Latent class models for testing monotonicity and invariant item ordering for polytomous items

Two assumptions that are relevant to many applications using item response theory are the assumptions of monotonicity (M) and invariant item ordering (IIO). A latent class model is proposed for ordinal items with inequality constraints on the class-specific item means. This model is used as a tool for testing for violations of M and IIO. A Gibbs sampling scheme is used for estimating the model parameters. It is shown that the deviance information criterion can be used as an overall test of M and IIO, while posterior predictive checks can be used to test these assumptions at the item level. A real data application illustrates a model-fitting strategy for detecting items that violate M and IIO.     

A multidimensional item response model: Constrained latent class analysis using the gibbs sampler and posterior predictive checks

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.This paper is supported by grant S40-645 of the Dutch Organization for Scientific Research (NWO).     

Model inference or model selection: discussion of Klugkist, Laudy, and Hoijtink (2005)

I. Klugkist, O. Laudy, and H. Hoijtink (2005) presented a Bayesian approach to analysis of variance models with inequality constraints. Constraints may play 2 distinct roles in data analysis. They may represent prior information that allows more precise inferences regarding parameter values, or they may describe a theory to be judged against the data. In the latter case, the authors emphasized the use of Bayes factors and posterior model probabilities to select the best theory. One difficulty is that interpretation of the posterior model probabilities depends on which other theories are included in the comparison. The posterior distribution of the parameters under an unconstrained model allows one to quantify the support provided by the data for inequality constraints without requiring the model selection framework.     

Bayesian Semiparametric Structural Equation Models with Latent Variables

Structural equation models (SEMs) with latent variables are widely useful for sparse covariance structure modeling and for inferring relationships among latent variables. Bayesian SEMs are appealing in allowing for the incorporation of prior information and in providing exact posterior distributions of unknowns, including the latent variables. In this article, we propose a broad class of semiparametric Bayesian SEMs, which allow mixed categorical and continuous manifest variables while also allowing the latent variables to have unknown distributions. In order to include typical identifiability restrictions on the latent variable distributions, we rely on centered Dirichlet process (CDP) and CDP mixture (CDPM) models. The CDP will induce a latent class model with an unknown number of classes, while the CDPM will induce a latent trait model with unknown densities for the latent traits. A simple and efficient Markov chain Monte Carlo algorithm is developed for posterior computation, and the methods are illustrated using simulated examples, and several applications.     

Multilevel IRT using dichotomous and polytomous response data

A structural multilevel model is presented where some of the variables cannot be observed directly but are measured using tests or questionnaires. Observed dichotomous or ordinal polytomous response data serve to measure the latent variables using an item response theory model. The latent variables can be defined at any level of the multilevel model. A Bayesian procedure Markov chain Monte Carlo (MCMC), to estimate all parameters simultaneously is presented. It is shown that certain model checks and model comparisons can be done using the MCMC output. The techniques are illustrated using a simulation study and an application involving students' achievements on a mathematics test and test results regarding management characteristics of teachers and principles.     

An Isotonic Partial Credit Model for Ordering Subjects on the Basis of Their Sum Scores

In practice, the sum of the item scores is often used as a basis for comparing subjects. For items that have more than two ordered score categories, only the partial credit model (PCM) and special cases of this model imply that the subjects are stochastically ordered on the common latent variable. However, the PCM is very restrictive with respect to the constraints that it imposes on the data. In this paper, sufficient conditions for the stochastic ordering of subjects by their sum score are obtained. These conditions define the isotonic (nonparametric) PCM model. The isotonic PCM is more flexible than the PCM, which makes it useful for a wider variety of tests. Also, observable properties of the isotonic PCM are derived in the form of inequality constraints. It is shown how to obtain estimates of the score distribution under these constraints by using the Gibbs sampling algorithm. A small simulation study shows that the Bayesian p-values based on the log-likelihood ratio statistic can be used to assess the fit of the isotonic PCM to the data, where model-data fit can be taken as a justification of the use of the sum score to order subjects.     

