I would like to compare how two variables are correlated within each of two between-subject conditions. To compare the correlations, I used online calculators to conduct a Fisher z-score comparison. I have an effect size as Cohen’s q. For rhetorical simplicity and consistency with the rest of the paper, I would like to be able to also report the percent of variance accounted for by condition (i.e., r squared). Is there a formula to convert Cohen’s q to r squared? Thank you for your insight. ~ Kevin
Hi, why not use a regression analysis where you include all factors instead? Assuming that your correlated variables are X and Y, regress Y against X, the groups, and the group-by-x -interaction. The interaction will tell you how the correlation between Y and X differ between the groups and you can calculate explained variance for that.
The difference in r^2 is given by the original correlations squared, I don't think you can go directly from q to to a difference in r^2 because it is on the wrong scale (it is a difference between Fisher z scores; in a sense it runs from -2 to +2 when converted back not -1 to 1).
To compare correlations there are various approaches - though I quite like Zou (2007)'s approach. For R code see the link (and comments include links to a package and web calculator).
Zou also gives a 95% CI for differences in R^2 (not implemented in these functions but easy to set up in Excel or R). However, these are for independent R^2 differences and the present example is for overlapping, dependent correlations. So I'd refer to the simple point estimate differences (not that the CI for the difference in r can't give you a CI for the difference in r^2 directly - see Zou's paper).
https://seriousstats.wordpress.com/2012/02/05/comparing-correlations/
Thank you for your feedback, Thomas and Thom. That’s definitely a good approach, Thomas, and I can use that to get change in r2. I would still like to be able to calculate r2 directly for stylistic reasons within the paper and for future use. Thom, I do have other data to incorporate into a formula from Cohen’s q to r2. I have other formulas for converting effect sizes. For example, I know that when the two samples have similar N and the sample size is large, we can calculate r2 from Cohen’s d with r2 = d2 / (d2 +4). And I have more elaborate formulas, with fewer assumptions, to convert Cohen’s d to r2. What I’m hoping is that there are similar formulas from Cohen’s q to r2. Thank you for your help! ~ Kevin
Hi Thom. Just wanted to let you know that my co-author Karl Wuensch and I have SPSS and SAS code for most of the common tests comparing correlations and regression coefficients, with confidence intervals where possible. See the links below. And perhaps mention us in the next edition of Serious Stats. ;-)
Cheers,
Bruce
https://sites.google.com/a/lakeheadu.ca/bweaver/Home/statistics/spss/my-spss-page/weaver_wuensch
http://core.ecu.edu/psyc/wuenschk/W&W/W&W-SAS.htm
https://link.springer.com/article/10.3758%2Fs13428-012-0289-7
BTW the R code for CI of differences in independent R^2 is now linked to from my blog. I'm not sure what the approach would be for overlapping differences. Possibly a bootstrap type approach.
https://seriousstats.wordpress.com/2017/05/25/ci-for-difference-between-independent-r-square-coefficients/
Kevin,
Though it's been some time since your question was posted, I wanted to comment.
If all you have to work with is the Cohen q:
Cohen's q is the difference between two Fisher-z converted correlations, and, as such, there is no unambiguous way to convert it back to a single correlation (or squared correlation). You can, of course, convert it to a difference in correlations by the following:
r(diff) = [exp(2q) - 1] / [exp(2q) + 1]
This is a nonlinear conversion (just as the Fisher-z is from r to z).
Good luck with your work.
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