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Designing Follow-Up Times

Studies of time-to-event are often conducted using follow-up sessions with subjects at risk. When these sessions must be widely spaced, their timing can significantly affect the efficiency of a study design. In this article we analyze the optimal timing of follow-up from a Bayesian decision theoretic standpoint. The article has two goals: (1) to develop the necessary distributional theory and computational approaches to determine optimal sequential and nonsequential follow-up schedules in the exponential case and (2) to demonstrate that unusually large gains in efficiency (more than threefold over the standard approach in the examples considered) can be achieved using our group sequential timing of follow-up. We hope that this striking illustration will encourage more systematic consideration of followup times in study designs. Specifically, we consider time-independent hazard rates. We derive posterior and predictive distributions in three scenarios: single follow-up time for the estimation of a single hazard rate, group-specific follow-up time for the comparison of hazard rates in two treatment groups or cohorts, and multiple follow-up times for a single hazard rate. We encounter a novel family of mixtures of gamma functions and characterize its moments, which play a critical role in the determination of optima. We then provide a solution to the optimal follow-up time in the single follow-up problem. We develop a practical and accurate approximation to the optimal solution as a function of prior hyperparameters that can be used to implement real time calculations and more complex sequential strategies. Finally, we consider the sequential choice of follow-up times. We discuss the general dynamic programming solution and illustrate it in the setting of a two-stage design.