Estimating individual treatment effects from responses and a predictive biomarker in a parallel group RCT

When being interested in administering the best of two treatments to an individual patient i, it is necessary to know the individual treatment effects (ITEs) of the considered subjects and the correlation between the possible responses (PRs) for two treatments. When data are generated in a parallel–group design RCT, it is not possible to determine the ITE for a single subject since we only observe two samples from the marginal distributions of these PRs and not the corresponding joint distribution due to the ’Fundamental Problem of Causal Inference’ [Holland, 1986, p. 947]. In this article, we present a counterfactual approach for estimating the joint distribution of two normally distributed responses to two treatments. This joint distribution can be estimated by assuming a normal joint distribution for the PRs and by using a normally distributed baseline biomarker which is defined to be functionally related to the sum of the ITE components. Such a functional relationship is plausible since a biomarker and the sum encode for the same information in a RCT, namely the variation between subjects. As a result of the interpretation of the biomarker as a proxy for the sum of ITE components, the estimation of the joint 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. As a consequence, a new light is thrown on the presence of treatment–biomarker interactions. We applied our approach to a classical medical data example on exercise and heart rate.