Sample size and threshold estimation for clinical trials with predictive biomarkers

With the increasing availability of newly discovered biomarkers personalized drug development is becoming more commonplace. Unless evidence of the dependence of clinical benefit on biomarker classification is a priori unequivocal, personalized drug development needs to jointly investigate treatments and biomarkers in clinical trials. Motivated by the development of contemporary cancer treatments, we propose targeting three main questions sequentially in order to determine (1) whether a drug is efficacious, (2) whether a biomarker can personalize treatment, and (3) how to define personalization. For time-to-event data satisfying the Cox proportional hazards model, we show that (1) and (2) may not directly involve the variance of an interaction term but of a contrast with smaller variance. An asymptotically exact covariance matrix for the parameter vector in the CPH model is derived to construct sample size formulae and an inference approach for thresholds of continuous biomarkers. The covariance matrix also reveals strategies for greater efficiency in trial design, for example, when the biomarker is binary or does not modulate the effect of treatment in the control arm. We motivate our approach by studying the outcome of a contemporary cancer study.