Beyond Pearson's Correlation: Modern Nonparametric Independence Tests for Psychological Research

When examining whether two continuous variables are associated, tests based on Pearson's, Kendall's, and Spearman's correlation coefficients are typically used. This paper explores modern nonparametric independence tests as an alternative, which, unlike traditional tests, have the ability to potentially detect any type of relationship. In addition to existing modern nonparametric independence tests, we developed and considered two novel variants of existing tests, most notably the Heller-Heller-Gorfine-Pearson (HHG-Pearson) test. We conducted a simulation study to compare traditional independence tests, such as Pearson's correlation, and the modern nonparametric independence tests in situations commonly encountered in psychological research. As expected, no test had the highest power across all relationships. However, the distance correlation and the HHG-Pearson tests were found to have substantially greater power than all traditional tests for many relationships and only slightly less power in the worst case. A similar pattern was found in favor of the HHG-Pearson test compared to the distance correlation test. However, given that distance correlation performed better for linear relationships and is more widely accepted, we suggest considering its use in place or additional to traditional methods when there is no prior knowledge of the relationship type, as is often the case in psychological research.

RANK-BASED INDICES FOR TESTING INDEPENDENCE BETWEEN TWO HIGH-DIMENSIONAL VECTORS

To test independence between two high-dimensional random vectors, we propose three tests based on the rank-based indices derived from Hoeffding's D , Blum-Kiefer-Rosenblatt's R and Bergsma-Dassios-Yanagimoto's τ * . Under the null hypothesis of independence, we show that the distributions of the proposed test statistics converge to normal ones if the dimensions diverge arbitrarily with the sample size. We further derive an explicit rate of convergence. Thanks to the monotone transformation-invariant property, these distribution-free tests can be readily used to generally distributed random vectors including heavily tailed ones. We further study the local power of the proposed tests and compare their relative efficiencies with two classic distance covariance/correlation based tests in high dimensional settings. We establish explicit relationships between D , R , τ * and Pearson's correlation for bivariate normal random variables. The relationships serve as a basis for power comparison. Our theoretical results show that under a Gaussian equicorrelation alternative, (i) the proposed tests are superior to the two classic distance covariance/correlation based tests if the components of random vectors have very different scales; (ii) the asymptotic efficiency of the proposed tests based on D , τ * and R are sorted in a descending order.