Bugni Federico A, Canay Ivan A, Shaikh Azeem M
Department of Economics, Duke University,
Department of Economics, Northwestern University,
J Am Stat Assoc. 2018;113(524):1784-1796. doi: 10.1080/01621459.2017.1375934. Epub 2018 Jun 28.
This paper studies inference for the average treatment effect in randomized controlled trials with covariate-adaptive randomization. Here, by covariate-adaptive randomization, we mean randomization schemes that first stratify according to baseline covariates and then assign treatment status so as to achieve "balance" within each stratum. Our main requirement is that the randomization scheme assigns treatment status within each stratum so that the fraction of units being assigned to treatment within each stratum has a well behaved distribution centered around a proportion as the sample size tends to infinity. Such schemes include, for example, Efron's biased-coin design and stratified block randomization. When testing the null hypothesis that the average treatment effect equals a pre-specified value in such settings, we first show the usual two-sample -test is conservative in the sense that it has limiting rejection probability under the null hypothesis no greater than and typically strictly less than the nominal level. We show, however, that a simple adjustment to the usual standard error of the two-sample -test leads to a test that is exact in the sense that its limiting rejection probability under the null hypothesis equals the nominal level. Next, we consider the usual -test (on the coefficient on treatment assignment) in a linear regression of outcomes on treatment assignment and indicators for each of the strata. We show that this test is exact for the important special case of randomization schemes with , but is otherwise conservative. We again provide a simple adjustment to the standard errors that yields an exact test more generally. Finally, we study the behavior of a modified version of a permutation test, which we refer to as the covariate-adaptive permutation test, that only permutes treatment status for units within the same stratum. When applied to the usual two-sample -statistic, we show that this test is exact for randomization schemes with and that additionally achieve what we refer to as "strong balance." For randomization schemes with , this test may have limiting rejection probability under the null hypothesis strictly greater than the nominal level. When applied to a suitably adjusted version of the two-sample -statistic, however, we show that this test is exact for all randomization schemes that achieve "strong balance," including those with . A simulation study confirms the practical relevance of our theoretical results. We conclude with recommendations for empirical practice and an empirical illustration·.
本文研究了具有协变量自适应随机化的随机对照试验中平均治疗效果的推断。在这里，协变量自适应随机化是指首先根据基线协变量进行分层，然后分配治疗状态，以便在每个层内实现“平衡”的随机化方案。我们的主要要求是，随机化方案在每个层内分配治疗状态，使得随着样本量趋于无穷大，每个层内被分配到治疗组的单位比例具有围绕某个比例的良好分布。这样的方案包括，例如，埃弗龙的偏硬币设计和分层区组随机化。在这种情况下检验平均治疗效果等于预先指定值的原假设时，我们首先表明通常的两样本检验是保守的，因为在原假设下它的极限拒绝概率不大于，并且通常严格小于名义水平。然而，我们表明对通常两样本检验的标准误差进行简单调整会得到一个精确检验，即其在原假设下的极限拒绝概率等于名义水平。接下来，我们考虑在治疗分配以及每个层的指标对结果的线性回归中通常的检验（关于治疗分配系数）。我们表明对于具有的随机化方案的重要特殊情况，这个检验是精确的，但在其他情况下是保守的。我们再次对标准误差进行简单调整，更一般地得到一个精确检验。最后，我们研究一种修改后的置换检验的行为，我们称之为协变量自适应置换检验，它只对同一层内的单位置换治疗状态。当应用于通常的两样本统计量时，我们表明对于具有且另外实现我们所称的“强平衡”的随机化方案，这个检验是精确的。对于具有的随机化方案，在原假设下这个检验的极限拒绝概率可能严格大于名义水平。然而，当应用于两样本统计量的适当调整版本时，我们表明对于所有实现“强平衡”的随机化方案，包括那些具有的方案，这个检验都是精确的。一项模拟研究证实了我们理论结果的实际相关性。我们最后给出了实证实践的建议和一个实证例证。
