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A multilevel discrete latent variable model for joint modeling of response accuracy and times
Luca Brusa1, Francesco Bartolucci2, Fulvia Pennoni1
1: University of Milano-Bicocca, Italy; 2: University of Perugia, Italy
In recent years, the widespread adoption of computer-based testing has produced large volumes of data on examinee behavior. Beyond traditional binary indicators of correct responses, these datasets now typically include item-level response times, providing a richer and more informative perspective on the performance. The joint modeling of response accuracy and response times has therefore attracted increasing attention, as the interaction between these two aspects can provide a clearer picture of the underlying phenomena. Furthermore, such data commonly exhibit a hierarchical structure, where, for example, students are nested within classes or schools. Individuals in the same cluster may share unobserved characteristics, inducing heterogeneity at both cluster and individual levels. Consequently, appropriately accounting for this nested structure is essential to ensure valid and unbiased inference. We propose a multilevel latent class response time model formulated using a normal-ogive parameterization—similar to the Rasch model—for the conditional probability of a correct response, while the conditional distribution of response times given the latent variables is assumed to be log-normal. To account for the multilevel structure of the data we assume discrete latent variables at both cluster and individual levels, and we adopt a multinomial logit parameterization to include covariates at both levels. Inference is carried out via a maximum likelihood approach using the Expectation–Maximization algorithm. We analyze data on dichotomous responses to a mathematical test and related item response times administered to a representative sample of Grade-10 students during the 2017-2018 school year by the Italian National Institute for the Evaluation of the Educational System. Through the proposed model, estimated with covariates at both student and class levels, we identify five distinct ability subpopulations of students characterized by different response times, with a nonlinear association between ability and speed. Low prior mathematics and anxiety emerge as significant covariates among others, being associated with both a lower probability of correct responses and longer response times. Anxiety is particularly influential on the performance of students with average ability. We also propose a model formulation within a hierarchical Bayesian framework. In this context, estimation is performed via a Markov chain Monte Carlo (MCMC) algorithm based on a data augmentation scheme. We aim to compare the two models and the related estimation methods in terms of computational efficiency, accuracy, and applied results.
