On the extensions of the Chatterjee-Spearman test

Technical Summary: On the Extensions of the Chatterjee-Spearman Test

Problem Statement
Chatterjee's correlation (2021) is a rank-based independence test that is consistent against all alternatives and possesses a simple asymptotic theory allowing for analytical $p$-value calculation. However, a significant limitation of Chatterjee's test is its low statistical power when detecting monotonic relationships compared to established rank tests like Spearman's or Kendall's correlations. While previous work (Zhang, 2024) proposed a "max-type" statistic combining Chatterjee's and Spearman's correlations to address this, several critical gaps remained:

1. Asymmetry: The original combined statistic is asymmetric; swapping the response ($Y$) and independent ($X$) variables can yield different results, which is problematic for applications like gene co-expression analysis where symmetric tests are preferred.
2. Limited Scope of Combination: The previous framework only combined Chatterjee's correlation with Spearman's. It was unclear how it would perform when combined with other rank correlations, such as Kendall's $\tau$ or quadrant correlation.
3. Multivariate Limitations: The existing test was restricted to random scalars. Extending this framework to random vectors (multivariate independence testing) required new theoretical developments.

Methodology
The paper proposes three key extensions to the Chatterjee-Spearman framework:

1. Symmetrization:
   The authors derive the asymptotic joint distribution of Spearman's correlation ($S_n$), Chatterjee's correlation ($\xi_n(X,Y)$), and the reversed Chatterjee's correlation ($\xi_n(Y,X)$) under the null hypothesis of independence. They prove that these three statistics are asymptotically joint normal and mutually independent. Based on this, they define a symmetric max-type statistic:
   $$\tilde{I}^S_n(X,Y) = \max\left\{ |S_n(X,Y)|, \sqrt{5/2}\xi_n(X,Y), \sqrt{5/2}\xi_n(Y,X) \right\}$$
   The asymptotic null distribution is derived analytically, allowing for direct $p$-value calculation without permutation.

2. Extension to Other Rank Correlations:
   The study investigates the asymptotic relationships between Chatterjee's correlation and Kendall's $\tau_n$ and the quadrant correlation $Q_n$. The authors establish that under independence, $\xi_n$ is asymptotically independent of both $\tau_n$ and $Q_n$. Consequently, they define two new symmetric tests:

   • Chatterjee-Kendall: $\tilde{I}^\tau_n(X,Y) = \max\left\{ |\tau_n(X,Y)|, \sqrt{5/2}\xi_n(X,Y), \sqrt{5/2}\xi_n(Y,X) \right\}$
   • Chatterjee-Quadrant: $\tilde{I}^Q_n(X,Y) = \max\left\{ 3/2|Q_n(X,Y)|, \sqrt{5/2}\xi_n(X,Y), \sqrt{5/2}\xi_n(Y,X) \right\}$
     Analytical $p$-values are derived for these statistics based on the joint normality results.

3. Multivariate Extensions:
   Two approaches are explored for multivariate data ($X \in \mathbb{R}^p, Y \in \mathbb{R}^q$):

   • Grothe et al. (2014) Formulas: Utilizing multivariate ranks to define $S_n$ and $\tau_n$. Since the asymptotic null distribution of the combined max-statistic is difficult to derive analytically due to the ratio form of multivariate Spearman/Kendall, the authors recommend a permutation test to estimate $p$-values and standard deviations.
   • Borel Isomorphism: Transforming random vectors into random scalars using a Borel merging function (e.g., binary expansion) to apply univariate results directly. However, simulations suggest this approach may result in low power for detecting monotonic relationships in multivariate settings.

Key Results

• Theoretical: The paper proves that under independence, Chatterjee's correlation is asymptotically independent of Spearman's, Kendall's, and quadrant correlations. It establishes the asymptotic joint normality of these statistics, enabling the derivation of analytical $p$-values for the symmetric univariate tests.
• Simulation (Univariate):
  • All proposed tests effectively control the Type I error rate.
  • The Chatterjee-Kendall test demonstrates superior statistical power compared to the Chatterjee-Spearman test, particularly for small sample sizes ($n=20$) and in linear/stepwise (monotonic) settings.
  • The combined tests outperform individual tests (Chatterjee, Spearman, Kendall) across diverse correlation patterns (linear, quadratic, sinusoidal, stepwise).
• Simulation (Multivariate):
  • The permutation-based approach using Grothe et al.'s formulas yields robust power.
  • The Borel isomorphism approach, while theoretically convenient, often fails to control Type I error rates when using analytical $p$-values for small samples and exhibits low power for monotonic associations in multivariate settings.
• Real Data Applications:
  • Yeast Transcriptomics: Applied to 4,381 genes, the Chatterjee-Kendall test identified the most significant genes (897) related to cell cycles, outperforming individual tests. It successfully detected both oscillatory (non-monotonic) and monotonic patterns missed by single tests.
  • Salespeople Data: Applied to multivariate associations between sales performance and reasoning scores, the permutation-based Chatterjee-Kendall test showed high sensitivity (95.1% rejection rate in small-sample simulations), outperforming the symmetrized Chatterjee test.

Significance and Claims
The paper claims to significantly broaden the applicability of rank-based combined independence tests. By symmetrizing the statistic, it removes the need to arbitrarily designate response and independent variables, making the method suitable for symmetric association analysis (e.g., gene co-expression). By demonstrating that Chatterjee's correlation is asymptotically independent of other major rank correlations, the paper validates the framework's flexibility, allowing the substitution of Spearman's correlation with Kendall's or quadrant correlation.

The authors modestly conclude that while the Chatterjee-Kendall test exhibits better empirical power, particularly for small samples and noisy data, there is currently no theoretical justification explaining why it outperforms the Chatterjee-Spearman test against local alternatives. They also note that the multivariate extensions based on Grothe et al.'s formulas are currently limited to low dimensions due to the sparsity of multivariate ranks in high-dimensional settings, suggesting that distance correlation methods remain preferable for high-dimensional independence testing.