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.     

A new generalized heterogeneous data model (GHDM) to jointly model mixed types of dependent variables

This paper formulates a generalized heterogeneous data model (GHDM) that jointly handles mixed types of dependent variables—including multiple nominal outcomes, multiple ordinal variables, and multiple count variables, as well as multiple continuous variables—by representing the covariance relationships among them through a reduced number of latent factors. Sufficiency conditions for identification of the GHDM parameters are presented. The maximum approximate composite marginal likelihood (MACML) method is proposed to estimate this jointly mixed model system. This estimation method provides computational time advantages since the dimensionality of integration in the likelihood function is independent of the number of latent factors. The study undertakes a simulation experiment within the virtual context of integrating residential location choice and travel behavior to evaluate the ability of the MACML approach to recover parameters. The simulation results show that the MACML approach effectively recovers underlying parameters, and also that ignoring the multi-dimensional nature of the relationship among mixed types of dependent variables can lead not only to inconsistent parameter estimation, but also have important implications for policy analysis.     

Estimation of multinomial probit-kernel integrated choice and latent variable model: comparison on one sequential and two simultaneous approaches

Integrated choice and latent variable (ICLV) model incorporates latent factors into standard discrete choice model with aim to provide greater explanatory power. Using simulated datasets, this study makes a comparison among three estimation approaches corresponding to the sequential approach and two simultaneous approaches including the maximum simulated likelihood with GHK estimator and maximum approximate composite marginal likelihood (MACML) approach, to evaluate their abilities to recover the underlying parameters of multinomial probit-kernel ICLV model. The results show that both simultaneous approaches outperform the sequential approach in terms of estimates accuracy and efficiency irrespective of the sample sizes, and the MACML approach is the most preferable due to its best performance on recovering true values of parameters with relatively small standard errors, especially when the sample size is large enough.     

A new estimation approach for the multiple discrete–continuous probit (MDCP) choice model

This paper develops a blueprint (complete with matrix notation) to apply Bhat’s (2011) Maximum Approximate Composite Marginal Likelihood (MACML) inference approach for the estimation of cross-sectional as well as panel multiple discrete–continuous probit (MDCP) models. A simulation exercise is undertaken to evaluate the ability of the proposed approach to recover parameters from a cross-sectional MDCP model. The results show that the MACML approach does very well in recovering parameters, as well as appears to accurately capture the curvature of the Hessian of the log-likelihood function. The paper also demonstrates the application of the proposed approach through a study of individuals’ recreational (i.e., long distance leisure) choice among alternative destination locations and the number of trips to each recreational destination location, using data drawn from the 2004 to 2005 Michigan statewide household travel survey.     

Introducing non-normality of latent psychological constructs in choice modeling with an application to bicyclist route choice

In the current paper, we propose the use of a multivariate skew-normal (MSN) distribution function for the latent psychological constructs within the context of an integrated choice and latent variable (ICLV) model system. The multivariate skew-normal (MSN) distribution that we use is tractable, parsimonious in parameters that regulate the distribution and its skewness, and includes the normal distribution as a special interior point case (this allows for testing with the traditional ICLV model). Our procedure to accommodate non-normality in the psychological constructs exploits the latent factor structure of the ICLV model, and is a flexible, yet very efficient approach (through dimension-reduction) to accommodate a multivariate non-normal structure across all indicator and outcome variables in a multivariate system through the specification of a much lower-dimensional multivariate skew-normal distribution for the structural errors. Taste variations (i.e., heterogeneity in sensitivity to response variables) can also be introduced efficiently and in a non-normal fashion through interactions of explanatory variables with the latent variables. The resulting model we develop is suitable for estimation using Bhat’s (2011) maximum approximate composite marginal likelihood (MACML) inference approach. The proposed model is applied to model bicyclists’ route choice behavior using a web-based survey of Texas bicyclists. The results reveal evidence for non-normality in the latent constructs. From a substantive point of view, the results suggest that the most unattractive features of a bicycle route are long travel times (for commuters), heavy motorized traffic volume, absence of a continuous bicycle facility, and high parking occupancy rates and long lengths of parking zones along the route.     