Likelihood inference on the underlying structure of IRT models

The assumptions underlying item response theory (IRT) models may be expressed as a set of equality and inequality constraints on the parameters of a latent class model. It is well known that the same assumptions imply that the parameters of the manifest distribution have to satisfy a more complicated set of inequality constraints which, however, are necessary but not sufficient. In this paper, we describe how the theory for likelihood-based inference under equality and inequality constraints may be used to test the underlying assumptions of IRT models. It turns out that the analysis based directly on the latent structure is simpler and more flexible. In particular, we indicate how several interesting extensions of the Rasch model may be obtained by partial relaxation of the basic constraints. An application to a data set provided by Educational Testing Service is used to illustrate the approach.We thank Dr. Gorman and Dr. Rogers of the Educational Testing Service for providing the data analyzed in Section 4. We also thank three reviewers for comments and suggestions.This revised article was published online in August 2005 with the PDF paginated correctly.     

Bayesian Inference for Growth Mixture Models with Latent Class Dependent Missing Data

Growth mixture models (GMMs) with nonignorable missing data have drawn increasing attention in research communities but have not been fully studied. The goal of this article is to propose and to evaluate a Bayesian method to estimate the GMMs with latent class dependent missing data. An extended GMM is first presented in which class probabilities depend on some observed explanatory variables and data missingness depends on both the explanatory variables and a latent class variable. A full Bayesian method is then proposed to estimate the model. Through the data augmentation method, conditional posterior distributions for all model parameters and missing data are obtained. A Gibbs sampling procedure is then used to generate Markov chains of model parameters for statistical inference. The application of the model and the method is first demonstrated through the analysis of mathematical ability growth data from the National Longitudinal Survey of Youth 1997 (Bureau of Labor Statistics, U.S. Department of Labor, 1997). A simulation study considering 3 main factors (the sample size, the class probability, and the missing data mechanism) is then conducted and the results show that the proposed Bayesian estimation approach performs very well under the studied conditions. Finally, some implications of this study, including the misspecified missingness mechanism, the sample size, the sensitivity of the model, the number of latent classes, the model comparison, and the future directions of the approach, are discussed.     

Inequality constrained analysis of variance: a Bayesian approach

Researchers often have one or more theories or expectations with respect to the outcome of their empirical research. When researchers talk about the expected relations between variables if a certain theory is correct, their statements are often in terms of one or more parameters expected to be larger or smaller than one or more other parameters. Stated otherwise, their statements are often formulated using inequality constraints. In this article, a Bayesian approach to evaluate analysis of variance or analysis of covariance models with inequality constraints on the (adjusted) means is presented. This evaluation contains two issues: estimation of the parameters given the restrictions using the Gibbs sampler and model selection using Bayes factors in the case of competing theories. The article concludes with two illustrations: a one-way analysis of covariance and an analysis of a three-way table of ordered means.     

Bayesian model selection for mixtures of structural equation models with an unknown number of components

This paper considers mixtures of structural equation models with an unknown number of components. A Bayesian model selection approach is developed based on the Bayes factor. A procedure for computing the Bayes factor is developed via path sampling, which has a number of nice features. The key idea is to construct a continuous path linking the competing models; then the Bayes factor can be estimated efficiently via grids in [0, 1] and simulated observations that are generated by the Gibbs sampler from the posterior distribution. Bayesian estimates of the structural parameters, latent variables, as well as other statistics can be produced as by‐products. The properties and merits of the proposed procedure are discussed and illustrated by means of a simulation study and a real example.     

A prior predictive loss function for the evaluation of inequality constrained hypotheses

In many types of statistical modeling, inequality constraints are imposed between the parameters of interest. As we will show in this paper, the DIC (i.e., posterior Deviance Information Criterium as proposed as a Bayesian model selection tool by Spiegelhalter, Best, Carlin, & Van Der Linde, 2002) fails when comparing inequality constrained hypotheses. In this paper, we will derive the prior DIC and show that it also fails when comparing inequality constrained hypotheses. However, it will be shown that a modification of the prior predictive loss function that is minimized by the prior DIC renders a criterion that does have the properties needed in order to be able to compare inequality constrained hypotheses. This new criterion will be called the Prior Information Criterion (PIC) and will be illustrated and evaluated using simulated data and examples. The PIC has a close connection with the marginal likelihood in combination with the encompassing prior approach and both methods will be compared. All in all, the main message of the current paper is: (1) do not use the classical DIC when evaluating inequality constrained hypotheses, better use the PIC; and (2) the PIC is considered a proper model selection tool in the context of evaluating inequality constrained hypotheses.     

