Bilker Warren B, Brensinger Colleen, Gur Ruben C
Multivariate Behav Res. 2004 Oct 1;39(4):565-94. doi: 10.1207/s15327906mbr3904_1.
Testing homogeneity of correlations with Fisher's Z is inappropriate when correlations are themselves correlated. Suppose measurements of brain activation and performance are taken before and during a verbal memory task. Of interest are changes in activity gradients in specific regions, R1, R2, R3, and performance, V. The "correlated correlations" of interest ρV,R1 , ρV,R2 , and ρV,R3 , have a single variable, V, in common. We wish to compare these correlations between males and females, across regions, and to assess an interaction of the correlation. Fisher's Z can compare pairs of correlations, and Olkin and Finn's (1990) method can test homogeneity of correlated correlations across a single within factor (based on asymptotic normality), but no current procedure can test a region by gender (within by between) interaction of correlations. We propose a nonparametric method for testing this interaction and both main effects. The procedure is analogous to two-way ANOVA, but hypotheses test homogeneity of correlations, not means. The null distributions are estimated with permutations, avoiding asymptotic distributional assumptions and enhancing applicability to smaller samples and non-normal data. Simulations demonstrated maintenance of correct level (power = alpha level under the null) for normal and non-normal data and small samples. The Olkin-Finn test had inflated level for non-normal data or small samples. The Fisher's Z had inflated level for non-normal data, but not for small samples. Our method had better efficiency across contrasts and data types and sizes. Applied to correlations between regional laterality of blood flow and verbal memory performance, the method showed sensitivity to a biologically meaningful sex by region interaction in these correlations. A SAS macro for CORANOVA is available.
当相关性本身存在关联时，使用费舍尔Z检验相关性的同质性是不合适的。假设在言语记忆任务之前和期间对大脑激活和表现进行测量。感兴趣的是特定区域R1、R2、R3的活动梯度变化以及表现V。感兴趣的“相关相关性”ρV,R1、ρV,R2和ρV,R3有一个共同的变量V。我们希望比较男性和女性在不同区域之间的这些相关性，并评估相关性的交互作用。费舍尔Z可以比较成对的相关性，奥尔金和芬恩（1990）的方法可以检验单个组内因素的相关相关性的同质性（基于渐近正态性），但目前没有程序可以检验相关性的区域×性别（组内×组间）交互作用。我们提出了一种用于检验这种交互作用和两个主效应的非参数方法。该程序类似于双向方差分析，但假设检验的是相关性的同质性，而不是均值。通过排列估计零分布，避免了渐近分布假设，并提高了对较小样本和非正态数据的适用性。模拟表明，对于正态和非正态数据以及小样本，该方法能维持正确的水平（零假设下的功效 = 显著性水平）。对于非正态数据或小样本，奥尔金 - 芬恩检验的水平会膨胀。对于非正态数据，费舍尔Z检验的水平会膨胀，但对于小样本则不会。我们的方法在各种对比以及数据类型和大小方面具有更好的效率。应用于血流区域偏侧性与言语记忆表现之间的相关性时，该方法显示出对这些相关性中具有生物学意义的性别×区域交互作用的敏感性。可获得用于CORANOVA的SAS宏。
