nuber of specifications tested for the user group and retained for the
H : User NonUser 0 User NonUser 14 Unique alternative specific constants are specified for users and non-users due to differences in variables User included in the data sets. For identification, the scale for user data source is set to one, i.e., 1.0. The relative scale for the non-user data is 2.104, which is significantly different fro 1.0 at a 0.001 level. The chi-squared statistic for the hypothesis of taste equality and data source variance inequality, which 2 copares Model 3 to Models 1 and 2 is 2 11023.5 3209.5 ( 14248.6) 31.2 which is 2 slightly less than the critical with 12 degrees of freedo at the 0.001 level of significance, 31.4. Thus, the null hypothesis cannot be rejected at this level of significance, although, it would be rejected at a lower level of confidence. This result is encouraging since previous studies suggest that the likelihood ratio test is very strict for cobining different data sources (Bradley and Kroes, 1992). 5.2 Accoodating Variance Heteroscedasticity within Data Sources Recent work in data cobination theory suggests the iportance of accoodating differential variances within data sources in addition to odeling the average variance difference between data sources (Swait and Adaowicz, 2001; Hensher et. al., 1999). In our epirical context, the possibility of non-constant error variances across observations ay arise due to any of the following reasons: 1. Respondents ay be ore thoughtful in evaluating the first choice alternative as copared to the second choice alternative. Therefore, lower error variances are expected for first choice observations relative to observations for second choice. 2. Respondent fatigue, learning) in SP experient replications which will be reflected in higher, lower) error variances with increasing experient nuber, and 3. Differences in perception variance for different trip lengths and/or socio-econoic characteristics of the decision aker. Table 2 presents the estiation results of the pooled odel with distinct error variance between users and non-users and three increental odels to test each of the above hypotheses regarding non-user and pooled odels except as needed due to data differences between users and non-users.
differential error variances across observations. Model 4 includes a scale paraeter for second choice 15 observations with an estiated value of 0.66, iplying higher error variance, or lower precision, for these observations relative to first choice observations for which the scale is fixed to 1.0. The odel rejects the reference odel at a significance level of 0.001, confiring our a priori expectation that respondents are less careful in aking decisions about their second choice than about their first choice. This result is consistent with the findings of Bradley and Daly (1994) in which they observed declining precision in paraeter estiates with increasing rank of the SP experient. [Place Table 2 around here] Model 5 accoodates ore extensive variance scaling by including experient specific scale paraeters for users and non-users to allow for respondent fatigue and/or learning effects in aking choices in the different SP scenarios. The results, illustrated in Figure 3, show a gradual decline in the agnitude of the user scale paraeter with increasing experient nuber copared to sall and nonsignificant differences for non-user. This suggests that the user respondents were getting fatigued, whereas the net effect on non-users, who presuably knew less about existing rail alternatives and ay have been learning with each successive experient, is neutral suggesting their eliination fro the odel. Differences in the coplexity of user and non-user designs ay have contributed to these differences; the user design was ore coplex than the non-user design, both in ters of the nuber of choice alternatives and the nuber of design variables. [Insert Figure 3 around here] Model 6 excludes experient specific scales for non-users but retains the experient specific user scales and accoodates differential variances for different trip length ranges to allow for variation in perceptions of the attributes describing the utility function along this diension (Gliebe et. al., 1999).
16 The results show that scale paraeters decline, (i.e., error variances increase) with increasing trip length, iplying that travelers are less sensitive to attributes describing the utility function for longer trips relative to shorter trips. This odel can be statistically copared to Models 3 and 4 using the likelihood ratio test and to Model 5 using the non-nested hypothesis test; it rejects each of these odels at a significance level of 0.001. Thus, allowing for heteroscedasticity across observations iproves the odel fit and interpretation. 5.3 Flexible Error Correlation Effective representation of choices in odels of intercity ode and service class choice can be iproved by allowing error correlation between pairs of alternatives beyond that provided by Nested Logit (NL) odels, such as the Generalized Nested Logit (GNL) odel. However, experience suggests that it is useful to do soe initial exploration with NL odel structures for the following two reasons. First, the estiation results of alternative NL odels can provide valuable insight into the error correlation structure, or substitution patterns, aongst the choice alternatives. These insights can then be used to develop hypotheses regarding siilarity relationships aong alternatives to be tested in a GNL fraework. Second, the NL odel(s) provide a benchark to evaluate the epirical superiority of ore advanced odel structures, an iportant consideration given the added coplexity of these odels both in ters of estiation and interpretation. 5.3.1. Nested Logit Models The large nuber of possible nesting structures discourages estiation of all possible nesting structures for even oderate nubers of alternatives. To liit the nuber of structures, we considered only 2-level and 3-level NL odels. The best of these odel structures, illustrated in Figure 4, were statistically superior to all other NL odels tested, and yielded logsu paraeters bounded by zero and one. The extensive nature of the search provides a high level of confidence in the statistical superiority of these odels over un-tested structures.
17 [Insert Figure 4 around here] Model 7 has a 2-level nesting structure with all rail alternatives grouped in a coon nest, iplying a higher substitution aongst the rail alternatives, relative to non-rail odes. In coparison, Model 8 provides additional differentiation in substitution patterns aongst rail alternatives by including an additional level of nesting. The upper level nest groups all rail alternatives together to reflect siilarities in the unobserved attributes aongst these alternatives; the lower level includes nests for coach-preiu coach and sleeper-econoy sleeper. Both odels include an auto-not travel nest suggesting that autoobile travelers are likely to forgo their leisure trip when auto costs increase or auto is not available. The estiation results for these two NL odels are reported along with the preferred MNL odel in Table 3. [Place Table 3 around here] Coparing the estiation results of the two NL odels, the 3-level odel structure rejects the 2- level odel at a significance level greater than 0.001. In addition to strong goodness-of-fit results, the 3- level nesting structure has a ore intuitive behavioral interpretation. It iplies an interediate level of error correlation/siilarity aongst all rail alternatives, and a higher level of error correlation between coach and preiu coach and between sleeper and econoy sleeper. The estiated logsu paraeter for the coach-preiu coach nest is saller than the sleeper-econoy sleeper nest, iplying that the two coach classes are ore substitutable with each other than the two sleeper classes. Given that cofort is the ain discriinator aongst the unobserved attributes of these two coach classes, it sees reasonable that for long distance trips the two coach classes are perceived to be ore siilar to each other than the two sleeper classes.
