On the Analysis of Fraction Subtraction Data: The DINA Model, Classification, Latent Class Sizes, and the Q-Matrix

Cognitive diagnostic models (CDMs) attempt to uncover latent skills or attributes that examinees must possess in order to answer test items correctly. The DINA (deterministic input, noisy ‘‘and’’) model is a popular CDM that has been widely used. It is shown here that a logistic version of the model can easily be fit with standard software for latent class analysis. A partly Bayesian approach to estimation, posterior mode estimation, is used as a simple alternative to a fully Bayesian approach via Markov chain Monte Carlo methods. A latent-class analysis of a widely analyzed data set, the fraction subtraction data of K. K. Tatsuoka, reveals some neglected problems with respect to the classification of examinees; for example, examinees who get all of the items incorrect are classified as having most of the skills. It is also noted that obtaining large estimates of the latent class sizes can indicate misspecification of the Q-matrix, such as the inclusion of an irrelevant skill. It is shown, analytically and via simulations, that the problems are largely associated with the structure of the Q-matrix.