Asian Journal of Probability and Statistics

Aims: The association between two variables is a fundamental aspect of data analysis across various research fields. Among the many measures of association, Pearson’s, Spearman’s, Kendall’s, Hoeffding’s, Distance Correlation, and Bergsma and Dassios' tau are commonly used in practice. The aim of this study is to compare the statistical power of these six correlation coefficients in detecting various types of relationships, including linear, non-linear, and non-monotonic associations, under different levels of noise and sample sizes.

Study Design and Methodology: We evaluate these methods through simulations involving nine specific relationship types (linear, parabolic, cubic, power, sine, exponential, circle, and step function), following the work of Simon and Tibshirani (2011) in their commentary on Reshef et al. (2011).

Results: The results demonstrate that Bergsma and Dassios' tau consistently outperforms all other methods for non-linear and non-monotonic relationships, particularly for small sample sizes and complex dependencies. Distance Correlation and Hoeffding’s D also show competitive power, especially for larger sample sizes, but remain generally less powerful than Bergsma and Dassios' tau in small-sample scenarios. For linear relationships, Pearson and Spearman correlations remain the most powerful.

Conclusion: These findings highlight the robustness and versatility of Bergsma and Dassios' tau, making it a preferred measure for studies in fields such as social sciences and biology, where small sample sizes and complex relationships are common. Distance Correlation remains a strong alternative for larger datasets, while Pearson and Spearman correlations continue to be effective choices for linear relationships.