On the identifiability of the Polytomous Local Independence Model (PoLIM)

In the last years, growing attention has been paid to the generalization of KST deterministic concepts to the case of polytomous items. As a consequence of this extension, a generalized version of the basic local independence model (BLIM) has been recently proposed, named polytomous local independence model (PoLIM). Some of the main features of this new model have been investigated, but, to date, nothing has been specifically stated about its identifiability. In this research we present the first theoretical results about the problem of identifiability of the PoLIM. Such results represent a generalization to this polytomous model of what has been proven about the identifiability of the BLIM, which is the most widely used probabilistic model in dichotomous knowledge space theory. The study of the identifiability of the BLIM produced several research articles in the last ten years, especially focusing on the relations between two particular kinds of gradation of the deterministic structure, called forward and backward gradedness, and the unidentifiability of the model when applied to such structures. Here we show that the same kind of gradedness happens to apply also to the case of polytomus structures, and further attention is paid to some properties of forward and backward gradedness in the case of polytomous structures. For instance, we show that in the polytomous case there is no need to distinguish between forward and backward gradedness, but it is possible to simply speak about gradedness. Moreover, we show how gradedness of the polytomous structure leads to the same kind of tradeoffs studied in the BLIM between the probability of knowledge states and the error parameters of the items in which the polytomous structure is graded. The tradeoff equations are displayed and further directions to study the identifiability of the PoLIM are discussed.