Effects Among the Affected

We propose a novel causal estimand that elucidates how response to an earlier treatment (e.g., treatment initiation) modifies the effect of a later treatment (e.g., treatment discontinuation), thus learning if there are effects among the (un)affected. Specifically, we consider a working marginal structural model summarizing how the average effect of a later treatment varies as a function of the (estimated) conditional average effect of an earlier treatment. We define the estimand to be a data-adaptive causal parameter, allowing for estimation of the conditional average treatment effect using machine learning without making strong smoothness assumptions. We show how a sequentially randomized design can be used to identify this causal estimand, and we describe a targeted maximum likelihood estimator for the resulting statistical estimand, with influence curve-based inference. We present simulation studies that evaluate the performance of this estimator under various finite-sample scenarios. Throughout, we use the "Adaptive Strategies for Preventing and Treating Lapses of Retention in HIV Care" trial (NCT02338739) as an illustrative example, showing that discontinuation of conditional cash transfers for HIV care adherence was most harmful among those who had an increase in benefit from them initially.