Building an identifiable latent class model with covariate effects on underlying and measured variables

In recent years, latent class models have proven useful for analyzing relationships between measured multiple indicators and covariates of interest. Such models summarize shared features of the multiple indicators as an underlying categorical variable, and the indicators' substantive associations with predictors are built directly and indirectly in unique model parameters. In this paper, we provide a detailed study on the theory and application of building models that allow mediated relationships between primary predictors and latent class membership, but that also allow direct effects of secondary covariates on the indicators themselves. Theory for model identification is developed. We detail an Expectation-Maximization algorithm for parameter estimation, standard error calculation, and convergent properties. Comparison of the proposed model with models underlying existing latent class modeling software is provided. A detailed analysis of how visual impairments affect older persons' functioning requiring distance vision is used for illustration.This work was supported by National Institute on Aging (NIA) Program Project P01-AG-10184-03 and National Institutes of Mental Health grant R01-MH-56639-01A1. Dr. Bandeen-Roche is a Brookdale National Fellow. The authors wish to thank Drs. Gary Rubin and Sheila West for kindly making the Salisbury Eye Evaluation data available. We also thank the Editor, the Associate Editor, and three referees for their valuable comments.     

Loglinear multidimensional IRT models for polytomously scored items

A loglinear IRT model is proposed that relates polytomously scored item responses to a multidimensional latent space. The analyst may specify a response function for each response, indicating which latent abilities are necessary to arrive at that response. Each item may have a different number of response categories, so that free response items are more easily analyzed. Conditional maximum likelihood estimates are derived and the models may be tested generally or against alternative loglinear IRT models.Hank Kelderman is currently affiliated with Vrije Universiteit, Amsterdam.We thank Linda Vodegel-Matzen of the Division of Developmental Psychology of the University of Amsterdam for making available the data used in the example in this article.     

Locally dependent latent trait model and the dutch identity revisited

In this paper, we propose a class of locally dependent latent trait models for responses to psychological and educational tests. Typically, item response models treat an individual's multiple response to stimuli as conditional independent given the individual's latent trait. In this paper, instead the focus is on models based on a family of conditional distributions, or kernel, that describes joint multiple item responses as a function of student latent trait, not assuming conditional independence. Specifically, we examine a hybrid kernel which comprises a component for one-way item response functions and a component for conditional associations between items given latent traits. The class of models allows the extension of item response theory to cover some new and innovative applications in psychological and educational research. An EM algorithm for marginal maximum likelihood of the hybrid kernel model is proposed. Furthermore, we delineate the relationship of the class of locally dependent models and the log-linear model by revisiting the Dutch identity (Holland, 1990). The work is supported by a research grant from the Marshall School of Business, University of Southern California. The author thanks the anonymous referees for their suggestions.     

Latent variable selection in multidimensional item response theory models using the expectation model selection algorithm

The aim of latent variable selection in multidimensional item response theory (MIRT) models is to identify latent traits probed by test items of a multidimensional test. In this paper the expectation model selection (EMS) algorithm proposed by Jiang et al. (2015) is applied to minimize the Bayesian information criterion (BIC) for latent variable selection in MIRT models with a known number of latent traits. Under mild assumptions, we prove the numerical convergence of the EMS algorithm for model selection by minimizing the BIC of observed data in the presence of missing data. For the identification of MIRT models, we assume that the variances of all latent traits are unity and each latent trait has an item that is only related to it. Under this identifiability assumption, the convergence of the EMS algorithm for latent variable selection in the multidimensional two-parameter logistic (M2PL) models can be verified. We give an efficient implementation of the EMS for the M2PL models. Simulation studies show that the EMS outperforms the EM-based L₁ regularization in terms of correctly selected latent variables and computation time. The EMS algorithm is applied to a real data set related to the Eysenck Personality Questionnaire.     

