Estimating choice probabilities among nested alternatives

In several travel choice situations (e.g. automobile ownership level and trip frequency) the alternatives available to an individual randomly chosen from the population exhibit some internal choice-related ranking: the choice of a given alternative implies that all lower-ranked alternatives have been chosen. Such alternatives are referred to as “nested”. This paper presents a model for estimating choice probabilities among nested alternatives. The model is devised from the well known logit model and uses existing logit maximum-likelihood estimation techniques (and computer packages). The approach is shown to be more attractive than the multinomial logit and linear regression models, from a theoretical point of view, yet cheaper than the multinomial probit model. The model is developed in a disaggregate, utility maximization framework. An example application, estimating probabilities of trip frequencies by elderly individuals is presented.     

A utility maximizing model of the demand for multi-destination non-work travel

The multinomial probit model of travel demand is considerably more general but much less tractable than the better-known multinomial logit model. In an effort to determine the effects of using the relatively simple logit model in situations where the assumptions of probit modeling are satisfied but those of logit modeling are not, the accuracy of the multinomial logit model as an approximation to a variety of three-alternative probit models has been evaluated. Multinomial logit can give highly erroneous estimates of the choice probabilities of multinomial probit models. However, logit models appear to give asymptotically accurate estimates of the ratios of the coefficients of the systematic components of probit utility functions, even when the logit choice probabilities differ greatly from the probit ones. Large estimation data sets are not necessarily needed to enable likelihood ratio tests to distinguish three-alternative probit models from logit models that give seriously erroneous estimates of the probit choice probabilities. Inclusion of alternative-specific dummy variables in logit utility functions cannot be relied upon to reduce significantly the errors of logit approximations to the choice probabilities of probit models whose utility functions do not contain the dummies.     

The accuracy of the multinomial logit model as an approximation to the multinomial probit model of travel demand

The multinomial probit model of travel demand is considerably more general but much less tractable than the better-known multinomial logit model. In an effort to determine the effects of using the relatively simple logit model in situations where the assumptions of probit modeling are satisfied but those of logit modeling are not, the accuracy of the multinomial logit model as an approximation to a variety of three-alternative probit models has been evaluated. Multinomial logit can give highly erroneous estimates of the choice probabilities of multinomial probit models. However, logit models appear to give asymptotically accurate estimates of the ratios of the coefficients of the systematic components of probit utility functions, even when the logit choice probabilities differ greatly from the probit ones. Large estimation data sets are not necessarily needed to enable likelihood ratio tests to distinguish three-alternative probit models from logit models that give seriously erroneous estimates of the probit choice probabilities. Inclusion of alternative-specific dummy variables in logit utility functions cannot be relied upon to reduce significantly the errors of logit approximations to the choice probabilities of probit models whose utility functions do not contain the dummies.     

Linear probit models: Statistical properties and improved estimation methods

The multinomial probit model is a statistical tool that is well suited to analyze some transportation problems. Modal split, gap acceptance, and route choice are some examples of application contexts. This paper presents an in-depth analysis of its statistical properties and an estimation method for the trinomial case. In the statistical part of the paper it is shown that for multinomial probit models with specifications that are linear in the parameters, the global maximum of the log-likelihood function is consistent if the data do not exhibit multicollinearity as defined in the text. For the special case with three alternatives, lack of multicollinearity is also shown to guarantee asymptotic efficiency and normality, and the uniqueness of any root of the likelihood equations. In addition, it is also shown that for the trinomial probit model certain goodness-of-it measures and test statistics can be easily calculated. The methods part of the paper introduces an estimation process that solves the likelihood equations using a special purpose table of the bivariate normal distribution and analytical derivatives of the log-likelihood function. The method is very accurate, can be applied to nonlinear specifications, and is considerably faster than current computer programs. For linear specifications, the method can be mathematically proven to converge if the log-likelihood equations have a root.     

Reconsidering the multinomial probit model

Recently developed computational methods have greatly reduced the difficulty of estimating multinomial probit models and may soon make multinomial probit a computationally feasible option in applied travel demand modeling. This paper discusses some of the benefits and costs that are associated with the use of multinomial probit in demand modeling. It is argued that although there are situations in which multinomial probit is essential for achieving a satisfactory model, most problems with existing demand models are unlikely to be mitigated by the use of multinomial probit.     

