In this paper it will be shown that a certain class of constrained latent class models may be interpreted as a special case of nonparametric multidimensional item response models. The parameters of this latent class model will be estimated using an application of the Gibbs sampler. It will be illustrated that the Gibbs sampler is an excellent tool if inequality constraints have to be taken into consideration when making inferences. Model fit will be investigated using posterior predictive checks. Checks for manifest monotonicity, the agreement between the observed and expected conditional association structure, marginal local homogeneity, and the number of latent classes will be presented.

Current modeling of response times on test items has been strongly influenced by the paradigm of experimental reaction-time research in psychology. For instance, some of the models have a parameter structure that was chosen to represent a speed-accuracy tradeoff, while others equate speed directly with response time. Also, several response-time models seem to be unclear as to the level of parametrization they represent. A hierarchical framework for modeling speed and accuracy on test items is presented as an alternative to these models. The framework allows a “plug-and-play approach” with alternative choices of models for the response and response-time distributions as well as the distributions of their parameters. Bayesian treatment of the framework with Markov chain Monte Carlo (MCMC) computation facilitates the approach. Use of the framework is illustrated for the choice of a normal-ogive response model, a lognormal model for the response times, and multivariate normal models for their parameters with Gibbs sampling from the joint posterior distribution.

In this paper, we study the identification of a particular case of the 3PL model, namely when the discrimination parameters are all constant and equal to 1. We term this model, 1PL-G model. The identification analysis is performed under three different specifications. The first specification considers the abilities as unknown parameters. It is proved that the item parameters and the abilities are identified if a difficulty parameter and a guessing parameter are fixed at zero. The second specification assumes that the abilities are mutually independent and identically distributed according to a distribution known up to the scale parameter. It is shown that the item parameters and the scale parameter are identified if a guessing parameter is fixed at zero. The third specification corresponds to a semi-parametric 1PL-G model, where the distribution G generating the abilities is a parameter of interest. It is not only shown that, after fixing a difficulty parameter and a guessing parameter at zero, the item parameters are identified, but also that under those restrictions the distribution G is not identified. It is finally shown that, after introducing two identification restrictions, either on the distribution G or on the item parameters, the distribution G and the item parameters are identified provided an infinite quantity of items is available.

In their seminal work on characterizing the manifest probabilities of latent trait models, Cressie and Holland give a theoretically important characterization of the marginal Rasch model. Because their representation of the marginal Rasch model does not involve any latent trait, nor any specific distribution of a latent trait, it opens up the possibility for constructing a Markov chain - Monte Carlo method for Bayesian inference for the marginal Rasch model that does not rely on data augmentation. Such an approach would be highly efficient as its computational cost does not depend on the number of respondents, which makes it suitable for large-scale educational measurement. In this paper, such an approach will be developed and its operating characteristics illustrated with simulated data.

A new class of parametric models that generalize the multivariate probit model and the errors-in-variables model is developed to model and analyze ordinal data. A general model structure is assumed to accommodate the information that is obtained via surrogate variables. A hybrid Gibbs sampler is developed to estimate the model parameters. To obtain a rapidly converged algorithm, the parameter expansion technique is applied to the correlation structure of the multivariate probit models. The proposed model and method of analysis are demonstrated with real data examples and simulation studies.

It is shown that measurement error in predictor variables can be modeled using item response theory (IRT). The predictor variables, that may be defined at any level of an hierarchical regression model, are treated as latent variables. The normal ogive model is used to describe the relation between the latent variables and dichotomous observed variables, which may be responses to tests or questionnaires. It will be shown that the multilevel model with measurement error in the observed predictor variables can be estimated in a Bayesian framework using Gibbs sampling. In this article, handling measurement error via the normal ogive model is compared with alternative approaches using the classical true score model. Examples using real data are given.

