Targeted maximum likelihood estimation for longitudinal two-stage designs with outcome subsampling
We consider efficient estimation of causal parameters in longitudinal two-stage designs with outcome subsampling, motivated by resampling designs in HIV-related mortality studies. In these studies, many participants become lost to follow-up; resampling designs address this by tracing a subset of lost individuals to ascertain their outcomes. Analyses often use inverse-probability-weighted Kaplan-Meier (wKM) estimators that discard longitudinal covariate information and suffer from efficiency losses. We note that resampling designs are an instance of a broader class: two-stage designs with outcome subsampling, in which a first stage collects some data on all participants and a second stage collects outcome information on a selected subset. This connection motivates two estimators. First, drawing on inverse probability of censoring weighted targeted maximum likelihood estimation (IPCW-TMLE) for two-stage designs, we develop its longitudinal extension, IPCW longitudinal TMLE (IPCW-LTMLE) and show that estimating and targeting the known second-stage sampling weights yields variance reductions of up to 36% over the use of known sampling probabilities. Second, given that inverse weighting sacrifices efficiency, we propose an LTMLE that incorporates the second-stage sampling indicator as an intervention node in the sequential regression framework, returning to plug-in estimation and avoiding inverse weighting entirely. Simulations across sample sizes show that LTMLE achieves up to 73% lower variance than wKM with known sampling weights, with reductions of 30-50% common across settings, while IPCW-LTMLE achieves consistent gains of 20-35%. We further demonstrate that cross-fitted variance estimation is essential for valid inference: standard variance estimators yield confidence interval coverage as low as 76%, while our cross-fitted variants consistently restore coverage to nominal levels.