How to relate potential outcomes: Estimating individual treatment effects under a given specified partial correlation

Abstract: In most medical research, the average treatment effect is used to evaluate a treatment's performance. However, precision medicine requires knowledge of individual treatment effects: What is the difference between a unit's measurement under treatment and control conditions? In most treatment effect studies, such answers are not possible because the outcomes under both experimental conditions are not jointly observed. This makes the problem of causal inference a missing data problem. We propose to solve this problem by imputing the individual potential outcomes under a specified partial correlation (SPC), thereby allowing for heterogeneous treatment effects. We demonstrate in simulation that our proposed methodology yields valid inferences for the marginal distribution of potential outcomes. We highlight that the posterior distribution of individual treatment effects varies with different specified partial correlations. This property can be used to study the sensitivity of optimal treatment outcomes under different correlation specifications. In a practical example on HIV-1 treatment data, we demonstrate that the proposed methodology generalises to real-world data. Imputing under the SPC, therefore, opens up a wealth of possibilities for studying heterogeneous treatment effects on incomplete data and the further adaptation of individual treatment effects.