Pearson correlation

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Understanding Pearson Correlation: Applications, Variability, and Robustness

Introduction to Pearson Correlation

The Pearson product-moment correlation coefficient (often denoted as ( r_p )) is a widely used statistic in psychological and biomedical research to measure the linear relationship between two continuous variables. It quantifies the degree to which a change in one variable is associated with a change in another variable, ranging from -1 to +1, where 0 indicates no linear relationship .

Variability and Bias in Pearson Correlation

Sample Size and Distribution Effects

The variability and bias of the Pearson correlation coefficient can be influenced by the sample size and the distribution of the data. For normally distributed variables, ( r_p ) and the Spearman rank correlation coefficient (( r_s )) have similar expected values, but ( r_s ) tends to be more variable, especially with strong correlations. When dealing with high kurtosis distributions, ( r_p ) becomes more variable than ( r_s ).

Standard Error Calculation

Calculating the standard error of the Pearson correlation is complex due to its sampling distribution. Various approximations exist, with Bonett's (2008) method being the least biased for small sample sizes. For larger samples (n ≥ 40), most estimators are unbiased.

Robustness to Outliers and Non-Normal Data

Sensitivity to Outliers

Pearson's correlation is sensitive to outliers, which can significantly distort the results. Robust methods, such as the percentage-bend correlation and skipped-correlations, provide better estimates by down-weighting or removing outliers, thus maintaining accurate false positive control without loss of power.

Non-Normal Data

When data are non-normally distributed, using Pearson's correlation can inflate Type I error rates and reduce power. Transforming data to a normal shape before assessing Pearson correlation can minimize these errors. Among transformation methods, rank-based inverse normal transformation is particularly effective. For small and extremely non-normal samples, permutation tests often outperform other methods.

Applications and Interpretations

Appropriate Use Cases

Pearson correlation is suitable for data that follow a bivariate normal distribution and exhibit a linear relationship. For non-normally distributed data, ordinal data, or data with significant outliers, Spearman's rank correlation is a better choice as it measures monotonic relationships .

Clinical and Psychological Research

In clinical and psychological research, Pearson correlation is frequently used to explore relationships between variables such as visual acuity, contrast sensitivity, and other biomedical measurements. However, it is crucial to report the normality of the data, ( r )-value, ( p )-value, and the extent of shared variance to ensure accurate interpretation.

Conclusion

Pearson correlation remains a fundamental tool for measuring linear associations between variables. However, its application requires careful consideration of data distribution, sample size, and the presence of outliers. By understanding these factors and employing robust methods when necessary, researchers can ensure more accurate and reliable results in their studies.