An integrated model for discrete and continuous decisions with application to vehicle ownership,type and usage choices

This paper proposes an integrated econometric framework for discrete and continuous choice dimensions. The model system is applied to the problem of household vehicle ownership, type and usage. A multinomial probit is used to estimate household vehicle ownership, a multinomial logit is used to estimate the vehicle type (class and vintage) choices, and a regression is used to estimate the vehicle usage decisions. Correlation between the discrete (number of vehicles) and the continuous (total annual miles traveled) parts is captured with a full variance–covariance matrix of the unobserved factors. The model system is estimated using Simulated Log-Likelihood methods on data extracted from the 2009 US National Household Travel Survey and a secondary dataset on vehicle characteristics. Model estimates are applied to evaluate changes in vehicle holding and miles driven, in response to the evolution of social societies, living environment and transportation policies.     

Comparative analysis of transit assignment: evidence from urban railway system in the Tokyo Metropolitan Area

This paper empirically compares the performance of six traffic assignment methods using the same empirical dataset of route choice. Multinomial logit (MNL), structured multinomial probit (SMNP), user equilibrium (UE), logit-based stochastic user equilibrium (SUE), probit-based SUE, and all-or-nothing (AON) assignment methods are applied to the comparative analysis. The investigated methods include those with three types of error components in their cost functions and two types of flow dependencies. Four methods of generating the route choice set are also compared for use as stochastic traffic assignment methods. The revealed preference data of urban rail route choice in the Tokyo Metropolitan Area are used for the case analysis. The empirical case analysis shows that probit-based SUE provides the best accuracy but requires the longest computation time. It also shows that the heuristics used to generate the choice set influence the method’s accuracy, while the incorporation of route commonality and in-vehicle congestion significantly improves its accuracy. Finally, the implications for practical rail planning are discussed on the basis of the analysis results.     

An application of a rank ordered probit modeling approach to understanding level of interest in autonomous vehicles

Surveys of behavior could benefit from information about people’s relative ranking of choice alternatives. Rank ordered data are often collected in stated preference surveys where respondents are asked to rank hypothetical alternatives (rather than choose a single alternative) to better understand their relative preferences. Despite the widespread interest in collecting data on and modeling people’s preferences for choice alternatives, rank-ordered data are rarely collected in travel surveys and very little progress has been made in the ability to rigorously model such data and obtain reliable parameter estimates. This paper presents a rank ordered probit modeling approach that overcomes limitations associated with prior approaches in analyzing rank ordered data. The efficacy of the rank ordered probit modeling methodology is demonstrated through an application of the model to understand preferences for alternative configurations of autonomous vehicles (AV) using the 2015 Puget Sound Regional Travel Study survey data set. The methodology offers behaviorally intuitive model results with a variety of socio-economic and demographic characteristics, including age, gender, household income, education, employment and household structure, significantly influencing preference for alternative configurations of AV adoption, ownership, and shared usage. The ability to estimate rank ordered probit models offers a pathway for better utilizing rank ordered data to understand preferences and recognize that choices may not be absolute in many instances.     

Parameter estimability in the multinomial probit model

A particular parameter estimability problem in the multinomial probit model is considered. Making use of a model discussed in the literature, some problems with the usual method of specification are discussed. Some general comments are made on the problem of selecting a normalisation when using the multinomial probit model.     

Individual characteristics and stated preferences for alternative energy sources and propulsion technologies in vehicles: A discrete choice analysis for Germany

With respect to the German goal of a transition to a lead market for electromobility within a short time period, this paper empirically examines the preferences for alternative energy sources or propulsion technologies in vehicles and particularly for electric vehicles. The data stem from a stated preference discrete choice experiment with 598 potential German car buyers. In order to simulate a realistic future purchase situation, seven vehicle types were incorporated in each of the six choice sets, i.e. hybrid, gas, biofuel, hydrogen, and electric vehicles besides common gasoline and diesel vehicles. The econometric analysis with flexible multinomial probit models reveals that potential car buyers in Germany currently have a low stated preference for electric, hydrogen, and hybrid vehicles. While our paper also discusses the impact of common vehicle attributes such as purchase price or service station availability, it particularly considers the effect of socio-demographic and environmental awareness variables. The estimation results reveal that younger potential car buyers have a higher stated preference for hydrogen and electric vehicles, males have a higher stated choice of hydrogen vehicles, and environmentally aware potential car buyers have a higher stated preference for hydrogen and electric vehicles. These results suggest that common policy instruments such as the promotion of research and development, taxation, or subsidization in the field of electromobility could be supplemented by strategies to increase the social acceptance of alternative vehicle types that are directly oriented to these population groups. Methodologically, our study highlights the importance of the inclusion of taste persistence across the choice sets and a high number of random draws in the Geweke–Hajivassiliou–Keane simulator in the simulated maximum likelihood estimation of the multinomial probit models.     

