Item Response Theory, Latent Classes and Rule Space

摘要

This study demonstrated that item response theory, latent class models, and the rule space model introduced by Tatsuoka (1985) and Tatsuoka and Tatsuoka (1987) are algebraically related. Specifically, it was shown (1) that IRT functions may actually be regarded as the conditional density functions of item scores for a special latent class representing the null state of knowledge (i.e., the state that would ideally produce a response vector of all zeros); and (2) that estimates of the item parameters of IRT functions can be determined from the union of several latent classes with the following property: when their response vectors are mapped into rule space, the centroids of these projections lie approximately along the first principal axis of the union set. Bug distributions, which are density function of the numbers of slips away from the ideal rule-generated response patterns, play an important role in interrelating IRT and latent-class models; they in fact hold the key to the development of a general theory of rule space that includes these two models as special cases. Furthermore, bug distributions form the basis for developing new indices that measure the stability of states or rules and the consistency with which a particular rule is applied with no intrusion of slips.