A necessary and sufficient condition for unique skill assessment

The skill-based extension of the theory of knowledge structures forms the framework for addressing the problem of whether it is possible to uniquely assess the skills underlying the solution behavior exhibited on some set of items. Technically speaking, the paper strives for characterizing the so-called conjunctive skill functions, assigning to each item a subset of skills sufficient for solving it, that allow for singling out a unique state of a given competence structure. While previously proposed properties turn out to be either sufficient, or necessary within this setting, the paper provides a necessary and sufficient condition. Possible extensions of this characterization to more general skill functions are discussed. The conclusions cover suggestions on how to extend a test so that it allows for unique skill assessment, as well as implications for the identifiability of probabilistic models defined on top of the deterministic framework.