# 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).