18 The above discussion clearly shows that the 3-level NL is statistically and behaviorally superior to the other odels presented thus far. However, this NL odel still iposes a nuber of restrictions on the siilarity relationships between pairs of alternatives that ay not necessarily hold in this epirical setting. First, it iposes equal substitution between coach and the two sleeper alternatives; arguably, coach alternatives are likely to have a higher degree of siilarity with the econoy sleeper than the regular sleeper. Second, it does not allow any siilarity in unobserved attributes of rail and non-rail alternatives a restriction that is contrary to the iplied relationships by the statistically significant coach-air nest in alternative NL odels (not reported here). Therefore, despite its statistical superiority over the 2-level NL structure, Model 8 fails to accoodate the coplex substitution patterns that are iplied by soe of the estiation results. 5.3.2 Generalized Nested Logit Model The liitations of the NL odel are rooted in the requireent that each alternative can be included in one and only one nest. The GNL odel addresses this proble through the use of paraeters,, which allocate proportions of alternatives to different nests. We illustrate the usefulness of this flexibility by estiating a GNL odel with four siilarity nests based on the insights gained fro results of alternative NL odels. These are a Rail nest, a Coach-Preiu Coach nest, an Auto-Not Travel nest, and a Coach- Air nest. Why not include a sleeper-es nest? The siilarity nests include estiated portions of alternatives grouped together to represent different siilarity relationships. In addition a dis-siilarity nest is specified which includes portions of all the alternatives that are not allocated to any of the siilarity nest(s). The logsu paraeter for the dissiilarity nest is restricted to 1.0 and in essence provides an MNL type relationship for the portions of alternatives included in this nest. The tree structure for this GNL odel (Model 9) is shown in Figure 5.
Table 4 copares the estiation results of this GNL odel with the best NL, Model 8); since the 19 paraeters in the systeatic coponent of the utility are very siilar for these two odels, only the logsu, scale, and allocation paraeters are reported in the interest of brevity. All logsu and allocation paraeters in the GNL odel, Model 9) are significantly different fro one and zero, respectively. The ost iportant difference between the GNL and the 3-level NL odel structure in this case is its flexibility to include rail coach alternative in two nests, rail and coach-air, and to allow allocation of parts of alternatives to the dis-siilarity nest. This flexibility is reflected in its substantially better loglikelihood relative the 3-level NL, rejecting it, and all preceding odels at a high level of significance. These epirical results support our earlier conjecture that correlation in unobserved coponents of utility exists along a nuber of diensions, which ay lead to acceptance of a nuber of different nesting structures. By allowing alternatives to appear in ultiple nests, the GNL odel provides the flexibility to accoodate differential siilarities between pairs of alternatives within the closed-for GEV fraework. [Place Table 4 around here] 5.4 Accoodating Covariance Heterogeneity The GNL odel in the preceding section allows for coplex patterns of substitution aongst alternatives, but assues the degree of substitution to be constant across different individuals. This assuption of covariance hoogeneity ay be too restrictive in any epirical situations. We hypothesize that trip distance ay play an iportant role in deterining the copetitive relationships aongst alternatives; in which case, the logsu paraeter for each nest in the GNL odel should vary by trip length. The following hypotheses are developed for each nest. 1) Coach-Air Nest - Coach and air are likely to copete ore strongly in relatively shorter distance arkets where differences in total travel tie, su of access, terinal and in-vehicle tie) ay not be substantial. For longer distance trips the differences in total travel tie ay be large enough to
discourage copetition between the two iplying that the logsu coefficient will increase with 20 distance. 2) Coach-Preiu Coach Nest - The priary difference between the coach and preiu coach alternatives is the higher level of cofort offered by the preiu class. 3) Autoobile-Not Travel Nest - Travelers choosing autoobile for long trips, despite the increased discofort of driving, are unlikely to consider coon carrier odes (air or rail) as substitutes, given the uniqueness of the autoobile ode, (e.g., driving through scenic ountains, schedule flexibility). Therefore, a higher degree of substitution is expected between autoobile and not travel alternatives for longer distance trips, relative to short trips. This relationship can be reflected in the odel by a logsu that decreases with increasing trip distance. 4) Rail Nest The large nuber of alternatives in this nest; coach, preiu coach, sleeper, and econoy sleeper; akes it difficult to hypothesize an unabiguous relationship between the logsu paraeter and trip distance. For exaple, the substitution between sleeper and econoy sleeper is likely to decline with increasing trip distance; this expectation is based on the notion that sleeper users are ore sensitive to cofort and are unlikely to switch to econoy version for longer distance trips. However, as discussed previously, coach and preiu coach are ore likely to be substitutable with increasing trip distance. To avoid this confound, we do not allow for covariance heterogeneity in the Rail nest. The heterogeneity hypotheses are forulated as a function of trip distance, as follows: 1 1 exp( (Trip Distance)) f (11)
The hypothesis of hoogeneity is equivalent to requiring that equal to zero. That is, 21 H : = 0 0, a Coach Air H : = 0 0, b CoachPreiuCoach (12) H : = 0 0, c AutoNot Travel When the null hypothesis is true, the paraeter can be set so that the results are equivalent of equal to 0.05 is = 2.9444 for the rail nest. Finally, f is a specified function of trip distance (linear, square root, log, power, etc.). The use of the logistic function ensures the logsus lie in the zero-one range. Alternative functional fors of trip distance were tested for the f function described above; in this case, the logarith provided the best statistical fit. Table 5 shows the estiation results of the covariance heterogeneous GNL odel (Model 10) and the corresponding odel without covariance heterogeneity (Model 9). [Insert Table 5 around here.] As expected, Model 10 yields a substantially iproved goodness of fit over its restricted version, Model 9); the log-likelihood ratio test rejects the restricted odel at a confidence level greater than 99.9%. The large positive paraeters, 2 and 3, iply a high degree of covariance heterogeneity for coach-preiu coach and auto-not travel nests, respectively. Substituting these paraeters in the logistic function in equation 8 shows these logsu paraeters to decline with increasing distance, a result that is consistent with expectations stated above. The relationships iplied in Model 10 are illustrated in Figure 6. [Insert Figure 6 around here]
22 Figure 6 shows that in Model 10, the degree of siilarity, which is inversely related to the logsu paraeter,, and defined here as 1- between coach-preiu coach, and auto-not travel increases with trip length in the odel with covariance heterogeneity. This interpretation is different and behaviorally ore appealing than the constant siilarity relationship iplied by Model 9. Finally, the allocation and utility function paraeter estiates show no eaningful differences between the two odels, suggesting that incorporation of covariance heterogeneity, or lack thereof, has no significant ipact on other odel coponents. The failure to reject covariance hoogeneity hypothesis for the coach-air nest suggests that Model 10 could be siplified by eliinating this paraeter. Which we could do as suggested above. 5.4 Suary Coparison of Estiation Results Table 6 provides a suary coparison of the estiation results at each stage of odel developent. The table shows for each stage, the odel structure adopted, the data set(s) used, the types of flexibility incorporated, the nuber of estiated paraeters in the odel, the log-likelihood value at convergence, the 2 test statistic for coparing each odel with the preceding sipler odel, and the significance level at which the sipler odel is rejected. The results show draatic iproveents in the goodness-of-fit results at each stage of odel refineent 5 and rejection of the sipler odel. The increase in odeling coplexity at each stage is reflected by the differences in the nuber of estiated paraeters and the type of odel structure adopted. [Place Table 6 around here] 6. CONCLUSIONS This research integrates the considerable progress that has been ade in relaxing the assuption of independence across alternatives and the hoogeneity of error variance/covariance across observations 5 The drop in log-likelihood value fro base odels to stage 1 is due to the larger data set resulting fro cobining the user and non-user data sets.
