The stratum-specific treatment effect function is a random variable giving the average treatment effect (ATE) for a randomly drawn stratum of potential confounders a clinician may use to assign treatment. In addition to the ATE, the variance of the stratum-specific treatment effect function is fundamental in determining the heterogeneity of treatment effect values. We offer a non-parametric plug-in estimator, the targeted maximum likelihood estimator (TMLE) and the cross-validated TMLE (CV-TMLE), to simultaneously estimate both the average and variance of the stratum-specific treatment effect function. The CV-TMLE is preferable because it guarantees asymptotic efficiency under two conditions without needing entropy conditions on the initial fits of the outcome model and treatment mechanism, as required by TMLE. Particularly, in circumstances where data adaptive fitting methods are very important to eliminate bias but hold no guarantee of satisfying the entropy condition, we show that the CV-TMLE sampling distributions maintain normality with a lower mean squared error than TMLE. In addition to verifying the theoretical properties of TMLE and CV-TMLE through simulations, we highlight some of the challenges in estimating the variance of the treatment effect, which lack double robustness and might be biased if the true variance is small and sample size insufficient.

中文翻译：

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治疗效果异质性的基本量度

特定于层的治疗效果函数是一个随机变量，可为临床医生可能用于分配治疗的潜在混杂因素的随机绘制层提供平均治疗效果（ATE）。除ATE外，特定于层的治疗效果函数的方差对于确定治疗效果值的异质性至关重要。我们提供非参数插件估算器，目标最大似然估算器（TMLE）和交叉验证的TMLE（CV-TMLE），以同时估算特定于分层的治疗效果函数的平均值和方差。CV-TMLE是可取的，因为它可以确保在两种条件下的渐近效率，而无需像TMLE所要求的那样对结果模型和治疗机制的初始拟合进行熵条件运算。特别，在数据自适应拟合方法对于消除偏差非常重要但不能保证满足熵条件的情况下，我们证明CV-TMLE采样分布保持正态性，且均方差比TMLE低。除了通过仿真验证TMLE和CV-TMLE的理论特性外，我们重点介绍了估算治疗效果方差方面的一些挑战，这些挑战缺乏双重鲁棒性，如果真实方差很小且样本量不足，则可能会产生偏差。