Rule Space Decision Rules and Attribute Mastery Probabilities
The two variables q and the numerator of z, denoted by f(X,q), are not only continuous variables but also mutually uncorrelated, as has been shown (Tatsuoka, 1985). An ideal item score pattern R associated with knowledge state πR can be mapped to an ordered pair of the estimated IRT ability value of qR and the covariance of two residuals, f(R, qR), that is denoted by YR = (qR, f(R, qR)). The response patterns deviating from Rl
with different numbers of slips will cluster around YRl and are assumed to follow a multivariate normal distribution with the centroid YRl and the covariance matrix SRl (Tatsuoka & Tatsuoka, 1987).