Estimating the distribution of heterogeneous treatment effects from treatment responses and from a predictive biomarker in a parallel-group RCT: A structural model approach.

When the objective is to administer the best of two treatments to an individual, it is necessary to know his or her individual treatment effects (ITEs) and the correlation between the potential responses (PRs) Y i 1 and Y i 0 under treatments 1 and 0. Data that are generated in a parallel-group design RCT does not allow the ITE to be determined because only two samples from the marginal distributions of these PRs are observed and not the corresponding joint distribution. This is due to the "fundamental problem of causal inference." Here, we present a counterfactual approach for estimating the joint distribution of two normally distributed responses to two treatments. This joint distribution of the PRs Y i 1 and Y i 0 can be estimated by assuming a bivariate normal distribution for the PRs and by using a normally distributed baseline biomarker Z i functionally related to the sum Y i 1 + Y i 0 . Such a functional relationship is plausible since a biomarker Z i and the sum Y i 1 + Y i 0 encode for the same information in an RCT, namely the variation between subjects. The estimation of the joint trivariate distribution is subjected to some constraints. These constraints can be framed in the context of linear regressions with regard to the proportions of variances in the responses explained and with regard to the residual variation. This presents new insights on the presence of treatment-biomarker interactions. We applied our approach to example data on exercise and heart rate and extended the approach to survival data.