On initial item selection in cognitive diagnostic computerized adaptive testing

There has recently been much interest in computerized adaptive testing (CAT) for cognitive diagnosis. While there exist various item selection criteria and different asymptotically optimal designs, these are mostly constructed based on the asymptotic theory assuming the test length goes to infinity. In practice, with limited test lengths, the desired asymptotic optimality may not always apply, and there are few studies in the literature concerning the optimal design of finite items. Related questions, such as how many items we need in order to be able to identify the attribute pattern of an examinee and what types of initial items provide the optimal classification results, are still open. This paper aims to answer these questions by providing non-asymptotic theory of the optimal selection of initial items in cognitive diagnostic CAT. In particular, for the optimal design, we provide necessary and sufficient conditions for the Q-matrix structure of the initial items. The theoretical development is suitable for a general family of cognitive diagnostic models. The results not only provide a guideline for the design of optimal item selection procedures, but also may be applied to guide item bank construction.