23 within the context of closed for extensions of the MNL/NL odels. To accoplish this objective, an increental odeling approach was adopted, i.e., we started fro the siple MNL structure and sequentially relaxed soe of its restrictive assuptions to estiate progressively ore flexible odel(s). The statistical fit and behavioral appeal of the estiated odels iproved substantially with each additional relaxation, strongly supporting the concept of cobining isolated generalizations in an integrated odeling fraework. The Heterogenous Generalized Nested Logit odel developed here allows for heterogeneity in error variance and covariance structure, thereby explicitly accounting for the role of error variance/covariance in the choice decision process. Such coprehensive treatent of the rando error ters in odel specification extends our ability to understand and represent coplex behavioral processes involved in ode/class choice decision-aking. These results suggest the iportance of considering extensions of odel specification to allow for coplex covariance between pairs of alternatives as their presence ay substantially influence odel goodness-of-fit and change odel forecasts in a variety of realistic scenarios. An iportant issue in odel developent is the order in which the increents of odel refineent are undertaken and presented. Since changes in later increents could ipact the decisions ade in earlier stages, it would be appropriate to revisit these decision. However, in this case, exaination of the final odels, indicates that earlier decisions about which variables to include in the utility function, the structure of the GNL odel and the heterogeneity between and within saples are supported by the high degree of consistency of those paraeters and their levels of significance across odels. This final testing of the odel coponents established in earlier stages is essential to the overall quality of the approach and the resultant odels.
REFERENCES 24 Beser, M. and Algers S., 2002. SAMPERS: The new Swedish national travel deand forecasting tool, in Lundqvist, L. and Mattsson, L.-G., eds.), National Transport Models: Recent Developents. Aptech Systes. Gauss Applications, 1995. Constrained Maxiu Likelihood, Aptech Systes. Inc., Maple Valley, WA, 1995. Bhat C.R., 1997. A Nested Logit Model with Covariance Heterogeneity, Transportation Research, Part B, Vol. 31, pp. 11-21. Bhat, C.R. and Koppelan, F.S., 1999. Activity-Based Modeling of Travel Deand, Handbook of Transportation Science, Randolph H., (ed.), Elsevier Publishing. Bradley, M.A. and Daly, A., 1994. Use of Logit Scaling Approach to Test for Rank-Order and Fatigue Effects in Stated Preference Data, Transportation, Vol. 21, pp. 167-184. Bradley, M. and Kroes, E., 1992. Forecasting Issues in Stated Preference Survey Research, in Elizabeth S. Apt, Anthony J. Richardson, and Arni H. Meyburg, (ed.), Selected Readings in Transport Survey Methodology, Eucalyptus Press, Melbourne, Australia, pp. 89-107. Daganzo, C., 1979, Multinoial Probit: The Theory and its Application to Deand Forecasting. Acadeic Press, New York. Forinash, C.V. and Koppelan, F.S., 1993. Application and interpretation of nested logit odels of intercity ode choice, Transportation Research Record, No. 1413, 98-106. Grayson, A., 1979. Disaggregate Model of Mode Choice in Intercity Travel, Transportation Research Record, No. 835, pp. 36-42. Hensher, D.A., 1998. Intercity Rail Services: A Nested Logit Stated Choice Analysis of Pricing Options, Journal of Advanced Transportation, Vol. 32, No. 2, Suer 1998, pp. 130-181. Hensher, D., Louviere, J., and Swait, J., 1999. Cobining Sources of Preference Data, Journal of Econoetrics, Vol. 89, pp. 197-221. Doencich, T. A., D. McFadden, 1975. Urban Travel Deand A Behavioral Analysis. Aerican Elsevier Publishing Copany, Inc., New York McFadden, D., 1975. On Independence, Structure, and Siultaneity in Transportation Deand Analysis, Working Paper No. 7511, Urban Travel Deand Forecasting Project, Institute of Transportation and Traffic Engineering, University of California, Berkeley. Papola, A., 2004. Soe Developents on the Cross-Nested Logit Model, Transportation Research-B, V.38, N.9, pp.833-851.
25 Stopher, P.R., and Prashker, J.N. Intercity Passenger Forecasting: The Use of Current Travel Forecasting Procedures, Proceeding, Transportation Research Foru: 17th Annual Meeting, Vol. 17, No. 1, pp. 67-75. Swait, J. and Adaowicz, W. Choice Environent, Market Coplexity, and Consuer Behavior: A Theoretical and Epirical Approach for Incorporating Decision Coplexity into Models of Consuer Choice, Organizational Behavior and Huan Decision Processes, V. 86, N.2, 2001, Pages 141-167. Swait, J., 2001. Choice Set Generation within the Generalized Extree Value Faily of Discrete Choice Models, Transportation Research-B, V.35, N.7, pp. 643-666. Wen, C.H. and Koppelan, F.S., 2001. The Generalized Nested Logit Model, Transportation Research- B, V.35, N.7, pp. 627-641. Willias, H.C.W.L. and Ortuzar, J.D., 1982. Behavioral Theories of Dispersion and the Mis- Specification of Travel Deand Models, Transportation Research-B, V.16, N.3, pp167-219.
