[R-meta] Meta Analysis of Interaction Effects

[Viechtbauer, Wolfgang (NP)]

Dear Ying,

If the studies use different scales, then you cannot directly compare / aggregate unstandardized regression coefficients, so yes, you will have to consider some kind of standardized measure. For regular regression coefficients (not reflecting an interaction), I am quite partial (pun intended) to using partial correlation coefficients (or their r-to-z transformed version). The nice thing about these is that they can be easily computed if all that you know if the test statistic (t-statistic) for the regression coefficient, the number of predictors in the model, and the total sample size. See:

https://wviechtb.github.io/metafor/reference/escalc.html#partial-and-semi-partial-correlations

This information is typically available when study authors are reporting the results from a linear regression model, making this measure a practicable choice. In principle, one could also compute a partial correlation coefficient for an interaction term, although I have never seen anybody do this.

Mixing together continuous-continuous and continuous-categorical interaction coefficients (of whatever type) raises other issues. Doing so could make sense if the categorical variable is a categorized version of what the corresponding continuous from continuous-continuous interactions reflects. However, then one has to 'undo' this kind of categorization, which is analogous to what happens when we do a meta-analysis of correlation coefficients, but some studies only provide point-biserial correlations. See:

Jacobs, P., & Viechtbauer, W. (2017). Estimation of the biserial correlation and its sampling variance for use in meta-analysis. Research Synthesis Methods, 8(2), 161-180. https://doi.org/10.1002/jrsm.1218

I am not familiar with some analogous procedures for turning a 'point-biserial partial correlation coefficient' into a 'biserial partial correlation coefficient'. I certainly would not simply use equation (8) from the article above on a 'point-biserial partial correlation coefficient' unless there is some evidence (empirical or analytical) that this makes sense. Even then, that still leaves the issue of how to compute the sampling variance of the resulting value.

Finally, for the case where two simple slopes are reported: That seems also very tricky to me how to turn this into something that would be meaningfully comparable to a partial correlation coefficient. This vaguely relates to methods discussed in this article:

Pustejovsky, J. E. (2014). Converting from d to r to z when the design uses extreme groups, dichotomization, or experimental control. Psychological Methods, 19(1), 92-112. https://doi.org/10.1037/a0033788

But this is not about partial correlations and interaction terms, so while a connection is there, it is not immediate.

In essence, I think it will be very difficult to compute some commensurable measure from all these different cases.

Best,
Wolfgang

> -----Original Message-----
> From: R-sig-meta-analysis <r-sig-meta-analysis-bounces using r-project.org> On Behalf
> Of Ying Cheng via R-sig-meta-analysis
> Sent: Monday, June 24, 2024 20:10
> To: r-sig-meta-analysis using r-project.org
> Cc: Ying Cheng <Ying.Cheng using csusb.edu>
> Subject: [R-meta] Meta Analysis of Interaction Effects
>
> Dear Meta Community,
>
> I hope everyone has enjoyed the summer so far!
>
> When conducting a meta-analysis, I am wondering about the appropriate way to
> extract the effect size for an interaction term (e.g., continuous variable x
> continuous variable or continuous variable x categorical variable) stemming from
> regressions.  Many studies reported unstandardized coefficients of the
> interaction terms. However, given they are using different scales, I assume
> standardized coefficients will be more appropriate (?). If so, is there any way
> to transform an unstandardized coefficient of an interaction term from
> regression to a standardized coefficient? Additionally, some studies also
> reported simple slopes for the independent variable above and below 1 SD of the
> mean. Would this information be helpful to calculate the effect size for the
> interaction?
>
> Another related question would be the way to obtain the sampling variance of
> such an interaction effect.
>
> Any information or recommended reading will be greatly valued!
>
> P.S. I appreciate a previous post discussing the effect size for an interaction
> term stemming from 2*2 ANOVA (https://stat.ethz.ch/pipermail/r-sig-meta-
> analysis/2018-February/000655.html).
>
> Best,
> Ying
>
> Ying Cheng, PhD
> Associate Professor
> Department of Management
> California State University, San Bernardino