A latent class model for rating data

A latent class model for rating data is presented which is the analogue of Andrich's binomial Rasch model for Lazarsfeld's latent class analysis (LCA). The response probabilities for the rating categories follow a binomial distribution and depend on class-specific item parameters. The EM-algorithm for parameter estimation as well as goodness of fit tests for the model are described. An example using questionnaire items on interest in physics illustrates the use of the model as an alternative to the latent trait approach of analyzing test data.I would like to thank Clifford Clogg and the anonymous reviewers for their helpful comments.     

A generalized longitudinal mixture IRT model for measuring differential growth in learning environments

This article describes a generalized longitudinal mixture item response theory (IRT) model that allows for detecting latent group differences in item response data obtained from electronic learning (e-learning) environments or other learning environments that result in large numbers of items. The described model can be viewed as a combination of a longitudinal Rasch model, a mixture Rasch model, and a random-item IRT model, and it includes some features of the explanatory IRT modeling framework. The model assumes the possible presence of latent classes in item response patterns, due to initial person-level differences before learning takes place, to latent class-specific learning trajectories, or to a combination of both. Moreover, it allows for differential item functioning over the classes. A Bayesian model estimation procedure is described, and the results of a simulation study are presented that indicate that the parameters are recovered well, particularly for conditions with large item sample sizes. The model is also illustrated with an empirical sample data set from a Web-based e-learning environment.     

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.     

Testing for DIF in a model with single peaked item characteristic curves: The parella model

The PARELLA model is a probabilistic parallelogram model that can be used for the measurement of latent attitudes or latent preferences. The data analyzed are the dichotomous responses of persons to items, with a one (zero) indicating agreement (disagreement) with the content of the item. The model provides a unidimensional representation of persons and items. The response probabilities are a function of the distance between person and item: the smaller the distance, the larger the probability that a person will agree with the content of the item. This paper discusses how the approach to differential item functioning presented by Thissen, Steinberg, and Wainer can be implemented for the PARELLA model. Requests for the PARELLA software should be sent to Iec Progamma PO Box 841, 9700 AV Groningen, The Netherlands.     

A latent topic model with Markov transition for process data

We propose a latent topic model with a Markov transition for process data, which consists of time-stamped events recorded in a log file. Such data are becoming more widely available in computer-based educational assessment with complex problem-solving items. The proposed model can be viewed as an extension of the hierarchical Bayesian topic model with a hidden Markov structure to accommodate the underlying evolution of an examinee's latent state. Using topic transition probabilities along with response times enables us to capture examinees' learning trajectories, making clustering/classification more efficient. A forward-backward variational expectation-maximization (FB-VEM) algorithm is developed to tackle the challenging computational problem. Useful theoretical properties are established under certain asymptotic regimes. The proposed method is applied to a complex problem-solving item in the 2012 version of the Programme for International Student Assessment (PISA).     

Latent Class Models for Diary Method Data: Parameter Estimation by Local Computations

The increasing use of diary methods calls for the development of appropriate statistical methods. For the resulting panel data, latent Markov models can be used to model both individual differences and temporal dynamics. The computational burden associated with these models can be overcome by exploiting the conditional independence relations implied by the model. This is done by associating a probabilistic model with a directed acyclic graph, and applying transformations to the graph. The structure of the transformed graph provides a factorization of the joint probability function of the manifest and latent variables, which is the basis of a modified and more efficient E-step of the EM algorithm. The usefulness of the approach is illustrated by estimating a latent Markov model involving a large number of measurement occasions and, subsequently, a hierarchical extension of the latent Markov model that allows for transitions at different levels. Furthermore, logistic regression techniques are used to incorporate restrictions on the conditional probabilities and to account for the effect of covariates. Throughout, models are illustrated with an experience sampling methodology study on the course of emotions among anorectic patients. Frank Rijmen was partly supported by the Fund for Scientific Research Flanders (FWO).     

Bayesian estimation and model selection in ordered latent class models for polytomous items

In a latent class IRT model in which the latent classes are ordered on one dimension, the class specific response probabilities are subject to inequality constraints. The number of these inequality constraints increase dramatically with the number of response categories per item, if assumptions like monotonicity or double monotonicity of the cumulative category response functions are postulated. A Markov chain Monte Carlo method, the Gibbs sampler, can sample from the multivariate posterior distribution of the parameters under the constraints. Bayesian model selection can be done by posterior predictive checks and Bayes factors. A simulation study is done to evaluate results of the application of these methods to ordered latent class models in three realistic situations. Also, an example of the presented methods is given for existing data with polytomous items. It can be concluded that the Bayesian estimation procedure can handle the inequality constraints on the parameters very well. However, the application of Bayesian model selection methods requires more research.     

A Composite Likelihood Inference in Latent Variable Models for Ordinal Longitudinal Responses

The paper proposes a composite likelihood estimation approach that uses bivariate instead of multivariate marginal probabilities for ordinal longitudinal responses using a latent variable model. The model considers time-dependent latent variables and item-specific random effects to be accountable for the interdependencies of the multivariate ordinal items. Time-dependent latent variables are linked with an autoregressive model. Simulation results have shown composite likelihood estimators to have a small amount of bias and mean square error and as such they are feasible alternatives to full maximum likelihood. Model selection criteria developed for composite likelihood estimation are used in the applications. Furthermore, lower-order residuals are used as measures-of-fit for the selected models.     