26 LIST OF FIGURES FIGURE 1: OVERVIEW OF THE ORIGIN OF DIFFERENT RANDOM UTILITY MODELS FIGURE 2: MODEL DEVELOPMENT APPROACH FIGURE 3: USER AND NON-USER SCALE AS A FUNCTION OF SP EXPERIMENT NUMBER FIGURE 4: TREE STRUCTURE FOR ALTERNATIVE NESTED LOGIT MODELS FIGURE 5: TREE STRUCTURE FOR GENERALIZED NESTED LOGIT MODEL FIGURE 6: VARIATION OF LOGSUM PARAMETERS AS A FUNCTION OF DISTANCE LIST OF TABLES TABLE 1: ESTIMATION RESULTS FOR USER, NON-USER AND POOLED MNL MODELS TABLE 2: MODELS WITH UNEQUAL ERROR VARIANCES WITHIN DATA SOURCES TABLE 3: MNL AND ALTERNATTIVE NESTED LOGIT MODELS WITH VARIANCE HETEROGENEITY TABLE 4: COMPARISON OF ESTIMATION RESULTS OF BEST NL AND GNL MODEL TABLE 5: GNL MODEL WITH AND WITHOUT COVARIANCE HETEROGENEITY TABLE 6: SUMMARY COMPARISON OF ESTIMATION RESULTS
27 FIGURE 1: Overview of the Origin of Different Rando Utility Models Rando Utility Models Negative Exponential Distribution Model Multinoial Logit Model Multinoial Probit Model I. Include Attributes of Copeting Alternatives II. Relax IID of Error Ters Across Alternatives III. Relax IID of Error Ters Across Observations Non-independent Errors Non-identical Errors Variance Relaxation Covariance Relaxation Mother Logit Dogit Model C-Logit Model Generalized Extree Value Models Nested Logit Product Differentiation Logit Paired Cobinatorial Logit Cross-Nested Logit Generalized Nested Logit Generalized MNL Ordered GEV 1. Closed For Models Oddball Alternative Paraeterized HMNL 2. Open For Models Heteroscedastic Extree Value Heteroscedastic MNL Covariance Heterogenous Nested Logit Mixed Logit
Vaneet Sethi and Frank S. Koppelan 28 FIGURE 2: Model Developent Approach Base Model Reference MNL Model User Non-user Pooled User and Non-user Stage 1 Extensions Variance Heteroscedasticity Within Data Sources Rank Order effects Fatigue effects Trip Context Stage 2 Extensions Flexible Correlation Structure Nested Logit Generalized Nested Logit Stage 3 Extensions Correlation Heterogeneity Paraeterize Logsu Trip Distance
Vaneet Sethi and Frank S. Koppelan 29 FIGURE 3: User and Non-User Scale as a Function of SP Experient Nuber 2.50 Users - 1st Choice Non-Users - 1st Choice Users - 2nd Choice Non-Users - 2nd Choice 2.14 Scale Paraeter 1.41 1.00 0.66 0.40 1 2 3 4 5 6 7 8 9 Experient Nuber
Vaneet Sethi and Frank S. Koppelan 30 FIGURE 4: Tree Structure for Alternative Nested Logit Models Model 7 - Rail and Auto-Not Travel Nest 0.66 0.57 Coach Sleeper PC ES Auto Not Travel Air Model 8-3 Level Rail Nest, and Auto-Not Travel Nest 0.70 0.46 0.56 0.57 Coach PC Sleeper ES Auto Not Travel Air FIGURE 5: Tree Structure for Generalized Nested Logit Model Model 9 - Rail, Coach-PC, Coach-Air, Auto-Not Travel Nests 0.05 0.32 0.05 0.31 1.0 Coach Sleeper PC ES Coach PC Coach Air Auto Not Coach Travel PC Sleeper ES Air Auto Not Travel
Vaneet Sethi and Frank S. Koppelan 31 FIGURE 6: Siilarity Relationship between Alternatives as a Function of Distance 1 0.9 0.8 Siilarity Index (1-Logsu) 0.7 0.6 0.5 0.4 0.3 0.2 0.1 Rail Nest Auto-Not Travel Nest Coach-Preiu Coach Nest Coach-Air Nest 0 0 500 1000 1500 2000 2500 Distance (iles)
Vaneet Sethi and Frank S. Koppelan 32 Inertia Rail Cost Schedule Convenience Overnight Duy Quality of Service Group Size Incoe Distance Alternative Specific Constants - Users Alternative Specific Constants - Non-Users Table 1: Estiation Results for User, Non-user and Pooled MNL Models Variable Naes Model 1 Preferred User Model Model 2 Constrained Non-User Model Model 3 Pooled Model with Unequal Variance Inertia_Coach 1.7284 (24.1) 1.7149 (25.1) Inertia_Sleeper 1.0785 (12.6) 1.1223 15.2) Scaled Rail Transportation Cost ($100) -0.1387 (-9.6) -0.2909 (-12.8) -0.1395-16.6) Scaled Rail Upgrade Cost_Low Incoe ( $100) -0.0237 (12.4) -0.0495 (-12.8) -0.0237 (-16.6) Scaled Rail Upgrade Cost_High Incoe ( $100) -0.0192 (-10.0) -0.0407-12.8) -0.0195 (-16.6) Bad Departure Hour Duy_Rail -0.2679 (-3.9) -0.2652 (-3.9) Bad Arrival Hour Duy_Rail -0.3161 (-4.5) -0.3159 (-4.5) Bad Departure Hour Duy_Not Travel 0.2974 (3.4) 0.3075 (3.5) Bad Arrival Hour Duy_Not Travel 0.5273 (6.0) 0.5330 (6.0) Overnight Rail Trip Duy_Sleeper & Econoy Sleeper 0.1243 (1.7) 0.1478 (2.0) Delay Duy_Rail -0.2286 (-4.1) -0.2350 (-4.3) Quality Rating_Rail 0.0906 (7.0) 0.1520 (8.7) 0.0770 (9.3) Group Size 1 Duy_Auto -0.2611-2.2) -0.6477 (-4.0) -0.2675 (-4.2) Group Size 2 Duy_Auto -0.3106 (2.5) -0.3066 (-2.5) -0.1776 (-3.2) GroupSize 1 Duy_Sleeper & Econoy Sleeper -0.7821 (-8.0) -0.5915 (-3.0) -0.5003 (-7.2) Group Size 2 Duy_Sleeper & Econoy Sleeper -0.6112 (-6.1) -0.4746 (-2.6) -0.3773 (-5.5) Incoe_Sleeper (in $1000's) 0.0069 (5.6) 0.0072 (7.2) Incoe_Air (in $1000's) 0.0074 (7.4) 0.0105 (2.9) 0.0068 (7.7) Incoe_Auto (in $1000's) 0.0020 (1.3) 0.0037 (1.4) 0.0020 (2.1) Distance_Sleeper & Econoy Sleeper 0.0002 (5.3) 0.0002 (5.5) Distance_Air 0.0001 (3.0) 0.0004 (3.3) 0.0002 (4.8) Distance_Auto -0.0003 (-4.4) -0.0003 (-3.5) -0.0002 (-4.8) Constant_Sleeper 0.6786 (4.1) 0.3950 (2.7) Constant_Preiu Coach -0.1086 (-2.3) -0.1061 (-2.3) Constant_Econoy Sleeper 1.3030 (9.5) 1.0412 (8.8) Constant_Air 0.2374 (1.4) 0.1125 (0.9) Constant_Auto 1.3866 (6.9) 1.1197 (8.5) Constant_Not Travel -0.1255 (-1.0) -0.2486 (-2.7) Constant_Sleeper 0.9015 (3.9) 0.5885 (7.5) Constant_Preiu Coach 0.4058 (4.7) 0.1893 (4.4) Constant_Econoy Sleeper 0.1653 (0.7) 0.2196 (2.6) Constant_Air 1.4754 (4.3) 0.6703 (4.8) Constant_Auto 2.2424 (9.7) 1.1218 (8.5) Constant_Not Travel -0.1205 (-0.9) -0.0289 (-0.6) Scale for Non-Users*** 2.1040 (13.9) Log likelihood at zero -12960.0-4923.7-17883.6 Log likelihood at convergence -11023.5-3209.5-14248.6 * Scaled Rail Transportation Cost = Transportation Cost/(1-exp(-0.001*Distance)) ** Scaled Rail Upgrade Cost = (Rail Cost - Transportation Cost)/(1-exp(-0.0005*Distance)) *** T-Statistic vs. 1.0
Vaneet Sethi and Frank S. Koppelan 33 TABLE 2: Models with Unequal Error Variances within Data Sources Variable Naes Model 3 Base Pooled Model Model 4 Scaling for 2 nd Choice Model 5 Scaling for 2 nd Choice and Fatigue/Learning Model 6 Scaling for 2 nd Choice, Fatigue and Distance Inertia Rail Cost Schedule Convenence Overnight Duy Inertia_Coach 1.7149 (25.1) 1.9489 (23.9) 2.3361 (17.4) 2.5695 (15.5) Inertia_Sleeper 1.1223 (15.2) 1.2890 (15.1) 1.5135 (12.9) 1.7718 (11.7) Scaled Rail Transportation Cost ($100)* -0.1395 (-16.6) -0.1635 (-16.5) -0.1958 (-13.6) -0.2139 (-12.9) Scaled Rail Upgrade Cost_Low Incoe ($100)** -0.0237 (-16.6) -0.0278 (-16.7) -0.0333 (-13.6) -0.0364 (-12.9) Scaled Rail Upgrade Cost_High Incoe ($100)** -0.0195 (-16.6) -0.0229 (-16.5) -0.0274 (-13.6) -0.0299 (-12.9) Bad Departure Hour Duy_Rail -0.2652 (-3.9) -0.3411 (-4.5) -0.4168 (-4.4) -0.4723 (-4.4) Bad Arrival Hour Duy_Rail -0.3159 (-4.5) -0.3822 (-4.9) -0.4386 (-4.5) -0.4811 (-4.3) Bad Departure Hour Duy_Not Travel 0.3075 (3.5) 0.4092 (4.0) 0.5406 (4.0) 0.6004 (3.9) Bad Arrival Hour Duy_Not Travel 0.5330 (6.0) 0.6416 (6.2) 0.8647 (6.2) 0.9702 (6.1) Overnight Rail Trip Duy_Sleeper & Econoy Sleeper 0.1478 (2.0) 0.1843 (2.1) 0.2487 (2.3) 0.1857 (1.5) Quality of Service Group Size Incoe Distance Alternative Specific Constants - Users Alternative Specific Constants - Non-Users Delay Duy_Rail -0.2350 (-4.3) -0.2933 (-4.6) -0.3562 (-4.5) -0.3749 (-4.2) Quality Rating_Rail 0.0770 (9.3) 0.0917 (9.0) 0.1156 (8.5) 0.1295 (8.1) Group Size 1 Duy_Auto -0.2675 (-4.2) -0.1842 (-2.6) -0.2426 (-2.8) -0.2200 (-2.3) Group Size 2 Duy_Auto -0.1776 (-3.2) -0.1463 (-2.5) -0.1823 (-2.5) -0.1708 (-2.1) GroupSize 1 Duy_Sleeper & Econoy Sleeper -0.5003 (-7.2) -0.5928 (-7.0) -0.7077 (-6.6) -0.7445 (-6.2) Group Size 2 Duy_Sleeper & Econoy -0.3773 (-5.5) -0.4852 (-5.7) -0.5858 (-5.5) -0.5930 (-5.0) Sl Incoe_Sleeper (in $1000's) 0.0072 (7.2) 0.0082 (7.1) 0.0099 (6.9) 0.0106 (6.4) Incoe_Air (in $1000's) 0.0068 (7.7) 0.0071 (7.0) 0.0088 (6.8) 0.0096 (6.5) Incoe_Auto (in $1000's) 0.0020 (2.1) 0.0020 (1.9) 0.0024 (1.9) 0.0022 (1.6) Distance_Sleeper & Econoy Sleeper (iles) 0.0002 (5.5) 0.0003 (5.0) 0.0003 (4.9) 0.0004 (4.8) Distance_Air (iles) 0.0002 (4.8) 0.0002 (4.8) 0.0002 (4.8) 0.0003 (5.0) Distance_Auto (iles) -0.0002 (-4.8) -0.0002 (-4.5) -0.0002 (-4.5) -0.0002 (-3.2) Constant_Sleeper 0.3950 (2.7) 0.5594 (3.3) 0.6831 (3.2) 0.7449 (3.1) Constant_Preiu Coach -0.1061 (-2.3) -0.1367 (-2.6) -0.1106 (-1.8) -0.1431 (-2.0) Constant_Econoy Sleeper 1.0412 (8.8) 1.2146 (8.5) 1.4992 (8.0) 1.6474 (7.8) Constant_Air 0.1125 (0.9) 0.2241 (1.6) 0.3302 (1.9) 0.3295 (1.6) Constant_Auto 1.1197 (8.5) 1.2484 (8.1) 1.6068 (7.9) 1.7193 (7.5) Constant_Not Travel -0.2486 (-2.7) -0.3612 (-3.1) -0.4728 (-3.2) -0.5467 (-3.2) Constant_Sleeper 0.5885 (7.5) 0.6456 (6.7) 0.7567 (6.2) 0.7945 (5.8) Constant_Preiu Coach 0.1893 (4.4) 0.2648 (4.9) 0.3215 (4.7) 0.3938 (5.0) Constant_Econoy Sleeper 0.2196 (2.6) 0.2122 (1.9) 0.2331 (1.7) 0.1908 (1.2) Constant_Air 0.6703 (4.8) 0.7410 (4.7) 0.9341 (4.6) 1.0765 (4.5) Constant_Auto 1.1218 (8.5) 1.1869 (7.7) 1.5029 (7.3) 1.5902 (7.0) Constant_Not Travel -0.0289 (-0.6) -0.0842 (-1.2) -0.0622 (-0.7) -0.0629 (-0.7)
