Keywords: Bayesian, Hierarchical modelling, Item response theory, Latent variable model, Markov chain Monte Carlo, Sports statistics
Item response theory (IRT) models are routinely used to measure student abilities based on tests, taking into account variation in difficulty of test questions. Our interest is in assessing individual ability in terms of their contribution to team performance. This is of substantial interest in many applications, as in real world situations individuals seldom work in isolation but instead work together in groups to accomplish specific tasks. We are particularly motivated by settings in which teams compete against each other; this can include sports, industry and military applications among others. With this motivation, we define a new class of Team IRT (T-IRT) models, which give each team an ability score based on the sum of the individual teammates scores. These scores do not represent an individuals' abilities in isolation but instead their contribution to the team. Based on the outcomes of a series of encounters between teams, we define a hierarchical model that relates team and individual abilities to outcomes. Importantly, we jointly model the abilities of individuals belonging to both teams for these encounters. We develop a Bayesian approach to inference, and assess the approach through simulation studies. We also consider an application to basketball data from the NBA, contrasting the results to current player ability scores.