Bayesian estimation of the proportion of treatment effect captured by a surrogate marker

Surrogate endpoints in clinical trials are biological markers or events observable earlier than the clinical endpoints (such as death) that are actually of primary interest. The "proportion of treatment effect captured" by a surrogate endpoint (PTE) is a frequentist measure intended to address the question of whether trials based on a surrogate endpoint reach the same conclusions as would have been reached using the true endpoint. The question of inferential interest is whether PTE for a given marker exceeds some threshold value, say 0.5. Calculating PTE requires fitting two different models to the same data. We develop a Markov chain Monte Carlo based method for estimating the Bayesian posterior distribution of PTE. The new method conditions on the truth of a single model. Obtaining the full posterior distribution enables direct statements such as "the posterior probability that PTE >0.5 is 0.085". Furthermore, credible sets do not depend on asymptotic approximations and can be computed using data sets for which the frequentist methods may be inaccurate or even impossible to apply. We illustrate with Bayesian proportional hazards models for clinical trial data. As a by-product of developing the Bayesian method, we show that the frequentist estimate of PTE also may be computed from quantities in a single model and calculate frequentist confidence intervals for PTE that tend to be narrower than those produced by standard methods but that provide equally good coverage.