# Bayesian Inference about Pearson Correlation

This feature requires SPSS® Statistics Standard Edition or the Advanced Statistics option.

Pearson correlation coefficient measures the linear relation between two scale variables jointly following a bivariate normal distribution. The conventional statistical inference about the correlation coefficient has been broadly discussed, and its practice has long been offered in IBM® SPSS Statistics. The design of the Bayesian inference about Pearson correlation coefficient allows users to draw Bayesian inference by estimating Bayes factors and characterizing posterior distributions.

- From the menus choose:
- Select the appropriate Test Variables to use for pairwise correlation inference from the Available Variables list. At least two source variables must be selected. When more than two variables are selected, the analysis is run on all of the selected variables' pairwise combinations.
- Select the desired Bayesian Analysis:
- Characterize Posterior Distribution: When selected, the Bayesian inference is made from a perspective that is approached by characterizing posterior distributions. You can investigate the marginal posterior distribution of the parameter(s) of interest by integrating out the other nuisance parameters, and further construct credible intervals to draw direct inference. This is the default setting.
- Estimate Bayes Factor: When selected, estimating Bayes factors (one of
the notable methodologies in Bayesian inference) constitutes a natural ratio to compare the marginal
likelihoods between a null and an alternative hypothesis.
Table 1. Commonly used thresholds to define significance of evidence Bayes Factor Evidence Category Bayes Factor Evidence Category Bayes Factor Evidence Category >100 Extreme Evidence for H1 1-3 Anecdotal Evidence for H1 1/30-1/10 Strong Evidence for H0 30-100 Very Strong Evidence for H1 1 No Evidence 1/100-1/30 Very Strong Evidence for H0 10-30 Strong Evidence for H1 1/3-1 Anecdotal Evidence for H0 1/100 Extreme Evidence for H0 3-10 Moderate Evidence for H1 1/10-1/3 Moderate Evidence for H0 *H0: Null Hypothesis**H1: Alternative Hypothesis* - Use Both Methods: When selected, both the Characterize Posterior Distribution and Estimate Bayes Factor inference methods as used.

- Specify the Maximum number of plots to see in the output. A set of plots can contain 3 plots on the same pane. The plots are generated in order from the first variable versus the remaining variables, then the second variable versus the remaining variables, and so on. The defined integer value must be between 0 and 50. By default, 10 sets of plots are output to accommodate five variables. This option is not available when Estimate Bayes Factor is selected.
- You can optionally click Criteria to specify Bayesian Pearson Correlation: Criteria settings (credible interval percentage, missing values
options, and numerical method settings), click Priors to specify Bayesian Pearson Correlation: Prior Distribution settings (value
*c*for the prior*p*(*ρ*) ∝ (1 -*ρ*^{2})^{c}, or click Bayes Factor to specify Bayesian Independent - Sample Inference: Estimate Bayes Factor settings.

^{1}Lee, M.D., and Wagenmakers, E.-J. 2013.

*Bayesian Modeling for Cognitive Science: A Practical Course*. Cambridge University Press.

^{2}Jeffreys, H. 1961.

*Theory of probability*. Oxford University Press.