Abstract
This review introduces classical item response theory (IRT) models as well as more contemporary extensions to the case of multilevel, multidimensional, and mixtures of discrete and continuous latent variables through the lens of discrete multivariate analysis. A general modeling framework is discussed, and the applications of this framework in diverse contexts are presented, including large-scale educational surveys, randomized efficacy studies, and diagnostic measurement. Other topics covered include parameter estimation and model fit evaluation. Both classical (numerical integration based) and more modern (stochastic) parameter estimation approaches are discussed. Similarly, limited information goodness-of-fit testing and posterior predictive model checking are reviewed and contrasted. The review concludes with a discussion of some emerging strands in IRT research such as response time modeling, crossed random effects models, and non-standard models for response processes.
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