Longitudinal Causal Inference with Selective Eligibility

Dropout poses a significant challenge to causal inference in longitudinal studies with time-varying treatments. However, existing research does not simultaneously address dropout and time-varying treatments. We examine selective eligibility, an important yet overlooked source of non-ignorable dropout in such settings. This problem arises when a unit's prior treatment history influences its eligibility for subsequent treatments, a common scenario in medical and other settings. We propose a general methodological framework for longitudinal causal inference with selective eligibility. By focusing on a subgroup of units who would become eligible for treatment given a specific past treatment sequence, we define the time-specific eligible treatment effect and expected number of outcome events under a treatment sequence of interest. Under a generalized version of sequential ignorability, we derive two nonparametric identification formulae, each leveraging different parts of the observed data distribution. We then derive the efficient influence function of each causal estimand, yielding the corresponding doubly robust estimator. Finally, we apply the proposed methodology to an impact evaluation of a pre-trial risk assessment instrument in the criminal justice system, in which selective eligibility arises due to recidivism.