The Relative Effect and Adjusted Association

Definition

In view of the fundamental problems with the PE (see Section 3.3.3), Buyse and Molenberghs (1998) proposed two new quantities to assess surrogacy, the so-called Relative Effect (RE) and the adjusted association (p_z). In the setting where both S and T are continuous normally distributed endpoints, these quantities are:

The RE is the ratio of the effect of Z on T and the effect of Z on S. Thus, it is a factor that allows “translating” the effect of Z on S into the effect of Z on T. Notice that, in contrast to the PE, the treatment effects involved in RE are not adjusted by post-randomization variables and thus these measures have a direct causal interpretation. Indeed, a and в are simply the average causal effects of the treatment on S and T, respectively (Alonso et al., 2014). The adjusted association p_z quantifies how strongly S and T are associated at the level of the individual patients after accounting for the treatment effect. If p_z = 1, there exists a deterministic relationship between S and T — and thus the true endpoint for an individual patient can be perfectly predicted based on his/her surrogate endpoint and the administered treatment. If p_z = 0, knowledge of S does not improve the prediction of T in an individual patient.

The RE is the ratio of two parameters so its confidence interval can be computed by using the delta method or Fieller's theorem. A confidence interval for p_z can be computed based on the general Fisher transformation procedure for correlations or by bootstrapping (Burzykowski, Molenberghs, and Buyse, 2005).