Multilevel and Latent Variable Modeling with Composite Links and Exploded Likelihoods

Composite links and exploded likelihoods are powerful yet simple tools for specifying a wide range of latent variable models. Applications considered include survival or duration models, models for rankings, small area estimation with census information, models for ordinal responses, item response models with guessing, randomized response models, unfolding models, latent class models with random effects, multilevel latent class models, models with log-normal latent variables, and zero-inflated Poisson models with random effects. Some of the ideas are illustrated by estimating an unfolding model for attitudes to female work participation. We wish to thank The Research Council of Norway for a grant supporting our collaboration.     

Consistent estimation in the rasch model based on nonparametric margins

Consider the class of two parameter marginal logistic (Rasch) models, for a test ofm True-False items, where the latent ability is assumed to be bounded. Using results of Karlin and Studen, we show that this class of nonparametric marginal logistic (NML) models is equivalent to the class of marginal logistic models where the latent ability assumes at most (m + 2)/2 values. This equivalence has two implications. First, estimation for the NML model is accomplished by estimating the parameters of a discrete marginal logistic model. Second, consistency for the maximum likelihood estimates of the NML model can be shown (whenm is odd) using the results of Kiefer and Wolfowitz. An example is presented which demonstrates the estimation strategy and contrasts the NML model with a normal marginal logistic model.This research was supported by NIMH traning grant, 2 T32 MH 15758-06 and by ONR contract N00014-84-K-0588. The author would like to thank Diane Lambert, John Rolph, and Stephen Fienberg for their assistance. Also, the comments of the referees helped to substantially improve the final version of this paper.     

Application of the pi* goodness-of-fit index to latent structure models

Testing model fit for latent structure models (latent trait models and latent class models) is difficult because of the lack of goodness-of-fit statistics with a known distribution. This paper describes the application of the pi* goodness-of-fit statistic to latent structure models. The statistic pi* is based on the concept of latent classes and has a natural interpretation when applied to these models. This statistic assumes that the population is made up of several classes that follow a parametric model, and a residual class outside the model. The value of pi* is the population proportion in the residual class. This paper describes the estimation algorithms of pi* for latent trait and latent class models and an empirical example with a scale of study habits. There are two latent classes in these data: bad and regular students, which are related to the student s responsibility.     

Invariance Properties for General Diagnostic Classification Models

In item response theory (IRT), the invariance property states that item parameter estimates are independent of the examinee sample, and examinee ability estimates are independent of the test items. While this property has long been established and understood by the measurement community for IRT models, the same cannot be said for diagnostic classification models (DCMs). DCMs are a newer class of psychometric models that are designed to classify examinees according to levels of categorical latent traits. We examined the invariance property for general DCMs using the log-linear cognitive diagnosis model (LCDM) framework. We conducted a simulation study to examine the degree to which theoretical invariance of LCDM classifications and item parameter estimates can be observed under various sample and test characteristics. Results illustrated that LCDM classifications and item parameter estimates show clear invariance when adequate model data fit is present. To demonstrate the implications of this important property, we conducted additional analyses to show that using pre-calibrated tests to classify examinees provided consistent classifications across calibration samples with varying mastery profile distributions and across tests with varying difficulties.     

Bayesian inference for graphical factor analysis models

We generalize factor analysis models by allowing the concentration matrix of the residuals to have nonzero off-diagonal elements. The resulting model is named graphical factor analysis model. Allowing a structure of associations gives information about the correlation left unexplained by the unobserved variables, which can be used both in the confirmatory and exploratory context. We first present a sufficient condition for global identifiability of this class of models with a generic number of factors, thereby extending the results in Stanghellini (1997) and Vicard (2000). We then consider the issue of model comparison and show that fast local computations are possible for this purpose, if the conditional independence graphs on the residuals are restricted to be decomposable and a Bayesian approach is adopted. To achieve this aim, we propose a new reversible jump MCMC method to approximate the posterior probabilities of the considered models. We then study the evolution of political democracy in 75 developing countries based on eight measures of democracy in two different years. We acknowledge support from M.U.R.S.T. of Italy and from the European Science Foundation H.S.S.S. Network. We are grateful to the referees and the Editor for many useful suggestions and comments which led to a substantial improvement of the paper. We also thank Nanny Wermuth for stimulating discussions and Kenneth A. Bollen for kindly providing us with the data-set.     

