Evaluating candidate principal surrogate endpoints

Abstract

Frangakis and Rubin (2002, Biometrics 58, 21-29) proposed a new definition of a surrogate endpoint (a "principal" surrogate) based on causal effects. We introduce an estimand for evaluating a principal surrogate, the causal effect predictiveness (CEP) surface, which quantifies how well causal treatment effects on the biomarker predict causal treatment effects on the clinical endpoint. Although the CEP surface is not identifiable due to missing potential outcomes, it can be identified by incorporating a baseline covariate(s) that predicts the biomarker. Given case-cohort sampling of such a baseline predictor and the biomarker in a large blinded randomized clinical trial, we develop an estimated likelihood method for estimating the CEP surface. This estimation assesses the "surrogate value" of the biomarker for reliably predicting clinical treatment effects for the same or similar setting as the trial. A CEP surface plot provides a way to compare the surrogate value of multiple biomarkers. The approach is illustrated by the problem of assessing an immune response to a vaccine as a surrogate endpoint for infection.

Figure 1

Example CEP^R(υ₁, υ₀) = h(R₍₁₎( υ₁, υ₀), R₍₀₎( υ₁, υ₀)) surfaces, with h(x, y) = 1 − x/y. The surface in (i) reflects a biomarker with no surrogate value, wherein the clinical treatment effect is the same for all treatment effects on the biomarker. The surface in (ii) reflects a biomarker with high surrogate value, wherein the average causal effect on the clinical endpoint is zero for υ₁ = υ₀ and has a large increase in υ₁ − υ₀ for υ₁ > υ₀. Because CEP^R(υ₁, υ₀) = 0 for υ₁ = υ₀, both biomarkers satisfy average causal necessity. Furthermore, because CEP^R( υ₁, υ₀) > 0 for all υ₁ > υ₀, the biomarker in (ii) satisfies one-sided average causal sufficiency. This figure appears in color in the electronic version of this article.

Figure 2

For case CB with S_i(0) = c for all i with c = L the lower bound of S, biomarkers S that have no (horizontal solid line), modest (dashed line), moderate (dotted line), and high (hatched line) surrogate value. Here CEP^risk(s₁, c) = h(risk₍₁₎(s₁, c), risk₍₀₎(s₁, c)) with h(x, y) = 1 − x/y. Because CEP^risk(c, c) = 0 and CEP^risk(s₁, c) > 0 for all s₁ > c, the latter two S’s satisfy average causal necessity and average causal sufficiency, and hence are principal surrogates. This figure appears in color in the electronic version of this article.