Assume there are two randomized studies of a treatment effect, a prior study (Study A) and a current study (Study B). Study A was completed up to some time t, while Study B was stopped at time \(t_0<t\). In both studies, a surrogate marker was measured at time \(t_0\) for individuals still observable at \(t_0\). Let \(G\) be the binary treatment indicator with \(G=1\) for treatment and \(G=0\) for control and we assume throughout that subjects are randomly assigned to a treatment group at baseline. Let \(T_K^{(1)}\) and \(T_K^{(0)}\) denote the time of the primary outcome of interest, death for example, under the treatment and under the control, respectively, in Study K. Let \(S_K^{(1)}\) and \(S_K^{(0)}\) denote the surrogate marker measured at time \(t_0\) under the treatment and the control, respectively, in Study K.
The treatment effect quantity of interest, \(\Delta_K(t)\), is the difference in survival rates by time \(t\) under treatment versus control,
$$ \Delta_K(t)=E\{ I(T_K^{(1)}>t)\} - E\{I(T_K^{(0)}>t)\} = P(T_K^{(1)}>t) - P(T_K^{(0)}>t)$$
where \(t>t_0\). Here, we recover an estimate of \(\Delta_B(t)\) using Study B information (which stopped follow-up at time \(t_0<t\)) and Study A information (which has follow-up information through time t). The estimate is obtained as $$\hat{\Delta}_{EB}(t,t_0)/ \hat{R}_{SA}(t,t_0)$$
where \(\hat{\Delta}_{EB}(t,t_0)\) is the early treatment effect estimate in Study B, described in the early.delta.test documention, and \(\hat{R}_{SA}(t,t_0)\) is the proportion of treatment effect explained by the surrogate marker information at \(t_0\) in Study A. This proportion is calculated as \(\hat{R}_{SA}(t,t_0) =\hat{\Delta}_{EA}(t,t_0)/\hat{\Delta}_A(t)\)
where $$\hat{\Delta}_A(t)=n_{A1}^{-1}\sum_{i=1}^{n_{A1}}\frac{I(X_{Ai}^{(1)}>t)}{\hat{W}_{A1}^C(t)}-n_{A0}^{-1}\sum_{i=1}^{n_{A0}}\frac{I(X_{Ai}^{(0)}>t)}{\hat{W}_{A0}^C(t)},$$
and \(\hat{\Delta}_{EA}(t,t_0)\) is parallel to \(\hat{\Delta}_{EB}(t,t_0)\) except replacing
\(n_{A0}^{-1} \sum_{i=1}^{n_{A0}} \hat{r}_A^{(0)}(t|S_{Ai}^{(0)}, t_0) \frac{I(X_{Ai}^{(0)} > t_0)}{\hat{W}_{A0}^C(t_0)}\)
by \(n_{A0}^{-1}\sum_{i=1}^{n_{A0}}\hat{W}_{A0}^C(t)^{-1}I(X_{Ai}^{(0)}>t),\) and \(\hat{W}^C_{Ag}(\cdot)\) is the Kaplan-Meier estimator of the survival function for \(C_{A}^{(g)}\) for \(g=0,1\).
Perturbation resampling is used to provide a standard error estimate for the estimate of \(\Delta_B(t)\) and a confidence interval.