A Note on “Constant Latent Odds-Ratios Models and the Mantel–Haenszel Null Hypothesis” Hessen, 2005

In a recent paper, Hessen (Psychometrika 70(3):497–516, 2005) introduces the class of constant latent odds-ratios models as an extension of the Rasch model for which the sum score is still the sufficient statistic for ability. In this paper the relation between both the general and the general parametric constant latent odds-ratios model and the Rasch model is considered.     

The asymptotic posterior normality of the latent trait for polytomous IRT models

Chang and Stout (1993) presented a derivation of the asymptotic posterior normality of the latent trait given examinee responses under nonrestrictive nonparametric assumptions for dichotomous IRT models. This paper presents an extention of their results to polytomous IRT models in a fairly straightforward manner. In addition, a global information function is defined, and the relationship between the global information function and the currently used information functions is discussed. An information index that combines both the global and local information is proposed for adaptive testing applications.This research was partially supported by Educational Testing Service Allocation Project No. 79424. The author wishes to thank Charles Davis, Xuming He, Frank Jenkins, Spence Swinton, William Stout, Ming-Mai Wang, and Zhiliang Ying for their helpful comments and discussions. The author particularly wishes to thank the Editor, Shizuhiko Nishisato, the Associate Editor, and three anonymous reviewers for their thoroughness and thoughtful suggestions.     

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.     

Latent Variable Selection for Multidimensional Item Response Theory Models via $$L_{1}$$ Regularization

We develop a latent variable selection method for multidimensional item response theory models. The proposed method identifies latent traits probed by items of a multidimensional test. Its basic strategy is to impose an \(L_{1}\) penalty term to the log-likelihood. The computation is carried out by the expectation–maximization algorithm combined with the coordinate descent algorithm. Simulation studies show that the resulting estimator provides an effective way in correctly identifying the latent structures. The method is applied to a real dataset involving the Eysenck Personality Questionnaire.     

Testing unidimensionality in polytomous Rasch models

A fundamental assumption of most IRT models is that items measure the same unidimensional latent construct. For the polytomous Rasch model two ways of testing this assumption against specific multidimensional alternatives are discussed. One, a marginal approach assuming a multidimensional parametric latent variable distribution, and, two, a conditional approach with no distributional assumptions about the latent variable. The second approach generalizes the Martin-Löf test for the dichotomous Rasch model in two ways: to polytomous items and to a test against an alternative that may have more than two dimensions. A study on occupational health is used to motivate and illustrate the methods.The authors would like to thank Niels Keiding, Klaus Larsen and the anonymous reviewers for valuable comments to a previous version of this paper. This research was supported by a grant from the Danish Research Academy and by a general research grant from Quality Metric, Inc.     

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.     

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.     

To Weight or Not to Weight? Balancing Influence of Initial Items in Adaptive Testing

It has been widely reported that in computerized adaptive testing some examinees may get much lower scores than they would normally if an alternative paper-and-pencil version were given. The main purpose of this investigation is to quantitatively reveal the cause for the underestimation phenomenon. The logistic models, including the 1PL, 2PL, and 3PL models, are used to demonstrate our assertions. Our analytical derivation shows that, under the maximum information item selection strategy, if an examinee failed a few items at the beginning of the test, easy but more discriminating items are likely to be administered. Such items are ineffective to move the estimate close to the true θ, unless the test is sufficiently long or a variable-length test is used. Our results also indicate that a certain weighting mechanism is necessary to make the algorithm rely less on the items administered at the beginning of the test. This research was partially supported by the NSF Grants SES0241020 and SES0613025. The authors thank the Editor, Associate Editor and two anonymous reviewers for their comments and suggestions. Send further information to Hua-Hua Chang, Department of Psychology, 603 E. Daniel Street, M/C 716, Champaign, IL 61820.