A positivity robust strategy to study effects of switching treatment

In studies of medical treatments, individuals often experience post-treatment events that predict their future outcomes. In this work, we study how to use initial observations of a recurrent event - a type of post-treatment event - to offer updated treatment recommendations in settings where no, or few, individuals are observed to switch between treatment arms. Specifically, we formulate an estimand quantifying the average effect of switching treatment on subsequent events. We derive bounds on the value of this estimand under plausible conditions and propose non-parametric estimators of the bounds. Furthermore, we define a value and regret function for a dynamic treatment-switching regime, and use these to determine 3 types of optimal regimes under partial identification: the pessimist (maximin value), optimist (maximax value), and opportunist (minimax regret) regimes. The pessimist regime is guaranteed to perform at least as well as the standard of care. We apply our methods to data from the Systolic Blood Pressure Intervention Trial.