Assessing students’ abilities: a hybrid archetypal analysis and IRT approach

Item response theory (IRT, Hambleton & Swaminathan, 1985) measures latent traits from one or more sets of manifest variables, namely items, by defining the relations between the observed variables (e.g., item responses to a test) and the latent variables. Three of the five higher education items that refer to a student’s abilities, as defined by the Dublin descriptors, are considered in this proposal: knowledge, application, and judgment. Moreover, IRT assumes that students belong to homogeneous groups concerning these abilities. The semi-parametric multivariate latent class IRT models (MultiLCIRT, Bartolucci (2007), Bacci et al. (2014)) represent a practical approach to finding groups by aggregating the units with respect to the group’s average abilities. However, assessors generally want to discover “extreme” groups of students: the most skilled, but especially those profiles that have peculiar deficits for one or more learning abilities, to define a recommendation system helping the student to fill the gaps. Archetypical analysis (AA) represents an effective data partitioning alternative to the clustering approaches around the means. The archetypes are observed or unobserved extreme points lying on the convex hull, minimizing the sum of the squared distances from all points. The algorithm computes a membership vector for each unit with respect to each archetype. This proposal integrates the LC-IRT model with the probabilistic archetypal analysis (PAA, Seth & Eugster, 2016). It presents a hybrid estimation algorithm for the multidimensional LC-IRT model, which iteratively computes the latent variables and the units’ memberships to a set of k archetypes, where k is assumed to be known.