Optimal designs in longitudinal trials with varying treatment effects and discrete-time survival endpoints

It is plausible to assume that the treatment effect in a longitudinal study will vary over time. It can become either stronger or weaker as time goes on. Here, we extend previous work on optimal designs for discrete-time survival analysis to trials with the treatment effect varying over time. In discrete-time survival analysis, subjects are measured in discrete time intervals, while they may experience the event at any point in time. We focus on studies where the width of time intervals is fixed beforehand, meaning that subjects are measured more often when the study duration increases. The optimal design is defined as the optimal combination of the number of subjects, the number of measurements for each subject, and the optimal proportion of subjects assigned to the experimental condition. We study optimal designs for different optimality criteria and linear cost functions. We illustrate the methodology of finding optimal designs using a clinical trial that studies the effect of an outpatient mental health program on reducing substance abuse among patients with severe mental illness. We observe that optimal designs depend to some extent on the rate at which group differences vary across time intervals and the direction of these changes over time. We conclude that an optimal design based on the assumption of a constant treatment effect is not likely to be efficient if the treatment effect varies across time.