Biometrics 61, 1102–1111 December 2005 DOI: 10.1111/j.1541-0420.2005.00380.x Counterfactual Links to the Proportion of Treatment Effect Explained by a Surrogate Marker Jeremy M. G. Taylor, 1, ∗ Yue Wang, 2 and Rodolphe Thi´ ebaut 3 1 Department of Biostatistics, University of Michigan, Ann Arbor, Michigan 48109, U.S.A. 2 Research Laboratory, Merck & Co., West Point, Pennsylvania 19486, U.S.A. 3 INSERM E0338 Biostatistics, ISPED, Bordeaux 2 University, Bordeaux 33076, France ∗ email: jmgt@umich.edu Summary. In a randomized clinical trial, a statistic that measures the proportion of treatment effect on the primary clinical outcome that is explained by the treatment effect on a surrogate outcome is a useful concept. We investigate whether a statistic proposed to estimate this proportion can be given a causal interpretation as defined by models of counterfactual variables. For the situation of binary surrogate and outcome variables, two counterfactual models are considered, both of which include the concept of the proportion of the treatment effect, which acts through the surrogate. In general, the statistic does not equal either of the two proportions from the counterfactual models, and can be substantially different. Conditions are given for which the statistic does equal the counterfactual model proportions. A randomized clinical trial with potential surrogate endpoints is undertaken in a scientific context; this context will naturally place constraints on the parameters of the counterfactual model. We conducted a simulation experiment to investigate what impact these constraints had on the relationship between the proportion explained (PE) statistic and the counterfactual model proportions. We found that observable constraints had very little impact on the agreement between the statistic and the counterfactual model proportions, whereas unobservable constraints could lead to more agreement. Key words: Causal effects; Clinical trial; Counterfactual model; Direct effect; Indirect effect; Surrogate marker. 1. Introduction Clinical trials with rare primary endpoints or long duration times often require large sample sizes and extensive periods of follow-up. Because of this, there has been increasing inter- est in using surrogate endpoints in lieu of the primary end- points in these situations. A number of statistical articles con- cerned with evaluating surrogate markers have been written (Prentice, 1989; Freedman, Graubard, and Schatzkin, 1992; Daniels and Hughes, 1997; Buyse et al., 2000; Li, Meredith, and Hoseyni, 2001; Wang and Taylor, 2002; Ditlevsen et al., 2005). Surrogate endpoints are usually intermediate biomark- ers in disease development, which can be assessed earlier and more easily. They are generally proposed based on the bio- logical process of a disease and their strong associations with the primary endpoint. Prentice (1989) proposed a formal definition of surrogate endpoints and gave general operational criteria for validation of surrogate endpoints. Prentice’s criteria leads to considera- tion of a model for the treatment effect on the primary end- point adjusting for the surrogate marker and statistical tests for τ adj. = 0, where τ adj. is the adjusted treatment effect in the model. Prentice’s criteria, which requires a surrogate endpoint to fully capture the treatment effect on the primary endpoint, is rather too stringent. In practice, it is more likely that a sur- rogate endpoint may explain part but not all the treatment effect. Thus, a quantitative measure of the proportion of the treatment effect that is explained by the surrogate marker was proposed by Freedman et al. (1992). This measure was given by P =(τ unadj. − τ adj. )/(τ unadj. ), where τ unadj. is the treatment effect on the primary outcome without adjusting for the marker. The properties of this statistic are reviewed in Wang and Taylor (2002) and two alternative statistics F and F ′ for assessing the proportion of the treatment effect explained were proposed. We use the following notation: T and S denote the pri- mary endpoint and surrogate marker, respectively. They are assumed to be binary. Z is the treatment variable, with Z = 1 for treatment (or new treatment) and Z = 0 for placebo (or standard treatment). We assume a positive effect of the treat- ment, with T = 1 and S = 1 representing better outcomes. In a randomized clinical trial, a perfect surrogate occurs when S captures all the dependence of T on Z, that is, P (T | Z, S)= P (T | S). A useless surrogate can occur when, conditional on the treatment, the surrogate is independent of the primary endpoint, that is, P (T | Z, S)= P (T | Z), or when S is inde- pendent of the treatment group, that is, P (S | Z)= P (S). An alternative approach to the consideration of surrogate markers is through models of counterfactual variables. Such models are frequently used in the statistical literature on causal inference. The general idea of a counterfactual model 1102