The effect sizes of Pearson correlation coefficient

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Understanding Effect Sizes of Pearson Correlation Coefficient

Introduction to Pearson Correlation Coefficient

The Pearson correlation coefficient (PCC), denoted as ( r ), is a measure of the linear relationship between two variables. It ranges from -1 to 1, where 1 indicates a perfect positive linear relationship, -1 indicates a perfect negative linear relationship, and 0 indicates no linear relationship. Understanding the effect sizes of PCC is crucial for interpreting the strength and significance of these relationships in various research contexts.

Traditional Guidelines for Effect Sizes

Cohen (1988) provided widely accepted guidelines for interpreting the magnitude of correlation coefficients: ( r = 0.10 ) as small, ( r = 0.30 ) as medium, and ( r = 0.50 ) as large. These guidelines, however, were based on qualitative impressions rather than systematic quantitative analysis.

Empirical Re-evaluation of Cohen's Guidelines

Recent studies have re-evaluated Cohen's guidelines using large samples of meta-analytically derived correlations. One study analyzed 708 meta-analytically derived correlations and found that the 25th, 50th, and 75th percentiles corresponded to correlations of 0.11, 0.19, and 0.29, respectively. This suggests that Cohen's guidelines might be too stringent, and researchers should consider correlations of 0.10, 0.20, and 0.30 as relatively small, typical, and relatively large.

Influence of Sample Size and Variability

The effect of sample size and variability on the PCC is significant. Variability in excess of 10% of the range for each variable can result in a mean reduction of the shared variance by 50% or greater. While sample size does not affect the mean PCC, it dramatically affects extreme percentile values, producing unreliable results. Therefore, small PCC values can sometimes be artifacts of variability, and researchers should be cautious when interpreting these values without considering associated variabilities.

Comparison with Other Correlation Coefficients

The Pearson correlation coefficient is often compared with other correlation measures like Spearman's rank correlation coefficient. For normally distributed variables, both coefficients have similar expected values, but Spearman's is more variable, especially when the correlation is strong. However, for variables with high kurtosis, Pearson's is more variable. Thus, Pearson's is suitable for light-tailed distributions, while Spearman's is preferable for heavy-tailed distributions or when outliers are present.

Base Rate Dependence

The magnitude of the Pearson correlation can be influenced by the base rate when applied to dichotomous or ordinal data. Correlations tend to decrease when the dichotomous variable does not have a 50% base rate. This sensitivity to base rates suggests that alternative effect size statistics like AUCs, Pearson/Thorndike adjusted correlations, Cohen's d, or polychoric correlations might be more robust in certain contexts.

Common Language Effect Size

The Pearson correlation coefficient can be translated into a common language effect size, which shows the probability of obtaining a certain value on one variable given the value on the other variable. This translation makes the size of a correlation coefficient more understandable to laypeople, enhancing the interpretability of statistical results.

Conclusion

The interpretation of Pearson correlation coefficients is nuanced and context-dependent. While Cohen's traditional guidelines provide a starting point, empirical evidence suggests that these guidelines may overestimate effect sizes in certain fields. Researchers should consider the influence of sample size, variability, and base rates when interpreting PCC values and may benefit from using alternative effect size statistics in specific scenarios. Understanding these factors ensures more accurate and meaningful interpretations of correlation coefficients in research.