Vaneet Sethi and Frank S. Koppelan 34 TABLE 2 (cont.): Models with Unequal Error Variances within Data Sources Variable Naes Model 3 Base Pooled Model Model 4 Scaling for 2 nd Choice Model 5 Model 6 Scaling for 2 nd Scaling for 2 nd Choice and Choice, Fatigue Fatigue/ and Distance Learning Scale for Non-Users 2.1040 (13.9) 2.0926 (12.9) 1.5887 (4.8) 1.6864 (7.0) Scale for Second Choice 0.6634 (-9.6) 0.6623 (-9.6) 0.6601 (-9.7) Scale_User Experient 2 0.9322 (-1.2) 0.9338 (-1.1) Scale_User Experient 3 0.8808 (-2.0) 0.8764 (-2.0) Scale_User Experient 4 0.8208 (-2.9) 0.8154 (-2.9) Scale_User Experient 5 0.8640 (-4.1) 0.8651 (-4.0) Variance Scale Paraeters*** Scale_User Experient 6 0.7915 (-3.3) 0.7960 (-3.2) Scale_User Experient 7 0.5915 (-4.5) 0.5920 (-4.5) Scale_User Experient 8 0.6953 (-3.4) 0.6942 (-3.4) Scale_NonUser Experient 2 1.0994 (1.0) Scale_NonUser Experient 3 0.9832 (-0.2) Scale_NonUser Experient 4 1.1195 (1.2) Scale_NonUser Experient 5 1.0004 (0.0) Scale_NonUser Experient 6 1.0987 (1.0) Scale_NonUser Experient 7 1.1183 (1.2) Scale_NonUser Experient 8 1.0892 (0.9) Scale_NonUser Experient 9 1.1612 (1.6) Scale_Distance 500-1000 iles 0.9202 (-2.0) Scale_Distance 1000-1500 iles 0.9158 (-1.8) Scale_Distance>1500 iles 0.8350 (-4.1) Log likelihood at zero -17883.6-17883.6-17883.6-17883.6 Log likelihood at convergence -14248.6-14170.4-14138.8-14132.3 * Scaled Rail Transportation Cost = Transportation Cost/(1-exp(-0.001*Distance)) ** Scaled Rail Upgrade Cost = (Rail Cost - Transportation Cost)/(1-exp(-0.0005*Distance)) *** T-Statistic vs. 1.0
Vaneet Sethi and Frank S. Koppelan 35 TABLE 3: MNL and Alternative Nested Logit Models with Variance Heteroscedasticity Variable Naes Model 6 Heteroscedastic MNL Model Model 7 "Best" 2-Level Heteroscedastic NL Model Model 8 "Best" 3-Level Heteroscedastic NL Model Inertia Rail Cost Schedule Conveneince Inertia_Coach 2.5695 (15.5) 1.8181 (12.5) 1.8433 (12.6) Inertia_Sleeper 1.7718 (11.7) 1.0566 (7.6) 1.0874 (7.7) Scaled Rail Transportation Cost (in $100's) -0.2139 (-12.9) -0.1521 (-10.9) -0.1479 (-10.7) Scaled Rail Upgrade Cost_Low Incoe (in $100's) -0.0364 (-12.9) -0.0259 (-10.9) -0.0251 (-10.7) Scaled Rail Upgrade Cost_High Incoe (in $100's) -0.0299 (-12.9) -0.0213 (-10.9) -0.0207 (-10.7) Bad Departure Hour Duy_Rail -0.4723 (-4.4) -0.4282 (-4.5) -0.4309 (-4.5) Bad Arrival Hour Duy_Rail -0.4811 (-4.3) -0.4179 (-4.3) -0.4168 (-4.3) Bad Departure Hour Duy_Not Travel 0.6004 (3.9) 0.4277 (3.7) 0.4331 (3.7) Bad Arrival Hour Duy_Rail 0.9702 (6.1) 0.8064 (6.7) 0.8140 (6.7) Overnight Duy Overnight Rail Trip Duy_Sleeper & Econoy Sleeper 0.1857 (1.5) 0.2309 (2.6) 0.2433 (2.7) Quality of Service Group Size Incoe Distance Alternative Specific Constants - Users Alternative Specific Constants - Non-Users Delay Duy_Rail -0.3749 (-4.2) -0.3214 (-3.9) -0.3264 (-4.0) Quality Rating_Rail 0.1295 (8.1) 0.0996 (7.3) 0.0967 (7.2) Group Size 1 Duy_Auto -0.2200 (-2.3) -0.2491 (-3.3) -0.2418 (-3.2) Group Size 2 Duy_Auto -0.1708 (-2.1) -0.1969 (-3.0) -0.1885 (-3.0) GroupSize 1 Duy_Sleeper & Econoy Sleeper -0.7445 (-6.2) -0.4919 (-5.9) -0.4718 (-5.6) Group Size 2 Duy_Sleeper & Econoy Sleeper -0.5930 (-5.0) -0.4313 (-5.2) -0.4116 (-4.9) Incoe_Sleeper (in $1000's) 0.0106 (6.4) 0.0079 (6.5) 0.0077 (6.4) Incoe_Air (in $1000's) 0.0096 (6.5) 0.0090 (6.9) 0.0088 (6.7) Incoe_Auto (in $1000's) 0.0022 (1.6) 0.0018 (1.6) 0.0017 (1.7) Distance_Sleeper&Econoy Sleeper (iles) 0.0004 (4.8) 0.0002 (4.6) 0.0002 (5.0) Distance_Air (iles) 0.0003 (5.0) 0.0002 (4.3) 0.0002 (4.3) Distance_Auto (iles) -0.0002 (-3.2) -0.0002 (-4.2) -0.0002 (-4.1) Constant_Sleeper 0.7449 (3.1) 0.5506 (3.1) 0.4757 (2.5) Constant_Preiu Coach -0.1431 (-2.0) -0.0562 (-1.2) 0.0143 (0.3) Constant_Econoy Sleeper 1.6474 (7.8) 1.1945 (7.3) 1.1289 (6.8) Constant_Air 0.3295 (1.6) -0.2576 (-1.5) -0.2866 (-1.7) Constant_Auto 1.7193 (7.5) 0.9859 (5.1) 0.9407 (4.9) Constant_Not Travel -0.5467 (-3.2) -0.7786 (-4.7) -0.8284 (-5.0) Constant_Sleeper 0.7945 (5.8) 0.6519 (6.8) 0.6093 (6.3) Constant_Preiu Coach 0.3938 (5.0) 0.2197 (3.8) 0.2205 (3.9) Constant_Econoy Sleeper 0.1908 (1.2) 0.2096 (2.2) 0.1801 (1.8) Constant_Air 1.0765 (4.5) 0.8969 (4.3) 0.8500 (4.1) Constant_Auto 1.5902 (7.0) 1.3058 (6.9) 1.2629 (6.7) Constant_Not Travel -0.0629 (-0.7) 0.0713 (0.8) 0.0817 (1.0)