Implementing the maximum likelihood methodology to measure a driver’s critical gap

Most of the capacity calculation procedures for two-way stop-controlled (TWSC) intersections are based on gap acceptance models. Critical gap is one of the major parameters for gap acceptance models. The accuracy of capacity estimation is mainly determined by the accuracy of the critical gap. This paper focuses on the implementation of the maximum likelihood technique to measure a driver’s critical gap using field data. A methodology to define gap events is proposed, so that the accepted gaps and maximum rejected gaps required by the maximum likelihood technique could be obtained. Specific issues regarding multi-lane situations and major street right turn movement are discussed. Special conditions observed during the research are addressed when the proposed method cannot be applied directly, such as the existence of a mid-block refuge area where minor street drivers can seek gaps in a two-stage process, pedestrian blockage, and downstream queue spill back. The proposed method was adopted in measuring critical gap under US conditions during a research project, described by Kyte et al. (1996). ©     

Discrete mode-choice analysis of urban travel demand by the Analytic Hierarchy Process

This paper develops a new procedure for the problem of multimodal urban corridor travel demand estimation by using the Analytic Hierarchy Process (AHP). Certain conceptual and operational features of the AHP are common to the discrete choice theory-based modeling approach. Whereas the computational and data requirements of standard discrete choice models are immense, the proposed AHP approach deals efficiently with multidimensionality, nested demand structure and discrete travel decision making behavior. The paper concludes by summarizing the AHP-aided, step-by-step procedure for metropolitan travel demand (modal split) estimation.     

Statistical estimation of the parameters of Daganzo's gap acceptance model

In a recent article in Transportation Research, Daganzo (1981) described a model of gap acceptance that permits the mean of the gap acceptance function to vary among drivers and permits the duration of the shortest acceptable gap for each driver to vary among gaps. The model contains several constant parameters whose values must be estimated statistically from observations of drivers' behavior. The results of numerical experiments reported by Daganzo (1981) suggested that the values of the parameters cannot be estimated by the method of maximum likelihood, which is the most obvious estimation technique, and Daganzo proposed using a sequential estimation method instead. The sequential method appeared to yield reasonable numerical results. In this paper, it is shown that subject to certain reasonable assumptions concerning the true parameter values and the probability distribution of gap durations, the maximum likelihood method does, in fact, yield consistent estimates of the parameters of Daganzo's model, whereas the sequential method does not. Hence, maximum likelihood is the better estimation method for this model.     

Estimating “tree” logit models

The existence of the “tree” generalisation of the multinomial logit model, its consistency with theories of “rational” human behaviour, as well as the advantages offered in terms of more realistic modelling by this model form, have all been known for some time. Although a few studies have applied tree logit models, however, applications have been restricted by practical and theoretical difficulties in the estimation of these models, in particular with the sequential estimation that has normally been necessary. This paper presents a new estimation procedure, employing the maximum likelihood criterion, for the estimation of tree logit models. The procedure is based on a simple notation for the tree structure and is believed to be original. It offers the possibility of estimating simultaneously all the parameters of general tree logit structures, without restriction on the number of levels but allowing the user to impose constraints of parameter equality where required. Results are presented indicating both the usefulness of the method in deriving improved models, relative to the more familiar sequential estimations, and its practicality and speed of operation on main-frame, minicomputers and microcomputers.     

