Prentice (1989) defined a surrogate endpoint as “a response variable for which a test of the null hypothesis of no relationship to the treatment groups under comparison is also a valid test of the corresponding null hypothesis based on the true endpoint” (Prentice, 1989; p. 432). This definition essentially requires that the surrogate endpoint should capture any relationship between the treatment and the true endpoint (Lin et al., 1997). Symbolically, Prentice’s definition can be written as:
where f (S) and f (T) denote the probability distributions of the random variables S and T, and f (S | Z) and f (T | Z) denote the probability distributions
of S and T conditional on the value of Z, respectively. Note that this definition involves the triplet (T, S, Z), so S is a surrogate for T only with respect to the effect of some specific treatment Z and not (necessarily) for a different treatment (except when S is a perfect surrogate for T, i.e., except when S and T are deterministically related).
Based on his definition of a surrogate endpoint, Prentice formulated four operational criteria that should be fulfilled to validate a candidate surrogate endpoint:
Thus, the treatment Z should have a significant effect on S (see (3.1)), the treatment Z should have a significant effect on T (see (3.2)), S should have a significant effect on T (see (3.3)), and the effect of the treatment Z on T should be fully captured by S (see (3.4)).