# Bayesian Pearson Correlation Analysis

• Ton J. Cleophas
• Aeilko H. Zwinderman
Chapter

## Abstract

In studies with two continuous variables usually named the x-values and y-values a linear relation between the two variables can be assessed with the help of the Pearson correlation coefficient R. R is a measure of strength of association, and varies from −1 to +1. Instead of a traditional Pearson correlation analysis a Bayesian analysis of linear correlation is possible.

A traditional analysis of Pearson linear correlation analysis provided an r-value of 0.483 with an F statistic of 10047, p-value 0.003. A Bayesian Analysis of Pearson linear correlation provided support in favor of the traditional test with a Bayes factor of 0.105.

The maximum of the posterior likelihood distribution was 0.478 with 95% credible interval
$$0.183\kern0.5em \mathrm{to}\kern0.5em 0.685.$$

This was less wide than the 95% confidence interval of the traditional Pearson linear correlation which was

$$\begin{array}{l}0.483\pm 2\times \left[\left(4.83\times 0.553\right)/0.174\right]=\\ {}0.483\pm 2\times 0.1535=\\ {}\mathrm{between}\ 0.176\ \mathrm{and}\ 0.790.\end{array}}$$

Thus, the Bayesian analysis provided slightly better statistics than did the traditional Pearson correlation analysis.

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© Springer International Publishing AG, part of Springer Nature 2018

## Authors and Affiliations

• Ton J. Cleophas
• 1
• Aeilko H. Zwinderman
• 2
1. 1.Department Medicine Albert Schweitzer HospitalAlbert Schweitzer HospitalSliedrechtThe Netherlands
2. 2.Department Biostatistics and EpidemiologyAcademic Medical Center Department Biostatistics and EpidemiologyAmsterdamThe Netherlands