When Do Surrogate Metrics Work? A Finite-Sample Comparison Under Realistic Failure Modes

Abstract

Online experiments must often be evaluated before long-term outcomes mature. Under rolling enrollment, these outcomes are observed only for early enrollees, while short-term surrogates are available for everyone. We compare seven estimators across eleven data-generating processes, spanning partial mediation, drift, outcome sparsity, and enrollment-time labeling, with up to $R = 2,000$ replications over more than 500 method-by-scenario cells. We find a sharp robustness-efficiency tradeoff: the surrogate index delivers large efficiency gains when surrogacy holds but its coverage collapses under violations, while PPI-family methods stay asymptotically valid under random labeling at smaller gains. We give the finite-sample variance of PPI++ in the all-units parameterization for a fixed predictor, a joint asymptotic distribution for the two estimators under cross-fitting, and a Hausman-type estimator-disagreement diagnostic, then quantify the detection-damage gap: in the partial-mediation design, where we locate both edges, a band of violations destroys surrogate-index coverage yet is too small to detect on most datasets. On the 64,000-customer Hillstrom experiment the diagnostic rarely flags a violation that biases the surrogate index, and a Cauchy-kernel hybrid of the two estimators inherits 10.7% relative bias; on the 14-million-user Criteo experiment the violation is detected, and subsampling traces detection turning on with scale as damage persists. PPI++ has limits: at a rare-conversion $n = 30,000$ Criteo subsample its empirical coverage is 85.0%. We recommend prespecifying PPI++ with the exact variance as the primary analysis under random labeling and adequate labeled outcome counts, reading the diagnostic as a warning, not a certificate, and treating the surrogate index as a sensitivity analysis.