A Robust Spearman Correlation Coefficient Permutation Test

TL;DR

This paper introduces a new permutation test for Spearman's correlation that remains valid and robust across various distributions and small sample sizes, addressing limitations of traditional methods.

Contribution

We developed a studentized permutation test for Spearman's correlation that is theoretically valid under dependence and uncorrelatedness, improving robustness over existing tests.

Findings

The new test maintains proper type I error control across diverse distributions.

Traditional Spearman tests perform poorly when normality assumptions are violated.

The proposed method is effective in real-world biological data analysis.

Abstract

In this work, we show that Spearman's correlation coefficient test about $H_0:\rho_s=0$ found in most statistical software packages is theoretically incorrect and performs poorly when bivariate normality assumptions are not met or the sample size is small. The historical works about these tests make an unverifiable assumption that the approximate bivariate normality of original data justifies using classic approximations. In general, there is common misconception that the tests about $\rho_s=0$ are robust to deviations from bivariate normality. In fact, we found under certain scenarios violation of the bivariate normality assumption has severe effects on type I error control for the most commonly utilized tests. To address this issue, we developed a robust permutation test for testing the general hypothesis $H_0: \rho_s=0$. The proposed test is based on an appropriately studentized statistic. We will show that the test is theoretically asymptotically valid in the general setting when two paired variables are uncorrelated but dependent. This desired property was demonstrated across a range of distributional assumptions and sample sizes in simulation studies, where the proposed test exhibits robust type I error control across a variety of settings, even when the sample size is small. We demonstrated the application of this test in real world examples of transcriptomic data of the TCGA breast cancer patients and a data set of PSA levels and age.