The EM algorithm for latent class analysis with equality constraints

The EM algorithm is a popular iterative method for estimating parameters in the latent class model where at each step the unknown parameters can be estimated simply as weighted sums of some latent proportions. The algorithm may also be used when some parameters are constrained to equal given constants or each other. It is shown that in the general case with equality constraints, the EM algorithm is not simple to apply because a nonlinear equation has to be solved. This problem arises, mainly, when equality constrints are defined over probabilities indifferent combinations of variables and latent classes. A simple condition is given in which, although probabilities in different variable-latent class combinations are constrained to be equal, the EM algorithm is still simple to apply.The authors are grateful to the Editor and the anonymous reviewers for their helpful comments on an earlier draft of this paper. C. C. Clogg and R. Luijkx are also acknowledged for verifying our results with their computer programs MLLSA and LCAG, respectively.     

Distinguishing Between Latent Classes and Continuous Factors with Categorical Outcomes: Class Invariance of Parameters of Factor Mixture Models

Factor mixture models are latent variable models with categorical and continuous latent variables that can be used as a model-based approach to clustering. A previous article covered the results of a simulation study showing that in the absence of model violations, it is usually possible to choose the correct model when fitting a series of models with different numbers of classes and factors within class. The response format in the first study was limited to normally distributed outcomes. This article has 2 main goals, first, to replicate parts of the first study with 5-point Likert scale and binary outcomes, and second, to address the issue of testing class invariance of thresholds and loadings. Testing for class invariance of parameters is important in the context of measurement invariance and when using mixture models to approximate nonnormal distributions. Results show that it is possible to discriminate between latent class models and factor models even if responses are categorical. Comparing models with and without class-specific parameters can lead to incorrectly accepting parameter invariance if the compared models differ substantially with respect to the number of estimated parameters. The simulation study is complemented with an illustration of a factor mixture analysis of 10 binary depression items obtained from a female subsample of the Virginia Twin Registry.     

Local homogeneity in latent trait models. A characterization of the homogeneous monotone irt model

The stochastic subject formulation of latent trait models contends that, within a given subject, the event of obtaining a certain response pattern may be probabilistic. Ordinary latent trait models do not imply that these within-subject probabilities are identical to the conditional probabilities specified by the model. The latter condition is called local homogeneity. It is shown that local homgeneity is equivalent to subpopulation invariance of the model. In case of the monotone IRT model, local homogeneity implies absence of item bias, absence of item specific traits, and the possibility to join overlapping subtests. The following characterization theorem is proved: the homogeneous monotone IRT model holds for a finite or countable item pool if and only if the pool is experimentally independent and pairwise nonnegative association holds in every positive subpopulation.This research was supported by the Dutch Interuniversity Graduate School of Psychometrics and Sociometrics. The authors wish to thank two reviewers for their thorough comments.     

A latent trait model for dichotomous choice data

The PARELLA model is a probabilistic parallelogram model that can be used for the measurement of latent attitudes or latent preferences. The data analyzed are the dichotomous responses of persons to stimuli, with a one (zero) indicating agreement (disagreement) with the content of the stimulus. The model provides a unidimensional representation of persons and items. The response probabilities are a function of the distance between person and stimulus: the smaller the distance, the larger the probability that a person will agree with the content of the stimulus. An estimation procedure based on expectation maximization and marginal maximum likelihood is developed and the quality of the resulting parameter estimates evaluated.I gratefully acknowledge Ivo Molenaar and Wijbrandt van Schuur for their advice and encouragement during the course of the investigation, Derk-Jan Kiewiet who constructed the program for the ML estimator for the person parameter and Anne Boomsma, Wendy Post, Tom Snijders, and David Thissen for their comments on smaller aspects of the investigation.     

Eye movements reveal the on-line computation of lexical probabilities during reading

Skilled readers are able to derive meaning from a stream of visual input with remarkable efficiency. In this article, we present the first evidence that statistical information latent in the linguistic environment can contribute to an account of reading behavior. In two eye-tracking studies, we demonstrate that the transitional probabilities between words have a measurable influence on fixation durations, and using a simple Bayesian statistical model, we show that lexical probabilities derived by combining transitional probability with the prior probability of a word's occurrence provide the most parsimonious account of the eye movement data. We suggest that the brain is able to draw upon statistical information in order to rapidly estimate the lexical probabilities of upcoming words: a computationally inexpensive mechanism that may underlie proficient reading.     

On estimation of parameters in latent structure analysis

The latent structure model considered here postulates that a population of individuals can be divided intom classes such that each class is homogeneous in the sense that for the individuals in the class the responses toN dichotomous items or questions are statistically independent. A method is given for deducing the proportions of the population in each latent class and the probabilities of positive responses to each item for individuals in each class from knowledge of the probabilities of positive responses for individuals from the population as a whole. For estimation of the latent parameters on the basis of a sample, it is proposed that the same method of analysis be applied to the observed data. The method has the advantages of avoiding implicitly defined and unobservable quantities, and of using relatively simple computational procedures of conventional matrix algebra, but it has the disadvantages of using only a part of the available information and of using that part asymmetrically.Work supported by the RAND Corporation.     

