Bayesian Thurstonian models for ranking data using JAGS

A Thurstonian model for ranking data assumes that observed rankings are consistent with those of a set of underlying continuous variables. This model is appealing since it renders ranking data amenable to familiar models for continuous response variables—namely, linear regression models. To date, however, the use of Thurstonian models for ranking data has been very rare in practice. One reason for this may be that inferences based on these models require specialized technical methods. These methods have been developed to address computational challenges involved in these models but are not easy to implement without considerable technical expertise and are not widely available in software packages. To address this limitation, we show that Bayesian Thurstonian models for ranking data can be very easily implemented with the JAGS software package. We provide JAGS model files for Thurstonian ranking models for general use, discuss their implementation, and illustrate their use in analyses.     

The Latent Structure of Substance Use Among American Indian Adolescents: An Example Using Categorical Variables

Researchers frequently have only categorical data to analyze and cannot, for theoretical or methodological reasons, assume that the observed variables are discrete representations of an underlying continuous variable. We present latent class analysis as an alternative method of measuring latent variables in these circumstances. Latent class analysis does not require the assumptions of factor analyses about the nature of manifest and latent variables, but does allow the use of more precise model selection than techniques such as cluster analysis. We modeled the lifetime substance use of American Indian youth. The latent class model of American Indian teenagers' substance use had four classes: Abstaining, Predominantly Alcohol, Predominantly Alcohol and Marijuana, and Plural Substance. We then demonstrated the usefulness of this latent variable by using it to differentiate levels of several variables in a manner consistent with Social Cognitive Theory.     

Multilevel Mixture Factor Models

Factor analysis is a statistical method for describing the associations among sets of observed variables in terms of a small number of underlying continuous latent variables. Various authors have proposed multilevel extensions of the factor model for the analysis of data sets with a hierarchical structure. These Multilevel Factor Models (MFMs) have in common that—as in multilevel regression analysis—variation at the higher level is modeled using continuous random effects. In this article, we present an alternative multilevel extension of factor analysis which we call the Multilevel Mixture Factor Model (MMFM). It is based on the assumption that higher level units belong to latent classes that differ in terms of the parameters of the factor model specified for the lower level units. We demonstrate the added value of MMFM compared with MFM, both from a theoretical and applied perspective, and we illustrate the complementarity of the two approaches with an empirical application on students' satisfaction with the University of Florence. The multilevel aspect of this application is that students are nested within study programs, which makes it possible to cluster these programs based on their differences in students' satisfaction.     

Ordinal state-trait regression for intensive longitudinal data

In many psychological studies, in particular those conducted by experience sampling, mental states are measured repeatedly for each participant. Such a design allows for regression models that separate between- from within-person, or trait-like from state-like, components of association between two variables. But these models are typically designed for continuous variables, whereas mental state variables are most often measured on an ordinal scale. In this paper we develop a model for disaggregating between- from within-person effects of one ordinal variable on another. As in standard ordinal regression, our model posits a continuous latent response whose value determines the observed response. We allow the latent response to depend nonlinearly on the trait and state variables, but impose a novel penalty that shrinks the fit towards a linear model on the latent scale. A simulation study shows that this penalization approach is effective at finding a middle ground between an overly restrictive linear model and an overfitted nonlinear model. The proposed method is illustrated with an application to data from the experience sampling study of Baumeister et al. (2020, Personality and Social Psychology Bulletin, 46, 1631).     

Toward Computational Model of Emotion from Individual Difference in Perceiving Facial Expressions

We propose a computational model for identifying emotional state of a facial expression from appraisal scores given by human observers utilizing their differences in perception. The appraisal model of human emotion is adopted as the basis of this evaluation process with appraisal variables as output. We investigated the performance for both categorical and continuous representation of the variables appraised by human observers. Analysis of the data exhibits higher degree of agreement between estimated Indian ratings and the available reference when these are rated through continuous domain. We also observed that emotional state with negative valence are influential in the perception of hybrid emotional state like ‘Surprise’, only when appraisal variables are labeled through categories of emotions. Thus, the proposed method has implications in developing software to detect emotion using appraisal variables in continuous domain, perceived from facial expression of an agent (or human subject). Further, this model can be customized to include cultural variability in recognizing emotions.     

