Estimating flexible distributions of ideal-points with external analysis of preferences

Ideal-points are widely used to model choices when preferences are single-peaked. Ideal-point choice models have been typically estimated at the individual-level, or have been based on the assumption that ideal-points are normally distributed over the population of choice makers. We propose two probabilistic ideal-point choice models for the external analysis of preferences that allow for more flexible multimodal distributions of ideal-points, thus acknowledging the existence of subpopulations with distinct preferences. The first model extends the ideal-point probit model for heterogeneous preferences to accommodate a mixture of multivariate normal distributions of ideal-points. The second model assumes that ideal-points are uniformly distributed within finite ranges of the attribute space, leading to a more simplistic formulation and a more flexible distribution. The two models are applied to simulated and actual choice data, and compared to the ideal-point probit model.This research was funded by the Dean's Fund for Faculty Research of the Owen Graduate School of Management, Vanderbilt University.     

Multivariate normal maximum likelihood with both ordinal and continuous variables,and data missing at random

A novel method for the maximum likelihood estimation of structural equation models (SEM) with both ordinal and continuous indicators is introduced using a flexible multivariate probit model for the ordinal indicators. A full information approach ensures unbiased estimates for data missing at random. Exceeding the capability of prior methods, up to 13 ordinal variables can be included before integration time increases beyond 1 s per row. The method relies on the axiom of conditional probability to split apart the distribution of continuous and ordinal variables. Due to the symmetry of the axiom, two similar methods are available. A simulation study provides evidence that the two similar approaches offer equal accuracy. A further simulation is used to develop a heuristic to automatically select the most computationally efficient approach. Joint ordinal continuous SEM is implemented in OpenMx, free and open-source software.     

A generalized estimating equations approach to mixed‐effects ordinal probit models

Clustered ordinal responses, which are commonplace in behavioural and educational research, are often analysed using mixed‐effects ordinal probit models. Likelihood‐based inference for these models can be computationally burdensome, and may compromise the consistency of estimators if the model is misspecified. We propose an alternative inferential approach based on generalized estimating equations. We show that systems of estimating equations can be specified for mixed‐effects ordinal probit models that avoid the potentially heavy computational demands of maximum likelihood estimation, and can also provide inferences that are robust with respect to some forms of model misspecification—particularly serial effects in longitudinal data.     

A Bayesian Multi-Level Factor Analytic Model of Consumer Price Sensitivities Across Categories

Identifying price sensitive consumers is an important problem in marketing. We develop a Bayesian multi-level factor analytic model of the covariation among household-level price sensitivities across product categories that are substitutes. Based on a multivariate probit model of category incidence, this framework also allows the researcher to model overall price sensitivity (i.e., indicated by higher-order factor scores) as a function of household-level covariates. All model parameters are estimated simultaneously to circumvent the downward bias resulting from two-stage estimation. The modeling framework is illustrated using scanner panel data from multiple categories of instant coffee.     

A general model for the analysis of multilevel data

A general model is developed for the analysis of multivariate multilevel data structures. Special cases of the model include repeated measures designs, multiple matrix samples, multilevel latent variable models, multiple time series, and variance and covariance component models.We would like to acknowledge the helpful comments of Ruth Silver. We also wish to thank the referees for helping to clarify the paper. This work was partly carried out with research funds provided by the Economic and Social Research Council (U.K.).     

Four simultaneous component models for the analysis of multivariate time series from more than one subject to model intraindividual and interindividual differences

A class of four simultaneous component models for the exploratory analysis of multivariate time series collected from more than one subject simultaneously is discussed. In each of the models, the multivariate time series of each subject is decomposed into a few series of component scores and a loading matrix. The component scores series reveal the latent data structure in the course of time. The interpretation of the components is based on the loading matrix. The simultaneous component models model not only intraindividual variability, but interindividual variability as well. The four models can be ordered hierarchically from weakly to severely constrained, thus allowing for big to small interindividual differences in the model. The use of the models is illustrated by an empirical example.This research has been made possible by funding from the Netherlands Organization of Scientific Research (NWO) to the first author. The authors are obliged to Tom A.B. Snijders, Jos M.F. ten Berge and three anonymous reviewers for comments on an earlier version of this paper, and to Kim Shifren for providing us with her data set, which was collected at Syracuse University.     

