A Bayesian approach to nonlinear latent variable models using the Gibbs sampler and the metropolis-hastings algorithm

Nonlinear latent variable models are specified that include quadratic forms and interactions of latent regressor variables as special cases. To estimate the parameters, the models are put in a Bayesian framework with conjugate priors for the parameters. The posterior distributions of the parameters and the latent variables are estimated using Markov chain Monte Carlo methods such as the Gibbs sampler and the Metropolis-Hastings algorithm. The proposed estimation methods are illustrated by two simulation studies and by the estimation of a non-linear model for the dependence of performance on task complexity and goal specificity using empirical data.     

A latent class distance association model for cross‐classified data with a categorical response variable

In this paper we propose a latent class distance association model for clustering in the predictor space of large contingency tables with a categorical response variable. The rows of such a table are characterized as profiles of a set of explanatory variables, while the columns represent a single outcome variable. In many cases such tables are sparse, with many zero entries, which makes traditional models problematic. By clustering the row profiles into a few specific classes and representing these together with the categories of the response variable in a low‐dimensional Euclidean space using a distance association model, a parsimonious prediction model can be obtained. A generalized EM algorithm is proposed to estimate the model parameters and the adjusted Bayesian information criterion statistic is employed to test the number of mixture components and the dimensionality of the representation. An empirical example highlighting the advantages of the new approach and comparing it with traditional approaches is presented.     

Continuous and discrete latent structure models for item response data

Relations are examined between latent trait and latent class models for item response data. Conditions are given for the two-latent class and two-parameter normal ogive models to agree, and relations between their item parameters are presented. Generalizationss are then made to continuous models with more than one latent trait and discrete models with more than two latent classes, and methods are presented for relating latent class models to factor models for dichotomized variables. Results are illustrated using data from the Law School Admission Test, previously analyzed by several authors.     

On the explaining‐away phenomenon in multivariate latent variable models

Many probabilistic models for psychological and educational measurements contain latent variables. Well‐known examples are factor analysis, item response theory, and latent class model families. We discuss what is referred to as the ‘explaining‐away’ phenomenon in the context of such latent variable models. This phenomenon can occur when multiple latent variables are related to the same observed variable, and can elicit seemingly counterintuitive conditional dependencies between latent variables given observed variables. We illustrate the implications of explaining away for a number of well‐known latent variable models by using both theoretical and real data examples.     

A latent class unfolding model for analyzing single stimulus preference ratings

A multidimensional unfolding model is developed that assumes that the subjects can be clustered into a small number of homogeneous groups or classes. The subjects that belong to the same group are represented by a single ideal point. Since it is not known in advance to which group of class a subject belongs, a mixture distribution model is formulated that can be considered as a latent class model for continuous single stimulus preference ratings. A GEM algorithm is described for estimating the parameters in the model. The M-step of the algorithm is based on a majorization procedure for updating the estimates of the spatial model parameters. A strategy for selecting the appropriate number of classes and the appropriate number of dimensions is proposed and fully illustrated on some artificial data. The latent class unfolding model is applied to political science data concerning party preferences from members of the Dutch Parliament. Finally, some possible extensions of the model are discussed.The first author is supported as Bevoegdverklaard Navorser of the Belgian Nationaal Fonds voor Wetenschappelijk Onderzoek. Part of this paper was presented at the Distancia meeting held in Rennes, France, June 1992.     