Latent variable models for partially ordered responses and trajectory analysis of anger‐related feelings

A general framework is presented for the analysis of partially ordered set (poset) data. The work is motivated by the need to analyse poset data such as multi‐componential responses in psychological measurement and partially accomplished cognitive tasks in educational measurement. It is shown how the generalized loglinear model can be used to represent poset data that form a lattice and how latent‐variable models can be constructed by further specifying the canonical parameters of the loglinear representation. The approach generalizes a class of latent‐variable models for completely ordered data. We apply the methods to analyse data on the frequency and intensity of anger‐related feelings. Furthermore, we propose a trajectory analysis to gain insight into the response function of partially ordered emotional states.     

Using Latent Class Analysis to Model Temperament Types

Mixture models are appropriate for data that arise from a set of qualitatively different subpopulations. In this study, latent class analysis was applied to observational data from a laboratory assessment of infant temperament at four months of age. The EM algorithm was used to fit the models, and the Bayesian method of posterior predictive checks was used for model selection. Results show at least three types of infant temperament, with patterns consistent with those identified by previous researchers who classified the infants using a theoretically based system. Multiple imputation of group memberships is proposed as an alternative to assigning subjects to the latent class with maximum posterior probability in order to reflect variance due to uncertainty in the parameter estimation. Latent class membership at four months of age predicted longitudinal outcomes at four years of age. The example illustrates issues relevant to all mixture models, including estimation, multi-modality, model selection, and comparisons based on the latent group indicators.     

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.     

Inferring the structure of latent class

Present optimization techniques in latent class analysis apply the expectation maximization algorithm or the Newton-Raphson algorithm for optimizing the parameter values of a prespecified model. These techniques can be used to find maximum likelihood estimates of the parameters, given the specified structure of the model, which is defined by the number of classes and, possibly, fixation and equality constraints. The model structure is usually chosen on theoretical grounds. A large variety of structurally different latent class models can be compared using goodness-of-fit indices of the chi-square family, Akaike’s information criterion, the Bayesian information criterion, and various other statistics. However, finding the optimal structure for a given goodness-of-fit index often requires a lengthy search in which all kinds of model structures are tested. Moreover, solutions may depend on the choice of initial values for the parameters. This article presents a new method by which one can simultaneously infer the model structure from the data and optimize the parameter values. The method consists of a genetic algorithm in which any goodness-of-fit index can be used as a fitness criterion. In a number of test cases in which data sets from the literature were used, it is shown that this method provides models that fit equally well as or better than the models suggested in the original articles.     

Bayes Factors for Evaluating Latent Monotonicity in Polytomous Item Response Theory Models

The assumption of latent monotonicity is made by all common parametric and nonparametric polytomous item response theory models and is crucial for establishing an ordinal level of measurement of the item score. Three forms of latent monotonicity can be distinguished: monotonicity of the cumulative probabilities, of the continuation ratios, and of the adjacent-category ratios. Observable consequences of these different forms of latent monotonicity are derived, and Bayes factor methods for testing these consequences are proposed. These methods allow for the quantification of the evidence both in favor and against the tested property. Both item-level and category-level Bayes factors are considered, and their performance is evaluated using a simulation study. The methods are applied to an empirical example consisting of a 10-item Likert scale to investigate whether a polytomous item scoring rule results in item scores that are of ordinal level measurement.

     

多项式加工树模型次序约束的量化分析及其应用

多项式加工树（MPT）模型是一种认知测量模型,能够对潜在认知过程进行测量和检验。已有研究探讨了二链MPT模型次序约束的重新参数化问题,本研究探讨了MPT模型次序约束的量化分析方法并从二链推广到多链,同时归纳出MPT模型参数向量内和参数向量间两参数次序约束量化分析的结论。数据分析结果表明该方法不仅在MPT模型框架下验证了潜在参数次序关系,而且给出了约束的量化指标,为潜在认知测量提供更有意义的解释。     

A tutorial on Bayesian nonparametric models

A key problem in statistical modeling is model selection, that is, how to choose a model at an appropriate level of complexity. This problem appears in many settings, most prominently in choosing the number of clusters in mixture models or the number of factors in factor analysis. In this tutorial, we describe Bayesian nonparametric methods, a class of methods that side-steps this issue by allowing the data to determine the complexity of the model. This tutorial is a high-level introduction to Bayesian nonparametric methods and contains several examples of their application.     

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.