# Jeromy Anglim's Blog: Psychology and Statistics

## Monday, September 7, 2009

### Significance Tests on Correlations

OVERVIEW: I often speak to researchers wanting to compare the significance of two correlations. The two scenarios most commonly encountered are: 1) comparing dependent correlations; and 2) comparing independent correlations.

Two dependent correlations
This is the scenario when you have three variables, x, y, and z, and you want to compare the x-y correlation with the x-z correlation. It comes up often when you want to know which of two variables are more related to a third variable. In this sense it is often related to approaches that attempt to assess variable importance using multiple regression.

The main difficulty that most researchers that I talk to have (using SPSS) is that SPSS does not have an built-in tool to test for the statistical significance of a difference between correlations.

In a previous post, I discuss how to run a significance test with R, and in the comments for the post is a link to an SPSS macro that will do the same. For more information about the formulas follow this link.

If you want to obtain confidence intervals of the difference in two dependent correlations, read this.

Two independent correlations.
This is the scenario when two correlations are obtained from different samples and you want to test whether they are significantly different. An example is where a researcher wants to know whether intelligence test scores and performance are correlated the same in different social groups. In most cases, this is very similar to testing for a group by IV interaction effect. Thus, moderator regression is often a more appropriate means of testing the relationship. However, you can also run a specific test of statistical significance on the difference between the two correlations.

Other scenarios include: 3) testing whether a correlation is significantly different from some target value, usually zero, but possibly another value.; 4) a significance test on a correlation matrix; 5) Structural Equation Modelling software can also be used to test more general hypotheses about patterns in correlation matrices. But these things, I'll save for another post.

For scenarios 1, 2, and 3 above, Howell (Statistical Methods for Psychology) sets out the formulas in his chapter on Correlations.