This paper develops methods to test for associations between two variables with clustered data using a U-Statistic approach with a second-order approximation to the variance of the parameter estimate for the test statistic. The tests that are presented are for clustered versions of: Pearsons χ2${\chi }^2$${\chi }^2$ test, the Spearman rank correlation and Kendall's τ$\tau$$\tau$ for continuous data or ordinal data and for alternative measures of Kendall's τ$\tau$$\tau$ that allow for ties in the data. Shih and Fay use the U-Statistic approach but only consider a first-order approximation. The first-order approximation has inflated significance level in scenarios with small sample sizes. We derive the test statistics using the second-order approximations aiming to improve the type I error rates. The method applies to data where clusters have the same number of measurements for each variable or where one of the variables may be measured once per cluster while the other variable may be measured multiple times. We evaluate the performance of the test statistics through simulation with small sample sizes. The methods are all available in the R package cluscor.

中文翻译：

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小样本聚类数据的秩相关推断

本文开发了使用U统计方法测试两个变量与聚类数据之间关联的方法，该方法对测试统计量的参数估计方差进行了二阶近似。提供的测试适用于以下集群版本： Pearsonsχ2${\chi }^2$${\chi }^2$检验、Spearman 等级相关性和 Kendall 的τ$\tau$$\tau$对于连续数据或有序数据以及 Kendall 的替代度量τ$\tau$$\tau$允许在数据中建立联系。Shih 和 Fay 使用U -Statistic 方法，但只考虑一阶近似。在样本量较小的情况下，一阶近似值夸大了显着性水平。我们使用旨在提高 I 类错误率的二阶近似来推导测试统计量。该方法适用于集群对每个变量具有相同测量次数的数据，或者其中一个变量可以在每个集群中测量一次，而另一个变量可以测量多次。我们通过小样本模拟来评估测试统计的性能。这些方法都在 R 包cluscor中可用。