Tail-measurability in monotone latent variable models

We consider latent variable models for an infinite sequence (or universe) of manifest (observable) variables that may be discrete, continuous or some combination of these. The main theorem is a general characterization by empirical conditions of when it is possible to construct latent variable models that satisfy unidimensionality, monotonicity, conditional independence, andtail-measurability. Tail-measurability means that the latent variable can be estimated consistently from the sequence of manifest variables even though an arbitrary finite subsequence has been removed. The characterizing,necessary and sufficient, conditions that the manifest variables must satisfy for these models are conditional association and vanishing conditional dependence (as one conditions upon successively more other manifest variables). Our main theorem considerably generalizes and sharpens earlier results of Ellis and van den Wollenberg (1993), Holland and Rosenbaum (1986), and Junker (1993). It is also related to the work of Stout (1990).The main theorem is preceded by many results for latent variable modelsin general—not necessarily unidimensional and monotone. They pertain to the uniqueness of latent variables and are connected with the conditional independence theorem of Suppes and Zanotti (1981). We discuss new definitions of the concepts of true-score and subpopulation, which generalize these notions from the stochastic subject, random sampling, and domain sampling formulations of latent variable models (e.g., Holland, 1990; Lord & Novick, 1968). These definitions do not require the a priori specification of a latent variable model.The authors made equivalent contributions to the results of this article. Ellis' research was supported by the Dutch Interuniversitary Graduate School of Psychometrics and Sociometrics. Junker's research was supported by ONR Grant N00014-87-K-0277, NIMH Grant MH15758, and a Carnegie Mellon University Faculty Development Grant. In addition Junker would like to acknowledge the hospitality of the Nijmegen Institute for Cognition and Information during his visit to the University of Nijmegen in August 5–10, 1993.     

A maximum likelihood method for latent class regression involving a censored dependent variable

The standard tobit or censored regression model is typically utilized for regression analysis when the dependent variable is censored. This model is generalized by developing a conditional mixture, maximum likelihood method for latent class censored regression. The proposed method simultaneously estimates separate regression functions and subject membership in K latent classes or groups given a censored dependent variable for a cross-section of subjects. Maximum likelihood estimates are obtained using an EM algorithm. The proposed method is illustrated via a consumer psychology application.     

Latent variable models for analyzing children's processing ability

In this paper, we develop a latent processing ability model to analyze the speed of processing ability data. Our approach can not only effectively evaluate the effects of covariates on the latent processing ability, but also estimate the latent trait of each child by calculating its posterior mean. In addition, we derive the correlations structure of latent traits among different age groups. Simulations are conducted to evaluate the performance of our proposed model. The results indicated that the estimation of model parameters is satisfactory overall. The method is evaluated using real data from children aged 4–7 years in Changchun, China.     

Modelling pedestrian crossing behaviour in urban roads: A latent variable approach

As pedestrians are the most exposed and vulnerable road users to traffic accidents, urban planners frequently propose alternatives to improve their safety. However, some solutions, such as pedestrian bridges and crosswalks at signalized intersections, usually imply longer walking distances compared to the direct crossing alternative which, in its turn, involves a higher risk.In this article, a hybrid framework is proposed to analyse the pedestrians’ choice on how to cross an urban road where three crossing options are available: crossing directly, crossing by using a pedestrian bridge or using a crosswalk at a signalized intersection. The decision process is modelled as a discrete choice model incorporating latent variables to consider perceptions and psychological factors, using stated preference data coming from a survey applied in Bogotá, Colombia.Results show that the latent variables security/safety and attractiveness of each crossing alternative are relevant to understand the pedestrian crossing behaviour. These latent variables are strongly determined by socioeconomic characteristics of the individual (age, gender, level of study) and conditioned by the circumstances of the trip (main mode of transport, walking or not with children). It was found that a longer walking distance to a pedestrian bridge or a signalized crosswalk increases the probability of direct crossing, having a more relevant effect in the case of the pedestrian bridge.     

A latent class approach to fitting the weighted Euclidean model,clascal

A weighted Euclidean distance model for analyzing three-way proximity data is proposed that incorporates a latent class approach. In this latent class weighted Euclidean model, the contribution to the distance function between two stimuli is per dimension weighted identically by all subjects in the same latent class. This model removes the rotational invariance of the classical multidimensional scaling model retaining psychologically meaningful dimensions, and drastically reduces the number of parameters in the traditional INDSCAL model. The probability density function for the data of a subject is posited to be a finite mixture of spherical multivariate normal densities. The maximum likelihood function is optimized by means of an EM algorithm; a modified Fisher scoring method is used to update the parameters in the M-step. A model selection strategy is proposed and illustrated on both real and artificial data.The second author is supported as Bevoegdverklaard Navorser of the Belgian Nationaal Fonds voor Wetenschappelijk Onderzoek.     

