DINA Models for Multiple-Choice Items With Few Parameters: Considering Incorrect Answers.

The deterministic-input, noisy "and" gate (DINA) model can judge whether an individual examinee has mastered each skill that is needed to answer an item correctly. This information is useful for students to know their deficits and for teachers to teach effectively. The DINA model is a statistical model for binary (correct or incorrect) data. However, recently a DINA model for multiple-choice items was developed by de la Torre. The model is aimed at obtaining information about students' skills from incorrect answers. In the present study, new DINA models for multiple-choice items are developed that need many fewer parameters while still being able to express various answering probabilities without any restrictions on the form of the Q-matrix. Simulations using a Markov chain Monte Carlo method are performed to demonstrate the efficacies of the proposed models compared with the DINA model for binary data and the model of de la Torre for multiple-choice items, if appropriate starting values are set.