Modelling time to maximum competency in medical student progress tests

The current paper is motivated by longitudinal progress tests given to medical students in the United Kingdom, which are used to assess students' applied medical knowledge during their learning programme. The main analytic interest is the maximum competency each student achieves on the assessment and the point in the programme at which each student attains this competency. Direct estimates of maximum competency and the time at which students realised this competency are useful for optimising allocation of classroom and hands-on experiences, as well as to inform curriculum development. Models have been developed for estimating the timing of a threshold or cut-off common across people or for estimating different rates of change that occur for different phases of time. However, less attention has been paid to models interested in the timing of a value that can change across people—such as maximum competency—and where growth is flat in some phases of time. In this paper, we build a model that borrows pieces from various existing methods such as reparameterisations of polynomial models, splines for ceiling effects, time-to-criterion models, dynamic measurement and non-linear mixed-effect models to allow the motivating questions to be addressed from these data.