Plausibility of Finding a Valid Surrogate: Trial Level
Fano's inequality relates prediction accuracy with different information- theoretic concepts and, when applied to the evaluation of surrogate endpoints at the trial level, this inequality sets a limit for our capacity to successfully predict the expected causal treatment effect on the true endpoint using the expected causal treatment effect on the surrogate (Cover and Tomas, 1991; Alonso and Molenberghs, 2007). In fact, applying Fano’s inequality to the validation problem at the trial level leads to
This inequality raises some interesting issues. First, note that nothing has been assumed about the distribution of (a, в) and no specific form has been considered for the prediction function g. Second, (9.2) clearly shows that the quality of the prediction strongly depends on the characteristics of the true endpoint and the treatments under study, more specifically, on the power- entropy Е? (в). Essentially, Fano’s inequality defines a lower bound for the prediction error and this lower bound can be decomposed in two different elements. The second element on the right side of (9.2) depends on the surrogate through the value of t; the first element, however, is an intrinsic characteristic of the true-endpoint-treatment pair and it is independent of the surrogate. It is clear from (9.2) that the prediction error increases with Е? (в). Consequently, if the true endpoint has a large power-entropy at the trial level, then a surrogate should produce an t close to one to have some predictive value. In other words, the expected causal treatment effect on the surrogate would need to be almost deterministically related to the expected causal treatment effect on the true endpoint to have some predictive power. Essentially, this inequality hints at the fact that, for some true endpoints and treatments, the search for a valid surrogate at the trial level may not be viable.
When f (а,в) is a bivariate normal distribution Fano’s lower bound is reached, i.e., (9.2) becomes an equality and the right-hand side takes the simpler form (1 — R2rial). Based on this equality, Burzykowski and Buyse (2006) introduced a new measure of surrogacy at the trial level for normally distributed (а, в), the so-called surrogate threshold effect (STE). For details, see Section 4.5.