Surrogate markers and joint models for longitudinal and survival data.

There is increasing interest in the use of surrogate marker endpoints in comparative clinical trials to make decisions about treatment efficacy. They are attractive because a trial using a surrogate endpoint is typically smaller, faster, and cheaper than a trial using a clinical endpoint. In seminal work, Prentice gave a framework for studying surrogate endpoints and a formal definition of conditions that a marker should satisfy to be a valid surrogate endpoint. These definitions involve consideration of the joint distribution of the marker and the clinical endpoint. It is well recognized that the formal definition is unlikely to be satisfied in practice, and thus quantifying the proportion of treatment explained (PTE) by a surrogate marker is an intuitively appealing concept. Freedman et al. suggested a statistic to quantify the PTE. In the situation of a censored clinical event time and a longitudinal marker, calculation of this statistic requires fitting two different survival models. We show that for these two models to be consistent with one another requires a specific assumption about the joint model for the longitudinally measured marker and the clinical endpoint. Furthermore, in a simulation study we show that the Freedman et al. PTE statistic and other measures of surrogacy, motivated by the Prentice framework, can be estimated using the joint model. Thus to evaluate a marker as a potential surrogate endpoint it is crucial to understand the joint distribution of the marker and the clinical endpoint.