Covariate adjustment can improve precision in estimating treatment effects from randomized experiments. With fully observed data, regression adjustment and propensity score weighting are two asymptotically equivalent methods for covariate adjustment in randomized experiments. We show that this equivalence breaks down in the presence of missing outcomes, with regression adjustment no longer ensuring efficiency gain when the true outcome model is not linear in covariates. Propensity score weighting, in contrast, still guarantees efficiency over unadjusted analysis, and including more covariates in adjustment never harms asymptotic efficiency. Moreover, we establish the value of using partially observed covariates to secure additional efficiency. Based on these findings, we recommend a simple double-weighted estimator for covariate adjustment with incomplete outcomes and covariates: (i) impute all missing covariates by zero, and use the union of the completed covariates and corresponding missingness indicators to estimate the probability of treatment and the probability of having observed outcome for all units; (ii) estimate the average treatment effect by the coefficient of the treatment from the least-squares regression of the observed outcome on the treatment, where we weight each unit by the inverse of the product of these two estimated probabilities.

翻译：协变量调整可提高随机化实验中治疗效应估计的精度。在完全观测数据下，回归调整和倾向性评分加权是随机化实验中两种渐近等价的协变量调整方法。我们证明，当存在缺失结局时，这种等价性被打破：当真实结局模型不是协变量的线性函数时，回归调整不再保证效率提升。相比之下，倾向性评分加权仍能保证比未调整分析更高效，且调整中包含更多协变量不会损害渐近效率。此外，我们确立了使用部分观测协变量来获取额外效率的价值。基于这些发现，我们推荐一种简单的双重加权估计器，用于处理不完整结局和协变量时的协变量调整：（i）将所有缺失协变量用零插补，并使用完整协变量与相应缺失指示变量的并集来估计所有单元的治疗分配概率和结局观测概率；（ii）通过将观测结局对治疗变量进行最小二乘回归，并使用每个单元两种估计概率乘积的倒数作为权重，得到治疗后估计的平均治疗效应系数。