On a Reparameterization of the MC-DINA Model

The MC-DINA model is a cognitive diagnosis model (CDM) for multiple-choice items that was introduced by de la Torre (2009). The model extends the usual CDM in two basic ways: it allows for nominal responses instead of only dichotomous responses, and it allows skills to affect not only the choice of the correct response but also the choice of distractors. Here it is shown that the model can be re-expressed as a multinomial logit model with latent discrete predictors, that is, as a multinomial mixture model; a signal detection-like parameterization is also used. The reparameterization clarifies details about the structure and assumptions of the model, especially with respect to distractors, and helps to reveal parameter restrictions, which in turn have implications for psychological interpretations of the data and for issues with respect to statistical estimation. The approach suggests parsimonious models that are useful for practical applications, particularly for small sample sizes. The restrictions are shown to appear for items from the TIMSS 2007 fourth grade exam.