Test Surmise Relations, Test Knowledge Structures, and Their Characterizations

This paper investigates natural, left-, right-, and total-covering test surmise relations on a set of tests partitioning the domain of a knowledge structure. The properties of reflexivity, transitivity, and antisymmetry are examined. In particular, it is shown that the property of antisymmetry is satisfied for the left-, right-, and total-covering test surmise relations when the underlying knowledge structure is discriminative and the domain is finite. This paper also investigates natural, l-, r-, and c-type test knowledge structures. The concepts of a test surmise relation and test knowledge structure respectively generalize the concepts of a surmise relation and knowledge structure in knowledge space theory. The main thrust of this paper is an examination of characterizations of these models. Unlike at the level of items, at the level of tests, the test surmise relations and test knowledge structures may not necessarily be (uniquely) derived from each other. (a) Each can be characterized by the underlying surmise relation and knowledge structure, (b) the test surmise Preprint submitted to Journal of Mathematical Psychology 16 August 2007 relations can be characterized by the test knowledge structures, and (c) the test knowledge structures can, at least under some condition, be characterized by the test surmise relations.