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Diagnostic classification tests are designed to assess examinees’ discrete mastery status on a set of skills or attributes. Such tests have gained increasing attention in educational and psychological measurement. We review diagnostic classification models and their applications to testing and learning, discuss their statistical and machine learning connections and related challenges, and introduce some contemporary and future extensions.

 

Item response theory (IRT) has become one of the most popular statistical models for psychometrics, a field of study concerned with the theory and techniques of psychological measurement. The IRT models are latent factor models tailored to the analysis, interpretation, and prediction of individuals’ behaviors in answering a set of measurement items that typically involve categorical response data. Many important questions of measurement are directly or indirectly answered through the use of IRT models, including scoring individuals’ test performances, validating a test scale, linking two tests, among others. This paper provides a review of item response theory, including its statistical framework and psychometric applications. We establish connections between item response theory and related topics in statistics, including empirical Bayes, nonparametric methods, matrix completion, regularized estimation, and sequential analysis. Possible future directions of IRT are discussed from the perspective of statistical learning.