Generalized linear models with ordinally‐observed covariates

An ordinally‐observed variable is a variable that is only partially observed through an ordinal surrogate. Although statistical models for ordinally‐observed response variables are well known, relatively little attention has been given to the problem of ordinally‐observed regressors. In this paper I show that if surrogates to ordinally‐observed covariates are used as regressors in a generalized linear model then the resulting measurement error in the covariates can compromise the consistency of point estimators and standard errors for the effects of fully‐observed regressors. To properly account for this measurement error when making inferences concerning the fully‐observed regressors, I propose a general modelling framework for generalized linear models with ordinally‐observed covariates. I discuss issues of model specification, identification, and estimation, and illustrate these with examples.     

Generalized latent trait models

In this paper we discuss a general model framework within which manifest variables with different distributions in the exponential family can be analyzed with a latent trait model. A unified maximum likelihood method for estimating the parameters of the generalized latent trait model will be presented. We discuss in addition the scoring of individuals on the latent dimensions. The general framework presented allows, not only the analysis of manifest variables all of one type but also the simultaneous analysis of a collection of variables with different distributions. The approach used analyzes the data as they are by making assumptions about the distribution of the manifest variables directly.     

Dimension in latent variable models

Psychologists would like to say that a probability distribution on {0,1}ⁿ is d-dimensional if (1) the distribution can be represented by some smooth d-dimensional latent variable model and (2) the distribution cannot be represented by any smooth d−1 dimensional model. This does not work out because for d>1, every distribution that can be represented by a smooth d-dimensional model can also be represented by a smooth one-dimensional model. A proof and discussion of some implications of this mathematical result follow.     

Logistic latent trait models with linear constraints

Two linearly constrained logistic models which are based on the well-known dichotomous Rasch model, the ‘linear logistic test model’ (LLTM) and the ‘linear logistic model with relaxed assumptions’ (LLRA), are discussed. Necessary and sufficient conditions for the existence of unique conditional maximum likelihood estimates of the structural model parameters are derived. Methods for testing composite hypotheses within the framework of these models and a number of typical applications to real data are mentioned.     

Modelling non‐ignorable missing‐data mechanisms with item response theory models

A model‐based procedure for assessing the extent to which missing data can be ignored and handling non‐ignorable missing data is presented. The procedure is based on item response theory modelling. As an example, the approach is worked out in detail in conjunction with item response data modelled using the partial credit and generalized partial credit models. Simulation studies are carried out to assess the extent to which the bias caused by ignoring the missing‐data mechanism can be reduced. Finally, the feasibility of the procedure is demonstrated using data from a study to calibrate a medical disability scale.     

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 general class of latent variable models for ordinal manifest variables with covariate effects on the manifest and latent variables

Previous work on a general class of multidimensional latent variable models for analysing ordinal manifest variables is extended here to allow for direct covariate effects on the manifest ordinal variables and covariate effects on the latent variables. A full maximum likelihood estimation method is used to estimate all the model parameters simultaneously. Goodness‐of‐fit statistics and standard errors are discussed. Two examples from the 1996 British Social Attitudes Survey are used to illustrate the methodology.     

Compensatory and non‐compensatory multidimensional randomized item response models

Randomized response (RR) models are often used for analysing univariate randomized response data and measuring population prevalence of sensitive behaviours. There is much empirical support for the belief that RR methods improve the cooperation of the respondents. Recently, RR models have been extended to measure individual unidimensional behaviour. An extension of this modelling framework is proposed to measure compensatory or non‐compensatory multiple sensitive factors underlying the randomized item response process. A confirmatory multidimensional randomized item response theory model (MRIRT) is proposed for the analysis of multivariate RR data by modelling the response process and specifying structural relationships between sensitive behaviours and background information. A Markov chain Monte Carlo algorithm is developed to estimate simultaneously the parameters of the MRIRT model. The model extension enables the computation of individual true item response probabilities, estimates of individuals’ sensitive behaviour on different domains, and their relationships with background variables. An MRIRT analysis is presented of data from a college alcohol problem scale, measuring alcohol‐related socio‐emotional and community problems, and alcohol expectancy questionnaire, measuring alcohol‐related sexual enhancement expectancies. Students were interviewed via direct or RR questioning. Scores of alcohol‐related problems and expectancies are significantly higher for the group of students questioned using the RR technique. Alcohol‐related problems and sexual enhancement expectancies are positively moderately correlated and vary differently across gender and universities.     

Information-theoretic latent distribution modeling: distinguishing discrete and continuous latent variable models

Distinguishing between discrete and continuous latent variable distributions has become increasingly important in numerous domains of behavioral science. Here, the authors explore an information-theoretic approach to latent distribution modeling, in which the ability of latent distribution models to represent statistical information in observed data is emphasized. The authors conclude that loss of statistical information with a decrease in the number of latent values provides an attractive basis for comparing discrete and continuous latent variable models. Theoretical considerations as well as the results of 2 Monte Carlo simulations indicate that information theory provides a sound basis for modeling latent distributions and distinguishing between discrete and continuous latent variable models in particular.     

