A nonlinear mixed model framework for item response theory.

Abstract

Mixed models take the dependency between observations based on the same cluster into account by introducing 1 or more random effects. Common item response theory (IRT) models introduce latent person variables to model the dependence between responses of the same participant. Assuming a distribution for the latent variables, these IRT models are formally equivalent with nonlinear mixed models. It is shown how a variety of IRT models can be formulated as particular instances of nonlinear mixed models. The unifying framework offers the advantage that relations between different IRT models become explicit and that it is rather straightforward to see how existing IRT models can be adapted and extended. The approach is illustrated with a self-report study on anger.Mixed models (see, e.g., Diggle, Heagerty, Liang, & Zeger, 2002;Goldstein, 1995;Longford, 1993;Verbeke & Molenberghs, 2000) are a collection of statistical tools that are well suited for analyzing clustered data, such as, for example, data from students nested within schools or repeated measurement data (measurements nested within participants). Modeling clustered data under a model that assumes independent observations is inappropriate because the observations on the subunits (students, measurements) of the same unit (school, participant) tend to be more homogeneous than the observations on subunits of different units. The heterogeneity between units can be taken into account by assuming that (some of) the parameters of the model follow some random distribution over the population of units. Hence, (some of) the parameters of the model are random variables, and the model is a random effects model, or mixed model.The random effects represent the unit effects. The parameters that are not assumed to be random realizations from a distribution, and hence are not considered to be unit specific, are called fixed effects.Mixed models and related methods were first developed in the context of analysis of variance and regression analysis, leading to the linear mixed model. Other commonly used terms are multilevel models (Goldstein, 1995), hierarchical models (Raudenbush & Bryk, 2002), and random coefficient models (Longford, 1993). More recently, nonlinear mixed models were also developed. A nonlinear mixed model is a model with random coefficients in which the fixed and/or random effects enter the model nonlinearly (Davidian & Giltinan, 1995). As we explain below, a subset of nonlinear mixed models is the class of generalized linear mixed models (McCulloch & Searle, 2001).The main purpose of this article is to explain how item response theory (IRT) models can be conceptualized as nonlinear mixed models and to provide a framework for this conceptualization. There are four important assets of this approach. First, this conceptualization relates IRT to the broad statistical literature on mixed models. Second, applying the same framework to different IRT models can help in the understanding of their differences and commonalities. Third, using this framework, one can read...

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