S and T Both Continuous
Let us recall the setting in which both endpoints are continuous and normally distributed random variables. In this setting, Tj and Sj denote the true and surrogate endpoints for patient j in trial i, respectively, and Zj is the treatment indicator variable taking values 0/1 for the control/treated groups. The random treatment allocation in a clinical trial context naturally leads to the following bivariate model (Buyse et al., 2000)
where and pSi are trial-specific intercepts quantifying the average response
in the control group, Д and a are trial-specific expected causal treatment effects and eT j and eSij are correlated error terms, assumed to be zero-mean normally distributed with covariance matrix
i.e., (10.3) denotes the within-trial covariance matrix of T and S after adjusting by treatment and considering the patient as the level of analysis. In this setting, Buyse et al. (2000) proposed to assess the individual-level surrogacy using the coefficient of determination
Given the bivariate normality of (eS,eT) it seems sensible to quantify the association at the individual level using the SICC, i.e.,
with I(eS,eT) denoting the mutual information between both residuals. Furthermore, for the normal distribution, I(es,eT) = — log(l — p^s) and, therefore, Rhindiv = Pts = Rfndiv. This result, together with the ones presented in Section 9.3 for the trial level, show that the methodology introduced by Buyse et al. (2000) can basically be seen as a special case of the information-theoretic approach at the trial and individual level.