Analysis of parking cruising behaviour and parking location choice

ABSTRACT

This paper explores car drivers’ cruising behaviour and location choice for curb parking in areas with insufficient parking space based on a survey of car drivers in Beijing, China. Preliminary analysis of the data show that car drivers’ cruising behaviour is closely related to their parking duration and parking location. A multinomial probit (MNP) model is used to analyse cruising behaviour and the results show that the closer to the destination car drivers are, the more likely they choose to park on the curb. The adjacent locations are the basis of car drivers’ sequential parking decisions at different locations. The research results provide a better understanding of cruising behaviour for parking and recommendations for reducing cruising for parking. The provision of parking information can help regulate the parking demand distribution.     

A behavioral housing search model: Two-stage hazard-based and multinomial logit approach to choice-set formation and location selection

Residential location search has become an important topic to both practitioners and researchers as more detailed and disaggregate land-use and transportation demand models are developed which require information on individual household location decisions. The housing search process starts with an alternative formation and screening stage. At this level households evaluate all potential alternatives based on their lifestyle, preferences, and utilities to form a manageable choice set with a limited number of plausible alternatives. Then the final residential location is selected among these alternatives. This two-stage decision making process can be used for both aggregate zone-level selection as well as searching disaggregate parcel or building-based housing markets for potential dwellings. In this paper a zonal level household housing search model is developed. Initially, a household specific choice set is drawn from the entire possible alternatives in the area based on the average household work distance to each alternative. Following the choice set formation step, a discrete choice model is utilized for modeling the final residential zone selection of the household. A hazard-based model is used for the choice set formation module while the final choice selection is modeled using a multinomial logit formulation with a deterministic sample correction factor. The approach presented in the paper provides a remedy for the large choice set problem typically faced in housing search models.     

The dogit model

This paper presents the dogit model. That model is flexible enough to permit the choice among specific pairs of alternatives to be consistent with the independence from irrelevant alternatives axiom, as in a logit model, but it simultaneously allows the choice among other pairs not to be. Dogit parameters add an “income effect” to the “substitution effect” already built into the logit model; alternatively, they allow for the joint presence of compulsive and discretionary elements in consumer behavior, or for the identification of captive markets. Eventual estimation of the values of the parameters of the dogit model appears simpler than for the probit model.     

Does the choice model method and/or the data matter?

The opportunity to have seven data sets associated with a stated choice experiment that are very similar in content and design is rare, and provides an opportunity to look in detail at the empirical evidence within and between each data set in the context of a range of discrete choice estimation methods, from multinomial logit to latent class to scale multinomial logit to mixed logit, and the most general model, generalized mixed multinomial logit that accounts for preference and scale heterogeneity. Given the problems associated with data from different countries and time periods, we estimate separate models for each data set, obtaining values of travel time savings that are then updated post estimation to a common dollar for comparative purposes. We also pooled all data sets for a scaled MNL model, treating each data set as a set of three separate utility expressions, but linked to the other data sets through scale heterogeneity. This is not behaviourally appropriate with MNL, latent class or mixed logit. The main question investigated is whether there exists greater synergy in the willingness to pay evidence within model form across data sets compared to across model forms within data sets. The evidence suggests that there is a relatively greater convergence of evidence across the choice models, with the exception of generalized mixed logit, after controlling for data set differences; and there is strong evidence to suggest that differences between data sets do matter.     

A simulation evaluation of the maximum approximate composite marginal likelihood (MACML) estimator for mixed multinomial probit models

This paper evaluates the ability of the maximum approximate composite marginal likelihood (MACML) estimation approach to recover parameters from finite samples in mixed cross-sectional and panel multinomial probit models. Comparisons with the maximum simulated likelihood (MSL) estimation approach are also undertaken. The results indicate that the MACML approach recovers parameters much more accurately than the MSL approach in all model structures and covariance specifications. The MACML inference approach also estimates the parameters efficiently, with the asymptotic standard errors being, in general, only a small proportion of the true values. As importantly, the MACML inference approach takes only a very small fraction of the time needed for MSL estimation. In particular, the results suggest that, for the case of five random coefficients, the MACML approach is about 50 times faster than the MSL for the cross-sectional random coefficients case, about 15 times faster than the MSL for the panel inter-individual random coefficients case, and about 350 times or more faster than the MSL for the panel intra- and inter-individual random coefficients case. As the number of alternatives in the unordered-response model increases, one can expect even higher computational efficiency factors for the MACML over the MSL approach. Further, as should be evident in the panel intra- and inter-individual random coefficients case, the MSL is all but practically infeasible when the mixing structure leads to an explosion in the dimensionality of integration in the likelihood function, but these situations are handled with ease in the MACML approach. It is hoped that the MACML procedure will spawn empirical research into rich model specifications within the context of unordered multinomial choice modeling, including autoregressive random coefficients, dynamics in coefficients, space-time effects, and spatial/social interactions.     