Ordinary Least Squares Estimation of Parameters in Exploratory Factor Analysis With Ordinal Data

Exploratory factor analysis (EFA) is often conducted with ordinal data (e.g., items with 5-point responses) in the social and behavioral sciences. These ordinal variables are often treated as if they were continuous in practice. An alternative strategy is to assume that a normally distributed continuous variable underlies each ordinal variable. The EFA model is specified for these underlying continuous variables rather than the observed ordinal variables. Although these underlying continuous variables are not observed directly, their correlations can be estimated from the ordinal variables. These correlations are referred to as polychoric correlations. This article is concerned with ordinary least squares (OLS) estimation of parameters in EFA with polychoric correlations. Standard errors and confidence intervals for rotated factor loadings and factor correlations are presented. OLS estimates and the associated standard error estimates and confidence intervals are illustrated using personality trait ratings from 228 college students. Statistical properties of the proposed procedure are explored using a Monte Carlo study. The empirical illustration and the Monte Carlo study showed that (a) OLS estimation of EFA is feasible with large models, (b) point estimates of rotated factor loadings are unbiased, (c) point estimates of factor correlations are slightly negatively biased with small samples, and (d) standard error estimates and confidence intervals perform satisfactorily at moderately large samples.     

Maximum Likelihood Analysis of Nonlinear Structural Equation Models With Dichotomous Variables

In this article, a maximum likelihood approach is developed to analyze structural equation models with dichotomous variables that are common in behavioral, psychological and social research. To assess nonlinear causal effects among the latent variables, the structural equation in the model is defined by a nonlinear function. The basic idea of the development is to augment the observed dichotomous data with the hypothetical missing data that involve the latent underlying continuous measurements and the latent variables in the model. An EM algorithm is implemented. The conditional expectation in the E-step is approximated via observations simulated from the appropriate conditional distributions by a Metropolis-Hastings algorithm within the Gibbs sampler, whilst the M-step is completed by conditional maximization. Convergence is monitored by bridge sampling. Standard errors are also obtained. Results from a simulation study and a real example are presented to illustrate the methodology.     

Alternative to the wavelength interpretation of the auditory kappa effect

We argue that the simple wavelength interpretation of the auditory kappa effect proposed by Yoblick and Salvendy in 1970 is inadequate since only the frequency of an incoming soundwave is preserved at the tympanic membrane. Alternative explanations are proposed in terms of psychological variables. The kappa effect with tonal intervals is explained using an imputed velocity model in which frequency differences are equated to phenomenological "distances." The continuous tone effect is explained in terms of previously observed correlations between frequency and the volume or " bigness " of a tone and between size and perceived duration.     

Discriminant Analysis Using Mixed Continuous,Dichotomous, and Ordered Categorical Variables

This article proposes an intuitive approach for predictive discriminant analysis with mixed continuous, dichotomous, and ordered categorical variables that are defined via an underlying multivariate normal distribution with a threshold specification. The classification rule is based on the comparison of the observed data logarithm probability density functions. To reduce the computational burden, the analysis is conducted in the context of a confirmatory factor analysis model with independent error measurements. Identification of the dichotomous and ordered categorical variables is discussed. Results are obtained by implementations of a Monte Carlo expectation maximization (MCEM)algorithm and a path sampling procedure. Probabilities of misclassification are estimated via the idea of the “jackknife” method. A real example is given to illustrate the proposed method.     

