In this article, we propose a series of latent trait models for the responses and the response times on low stakes tests where some test takers respond preliminary without making full effort to solve the items. The models consider individual differences in capability and persistence. Core of the models is a race between the solution process and a process of disengagement that interrupts the solution process. The different processes are modeled with the linear ballistic accumulator model. Within this general framework, we develop different model variants that differ in the number of accumulators and the way the response is generated when the solution process is interrupted. We distinguish no guessing, random guessing and informed guessing where the guessing probability depends on the status of the solution process. We conduct simulation studies on parameter recovery and on trait estimation. The simulation study suggests that parameter values and traits can be recovered well under certain conditions. Finally, we apply the model variants to empirical data.
