Dichotomous and polytomous Q matrix theory

Abstract

Besides specifying the Q matrix correctly, an extended Q matrix theory should provide some answers to following questions: (a) how to find the universal set of knowledge states (KSs), (b) how to use an algorithm to calculate the set of ideal response patterns (IRPs), (c) how to construct the test Q matrix which establishes one-to-one mapping from the set of KSs to the set of IRPs, and (d) how to mine the hierarchy relation hidden in the Q matrix. In this note, a dichotomous and polytomous Q matrix theory is established and the properties of the polytomous Q matrix are explored to help address the above questions.