Model-free approach to quantifying the proportion of treatment effect explained by a surrogate marker

SummaryIn randomized clinical trials, the primary outcome, $Y$, often requires long-term follow-up and/or is costly to measure. For such settings, it is desirable to use a surrogate marker, $S$, to infer the treatment effect on $Y$, $\Delta$. Identifying such an $S$ and quantifying the proportion of treatment effect on $Y$ explained by the effect on $S$ are thus of great importance. Most existing methods for quantifying the proportion of treatment effect are model based and may yield biased estimates under model misspecification. Recently proposed nonparametric methods require strong assumptions to ensure that the proportion of treatment effect is in the range $[0,1]$. Additionally, optimal use of $S$ to approximate $\Delta$ is especially important when $S$ relates to $Y$ nonlinearly. In this paper we identify an optimal transformation of $S$, $g_{\tiny {\rm{opt}}}(\cdot)$, such that the proportion of treatment effect explained can be inferred based on $g_{\tiny {\rm{opt}}}(S)$. In addition, we provide two novel model-free definitions of proportion of treatment effect explained and simple conditions for ensuring that it lies within $[0,1]$. We provide nonparametric estimation procedures and establish asymptotic properties of the proposed estimators. Simulation studies demonstrate that the proposed methods perform well in finite samples. We illustrate the proposed procedures using a randomized study of HIV patients.