Bayesian analysis of mixtures in structural equation models with non‐ignorable missing data

Structural equation models (SEMs) have become widely used to determine the interrelationships between latent and observed variables in social, psychological, and behavioural sciences. As heterogeneous data are very common in practical research in these fields, the analysis of mixture models has received a lot of attention in the literature. An important issue in the analysis of mixture SEMs is the presence of missing data, in particular of data missing with a non‐ignorable mechanism. However, only a limited amount of work has been done in analysing mixture SEMs with non‐ignorable missing data. The main objective of this paper is to develop a Bayesian approach for analysing mixture SEMs with an unknown number of components and non‐ignorable missing data. A simulation study shows that Bayesian estimates obtained by the proposed Markov chain Monte Carlo methods are accurate and the Bayes factor computed via a path sampling procedure is useful for identifying the correct number of components, selecting an appropriate missingness mechanism, and investigating various effects of latent variables in the mixture SEMs. A real data set on a study of job satisfaction is used to demonstrate the methodology.     

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.     

Non‐linear structural equation models with correlated continuous and discrete data

Structural equation models (SEMs) have been widely applied to examine interrelationships among latent and observed variables in social and psychological research. Motivated by the fact that correlated discrete variables are frequently encountered in practical applications, a non‐linear SEM that accommodates covariates, and mixed continuous, ordered, and unordered categorical variables is proposed. Maximum likelihood methods for estimation and model comparison are discussed. One real‐life data set about cardiovascular disease is used to illustrate the methodologies.     

Bayesian latent variable models for the analysis of experimental psychology data

In this paper, we address the use of Bayesian factor analysis and structural equation models to draw inferences from experimental psychology data. While such application is non-standard, the models are generally useful for the unified analysis of multivariate data that stem from, e.g., subjects’ responses to multiple experimental stimuli. We first review the models and the parameter identification issues inherent in the models. We then provide details on model estimation via JAGS and on Bayes factor estimation. Finally, we use the models to re-analyze experimental data on risky choice, comparing the approach to simpler, alternative methods.     

Cogency,Warrant Transmission‐Increase and non‐Ideal Thinkers

Contemporary debates concerning warrant transmission take for granted this thesis: when warrant transmission fails the argument fails. I challenge this thesis. An argument with conclusion C, addressed to subject S, can be cogent in the sense that recognition that the premises entail (or make highly likely) C can rationally foster in S the belief in C, without the warrant for C necessarily being gained (or reinforced) by such recognition. A key idea is to accept that some arguments should be understood in a way that involves the abandonment of two characteristic idealizations imposed on rational thinkers by Bayesian modelling.     

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.     

Testing and modelling non‐normality within the one‐factor model

Maximum likelihood estimation in the one‐factor model is based on the assumption of multivariate normality for the observed data. This general distributional assumption implies three specific assumptions for the parameters in the one‐factor model: the common factor has a normal distribution; the residuals are homoscedastic; and the factor loadings do not vary across the common factor scale. When any of these assumptions is violated, non‐normality arises in the observed data. In this paper, a model is presented based on marginal maximum likelihood to enable explicit tests of these assumptions. In addition, the model is suitable to incorporate the detected violations, to enable statistical modelling of these effects. Two simulation studies are reported in which the viability of the model is investigated. Finally, the model is applied to IQ data to demonstrate its practical utility as a means to investigate ability differentiation.     

On models with variable universe

In this paper some parts of the model theory for logics based on generalised Kripke semantics are developed. Löwenheim-Skolem theorems and some applications of ultraproduct constructions for generalised Kripke models with variable universe are investigated using similar theorems of the model theory for classical logic. The results are generalizations of the theorems of [4].     

Using phantom and imaginary latent variables to parameterize constraints in linear structural models

The most widely-used computer programs for structural equation models analysis are the LISREL series of Jöreskog and Sörbom. The only types of constraints which may be made directly are fixing parameters at a constant value and constraining parameters to be equal. Rindskopf (1983) showed how these simple properties could be used to represent models with more complicated constraints, namely inequality constraints on unique variances. In this paper, two new concepts are introduced which enable a much wider variety of constraints to be made. The concepts, phantom and imaginary latent variables, allow fairly general equality and inequality constraints on factor loadings and structural model coefficients.During the preparation of this article, it was discovered that another researcher, Jack McArdle, had concurrently and independently discovered some of the techniques reported here. While he has chosen not to publish his research, I wish to acknowledge his work. I would like to thank Art Woodward for telling me about sort-of simple structure.     

Robustness of normal theory methods in the analysis of linear latent variate models

The structure of the covariance matrix of sample covariances under the class of linear latent variate models is derived using properties of cumulants. This is employed to provide a general framework for robustness of statistical inference in the analysis of covariance structures arising from linear latent variate models. Conditions for normal theory estimators and test statistics to retain each of their usual asymptotic properties under non-normality of latent variates are given. Factor analysis, LISREL and other models are discussed as examples.