Joint partial credit and time model

Joint modeling of responses and response times (RTs) has become a mainstream approach in psychological and educational measurement, offering a deeper understanding of test-taking behavior and more precise estimates of individual performance. However, despite its growing popularity, little is known about RT patterns under partial credit settings. When RTs vary across score categories, psychometric models may benefit from incorporating this heterogeneity to improve measurement accuracy. This article introduces the Joint Partial Credit and Time Model (JPCTM), a new measurement model for partial credit scores and RTs. The JPCTM builds upon the hierarchical framework (van der Linden, 2007) but employs the generalized partial credit model (Muraki, 1992) for scores and a log-normal distribution for RTs. Because item-specific intercepts and speed factors are not anticipated to fully account for heterogeneity in RTs across score categories, we extend the model with 1) varying intercepts (VI) across score categories and/or 2) cross loadings (CL) from the latent ability to RTs. A simulation study demonstrated that the JPCTM with VI and CL recovers its parameters well under a range of conditions. Empirical applications to two real-world datasets revealed that 1) RTs systematically differ across tests, items, and score categories, 2) the proposed model captures these differences, and 3) the standard errors of ability estimates can be reduced by jointly modeling partial credit scores and RTs. Overall, the JPCTM provides a comprehensive description of partial credit data with RTs and yields more stable measures of individual performance than partial credit models based solely on response scores.