Proactive route guidance to avoid congestion

We propose a proactive route guidance approach that integrates a system perspective: minimizing congestion, and a user perspective: minimizing travel inconvenience. The approach assigns paths to users so as to minimize congestion while not increasing their travel inconvenience too much. A maximum level of travel inconvenience is ensured and a certain level of fairness is maintained by limiting the set of considered paths for each Origin-Destination pair to those whose relative difference with respect to the shortest (least-duration) path, called travel inconvenience, is below a given threshold. The approach hierarchically minimizes the maximum arc utilization and the weighted average experienced travel inconvenience. Minimizing the maximum arc utilization in the network, i.e., the ratio of the number of vehicles entering an arc per time unit and the maximum number of vehicles per time unit at which vehicles can enter the arc and experience no slowdown due to congestion effects, is a system-oriented objective, while minimizing the weighted average experienced travel inconvenience, i.e., the average travel inconvenience over all eligible paths weighted by the number of vehicles per time unit that traverse the path, is a user-oriented objective. By design, to ensure computational efficiency, the approach only solves linear programming models. In a computational study using benchmark instances reflecting a road infrastructure encountered in many cities, we analyze, for different levels of maximum travel inconvenience and, the minimum maximum arc utilization and the weighted average experienced travel inconvenience. We find that accepting relatively small levels of maximum travel inconvenience can result in a significant reduction, or avoiding, of congestion.     

Interval estimation in the calibration of certain trip distribution models

This paper argues for interval, rather than point, estimation when calibrating some variants of the trip distribution “gravity” models. Analytic expressions are derived for the approximate asymptotic covariances of least squares and maximum likelihood estimates of the parameters in the impedance function under a variety of conditions. A comparative numerical example, and an application using migration flows, are also presented.     

Residential mobility and location choice: a nested logit model with sampling of alternatives

Residential mobility and relocation choice are essential parts of integrated transportation and land use models. These decision processes have been examined and modeled individually to a great extent but there remain gaps in the literature on the underlying behaviors that connect them. Households may partly base their decision to move from or stay at a current location on the price and quality of the available alternatives. Conversely, households that are on the market for a new location may evaluate housing choices relative to their previous residence. How and the degree to which these decisions relate to each other are, however, not completely understood. This research is intended to contribute to the body of empirical evidence that will help answer these questions. It is hypothesized that residential mobility and location choice are related household decisions that can be modeled together using a two-tier hierarchical structure. This paper presents a novel nested logit (NL) model with sampling of alternatives and a proposed procedure for sampling bias correction. The model was estimated using full-information maximum likelihood estimation methods. The results confirm the applicability of this NL model and support similar findings from other empirical studies in the residential mobility and location choice literature.     

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.     

Flexible long range planning using low cost information

Contemporary transport planning requires a flexible modelling approach which can be used to monitor the implementation of a long term plan checking regularly its short term performance with easily available data; the original model is periodically updated using low cost information and this allows the evaluation of the changes to the plan which may be required. Such an approach requires models suited to regular updating and to the use of data from different sources. Models to update trip matrices from traffic counts have been available for some time; however, the estimation and/or updating of other model stages with low cost data has escaped analytical treatment. The paper discusses this idea and formulates the updating problem for an example involving a joint destination/mode choice model under various assumptions about the nature of the available data. Analytical solutions are proposed as well as some general conclusions.requests for offprints     

Vehicle identification sensor models for origin–destination estimation

The traditional approach to origin–destination (OD) estimation based on data surveys is highly expensive. Therefore, researchers have attempted to develop reasonable low-cost approaches to estimating the OD vector, such as OD estimation based on traffic sensor data. In this estimation approach, the location problem for the sensors is critical. One type of sensor that can be used for this purpose, on which this paper focuses, is vehicle identification sensors. The information collected by these sensors that can be employed for OD estimation is discussed in this paper. We use data gathered by vehicle identification sensors that include an ID for each vehicle and the time at which the sensor detected it. Based on these data, the subset of sensors that detected a given vehicle and the order in which they detected it are available. In this paper, four location models are proposed, all of which consider the order of the sensors. The first model always yields the minimum number of sensors to ensure the uniqueness of path flows. The second model yields the maximum number of uniquely observed paths given a budget constraint on the sensors. The third model always yields the minimum number of sensors to ensure the uniqueness of OD flows. Finally, the fourth model yields the maximum number of uniquely observed OD flows given a budget constraint on the sensors. For several numerical examples, these four models were solved using the GAMS software. These numerical examples include several medium-sized examples, including an example of a real-world large-scale transportation network in Mashhad.     