Attitudinal models of modal choice: The multinomial case for selected nonwork trips

Attitudinal multinomial logit models of modal choice are presented for four nonwork activities: major grocery shopping, shopping for odds and ends, shopping for personal goods and visiting friends and acquaintances. Explanatory variables are individuals' beliefs about attributes of four modal alternatives: bus, car, taxi and walking. Factor analysis is employed to identify latent dimensions of perception of the modal alternatives and to eliminate problems of multicollinearities in model estimation. Models are estimated using data obtained for a sample of residents of Buffalo, New York. Planning implications of the methodology are assessed.This author is presently Systems Planner with Applied Resource Integration, Ltd., Boston, Massachusetts.     

Nested logit models for mixed-mode travel in urban corridors

The nested logit (NL) model is a generalisation of the well-known multinomial logit (MNL) model which copes with its “independence from irrelevant alternatives” problem, at the expense of more difficult calibration and use. Mixed-mode movements (i.e. park-and-ride) are by nature not independent of competing single-mode options and have, therefore, traditionally been inadequately modelled in most empirical applications. This paper reports on the specification, estimation, testing and comparison of MNL and NL models using disaggregate data of work trips in an urban corridor, where choice was among several alternatives including mixed-mode options. It was found that the more general NL model was more adequate, not only in theory but in practice. The paper concludes by comparing the disaggregate NL model with previously calibrated aggregate NL models for the same corridor using a different data set.     

Bayesian estimation of multinomial probit models of work trip choice

In this paper, we estimate a multinomial probit model of work trip mode choice in Seoul, Korea, using the Bayesian approach with Gibbs sampling. This method constructs a Markov chain Gibbs sampler that can be used to draw directly from the exact posterior distribution and perform finite sample likelihood inference. We estimate direct and cross-elasticities with respect to travel cost and the value of time. Our results show that travel demands are more sensitive to travel time than travel cost. The cross-elasticity results show that the bus has a greater substitute relation to the subway than the auto (and vice versa) and that an increase in the cost of an auto will increase the demand for bus transport more so than that of the subway.     

The maximum approximate composite marginal likelihood (MACML) estimation of multinomial probit-based unordered response choice models

The likelihood functions of multinomial probit (MNP)-based choice models entail the evaluation of analytically-intractable integrals. As a result, such models are usually estimated using maximum simulated likelihood (MSL) techniques. Unfortunately, for many practical situations, the computational cost to ensure good asymptotic MSL estimator properties can be prohibitive and practically infeasible as the number of dimensions of integration rises. In this paper, we introduce a maximum approximate composite marginal likelihood (MACML) estimation approach for MNP models that can be applied using simple optimization software for likelihood estimation. It also represents a conceptually and pedagogically simpler procedure relative to simulation techniques, and has the advantage of substantial computational time efficiency relative to the MSL approach. The paper provides a “blueprint” for the MACML estimation for a wide variety of MNP models.     

Strategic sampling for large choice sets in estimation and application

Many discrete choice contexts in transportation deal with large choice sets, including destination, route, and vehicle choices. Model estimation with large numbers of alternatives remains computationally expensive. In the context of the multinomial logit (MNL) model, limiting the number of alternatives in estimation by simple random sampling (SRS) yields consistent parameter estimates, but estimator efficiency suffers. In the context of more general models, such as the mixed MNL, limiting the number of alternatives via SRS yields biased parameter estimates. In this paper, a new, strategic sampling scheme is introduced, which draws alternatives in proportion to updated choice-probability estimates. Since such probabilities are not known a priori, the first iteration uses SRS among all available alternatives. The sampling scheme is implemented here for a variety of simulated MNL and mixed-MNL data sets, with results suggesting that the new sampling scheme provides substantial efficiency benefits. Thanks to reductions in estimation error, parameter estimates are more accurate, on average. Moreover, in the mixed MNL case, where SRS produces biased estimates (due to violation of the independence of irrelevant alternatives property), the new sampling scheme appears to effectively eliminate such biases. Finally, it appears that only a single iteration of the new strategy (following the initialization step using SRS) is needed to deliver the strategy’s maximum efficiency gains.