 Home Economics  # The Proportion of Treatment Effect Explained

## Definition

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.

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