Estimation of gap acceptance parameters within and across the population from direct roadside observation

This paper explores the feasibility of maximum likelihood as an approach to determine the parameters of gap acceptance functions when these functions vary from individual to individual. Specifically, it is shown that it is theoretically possible to estimate the average critical gap of a population of drivers (or pedestrians) and its variance, within and across individuals, from direct roadside observations. Although the Multinomial Probit Model provides a natural theoretical framework for the estimation of these parameters, the model seems not to be statistically estimable for this particular problem. It was shown, however, that if one of the parameters is known, the other two become estimable and a two-stage estimation process that takes into account this phenomenon can be utilized. The technique is demonstrated with the 203-driver data set included in Appendix A. The Multinomial Probit Model can also be used to determine simultaneously the mean critical gap, the mean critical lag (the first gap considered by a driver), and the variances of these. For the data set in Appendix A, the mean critical gap was significantly smaller than the mean critical lag, as one might expect. The techniques proposed in this paper have the further advantage of being statistically efficient with large data sets and of not requiring a panel of individuals to be observed under controlled conditions.     

Calibration of nonlinear car-following laws for traffic oscillation prediction

Frequency-domain analysis has been successfully used to (i) predict the amplification of traffic oscillations along a platoon of vehicles with nonlinear car-following laws and (ii) measure traffic oscillation properties (e.g., periodicity, magnitude) from field data. This paper proposes a new method to calibrate nonlinear car-following laws based on real-world vehicle trajectories, such that oscillation prediction (based on the calibrated car-following laws) and measurement from the same data can be compared and validated. This calibration method, for the first time, takes into account not only the driver’s car-following behavior but also the vehicle trajectory’s time-domain (e.g., location, speed) and frequency-domain properties (e.g., peak oscillation amplitude). We use Newell’s car-following model (1961) as an example and calibrate its parameters based on a penalty-based maximum likelihood estimation procedure. A series of experiments using Next Generation Simulation (NGSIM) data are conducted to illustrate the applicability and performance of the proposed approach. Results show that the calibrated car-following models are able to simultaneously reproduce observed driver behavior, time-domain trajectories, and oscillation propagation along the platoon with reasonable accuracy.     

Estimation of an integrated driving behavior model

This paper presents the methodology and results of estimation of an integrated driving behavior model that attempts to integrate various driving decisions. The model explains lane changing and acceleration decisions jointly and so, captures inter-dependencies between these behaviors and represents drivers’ planning capabilities. It introduces new models that capture drivers’ choice of a target gap that they intend to use in order to change lanes, and acceleration models that capture drivers’ behavior to facilitate the completion of a desired lane change using the target gap.The parameters of all components of the model are estimated simultaneously with the maximum likelihood method and using detailed vehicle trajectory data collected in a freeway section in Arlington, Virginia. The estimation results are presented and discussed in detail.     

Useful estimation procedures for critical gaps

Many different methods for the estimation of critical gaps at unsignalized intersections have been published in the international literature. This paper gives an overview of some of the more important methods. These methods are described by their characteristic properties. For comparison purposes a set of quality criteria has been formulated by which the usefulness of the different methods can be assessed. Among these one aspect seems to be of primary importance. This is the objective that the results of the estimation process should not depend on the traffic volume on the major street during the time of observation. Only if this condition is fulfilled, can the estimation be applied under all undersaturated traffic conditions at unsignalized intersections. To test the qualification of some of the estimation methods under this criterion, a series of comprehensive simulations has been performed. As a result, the maximum likelihood procedure (as it has been described by Troutbeck) and the method developed by Hewitt can be recommended for practical application. ©