Decompositions of Information Criteria in Joint Model Under Probit Link for Response

ABSTRACT The item response and response time are naturally encountered in variety of computerized testing, which are jointly analysed to achieve efficient model fitting. Several novel Bayesian assessment criteria, which quantify the enhancement in the fit of one dataset given the other, are proposed by utilizing decompositions of the deviance information criterion (DIC) and the logarithm of the pseudo‐marginal likelihood (LPML) under a logistic link. However, the calculation of these criteria under a probit link, a commonly employed alternative, remains unexplored. Moreover, decomposing DIC and LPML under probit link presents greater challenges than in the logistic model due to the intricate additional dimension integral. This paper addresses this gap by demonstrating the application of Bayesian model assessment criteria with a normal ogive model in the context of item response data within a joint modelling framework. Additionally, we also illustrate how to implement Bayesian sampling via a powerful R package ‘nimble’. Extensive simulation studies are conducted to evaluate the empirical performance of those criteria. Further, a detailed analysis of the Programme for International Student Assessment (PISA) data is carried out to illustrate the methodology.