Canonical correlation analysis (CCA) is a common method used to estimate the associations between two different sets of variables by maximizing the Pearson correlation between linear combinations of the two sets of variables. We propose a version of CCA for transelliptical distributions with an elliptical copula using pairwise Kendall’s tau to estimate a latent scatter matrix. Because Kendall’s tau relies only on the ranks of the data this method does not make any assumptions about the marginal distributions of the variables, and is valid when moments do not exist. We establish consistency and asymptotic normality for canonical directions and correlations estimated using Kendall’s tau. Simulations indicate that this estimator outperforms standard CCA for data generated from heavy tailed elliptical distributions. Our method also identifies more meaningful relationships when the marginal distributions are skewed. We also propose a method for testing for non-zero canonical correlations using bootstrap methods. This testing procedure does not require any assumptions on the joint distribution of the variables and works for all elliptical copulas. This is in contrast to permutation tests which are only valid when data are generated from a distribution with a Gaussian copula. This method’s practical utility is shown in an analysis of the association between radial diffusivity in white matter tracts and cognitive tests scores for six-year-old children from the Early Brain Development Study at UNC-Chapel Hill. An R package implementing this method is available at github.com/blangworthy/transCCA.

典型相关分析（CCA）是一种常用方法，用于通过最大化两组变量的线性组合之间的皮尔逊相关性来估计两组不同变量之间的关联。我们提出了一种用于跨椭圆分布的 CCA 版本，其椭圆联结函数使用成对的 Kendall's tau 来估计潜在的散布矩阵。由于 Kendall tau 仅依赖于数据的排名，因此该方法不会对变量的边际分布做出任何假设，并且在矩不存在时有效。我们为使用 Kendall tau 估计的规范方向和相关性建立了一致性和渐近正态性。模拟表明，对于从重尾椭圆分布生成的数据，该估计器的性能优于标准 CCA。当边际分布倾斜时，我们的方法还可以识别出更有意义的关系。我们还提出了一种使用引导方法测试非零规范相关性的方法。该测试过程不需要对变量的联合分布做出任何假设，并且适用于所有椭圆联结函数。这与排列测试相反，排列测试仅在从具有高斯联结函数的分布生成数据时才有效。该方法的实用性体现在对白质束径向扩散率与北卡罗来纳大学教堂山分校早期大脑发育研究中六岁儿童认知测试分数之间关系的分析中。实现此方法的R包可在 github.com/blangworthy/transCCA 上找到。