Conditional independence tests are crucial across various disciplines in determining the independence of an outcome variable $Y$ from a treatment variable $X$, conditioning on a set of confounders $Z$. The Conditional Randomization Test (CRT) offers a powerful framework for such testing by assuming known distributions of $X \mid Z$; it controls the Type-I error exactly, allowing for the use of flexible, black-box test statistics. In practice, testing for conditional independence often involves using data from a source population to draw conclusions about a target population. This can be challenging due to covariate shift -- differences in the distribution of $X$, $Z$, and surrogate variables, which can affect the conditional distribution of $Y \mid X, Z$ -- rendering traditional CRT approaches invalid. To address this issue, we propose a novel Covariate Shift Corrected Pearson Chi-squared Conditional Randomization (csPCR) test. This test adapts to covariate shifts by integrating importance weights and employing the control variates method to reduce variance in the test statistics and thus enhance power. Theoretically, we establish that the csPCR test controls the Type-I error asymptotically. Empirically, through simulation studies, we demonstrate that our method not only maintains control over Type-I errors but also exhibits superior power, confirming its efficacy and practical utility in real-world scenarios where covariate shifts are prevalent. Finally, we apply our methodology to a real-world dataset to assess the impact of a COVID-19 treatment on the 90-day mortality rate among patients.

翻译：条件独立性检验在多个学科领域中至关重要，用于确定结果变量$Y$在给定一组混杂变量$Z$的条件下是否独立于处理变量$X$。条件随机化检验（CRT）通过假设已知$X \mid Z$的分布，为此类检验提供了一个强大的框架；它能精确控制第一类错误，并允许使用灵活的黑盒检验统计量。在实践中，检验条件独立性通常涉及使用源群体的数据来推断目标群体的结论。由于协变量偏移——即$X$、$Z$以及代理变量的分布差异，这可能影响$Y \mid X, Z$的条件分布——使得传统的CRT方法失效，因此这具有挑战性。为解决此问题，我们提出了一种新颖的协变量偏移校正皮尔逊卡方条件随机化（csPCR）检验。该检验通过整合重要性权重并采用控制变量法来适应协变量偏移，从而降低检验统计量的方差并提升检验功效。理论上，我们证明了csPCR检验能渐近地控制第一类错误。通过模拟研究，我们实证表明，我们的方法不仅能控制第一类错误，还展现出更优的检验功效，这证实了其在协变量偏移普遍存在的实际场景中的有效性和实用性。最后，我们将该方法应用于一个真实世界数据集，以评估一种COVID-19治疗方法对患者90天死亡率的影响。