Clustering dichotomously scored items through functional k-means algorithm

In the educational field, it is common to analyze the probability of a correct response to a test item as a continuous function of the item parameters and the subject ability. This relation is given by the item response function. Since test data are expressed as curves, they can be analyzed through the functional data analysis approach. Indeed, several researchers suggest to estimate the shape of the item response function with a non-parametric approach in order to catch unusual or unforeseen features in the curve. On the contrary, item response theory models assume a specic parametric functional form for the item response function. In this paper, we propose an alternative method that combines the parametric specication of the common item response theory with the functional data analysis approach. In particular, we aim to classify the items through the functional k-means algorithm. The key idea is to transform the function space of the items in a convex space which guarantees desirable properties. Specically, we prove that, exploiting the convexity property, the functional centroids belong to the same function space as the item response functions. The applicability of our proposal in the educational filed, is demonstrated through a real data set concerning test data of the Italian Olympics of Statistics.