Modeling on the simulation writer interactive program

The Simulation Writer Interactive Program (SWIP) is an extension of the Michigan Experimental Simulation Supervisor program series, and is designed to permit individuals who are not skilled programmers to create numerical simulation models, primarily for instructional purposes. SWIP elicts from the user information about the manipulable (independent), central (intermediate), and observable (dependent) variables in the user’s model, as well as any necessary tables and control information. A powerful editor is available within SWIP for making modifications. SWIP deduces the structure of the user’s model from the information about the variables, and produces an interpreted version of the model which is used to generate simulated data. A variety of model structures are possible in SWIP, including multivariate models, models for repeated measures experiments, and timeseries models. A context-dependent question-answering facility is also available.     

What is the mechanism by which suicide attempts predispose to later suicide attempts? A mathematical model

A state dependence model of serial behavior suggests that each occurrence increases the subsequent likelihood of that behavior being repeated. A heterogeneity model, by contrast, suggests that the likelihood of a behavior occurring is predetermined, and uninfluenced by intervening occurrences. We have applied the random-effects probit model of Gibbons and Bock (1987) to examine the fit of the state dependence and heterogeneity models to longitudinal data on suicide attempts by 928 patients with affective disorder. Heterogeneity but not state dependence was required to model these data. The findings suggest that when considering patients with moderate to severe major affective disorder, the clinician should not interpret the absence of any recent suicide attempts to mean that the patient is at relatively low risk for attempting suicide in the future. An implication of the heterogeneity model is that suicide attempts made many years ago may have equal value to recent attempts when estimating an individual's "predisposition" to nonlethal attempts in the future.     

Trend in correlated proportions

A random effects probit model is developed for the case in which the same units are sampled repeatedly at each level of an independent variable. Because the observed proportions may be correlated under these conditions, estimating their trend with respect to the independent variable is no longer a standard problem for probit, logit or loglinear analysis. Using a qualitative analogue of a random regressions model, we employ instead marginal maximum likelihood to estimate the average latent trend line. Likelihood ratio tests of the hypothesis of no trend in the average line, and the hypothesis of no differences in average trend lines between experimental treatments, are proposed. We illustrate the model both with simulated data and with observed data from a clinical experiment in which psychiatric patients on two drug therapies are rated on five occasions for the presence or absence of symptoms.Supported by a grant from the MacArthur Foundation and National Science Foundation Grant BNS85-11774.The authors are indebted to James Heckman for calling our attention to the Clark algorithm.     

Some contributions to efficient statistics in structural models: Specification and estimation of moment structures

Current practice in structural modeling of observed continuous random variables is limited to representation systems for first and second moments (e.g., means and covariances), and to distribution theory based on multivariate normality. In psychometrics the multinormality assumption is often incorrect, so that statistical tests on parameters, or model goodness of fit, will frequently be incorrect as well. It is shown that higher order product moments yield important structural information when the distribution of variables is arbitrary. Structural representations are developed for generalizations of the Bentler-Weeks, Jöreskog-Keesling-Wiley, and factor analytic models. Some asymptotically distribution-free efficient estimators for such arbitrary structural models are developed. Limited information estimators are obtained as well. The special case of elliptical distributions that allow nonzero but equal kurtoses for variables is discussed in some detail. The argument is made that multivariate normal theory for covariance structure models should be abandoned in favor of elliptical theory, which is only slightly more difficult to apply in practice but specializes to the traditional case when normality holds. Many open research areas are described.     

A Robust Bayesian Approach for Structural Equation Models with Missing Data

In this paper, normal/independent distributions, including but not limited to the multivariate t distribution, the multivariate contaminated distribution, and the multivariate slash distribution, are used to develop a robust Bayesian approach for analyzing structural equation models with complete or missing data. In the context of a nonlinear structural equation model with fixed covariates, robust Bayesian methods are developed for estimation and model comparison. Results from simulation studies are reported to reveal the characteristics of estimation. The methods are illustrated by using a real data set obtained from diabetes patients.     

Quick approximation of bivariate functions

This paper presents two experiments where participants had to approximate function values at various generalization points of a square, using given function values at a small set of data points. A representative set of standard function approximation models was trained to exactly fit the function values at data points, and models' responses at generalization points were compared to those of humans. Then one defined a large class of possible models (including the best two identified predictors) and the class maximal possible prediction accuracy was evaluated. A new model of quick multivariate function approximation belonging to this class was proposed. Its prediction accuracy was close to the maximum possible, and significantly better than that of all other models tested. The new model also provided a significant account of human response variability. Finally, it was shown that this model is more particularly suitable for problems in which the visual system can perform some specific structuring of the data space. This model is therefore considered as a suitable starting point for further investigations into quick multivariate function approximation, which is to date an inadequately explored question in cognitive psychology.     

