Generalizing quasi-ordinal knowledge spaces to polytomous items

Abstract

Abstract Based on the celebrated Birkhoff theorem, Doignon and Falmagne (1985) establish a one-to-one correspondence between certain binary relations on a domain of (dichotomous) items and the quasi-ordinal knowledge spaces defined on it. It is shown that this construction can be generalized to apply to polytomous response formats, where response values are partially ordered so that they form a lattice. Polytomous knowledge states are defined for the case where these response scales are identical across items, as well as for item-specific response scales, a generalization which emerges quite naturally within the developed theoretical framework. Certain collections of polytomous knowledge states are characterized through precedence relations that form particular quasi-orders on an extended set of (virtual) items, including for each item several dichotomous instances representing all the possible response values. These kinds of polytomous knowledge structures can be built from data through applying to the extended item set procedures commonly used in knowledge structure theory. Confining consideration to finite sets of items and response values, the presented theory generalizes previous approaches in this direction. Its application is illustrated for data from the 2011 Trends in International Mathematics and Science Study (TIMSS).