Linear logistic latent class analysis (LCA) relates the item latent probabilities of LCA to basic parameters representing the effects of explanatory variables. Applications of this model to dichotomous data comprise paired comparisons (the compared objects are the explanatory variables); the measurement of change due to, for example, therapeutic interventions (the treatments are the explanatory variables); simple scaling models of the Rasch type (the item difficulties and the person abilities are the explanatory variables); and similar models that try to explain the item difficulties by a hypothesized item structure (the cognitive operations needed to solve the items are the explanatory variables). The two latter types of models approximate the Rasch model and the linear logistic test model (LLTM), respectively, and become equivalent to these models for a sufficiently large number of classes. The same principle of semiparametric maximum likelihood estimation allows one to approximate the mixed Rasch model with its class-specific item difficulties and the mixed LLTM with its class-specific operation difficulties. Generalizations of linear logistic LCA include models for dichotomous items having varying discriminatory power and/or being affected by guessing (three-parameter linear logistic LCA) and models for the analysis of polytomous data.
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