Multivariate stochastic processes compatible with “aspect” models of similarity and choice

Various recent works have developed feature or aspect models of similarity and preference. These models are more concerned with the fine detail of the judgment process than were prior models, but nevertheless they have not in general developed an underlying stochastic process compatible with the assumed structure. In this paper, we show that a particular class of multivariate stochastic processes, namely those associated with the Marshall-Olkin multivariate exponential distribution, generates several of these models. In particular, such stochastic processes (appropriately interpreted) yield Tversky's elimination by aspects model, Edgell and Geisler's (normal) additive random aspects model, and Shepard and Arabie's additive cluster model.This work was supported by Natural Science and Engineering Research Council of Canada Grant A8124 to A.A.J. Marley.     

Models for multivariate paired comparison experiments with ties

Three models for multivariate paired comparison experiments are developed using the Farlie-Gumbel-Morgenstern approach to constructing multivariate distributions. Model 3 is a direct extension of the multivariate paired comparison model with no ties due to Davidson and Bradley.     

Generating Nonnormal Multivariate Data Using Copulas: Applications to SEM

This article develops a procedure based on copulas to simulate multivariate nonnormal data that satisfy a prespecified variance-covariance matrix. The covariance matrix used can comply with a specific moment structure form (e.g., a factor analysis or a general structural equation model). Thus, the method is particularly useful for Monte Carlo evaluation of structural equation models within the context of nonnormal data. The new procedure for nonnormal data simulation is theoretically described and also implemented in the widely used R environment. The quality of the method is assessed by Monte Carlo simulations. A 1-sample test on the observed covariance matrix based on the copula methodology is proposed. This new test for evaluating the quality of a simulation is defined through a particular structural model specification and is robust against normality violations.     

Genetic evidence for multiple biological mechanisms underlying in-group favoritism

In-group favoritism is ubiquitous and associated with intergroup conflict, yet is little understood from a biological perspective. A fundamental question regarding the structure of favoritism is whether it is inflexibly directed toward distinct, "essentialist" categories, such as ethnicity and race, or is deployed in a context-sensitive manner. In this article, we report the first study (to our knowledge) of the genetic and environmental structure of in-group favoritism in the religious, ethnic, and racial domains. We contrasted a model of favoritism based on a single domain-general central affiliation mechanism (CAM) with a model in which each domain was influenced by specific mechanisms. In a series of multivariate analyses, utilizing a large, representative sample of twins, models containing only the CAM or essentialist domains fit the data poorly. The best-fitting model revealed that a biological mechanism facilitates affiliation with arbitrary groups and exists alongside essentialist systems that evolved to process salient cues, such as shared beliefs and ancestry.     

A Sandwich-Type Standard Error Estimator of SEM Models with Multivariate Time Series

Structural equation models are increasingly used as a modeling tool for multivariate time series data in the social and behavioral sciences. Standard error estimators of SEM models, originally developed for independent data, require modifications to accommodate the fact that time series data are inherently dependent. In this article, we extend a sandwich-type standard error estimator of independent data to multivariate time series data. One required element of this estimator is the asymptotic covariance matrix of concurrent and lagged correlations among manifest variables, whose closed-form expression has not been presented in the literature. The performance of the adapted sandwich-type standard error estimator is evaluated using a simulation study and further illustrated using an empirical example.     

Determining the Propensity for Academic Dishonesty Using Decision Tree Analysis

This article investigates the propensity for academic dishonesty by university students using the partitioning method of decision tree analysis. A set of prediction rules are presented, and conclusions are drawn. To provide context for the decision tree approach, the partition process is compared with results of more traditional probit regression models. Results of the decision tree analysis complement the probit models in terms of predictive accuracy and confirm results previously found in the literature. In particular, students’ moral character—whether they believe cheating is acceptable—is found to be the most important factor in determining the propensity for academic dishonesty.     

A multitrait-multimethod model with minimal assumptions

A new model of confirmatory factor analysis (CFA) for multitrait-multimethod (MTMM) data sets is presented. It is shown that this model can be defined by only three assumptions in the framework of classical psychometric test theory (CTT). All other properties of the model, particularly the uncorrelated-ness of the trait with the method factors are logical consequences of the definition of the model. In the model proposed there are as many trait factors as different traits considered, but the number of method factors is one fewer than the number of methods included in an MTMM study. The covariance structure implied by this model is derived, and it is shown that this model is identified even under conditions under which other CFA-MTMM models are not. The model is illustrated by two empirical applications. Furthermore, its advantages and limitations are discussed with respect to previously developed CFA models for MTMM data.