An SEM approach to continuous time modeling of panel data: relating authoritarianism and anomia

Panel studies, in which the same subjects are repeatedly observed at multiple time points, are among the most popular longitudinal designs in psychology. Meanwhile, there exists a wide range of different methods to analyze such data, with autoregressive and cross-lagged models being 2 of the most well known representatives. Unfortunately, in these models time is only considered implicitly, making it difficult to account for unequally spaced measurement occasions or to compare parameter estimates across studies that are based on different time intervals. Stochastic differential equations offer a solution to this problem by relating the discrete time model to its underlying model in continuous time. It is the goal of the present article to introduce this approach to a broader psychological audience. A step-by-step review of the relationship between discrete and continuous time modeling is provided, and we demonstrate how continuous time parameters can be obtained via structural equation modeling. An empirical example on the relationship between authoritarianism and anomia is used to illustrate the approach.     

Unidimensional factor models imply weaker partial correlations than zero-order correlations

In this paper we present a new implication of the unidimensional factor model. We prove that the partial correlation between two observed variables that load on one factor given any subset of other observed variables that load on this factor lies between zero and the zero-order correlation between these two observed variables. We implement this result in an empirical bootstrap test that rejects the unidimensional factor model when partial correlations are identified that are either stronger than the zero-order correlation or have a different sign than the zero-order correlation. We demonstrate the use of the test in an empirical data example with data consisting of fourteen items that measure extraversion.     

Inferring action structure and causal relationships in continuous sequences of human action

In the real world, causal variables do not come pre-identified or occur in isolation, but instead are embedded within a continuous temporal stream of events. A challenge faced by both human learners and machine learning algorithms is identifying subsequences that correspond to the appropriate variables for causal inference. A specific instance of this problem is action segmentation: dividing a sequence of observed behavior into meaningful actions, and determining which of those actions lead to effects in the world. Here we present a Bayesian analysis of how statistical and causal cues to segmentation should optimally be combined, as well as four experiments investigating human action segmentation and causal inference. We find that both people and our model are sensitive to statistical regularities and causal structure in continuous action, and are able to combine these sources of information in order to correctly infer both causal relationships and segmentation boundaries.     

Factor copula models for mixed data

We develop factor copula models to analyse the dependence among mixed continuous and discrete responses. Factor copula models are canonical vine copulas that involve both observed and latent variables, hence they allow tail, asymmetric and nonlinear dependence. They can be explained as conditional independence models with latent variables that do not necessarily have an additive latent structure. We focus on important issues of interest to the social data analyst, such as model selection and goodness of fit. Our general methodology is demonstrated with an extensive simulation study and illustrated by reanalysing three mixed response data sets. Our studies suggest that there can be a substantial improvement over the standard factor model for mixed data and make the argument for moving to factor copula models.     

The trade-off between accuracy and precision in latent variable models of mediation processes

Social psychologists place high importance on understanding mechanisms and frequently employ mediation analyses to shed light on the process underlying an effect. Such analyses can be conducted with observed variables (e.g., a typical regression approach) or latent variables (e.g., a structural equation modeling approach), and choosing between these methods can be a more complex and consequential decision than researchers often realize. The present article adds to the literature on mediation by examining the relative trade-off between accuracy and precision in latent versus observed variable modeling. Whereas past work has shown that latent variable models tend to produce more accurate estimates, we demonstrate that this increase in accuracy comes at the cost of increased standard errors and reduced power, and examine this relative trade-off both theoretically and empirically in a typical 3-variable mediation model across varying levels of effect size and reliability. We discuss implications for social psychologists seeking to uncover mediating variables and provide 3 practical recommendations for maximizing both accuracy and precision in mediation analyses.     

Generalized Processing Tree Models: Jointly Modeling Discrete and Continuous Variables

Multinomial processing tree models assume that discrete cognitive states determine observed response frequencies. Generalized processing tree (GPT) models extend this conceptual framework to continuous variables such as response times, process-tracing measures, or neurophysiological variables. GPT models assume finite-mixture distributions, with weights determined by a processing tree structure, and continuous components modeled by parameterized distributions such as Gaussians with separate or shared parameters across states. We discuss identifiability, parameter estimation, model testing, a modeling syntax, and the improved precision of GPT estimates. Finally, a GPT version of the feature comparison model of semantic categorization is applied to computer-mouse trajectories.     