Vaneet Sethi and Frank S. Koppelan 36 TABLE 3 (cont.): MNL and Alternative Nested Logit Models with Variance Heteroscedasticity Variance Scale Paraeters Logsu Paraeters Variable Naes Model 6 Heteroscedastic MNL Model Model 7 "Best" 2-Level Heteroscedastic NL Model Model 8 "Best" 3-Level Heteroscedastic NL Model Scale for Non-Users 1.6864 (7.0) 1.8098 (7.9) 1.8893 (8.4) Scale for Second Choice 0.6601 (-9.7) 0.7176 (-8.5) 0.7175 (-8.5) Scale_User Experient 2 0.9338 (-1.1) 0.9417 (-1.1) 0.9417 (-1.1) Scale_User Experient 3 0.8764 (-2.0) 0.8904 (-1.9) 0.8818 (-2.0) Scale_User Experient 4 0.8154 (-2.9) 0.8181 (-3.0) 0.8179 (-3.0) Scale_User Experient 5 0.8651 (-4.0) 0.8718 (-4.0) 0.8701 (-4.1) Scale_User Experient 6 0.7960 (-3.2) 0.8067 (-3.2) 0.8007 (-3.3) Scale_User Experient 7 0.5920 (-4.5) 0.5729 (-4.7) 0.5569 (-4.9) Scale_User Experient 8 0.6942 (-3.4) 0.6211 (-3.8) 0.6146 (-4.0) Scale_Distance 500-1000 iles 0.9202 (-2.0) 0.9159 (-2.3) 0.9192 (-2.2) Scale_Distance 1000-1500 iles 0.9158 (-1.8) 0.8885 (-2.5) 0.8835 (-2.6) Scale_Distance>1500 iles 0.8350 (-4.1) 0.9284 (-1.5) 0.9279 (-1.6) Rail Nest 0.6586 (-9.0) 0.7020 (-10.2) Coach-Sleeper Nest Coach-Preiu Coach Nest 0.4597 (-8.3) Coach-Econoy Sleeper Nest Coach-Air Nest Coach-Auto Nest Coach-Not Travel Nest Sleeper-Preiu Coach Nest Sleeper-Econoy Sleeper Nest 0.5585 (-7.5) Sleeper-Air Nest Sleeper-Auto Nest Sleeper-Not Travel Nest Preiu Coach-Air Nest Preiu Coach-Auto Nest Preiu Coach-Not Travel Nest Econoy Sleeper-Air Nest Econoy Sleeper-Auto Nest Econoy Sleeper-Not Travel Nest Air-Auto Nest Air-Not Travel Nest Auto-Not Travel Nest 0.5731 (-14.5) 0.5736 (-14.5) Log likelihood at zero -17883.6-17883.6-17883.6 Log likelihood at convergence -14132.3-14012.9-13999.5
Vaneet Sethi and Frank S. Koppelan 37 TABLE 4: Coparison of Estiation Results of Best NL and GNL Model Variable Naes Model 8 "Best" 3-Level Heteroscedastic NL Model Model 9 "Best" Heteroscedastic GNL Model Allocation Paraeters Logsu Paraeters Variance Scale Paraeters Scale for Non-Users 1.8893 (8.4) 1.6923 (6.1) Scale for Second Choice 0.7175 (-8.5) 0.6755 (-8.5) Scale_User Experient 2 0.9417 (-1.1) 0.9582 (-0.7) Scale_User Experient 3 0.8818 (-2.0) 0.8408 (-2.4) Scale_User Experient 4 0.8179 (-3.0) 0.7607 (-3.5) Scale_User Experient 5 0.8701 (-4.1) 0.8307 (-4.7) Scale_User Experient 6 0.8007 (-3.3) 0.7302 (-4.0) Scale_User Experient 7 0.5569 (-4.9) 0.5186 (-5.2) Scale_User Experient 8 0.6146 (-4.0) 0.5614 (-4.8) Scale_Distance 500-1000 iles 0.9192 (-2.2) 0.9438 (-1.5) Scale_Distance 1000-1500 iles 0.8835 (-2.6) 0.8493 (-3.1) Scale_Distance>1500 iles 0.9279 (-1.6) 0.9058 (-1.9) Rail Nest 0.7020 (-10.2) 0.0500 (*) Auto-Not Travel Nest 0.5736 (-14.5) 0.3094 (-7.5) Coach-Preiu Coach Nest 0.4597 (-8.3) 0.3243 (-9.9) Sleeper-Econoy Sleeper Nest 0.5585 (-7.5) Coach-Air Nest 0.0500 (*) Alpha Coach_Rail Nest 0.4255 (13.5) Alpha Sleeper_Rail Nest 0.5069 (18.7) Alpha Preiu Coach_Rail Nest 0.3417 (11.8) Alpha Econoy Sleeper_Rail Nest 0.4980 (15.0) Alpha Auto_Auto-Not Travel Nest 0.3721 (5.3) Alpha Not Travel_Auto-Not Travel Nest 0.9507 (10.9) Alpha Coach_Coach-Preiu Coach Nest 0.3576 (8.9) Alpha Preiu Coach_Coach-Preiu Coach Nest 0.6582 (22.7) Alpha Coach_Coach-Air Nest 0.2167 (6.1) Alpha Air_Coach-Air Nest 0.2712 (6.3) Log likelihood at zero -17883.6-17883.6 Log likelihood at convergence -13999.5-13925.3
Vaneet Sethi and Frank S. Koppelan 38 TABLE 5: GNL Model with and without Covariance Heterogeneity Variable Naes Model 9 "Best" Heteroscedastic GNL without Covariance Heterogeneity Model 10 Model 9 with Covariance Heterogeneity Inertia Rail Cost Schedule Conveneince Overnight Duy Quality of Service Group Size Incoe Distance Alternative Specific Constants - Users Alternative Specific Constants - Non-Users Inertia_Coach 1.3862 (10.2) 1.4386 (9.4) Inertia_Sleeper 1.1104 (8.3) 1.1806 (8.2) Scaled Rail Transportation Cost (in $100's) -0.1523 (-10.9) -0.1612 (-11.2) Scaled Rail Upgrade Cost_Low Incoe (in $100's) -0.0259 (-10.9) -0.0274 (-11.2) Scaled Rail Upgrade Cost_High Incoe (in $100's) -0.0213 (-10.9) -0.0226 (-11.2) Bad Departure Hour Duy_Rail -0.4732 (-5.2) -0.4778 (-4.7) Bad Arrival Hour Duy_Rail -0.4809 (-4.7) -0.4757 (-3.8) Bad Departure Hour Duy_Not Travel 0.4314 (3.7) 0.4234 (3.5) Bad Arrival Hour Duy_Rail 0.7724 (5.9) 0.7226 (5.1) Overnight Rail Trip Duy_Sleeper & Econoy Sleeper 0.3640 (3.5) 0.3397 (3.2) Delay Duy_Rail -0.3962 (-5.0) -0.4187 (-4.6) Quality Rating_Rail 0.1116 (6.7) 0.1211 (7.0) Group Size 1 Duy_Auto -0.2410 (-3.1) -0.2502 (-3.3) Group Size 2 Duy_Auto -0.2137 (-3.1) -0.2430 (-3.6) GroupSize 1 Duy_Sleeper & Econoy Sleeper -0.4450 (-5.3) -0.4678 (-5.2) Group Size 2 Duy_Sleeper & Econoy Sleeper -0.4394 (-5.1) -0.4659 (-5.2) Incoe_Sleeper (in $1000's) 0.0089 (7.4) 0.0093 (7.7) Incoe_Air (in $1000's) 0.0085 (6.5) 0.0092 (7.0) Incoe_Auto (in $1000's) 0.0016 (1.6) 0.0013 (1.3) Distance_Sleeper & Econoy Sleeper (iles) 0.0003 (8.2) 0.0003 (8.3) Distance_Air (iles) 0.0002 (5.1) 0.0003 (4.6) Distance_Auto (iles) -0.0001 (-2.4) -0.0001 (-2.9) Constant_Sleeper -0.1131 (-0.6) -0.0970 (-0.5) Constant_Preiu Coach 0.0626 (1.0) 0.0101 (0.1) Constant_Econoy Sleeper 0.6492 (3.4) 0.6958 (3.4) Constant_Air -0.4583 (-2.7) -0.5019 (-2.8) Constant_Auto 0.6834 (3.4) 0.8041 (4.0) Constant_Not Travel -1.3486 (-7.1) -1.3643 (-6.8) Constant_Sleeper 0.6468 (6.0) 0.6728 (5.9) Constant_Preiu Coach 0.2232 (4.0) 0.2319 (4.0) Constant_Econoy Sleeper 0.2317 (3.2) 0.2268 (2.9) Constant_Air 1.0698 (4.2) 1.1830 (4.2) Constant_Auto 1.4284 (6.2) 1.5769 (6.6) Constant_Not Travel 0.1736 (1.6) 0.1777 (1.6)
Vaneet Sethi and Frank S. Koppelan 39 Variance Scale Paraeters Logsu Paraeter Function (Covariance Heterogeneity) Allocation Paraeters Table 5 (continued): GNL Model with and without Covariance Heterogeneity Variable Naes Model 9 "Best" Heteroscedastic GNL without Covariance Heterogeneity Model 10 Model 9 with Covariance Heterogeneity Scale for Non-Users 1.6923 (6.1) 1.6255 (5.7) Scale for Second Choice 0.6755 (-8.5) 0.6788 (-8.9) Scale_User Experient 2 0.9582 (-0.7) 0.9692 (-0.5) Scale_User Experient 3 0.8408 (-2.4) 0.8494 (-2.4) Scale_User Experient 4 0.7607 (-3.5) 0.7620 (-3.8) Scale_User Experient 5 0.8307 (-4.7) 0.8283 (-5.0) Scale_User Experient 6 0.7302 (-4.0) 0.7248 (-4.3) Scale_User Experient 7 0.5186 (-5.2) 0.5434 (-5.0) Scale_User Experient 8 0.5614 (-4.8) 0.5798 (-4.8) Scale_Distance 500-1000 iles 0.9438 (-1.5) 0.8958 (-2.7) Scale_Distance 1000-1500 iles 0.8493 (-3.1) 0.7949 (-4.1) Scale_Distance>1500 iles 0.9058 (-1.9) 0.8602 (-2.7) Rail Nest - Constant 2.9444 (*) 2.9444 (*) - Log of Trip Distance -------- -------- Auto-Not Travel Nest - Constant 0.8029 (*) -6.0369 (*) - Log of Trip Distance -------- 1.0570 (3.1) Coach-Preiu Coach Nest - Constant 0.7341 (*) -12.8062 (*) - Log of Trip Distance -------- 2.0892 (11.4) Coach-Air Nest - Constant 2.9444 (*) 2.6708 (*) - Log of Trip Distance -------- 0.0060 (0.0) Alpha Coach_Rail Nest 0.4255 (13.5) 0.4161 (13.9) Alpha Sleeper_Rail Nest 0.5069 (18.7) 0.5051 (19.6) Alpha Preiu Coach_Rail Nest 0.3417 (11.8) 0.3482 (13.4) Alpha Econoy Sleeper_Rail Nest 0.4980 (15.0) 0.4957 (14.9) Alpha Auto_Auto-Not Travel Nest 0.3721 (5.3) 0.3598 (8.4) Alpha Not Travel_Auto-Not Travel Nest 0.9507 (10.9) 0.9999 (**) Alpha Coach_Coach-Preiu Coach Nest 0.3576 (8.9) 0.3458 (11.7) Alpha Preiu Coach_Coach-Preiu Coach Nest 0.6582 (22.7) 0.6517 (25.2) Alpha Coach_Coach-Air Nest 0.2167 (6.1) 0.2076 (4.4) Alpha Air_Coach-Air Nest 0.2712 (6.3) 0.2557 (5.5) Log likelihood at zero -17883.6 17883.6 Log likelihood at convergence -13925.3-13905.8 * T-statistics are not reported for the constants because the only reasonable test of the constant paraeter in this function is against negative infinity. ** T-statistics are not reported for allocation paraeters at the pre-set constraints.
Vaneet Sethi and Frank S. Koppelan 40 Table 6: Suary Coparison of Estiation Results Modeling Stage Model Structure Data Set User Non- User Variance Heterogeneity Between Within Data Data Sources Sources Correlation Heterogeneity Log- Likelihood Nuber of Paraeters Chi-square Statistic Significance to reject sipler odel Base Model Stage 1 Stage 2 MNL Yes -11023.5 28 NA NA MNL Yes -3209.5 18 NA NA MNL Yes Yes Yes -14248.6 31 31.2 <0.002 MNL Yes Yes Yes Yes -14132.3 42 232.6 <0.001 Nested Logit Yes Yes Yes Yes -13999.5 46 265.6 <0.001 Generalized Nested Logit Yes Yes Yes Yes -13925.3 56 148.4 <0.001 Stage 3 Generalized Nested Logit Yes Yes Yes Yes Yes -13905.8 58 39.0 <0.001