ABSTRACT
In the rule space, one of the motivations to express both the observed and ideal item score patterns by a set of ordered pairs of (θ, ζ)—this is a classification space—is to have a connection to the current psychometric theories, especially item response theory. Additionally, having discrete variables such as item scores as the coordinates in a classification space may cause computational problems. Kim (1989) used a discriminant analysis for a 20-item algebra test, in which binary item scores are independent variables and form a space with 19-dimensional item response space as a classification space. Kim classified a student response pattern into one of 12 predetermined knowledge states. He generated 1,200 response patterns from the 12 knowledge states, and classified observed item responses into one of the 12 knowledge states by using three different methods: discriminant analysis, RSM, and the Kth nearest method. Kim found that the covariance matrices for some knowledge states became singular, and he could not compute the inverses of these covariance matrices. The knowledge states having very high scores and very low scores often have singular covariance matrices. However, converting discrete variables (item scores) into continuous variables avoids this computational problem. Kim found that the rule space performed as well as the discriminant analysis, and much better than the Kth nearest method when the covariance matrices are not singular and hence invertible. Because the variables in the rule space are continuous, it is always possible to compute the squared Mahalanobis distances. In other words, classification results are always available. This is one of advantages for using RSM.
