Effect size for meta-regression

I have conducted a hierarchical meta-regression, and the results revealed a large R2 with all predictors in the model, which is common in meta-regressions. Some of the non-significant results might be the results of low statistical power and do not necessarily rule out moderator effects (Hedges & Pigott, 2004).

My question is whether there is a measure that will assist in interpreting the magnitude of the influence of each predictor in the model, to not rely solely on p-values. In primary studies it is not a problem. When you have multiple regression results you use the following formula to calculate semi-partial correlation rsp: Rsq is the square multiple correlation for the full model, tf is the t-test of the regression coefficient, n is the sample size and p is the number of predictors (Aloe & Becker, 2012, formula 2, p. 280).

I do not think, the formula can be used for the meta-regression results. The first problem you run into is that meta-regression results report z values, and not t values, and even if you replace t with z values, the results are not weighted as they should generally be in a meta-analysis.

Is there a similar measure that can be used for meta-regression results? None of the major meta-analysis books that I looked at mentioned it (e.g., Cooper, Hedges, & Valentine, 2009; Lipsey & Wilson, 2001; Pigott, 2012), neither any articles that I found. Those that I came across were mostly referring to power analyses in relation to meta-regressions (e.g., Hedges & Pigott, 2004) but not really to the question of how to calculate semi-partial or multivariate correlations. Note that this information was not provided by the primary studies.

References

Aloe, A. M., & Becker, B. J. (2012). An effect size for regression predictors in meta-analysis. Journal of Educational and Behavioral Statistics, 37, 278-297.

Cooper, H., Hedges, L. V., & Valentine, J. C. (Eds.), (2009). The handbook of research synthesis and meta-analysis (Vol. 2). New York: Russell Sage Foundation.

Hedges, L. V., & Pigott, T. D. (2004). The power of statistical tests for moderators in meta-analysis. Psychological Methods, 9, 426–445.

Lipsey, M. W., & Wilson, D. B. (2001). Practical meta-analysis. Thousand Oaks, CA: Sage Publications.

Pigott, T. E. (2012). Advances in meta-analysis. New York: Springer.

• You might want to look at some of the work outlined in jstatsoft.org/article/view/v017i01 and importance of variables in regression. I am not sure how directly it transfers to meta-regression. – mdewey Apr 23 '18 at 17:13
• Thank you. I will have a look. It might be actually not possible to transfer it into meta-regression. – Natalie Apr 24 '18 at 12:06