Managerial Work, Job Analysis, and Holland's RIASEC Vocational Environment Dimensions

We examined relations between the environmental dimensions underlying Holland's theory of vocational choice and skill requirements, context characteristics, and task frequency ratings for managerial jobs. The Holland environmental constructs were measured by the recently developedPosition Classification Inventory(PCI). The task, skill requirement, and context variables were measured using traditional job analysis surveys. Ten judges provided estimates of the expected correlations between the job analysis variables and the Holland constructs. The profile of observed correlations was generally consistent with the judges’ expectations based on Holland's theory, providing support for both that framework and the construct validity of the PCI. The one Holland dimension for which the data were least consistent with predictions was “Realistic.” Results provide a detailed picture of the work content, skills, and context variables within managerial work that are likely to be associated with the RIASEC dimensions. Implications for management development are discussed.     

Learning Continuous Probability Distributions with Symmetric Diffusion Networks

In this article we present symmetric diffusion networks, a family of networks that instantiate the principles of continuous, stochastic, adaptive and interactive propagation of information. Using methods of Markovion diffusion theory, we formalize the activation dynamics of these networks and then show that they can be trained to reproduce entire multivariate probability distributions on their outputs using the contrastive Hebbion learning rule (CHL). We show that CHL performs gradient descent on an error function that captures differences between desired and obtained continuous multivariate probability distributions. This allows the learning algorithm to go beyond expected values of output units and to approximate complete probability distributions on continuous multivariate activation spaces. We argue that learning continuous distributions is an important task underlying a variety of real-life situations that were beyond the scope of previous connectionist networks. Deterministic networks, like back propagation, cannot learn this task because they are limited to learning average values of independent output units. Previous stochastic connectionist networks could learn probability distributions but they were limited to discrete variables. Simulations show that symmetric diffusion networks can be trained with the CHL rule to approximate discrete and continuous probability distributions of various types.     

A general structural equation model with dichotomous,ordered categorical,and continuous latent variable indicators

A structural equation model is proposed with a generalized measurement part, allowing for dichotomous and ordered categorical variables (indicators) in addition to continuous ones. A computationally feasible three-stage estimator is proposed for any combination of observed variable types. This approach provides large-sample chi-square tests of fit and standard errors of estimates for situations not previously covered. Two multiple-indicator modeling examples are given. One is a simultaneous analysis of two groups with a structural equation model underlying skewed Likert variables. The second is a longitudinal model with a structural model for multivariate probit regressions.This research was supported by Grant No. 81-IJ-CX-0015 from the National Institute of Justice, by Grant No. DA 01070 from the U.S. Public Health Service, and by Grant No. SES-8312583 from the National Science Foundation. I thank Julie Honig for drawing the figures. Requests for reprints should be sent to Bengt Muthén, Graduate School of Education, University of California, Los Angeles, California 90024.     

Equivalence of a measurement model of cognitive abilities in U.S. standardization and Australian neuroscience samples

A psychological measurement model provides an explicit definition of (a) the theoretical and (b) the numerical relationships between observed scores and the latent variables that underlie the observed scores. Examination of the metric invariance of a measurement model involves testing the hypothesis that all components of the model relating observed scores to latent variables are equal across groups. The assumption of metric invariance is necessary for simple interpretation of scores. Establishing metric invariance also has implications for interpretation of convergent and divergent validity and patterns of deficit or disability. In this study the equivalence of the measurement model derived from the U.S. Wechsler Adult Intelligence Scale-III standardization sample was compared with a heterogeneous neurosciences sample in Australia. A pattern of strict metric invariance was observed across samples. These results provide evidence of the generality of the model underlying measurement of cognitive abilities.