Standard item response theory (IRT) models fit to dichotomous examination responses ignore the fact that sets of items (testlets) often come from a single common stimuli (e.g. a reading comprehension passage). In this setting, all items given to an examinee are unlikely to be conditionally independent (given examinee proficiency). Models that assume conditional independence will overestimate the precision with which examinee proficiency is measured. Overstatement of precision may lead to inaccurate inferences such as prematurely ending an examination in which the stopping rule is based on the estimated standard error of examinee proficiency (e.g., an adaptive test). To model examinations that may be a mixture of independent items and testlets, we modified one standard IRT model to include an additional random effect for items nested within the same testlet. We use a Bayesian framework to facilitate posterior inference via a Data Augmented Gibbs Sampler (DAGS; Tanner & Wong, 1987). The modified and standard IRT models are both applied to a data set from a disclosed form of the SAT. We also provide simulation results that indicates that the degree of precision bias is a function of the variability of the testlet effects, as well as the testlet design.

Multinomial processing tree models are widely used in many areas of psychology. A hierarchical extension of the model class is proposed, using a multivariate normal distribution of person-level parameters with the mean and covariance matrix to be estimated from the data. The hierarchical model allows one to take variability between persons into account and to assess parameter correlations. The model is estimated using Bayesian methods with weakly informative hyperprior distribution and a Gibbs sampler based on two steps of data augmentation. Estimation, model checks, and hypotheses tests are discussed. The new method is illustrated using a real data set, and its performance is evaluated in a simulation study.

The Gibbs sampler can be used to obtain samples of arbitrary size from the posterior distribution over the parameters of a structural equation model (SEM) given covariance data and a prior distribution over the parameters. Point estimates, standard deviations and interval estimates for the parameters can be computed from these samples. If the prior distribution over the parameters is uninformative, the posterior is proportional to the likelihood, and asymptotically the inferences based on the Gibbs sample are the same as those based on the maximum likelihood solution, for example, output from LISREL or EQS. In small samples, however, the likelihood surface is not Gaussian and in some cases contains local maxima. Nevertheless, the Gibbs sample comes from the correct posterior distribution over the parameters regardless of the sample size and the shape of the likelihood surface. With an informative prior distribution over the parameters, the posterior can be used to make inferences about the parameters underidentified models, as we illustrate on a simple errors-in-variables model.

A Bayesian procedure to estimate the three-parameter normal ogive model and a generalization of the procedure to a model with multidimensional ability parameters are presented. The procedure is a generalization of a procedure by Albert (1992) for estimating the two-parameter normal ogive model. The procedure supports analyzing data from multiple populations and incomplete designs. It is shown that restrictions can be imposed on the factor matrix for testing specific hypotheses about the ability structure. The technique is illustrated using simulated and real data.

A Bayesian approach is developed for analyzing nonlinear structural equation models with nonignorable missing data. The nonignorable missingness mechanism is specified by a logistic regression model. A hybrid algorithm that combines the Gibbs sampler and the Metropolis–Hastings algorithm is used to produce the joint Bayesian estimates of structural parameters, latent variables, parameters in the nonignorable missing model, as well as their standard errors estimates. A goodness-of-fit statistic for assessing the plausibility of the posited nonlinear structural equation model is introduced, and a procedure for computing the Bayes factor for model comparison is developed via path sampling. Results obtained with respect to different missing data models, and different prior inputs are compared via simulation studies. In particular, it is shown that in the presence of nonignorable missing data, results obtained by the proposed method with a nonignorable missing data model are significantly better than those that are obtained under the missing at random assumption. A real example is presented to illustrate the newly developed Bayesian methodologies.

Recently, it has been recognized that the commonly used linear structural equation model is inadequate to deal with some complicated substantive theory. A new nonlinear structural equation model with fixed covariates is proposed in this article. A procedure, which utilizes the powerful path sampling for computing the Bayes factor, is developed for model comparison. In the implementation, the required random observations are simulated via a hybrid algorithm that combines the Gibbs sampler and the Metropolis-Hastings algorithm. It is shown that the proposed procedure is efficient and flexible; and it produces Bayesian estimates of the parameters, latent variables, and their highest posterior density intervals as by-products. Empirical performances of the proposed procedure such as sensitivity to prior inputs are illustrated by a simulation study and a real example.

