Partially Observed Mixtures of IRT Models: An Extension of the Generalized Partial-Credit Model

Abstract

The generalized partial-credit model (GPCM) is used frequently in educational testing and in large-scale assessments for analyzing polytomous data. Special cases of the generalized partial-credit model are the partial-credit model-or Rasch model for ordinal data-and the two-parameter logistic (2PL) model. This article extends the GPCM to the class of discrete mixture distribution models. The developments presented here extend models such as the mixed Rasch model and dichotomous multiparameter item response theory (IRT) mixture models. In addition, the model proposed here allows estimation of multigroup models with partially missing grouping information. An application of the proposed partially observed mixture IRT model to a sparse matrix sample of item responses from a national large-scale assessment program is also presented. Index terms: item response theory (IRT), discrete mixture IRT models, multiple group IRT, generalized partial credit model (GPCM), missing data.Item response theory (IRT) models (Lord & Novick, 1968) idealize response behavior by a number of assumptions that commonly include local independence assumptions, monotonicity assumptions, and often specific functional forms modeling the dependency of the probability of correct item responses on ability. It is also common to include certain item and person homogeneity assumptions, which leads to very simple model structures. Most, if not all, of these simplifying assumptions can and have been relaxed in model extensions, which has led to models that are more appropriate for specific applications.Still, most extended models assume a continuous person variable, which may be unidimensional or multidimensional, and has-at least conceptually-indefinitely many different outcomes. In contrast to this continuity assumption, the actual estimation procedures that model the person variable as a nuisance parameter use a discrete approximation to the continuous person parameter distribution. For the Rasch model, conditional maximum likelihood (CML) estimation exploits the fact that the number of distinguishable ability estimates equals the number of sum scores. For the two-and three-parameter IRT models, the marginal maximum likelihood (MML) estimation uses a discrete approximation of the continuous integral to "marginalize" the likelihood. In this way, MML enables the estimation of item parameters without jointly estimating ability parameters.

Hierarchical General Diagnostic Models

“…Von Davier and Yamamoto (2004) pointed this out and described an extension of the GPCM for mixture versions, multiple group versions, and partially observed grouping versions, where the g n information is missing only for a portion of the sample.…”

Parameter Estimation with Mixture Item Response Theory Models: A Monte Carlo Comparison of Maximum Likelihood and Bayesian Methods

“…Use of the MixIRT model in a variety of contexts has been described in detail by a number of authors (Cohen & Bolt, 2005;von Davier & Yamamoto, 2004;von Davier & Rost, 1995;Mislevy & Verhelst, 1990;Rost, 1990;Yamamoto, 1987 classes) which are characterized by different item response models for a particular measure or instrument (Li, et al, 2009). In this context, psychometricians have used MixIRT to detect and characterize differential item functioning (DIF) (Cohen & Bolt, 2005;De Ayala, et al, 2002;Bolt, Cohen & Wollack, 2002, 2001).…”

Different Approaches to Covariate Inclusion in the Mixture Rasch Model

“…In spite of the differences between IRT and LCA, both these two types of models fall under a flexible framework of general diagnostic models (GDM; von Davier & Yamamoto, 2004;von Davier, 2005avon Davier, , 2008a which incorporate a variety of latent structure models that describe the probability of observed responses in terms of conditional probabilities given one or more latent variables (von Davier, 2009). …”