Efron Bradley
Department of Statistics, Sequoia Hall, 390 Serra Mall, Stanford, CA 94305-4065.
Ann Appl Stat. 2009 Jan 1;3(3):922-942. doi: 10.1214/09-AOAS236.
Having observed an m x n matrix X whose rows are possibly correlated, we wish to test the hypothesis that the columns are independent of each other. Our motivation comes from microarray studies, where the rows of X record expression levels for m different genes, often highly correlated, while the columns represent n individual microarrays, presumably obtained independently. The presumption of independence underlies all the familiar permutation, cross-validation, and bootstrap methods for microarray analysis, so it is important to know when independence fails. We develop nonparametric and normal-theory testing methods. The row and column correlations of X interact with each other in a way that complicates test procedures, essentially by reducing the accuracy of the relevant estimators.
在观察了一个(m\times n)矩阵(X)（其行可能相关）之后，我们希望检验列彼此独立的假设。我们的动机来自微阵列研究，其中(X)的行记录了(m)个不同基因的表达水平，这些基因通常高度相关，而列代表(n)个单独的微阵列，据推测是独立获得的。独立性假设是所有常见的微阵列分析排列、交叉验证和自助法的基础，所以了解独立性何时不成立很重要。我们开发了非参数和正态理论检验方法。(X)的行相关性和列相关性以一种使检验程序复杂化的方式相互作用，本质上是通过降低相关估计量的准确性来实现的。