The main purpose of this article is to develop a Bayesian approach for structural equation models with ignorable missing continuous and polytomous data. Joint Bayesian estimates of thresholds, structural parameters and latent factor scores are obtained simultaneously. The idea of data augmentation is used to solve the computational difficulties involved. In the posterior analysis, in addition to the real missing data, latent variables and latent continuous measurements underlying the polytomous data are treated as hypothetical missing data. An algorithm that embeds the Metropolis-Hastings algorithm within the Gibbs sampler is implemented to produce the Bayesian estimates. A goodness-of-fit statistic for testing the posited model is presented. It is shown that the proposed approach is not sensitive to prior distributions and can handle situations with a large number of missing patterns whose underlying sample sizes may be small. Computational efficiency of the proposed procedure is illustrated by simulation studies and a real example.

By means of more than a dozen user friendly packages, structural equation models (SEMs) are widely used in behavioral, education, social, and psychological research. As the underlying theory and methods in these packages are vulnerable to outliers and distributions with longer-than-normal tails, a fundamental problem in the field is the development of robust methods to reduce the influence of outliers and the distributional deviation in the analysis. In this paper we develop a maximum likelihood (ML) approach that is robust to outliers and symmetrically heavy-tailed distributions for analyzing nonlinear SEMs with ignorable missing data. The analytic strategy is to incorporate a general class of distributions into the latent variables and the error measurements in the measurement and structural equations. A Monte Carlo EM (MCEM) algorithm is constructed to obtain the ML estimates, and a path sampling procedure is implemented to compute the observed-data log-likelihood and then the Bayesian information criterion for model comparison. The proposed methodologies are illustrated with simulation studies and an example.

The basic theme of the EM algorithm, to repeatedly use complete-data methods to solve incomplete data problems, is also a theme of several more recent statistical techniques. These techniques—multiple imputation, data augmentation, stochastic relaxation, and sampling importance resampling—combine simulation techniques with complete-data methods to attack problems that are difficult or impossible for EM.

In this paper, we propose a Bayesian framework for estimating finite mixtures of the LISREL model. The basic idea in our analysis is to augment the observed data of the manifest variables with the latent variables and the allocation variables. The Gibbs sampler is implemented to obtain the Bayesian solution. Other associated statistical inferences, such as the direct estimation of the latent variables, establishment of a goodness-of-fit assessment for a posited model, Bayesian classification, residual and outlier analyses, are discussed. The methodology is illustrated with a simulation study and a real example.

In this article, a two-level regression model is imposed on the ability parameters in an item response theory (IRT) model. The advantage of using latent rather than observed scores as dependent variables of a multilevel model is that it offers the possibility of separating the influence of item difficulty and ability level and modeling response variation and measurement error. Another advantage is that, contrary to observed scores, latent scores are test-independent, which offers the possibility of using results from different tests in one analysis where the parameters of the IRT model and the multilevel model can be concurrently estimated. The two-parameter normal ogive model is used for the IRT measurement model. It will be shown that the parameters of the two-parameter normal ogive model and the multilevel model can be estimated in a Bayesian framework using Gibbs sampling. Examples using simulated and real data are given.

The full information item factor (FIIF) model is very useful for analyzing relations of dichotomous variables. In this article, we present a feasible procedure to assess local influence of minor perturbations for identifying influence aspects of the FIIF model. The development is based on a Q-displacement function which is closely related with the Monte Carlo EM algorithm in the ML estimation. In the E-step of this algorithm, the conditional expectations are approximated by sample means of observations simulated by the Gibbs sampler from the appropriate conditional distributions. It turns out that these observations can be utilized for computing the building blocks of the proposed diagnostic measures. The diagnoses are based on the conformal normal curvature that can be computed easily. A number of interesting perturbation schemes are considered. The methodology is illustrated with two real examples.

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.