The Proportion of Treatment Effect Explained
In view of the problems with the Prentice criteria (see Section 3.2.3), Freedman et al. (1992) proposed to quantify surrogacy as the proportion of the effect of treatment Z on T that is explained by S (the Proportion Explained, PE):
where f is the effect of treatment on T without correction for S and fS is the effect of the treatment on T with correction for S. The intuition behind the PE is that, if all treatment effect is mediated by S (i.e., if fS = 0), then PE = 1. On the other hand, if there is no mediation at all (i.e., if f = fS), then PE = 0 (but note that this intuitively appealing reasoning is flawed, see Section 3.3.3).
The PE is a ratio of parameters, so its confidence interval can be calculated using Fieller's theorem or the delta method (Burzykowski, Molenberghs, and Buyse, 2005). A good S should have a lower limit confidence interval value for PE that is close to 1. Note that the fourth Prentice criterion (which states that the effect of Z on T is fully captured by S, see (3.4)) is equivalent to the requirement that PE = 1.