论文标题
条件随机测试和仿冒品的功率分析
A Power Analysis of the Conditional Randomization Test and Knockoffs
论文作者
论文摘要
在许多科学问题中，研究人员试图将响应变量$ y $与一组潜在的解释变量联系起来$ x =（x_1，\ dots，x_p）$，首先尝试识别有助于这种关系的变量。用统计术语来说，可以提出这个目标，试图确定$ y $有条件地取决于的$ x_j $。有时，同时测试每个$ j $是有价值的，这通常称为变量选择。条件随机测试（CRT）和Model-X敲门词是最近提出的两种方法，分别对每个$ X_J $进行有条件的独立性测试和可变选择，计算出数据的任何测试统计量，并通过将其与$ X $ X $的$ X $分布的知识相比，通过将其与对合成的知识进行了对测试统计的比较来评估该测试统计量的重要性。我们的主要贡献是在高维线性模型中分析其功率，其中尺寸$ p $和样本量$ n $的比率收敛到正常数。我们给出了CRT的渐近功率的明确表达式，具有CRT $ P $值的可变选择和Model-X仿冒品，每种都基于边缘协方差，最小二乘系数或LASSO的测试统计量。我们分析的一种有用的应用是与CRT $ P $值和Model-X仿冒品的可变选择的渐近能力的直接理论比较；在与我们考虑的独立协变量的情况下，CRT可以占主导地位。我们还分析了当可用$ x $的分布的有限知识时，可以分析CRT中使用未标记的数据的功率增长，以及回顾性收集样品时CRT的功率。
In many scientific problems, researchers try to relate a response variable $Y$ to a set of potential explanatory variables $X = (X_1,\dots,X_p)$, and start by trying to identify variables that contribute to this relationship. In statistical terms, this goal can be posed as trying to identify $X_j$'s upon which $Y$ is conditionally dependent. Sometimes it is of value to simultaneously test for each $j$, which is more commonly known as variable selection. The conditional randomization test (CRT) and model-X knockoffs are two recently proposed methods that respectively perform conditional independence testing and variable selection by, for each $X_j$, computing any test statistic on the data and assessing that test statistic's significance by comparing it to test statistics computed on synthetic variables generated using knowledge of $X$'s distribution. Our main contribution is to analyze their power in a high-dimensional linear model where the ratio of the dimension $p$ and the sample size $n$ converge to a positive constant. We give explicit expressions of the asymptotic power of the CRT, variable selection with CRT $p$-values, and model-X knockoffs, each with a test statistic based on either the marginal covariance, the least squares coefficient, or the lasso. One useful application of our analysis is the direct theoretical comparison of the asymptotic powers of variable selection with CRT $p$-values and model-X knockoffs; in the instances with independent covariates that we consider, the CRT provably dominates knockoffs. We also analyze the power gain from using unlabeled data in the CRT when limited knowledge of $X$'s distribution is available, and the power of the CRT when samples are collected retrospectively.
