  Motivated by increasing pressure for decision makers to shorten the time
required to evaluate the efficacy of a treatment such that treatments deemed
safe and effective can be made publicly available, there has been substantial
recent interest in using an earlier or easier to measure surrogate marker, $S$,
in place of the primary outcome, $Y$. To validate the utility of a surrogate
marker in these settings, a commonly advocated measure is the proportion of
treatment effect on the primary outcome that is explained by the treatment
effect on the surrogate marker (PTE). Model based and model free estimators for
PTE have also been developed. While this measure is very intuitive, it does not
directly address the important questions of how $S$ can be used to make
inference of the unavailable $Y$ in the next phase clinical trials. In this
paper, to optimally use the information of surrogate S, we provide a framework
for deriving an optimal transformation of $S$, $g_{opt}(S)$, such that the
treatment effect on $g_{opt}(S)$ maximally approximates the treatment effect on
$Y$ in a certain sense. Based on the optimally transformed surrogate,
$g_{opt}(S)$, we propose a new measure to quantify surrogacy, the relative
power (RP), and demonstrate how RP can be used to make decisions with $S$
instead of $Y$ for next phase trials. We propose nonparametric estimation
procedures, derive asymptotic properties, and compare the RP measure with the
PTE measure. Finite sample performance of our estimators is assessed via a
simulation study. We illustrate our proposed procedures using an application to
the Diabetes Prevention Program (DPP) clinical trial to evaluate the utility of
hemoglobin A1c and fasting plasma glucose as surrogate markers for diabetes.