The EM algorithm for latent class analysis with equality constraints

The EM algorithm is a popular iterative method for estimating parameters in the latent class model where at each step the unknown parameters can be estimated simply as weighted sums of some latent proportions. The algorithm may also be used when some parameters are constrained to equal given constants or each other. It is shown that in the general case with equality constraints, the EM algorithm is not simple to apply because a nonlinear equation has to be solved. This problem arises, mainly, when equality constrints are defined over probabilities indifferent combinations of variables and latent classes. A simple condition is given in which, although probabilities in different variable-latent class combinations are constrained to be equal, the EM algorithm is still simple to apply.The authors are grateful to the Editor and the anonymous reviewers for their helpful comments on an earlier draft of this paper. C. C. Clogg and R. Luijkx are also acknowledged for verifying our results with their computer programs MLLSA and LCAG, respectively.     

A thurstonian pairwise choice model with univariate and multivariate spline transformations

A probabilistic choice model is developed for paired comparisons data about psychophysical stimuli. The model is based on Thurstone's Law of Comparative Judgment Case V and assumes that each stimulus is measured on a small number of physical variables. The utility of a stimulus is related to its values on the physical variables either by means of an additive univariate spline model or by means of multivariate spline model. In the additive univariate spline model, a separate univariate spline transformation is estimated for each physical dimension and the utility of a stimulus is assumed to be an additive combination of these transformed values. In the multivariate spline model, the utility of a stimulus is assumed to be a general multivariate spline function in the physical variables. The use of B splines for estimating the transformation functions is discussed and it is shown how B splines can be generalized to the multivariate case by using as basis functions tensor products of the univariate basis functions. A maximum likelihood estimation procedure for the Thurstone Case V model with spline transformation is described and applied for illustrative purposes to various artificial and real data sets. Finally, the model is extended using a latent class approach to the case where there are unreplicated paired comparisons data from a relatively large number of subjects drawn from a heterogeneous population. An EM algorithm for estimating the parameters in this extended model is outlined and illustrated on some real data.The first author is supported as Bevoegdverklaard Navorser of the Belgian Nationaal Fonds voor Wetenschappelijk Onderzoek. The authors are indebted to Ulf Böckenholt and Yoshio Takane for useful comments on an earlier draft of this paper.     

Constrained latent class analysis: Simultaneous classification and scaling of discrete choice data

A reparameterization of a latent class model is presented to simultaneously classify and scale nominal and ordered categorical choice data. Latent class-specific probabilities are constrained to be equal to the preference probabilities from a probabilistic ideal-point or vector model that yields a graphical, multidimensional representation of the classification results. In addition, background variables can be incorporated as an aid to interpreting the latent class-specific response probabilities. The analyses of synthetic and real data sets illustrate the proposed method.The authors thank Yosiho Takane, the editor and referees for their valuable suggestions. Authors are listed in reverse alphabetical order.     

A general latent trait model for response processes

The purpose of the current paper is to propose a general multicomponent latent trait model (GLTM) for response processes. The proposed model combines the linear logistic latent trait (LLTM) with the multicomponent latent trait model (MLTM). As with both LLTM and MLTM, the general multicomponent latent trait model can be used to (1) test hypotheses about the theoretical variables that underlie response difficulty and (2) estimate parameters that describe test items by basic substantive properties. However, GLTM contains both component outcomes and complexity factors in a single model and may be applied to data that neither LLTM nor MLTM can handle. Joint maximum likelihood estimators are presented for the parameters of GLTM and an application to cognitive test items is described.This research was partially supported by the National Institute of Education grant number NIE-6-7-0156 to Susan Embretson (Whitely), principal investigator. However the optinions expressed herein do not necessarily reflect the position or policy of the National Institute of Education, and no official endorsement by the National Institute of Education should be inferred.     

