Generalized pairwise comparisons summarize treatment effects in randomized experiments with ordinal, multivariate, or prioritized outcomes. We study optimal covariate adjustment under complete randomization, with potential outcomes and covariates fixed and treatment assignment as the sole source of randomness. We derive the exact randomization variance of the unadjusted estimator and characterize linear calibration adjustments using the between-arm difference in covariate means. The asymptotic variance is a strictly convex quadratic in the adjustment coefficient, yielding a closed-form optimum. The resulting estimator is asymptotically no less efficient than the unadjusted estimator, with an explicit precision gain that clarifies which covariates improve precision. However, natural pair-level and single-arm least squares adjustments generally use weights that differ from the allocation-dependent optimum and can substantially reduce efficiency. We also propose a within-arm variance estimator for asymptotically conservative inference, whose conservatism arises from unidentifiable joint variation in a unit's two potential scores, and show that residualizing each arm on its own projection coefficient saves the discrepancy that governs the efficiency loss of conventional adjustments. In the special case where the comparison function is the outcome difference, our proposed covariate-adjusted estimator is first-order equivalent to the fully interacted regression adjustment of Lin (Ann. Appl. Stat., 2013, 7, 295-318).

翻译：暂无翻译