A linear compensatory counterpart to and generalization of the DINA model

In applying cognitive diagnosis models (CDM), it is important to define how mastery of the multiple skills required by a test item combine to determine item performance. The two commonly used assumptions as to a combination rule lead to the distinction between conjunctive versus compensatory models. The present work develops methods to investigate the appropriateness of the conjunctive combination rule used in many CDMs such as the DINA model. First, a linear compensatory model is developed that is a counterpart model to DINA, differing only in the skill-combination rule. This new model is referred to as the "Linear Compensatory" (LC) model. Then, a new generalized model, the "Quasi-Compensatory" (or "QUIC") model, is described. Bayesian methods are developed for estimation of the three models. Two simulation studies are conducted investigating how accurately the parameters of the models can be recovered using the estimation methods. Study 1 assumes larger values for guessing parameters to represent a multiple-choice test and Study 2 simulates an open-ended test by setting a smaller value for the guessing parameters. Finally, in Study 3 the models were applied to the well-known mixed fraction subtraction data collected by K. Tatsuoka. Results of the three studies showed that while it is usually critical to use a model that assumes the correct combination rule, the QUIC model can distinguish conjunctive from compensatory items and fit both types of items and tests well, thus overcoming the need to specify the nature of the combination rule in advance