Shopping destination choice in Tehran: An integrated choice and latent variable approach

Integrated choice and latent variable (ICLV) models are increasingly considered in many fields as a means to gain a deeper understanding into the decision process of individuals as well as to potentially improve predictive ability. Literature has shown that the application of ICLV in context of shopping destination choice has not been conducted yet. This study uses data collected from the city of Tehran, Iran on shopping destination choice of 812 individuals. Then these attitudinal questions are used for the development of latent variables regarding the attitude and lifestyle of the participants. By including one latent variable (LV) reflecting the attitudes of clothing and lifestyle’s of grocery travelers the structural model reveals a sample distribution of this LV conditional on fundamental socio-economic characteristics. The results of our latent variable model clearly confirm that personal attitude toward clothing shopping center and lifestyle indeed impact destination choice. The results show that our experiment and the expanded hybrid choice model (HCM) allow appropriately identifying and investigating the effects of mixtures of latent attitudes on the intention shopping destination choice.     

Latent variable selection in multidimensional item response theory models using the expectation model selection algorithm

The aim of latent variable selection in multidimensional item response theory (MIRT) models is to identify latent traits probed by test items of a multidimensional test. In this paper the expectation model selection (EMS) algorithm proposed by Jiang et al. (2015) is applied to minimize the Bayesian information criterion (BIC) for latent variable selection in MIRT models with a known number of latent traits. Under mild assumptions, we prove the numerical convergence of the EMS algorithm for model selection by minimizing the BIC of observed data in the presence of missing data. For the identification of MIRT models, we assume that the variances of all latent traits are unity and each latent trait has an item that is only related to it. Under this identifiability assumption, the convergence of the EMS algorithm for latent variable selection in the multidimensional two-parameter logistic (M2PL) models can be verified. We give an efficient implementation of the EMS for the M2PL models. Simulation studies show that the EMS outperforms the EM-based L₁ regularization in terms of correctly selected latent variables and computation time. The EMS algorithm is applied to a real data set related to the Eysenck Personality Questionnaire.     

A flexible latent trait model for response times in tests

Latent trait models for response times in tests have become popular recently. One challenge for response time modeling is the fact that the distribution of response times can differ considerably even in similar tests. In order to reduce the need for tailor-made models, a model is proposed that unifies two popular approaches to response time modeling: Proportional hazard models and the accelerated failure time model with log–normally distributed response times. This is accomplished by resorting to discrete time. The categorization of response time allows the formulation of a response time model within the framework of generalized linear models by using a flexible link function. Item parameters of the proposed model can be estimated with marginal maximum likelihood estimation. Applicability of the proposed approach is demonstrated with a simulation study and an empirical application. Additionally, means for the evaluation of model fit are suggested.     

A latent trait model for dichotomous choice data

The PARELLA model is a probabilistic parallelogram model that can be used for the measurement of latent attitudes or latent preferences. The data analyzed are the dichotomous responses of persons to stimuli, with a one (zero) indicating agreement (disagreement) with the content of the stimulus. The model provides a unidimensional representation of persons and items. The response probabilities are a function of the distance between person and stimulus: the smaller the distance, the larger the probability that a person will agree with the content of the stimulus. An estimation procedure based on expectation maximization and marginal maximum likelihood is developed and the quality of the resulting parameter estimates evaluated.I gratefully acknowledge Ivo Molenaar and Wijbrandt van Schuur for their advice and encouragement during the course of the investigation, Derk-Jan Kiewiet who constructed the program for the ML estimator for the person parameter and Anne Boomsma, Wendy Post, Tom Snijders, and David Thissen for their comments on smaller aspects of the investigation.     

A note on parameter estimation for Lazarsfeld's latent class analysis

As the literature indicates, no method is presently available which takes explicitly into account that the parameters of Lazarsfeld's latent class analysis are defined as probabilities and are therefore restricted to the interval [0, 1]. In the present paper an appropriate transform on the parameters is performed in order to satisfy this constraint, and the estimation of the transformed parameters according to the maximum likelihood principle is outlined. In the sequel, a numerical example is given for which the basis solution and the usual maximum likelihood method failed. The different results are compared and the advantages of the proposed method discussed.