Comparing the Pearson and Spearman Correlation Coefficients Across Distributions and Sample Sizes: A Tutorial Using Simulations and Empirical Data (2408.15979v1)

Abstract: The Pearson product-moment correlation coefficient (rp) and the Spearman rank correlation coefficient (rs) are widely used in psychological research. We compare rp and rs on 3 criteria: variability, bias with respect to the population value, and robustness to an outlier. Using simulations across low (N = 5) to high (N = 1,000) sample sizes we show that, for normally distributed variables, rp and rs have similar expected values but rs is more variable, especially when the correlation is strong. However, when the variables have high kurtosis, rp is more variable than rs. Next, we conducted a sampling study of a psychometric dataset featuring symmetrically distributed data with light tails, and of 2 Likert-type survey datasets, 1 with light-tailed and the other with heavy-tailed distributions. Consistent with the simulations, rp had lower variability than rs in the psychometric dataset. In the survey datasets with heavy-tailed variables in particular, rs had lower variability than rp, and often corresponded more accurately to the population Pearson correlation coefficient (Rp) than rp did. The simulations and the sampling studies showed that variability in terms of standard deviations can be reduced by about 20% by choosing rs instead of rp. In comparison, increasing the sample size by a factor of 2 results in a 41% reduction of the standard deviations of rs and rp. In conclusion, rp is suitable for light-tailed distributions, whereas rs is preferable when variables feature heavy-tailed distributions or when outliers are present, as is often the case in psychological research.

• The paper compares Pearson (rp) and Spearman (rs) correlation coefficients based on variability, bias, and robustness using simulations and empirical data.
• Key findings indicate that rs is less variable and more robust than rp under non-normal or heavy-tailed data conditions, especially with outliers.
• The study recommends rs as a default measure for data likely to deviate from normality due to its enhanced robustness and reduced variability, while also stressing the importance of large sample sizes.

Insights Into Comparing Pearson and Spearman Correlation Coefficients

The paper "Comparing the Pearson and Spearman Correlation Coefficients Across Distributions and Sample Sizes: A Tutorial Using Simulations and Empirical Data," authored by De Winter, Gosling, and Potter, tackles an important analysis of two fundamental statistical measures widely used in psychological research—the Pearson product-moment correlation coefficient (r_p) and the Spearman rank correlation coefficient (r_s). By dissecting the relative performance of these coefficients across varying conditions, the paper provides a detailed tutorial for researchers, guiding them on which coefficient to prefer under differing scenarios.

Statistical Comparison and Simulation Design

The paper evaluates Pearson's and Spearman's coefficients through simulations and real-world datasets, focusing on three criteria: variability, bias in approximation to population values, and robustness to outliers. Through simulations with sample sizes ranging from N = 5 to N = 1,000, the paper finds that normally distributed variables reveal both r_p and r_s have similar expected values. However, r_s is less variable compared to r_p, particularly when the correlation is strong. In contrast, when analyzing variables with high kurtosis, r_s shows less variability and is generally more robust, especially in the presence of outliers, a common feature in psychological data.

Empirical Analysis and Practical Implications

The empirical portion of the paper utilized large datasets, including psychometric and Likert-scale surveys, to verify simulation results. For normally distributed datasets, like the ASVAB test scores, r_p demonstrated slightly lower variability compared to r_s. Conversely, for datasets exhibiting heavy tails or high kurtosis, r_s showed superior performance. Notably, the Spearman correlation often corresponded more accurately to the Pearson population coefficient (R_p) than r_p itself in heavily-tailed distributions, showcasing its robustness.

Theoretical and Practical Implications

Theoretical insights emphasize that the Spearman coefficient exhibits superior robustness and reduced variability in non-normal conditions, validating its application in situations where non-normality and outliers prevail. Practically, this outcome suggests researchers might prefer r_s when dealing with heavy-tailed or highly kurtotic datasets—a frequent occurrence in psychological and behavioral sciences—despite a slight loss of efficiency in the context of perfect bivariate normality.

Sample Size Considerations and Variability

The paper underscores the critical effect of sample size on correlation coefficient variability, drawing attention to the necessity of large sample sizes to achieve reliable estimates. With r_s reducing variability by approximately 20% compared to r_p, but with a sample size increase yielding a 41% reduction in standard deviation, the paper advocates for balancing sample size adjustments with the choice of correlation coefficient for optimal results.

Conclusion

This exploration into Pearson and Spearman correlation coefficients highlights the significant impact of distribution characteristics and sample size. While r_p aligns with assumptions of normality and linear relationships, r_s offers enhanced performance under non-normal, heavy-tailed conditions and in the presence of outliers. Consequently, r_s is recommended as a default measure for situations with anticipated deviations from normality, ensuring empirical findings are reliable and robust against unusual data distributions. Future research can expand on these results by exploring alternative robust estimation techniques and their applications across various empirical domains.