  In online experiments where the intervention is only exposed, or "triggered",
for a small subset of the population, it is critical to use variance reduction
techniques to estimate treatment effects with sufficient precision to inform
business decisions. Trigger-dilute analysis is often used in these situations,
and reduces the sampling variance of overall intent-to-treat (ITT) effects by
an order of magnitude equal to the inverse of the triggering rate; for example,
a triggering rate of $5\%$ corresponds to roughly a $20x$ reduction in
variance. To apply trigger-dilute analysis, one needs to know experimental
subjects' triggering counterfactual statuses, i.e., the counterfactual behavior
of subjects under both treatment and control conditions. In this paper, we
propose an unbiased ITT estimator with reduced variance applicable for
experiments where the triggering counterfactual status is only observed in the
treatment group. Our method is based on the efficiency augmentation idea of
CUPED and draws upon identification frameworks from the principal
stratification and instrumental variables literature. The unbiasedness of our
estimation approach relies on a testable assumption that the augmentation term
used for covariate adjustment equals zero in expectation. Unlike traditional
covariate adjustment or principal score modeling approaches, our estimator can
incorporate both pre-experiment and in-experiment observations. We demonstrate
through a real-world experiment and simulations that our estimator can remain
unbiased and achieve precision improvements as large as if triggering status
were fully observed, and in some cases can even outperform trigger-dilute
analysis.
