MIXTURE DISTRIBUTION DIAGNOSTIC MODELS

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ABSTRACTThis paper introduces the mixture general diagnostic model (MGDM), an extension of the general diagnostic model (GDM). The MGDM extension allows one to estimate diagnostic models for multiple known populations as well as discrete unknown, or not directly observed mixtures of populations. The GDM is based on developments that integrate located latent class models; multiple classification latent class models; and discrete, multidimensional item response models into a common framework. Models of this type express the probability of a response vector as a function of parameters that describe the individual item response variables in terms of required skills and of indirectly observed (latent) skill profiles of respondents. The skills required for solving the items are, as in most diagnostic models, represented as a design matrix that is often referred to as a Q‐matrix. This Q‐matrix consists of rows describing, for each item response, what combination of skills is needed to succeed or to obtain partial or full credit. The hypothesized Q‐matrix is either the result of experts rating items of an existing assessment (retrofitting) or comes directly out of the design of the assessment instrument, in which it served as a tool to design the items.The MGDM takes the GDM and integrates it into the framework of discrete mixture distribution models for item response data (see von Davier & Rost, 2006). This increases the utility of the GDM by allowing the estimation and testing of models for multiple populations. The MGDM allows for complex scale linkages that make assessments comparable across populations and makes it possible to test whether items function the same in different subpopulations. This can be done with known subpopulations (defined by grade levels, cohorts, etc.), as well as with unknown subpopulations that need to be identified by the model. In both cases, MGDMs make it possible to determine whether different sets of item‐by‐skill parameters and/or different skill distributions have to be assumed for different subpopulations. This amounts to a generalized procedure that can be used to test for differential item functioning (DIF) on one item or on multiple‐response variables using multiple‐group or mixture models. This procedure enables testing DIF models against models that allow additional skills for certain items in order to account for differences between subpopulations.