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.     

Assessing the Robustness of Mixture Models to Measurement Noninvariance

Abstract

Recent work reframes direct effects of covariates on items in mixture models as differential item functioning (DIF) and shows that, when present in the data but omitted from the fitted latent class model, DIF can lead to overextraction of classes. However, less is known about the effects of DIF on model performance—including parameter bias, classification accuracy, and distortion of class-specific response profiles—once the correct number of classes is chosen. First, we replicate and extend prior findings relating DIF to class enumeration using a comprehensive simulation study. In a second simulation study using the same parameters, we show that, while the performance of LCA is robust to the misspecification of DIF effects, it is degraded when DIF is omitted entirely. Moreover, the robustness of LCA to omitted DIF differs widely based on the degree of class separation. Finally, simulation results are contextualized by an empirical example.     

Comparing Latent Structures of the Grade of Membership,Rasch, and Latent Class Models

This paper focuses on model interpretation issues and employs a geometric approach to compare the potential value of using the Grade of Membership (GoM) model in representing population heterogeneity. We consider population heterogeneity manifolds generated by letting subject specific parameters vary over their natural range, while keeping other population parameters fixed, in the marginal space (based on marginal probabilities) and in the full parameter space (based on cell probabilities). The case of a 2 × 2 contingency table is discussed in detail, and a generalization to 2^J tables with J ≥ 3 is sketched. Our approach highlights the main distinction between the GoM model and the probabilistic mixture of classes by demonstrating geometrically the difference between the concepts of partial and probabilistic memberships. By using the geometric approach we show that, in special cases, the GoM model can be thought of as being similar to an item response theory (IRT) model in representing population heterogeneity. Finally, we show that the GoM item parameters can provide quantities analogous to more general logistic IRT item parameters. As a latent structure model, the GoM model might be considered a useful alternative for a data analysis when both classes of extreme responses, and additional heterogeneity that cannot be captured by those latent classes, are expected in the population. This work was supported by Award #1R03 AG18986-01 from the National Institute on Aging and NIH grant #1R01 CA94212-01. The presentation of the ideas in this paper owes much to discussions with Stephen Fienberg and Brian Junker, Carnegie Mellon University. The author thanks Jim Ramsay and two anonymous reviewers for their valuable comments on earlier drafts of this paper.     

Characterizing the manifest probabilities of latent trait models

The problem of characterizing the manifest probabilities of a latent trait model is considered. The item characteristic curve is transformed to the item passing-odds curve and a corresponding transformation is made on the distribution of ability. This results in a useful expression for the manifest probabilities of any latent trait model. The result is then applied to give a characterization of the Rasch model as a log-linear model for a 2 ^J -contingency table. Partial results are also obtained for other models. The question of the identifiability of “guessing” parameters is also discussed. The research reported here is collaborative in every respect and the order of authorship is alphabetical. Dr. Cressie was a Visiting Research Scientist at ETS during the Fall of 1980. His current address is: School of Mathematical Sciences, The Flinders University of South Australia, Bedford Park SA, 5042, AUSTRALIA. The preparation of this paper was supported, in part, by the Program Statistics Research Project in the Research Statistics Group at ETS.     

Hierarchial chunking in sentence processing

In order to evaluate a left-to-right hierarchical chunking model of sentence perception, Johnson’s Hierarchical Clustering Scheme (HCS) technique was applied to data obtained from sentence intelligibility tests. One hundred and twenty Ss listened to sentences disturbed by white noise. After each presentation they wrote down what they had heard. For each sentence, a table of conditional probabilities p(j/i) was computed, where p(j/i) is the probability that word j had been correctly identified. given correct identification of word i. This was done for all i’s and j’s from the sentence. HCS analysis of the off-diagonal submatrices for which words i precede words j (“forward conditional probabilities ”) yielded satisfactory results. Apparently there is a latent hierarchical structure to these data. The large chunks that appear from these analyses do generally correspond to major syntactic constituents. Minor constituents, however, are very often not reflected in the chunking pattern.     

Latent class models for nonmonotone dichotomous items

Starting from perfectly discriminating nonmonotone dichotomous items, a class of probabilistic models with or without response errors and with or without intrinsically unscalable respondents is described. All these models can be understood as simply restricted latent class analysis. Thus, the estimation and identifiability of the parameters (class sizes and item latent probabilities) as well as the chi-squared goodness-of-fit tests (Pearson and likelihood-ratio) are free of the problems. The applicability of the proposed variants of latent class models is demonstrated on real attitudinal data.This research was supported by the Kulturamt der Stadt Wien, Magistratsabteilung 7.The author wishes to thank the editor, Ivo W. Molenaar, as well as Clifford C. Clogg and the anonymous reviewers for their valuable comments on the earlier drafts of this paper.