A logistic approach to knowledge structures

Models for quantitative (or numerical) testing like e.g. educational testing have a relatively long tradition in psychology, while the qualitative (or nonnumerical) approach to psychometrics is more recent. The approach presented in this paper can be regarded as an attempt to integrate, to some extent, the numerical and nonnumerical fields. In numerical testing a subject is characterized by some real-valued parameter representing her level or ability. In the nonnumerical approach the knowledge state of an individual is represented by the subset of problems that the individual is capable of solving. We propose a model in which the relationship between the ability levels and the knowledge states is worked out on a probabilistic basis. The central idea is that the ability parameters and the knowledge states are not independent. A logistic model is derived which specifies the probabilities of the knowledge states conditional on the ability levels. We show that the Rasch model arises as a special case of the proposed model.