[R-meta] Including different correlation coefficients in a meta-analysis

[Alexandra Bagaïni]

Hello,
Firstly, thank you for creating this space to ask questions and share helpful resources on conducting a meta-analysis. I have already come across very helpful responses in previous threads (e.g., https://stat.ethz.ch/pipermail/r-sig-meta-analysis/2017-July/000016.html and https://stat.ethz.ch/pipermail/r-sig-meta-analysis/2022-February/003831.html ) related to combining different effect size metrics/correlations.  Based on these responses, and suggested resources, I found it extremely helpful to read Jacobs & Viechtbauer (2017) and Pustejovsky (2014) to better understand the nuances between different correlations (e.g., point-biserial vs. biserial, phi vs. tetrachoric) and how to combine them in a meta-analysis.
 However, I am not sure if I can properly apply the approaches discussed in those papers to the data that I am working with.
I am currently working on a meta-analysis (mainly using the metafor package) on age differences. I have outcomes from the following types of studies:

-        Studies where they had participants of all ages (e.g., aged between 18 and 80 years) complete the same task/measure from which I obtained (Pearson’s r) correlation coefficients.

-        Studies that used an “extreme group” design in which they recruited a group of younger adults (e.g., 18-30 years) and a group of older adults (e.g., 60-80 years) and had them complete the same task/measure, and for each group I have the means and SDs which I can use to compute Hedge’s g or a point-biserial correlation coefficient, but I am not sure if it is sound that I compute a biserial correlation coefficient to better compare it to Pearson’s r.

-        Studies that used an “extreme group” design, but that have a dichotomous outcome, therefore, I obtain a 2x2 frequency table, but here the outcomes are “naturally” dichotomous, so from my understanding it would be sounder that I compute a phi coefficient rather than potentially a tetrachoric correlation coefficient.
I read in the above-mentioned texts that to compute a biserial correlation coefficient, the underlying continuous variable (in this case: age) that was “dichotomized”, has to be normally distributed, but in my case, with the studies that used an extreme group design, the distribution would be somewhat bimodal. In none of the studies that I included have age groups been created based on an age threshold (i.e., recruiting participants aged 18-80, and splitting at age Y). Also, from what I understand (but I am perhaps mistaken), I cannot truly use the formulas (5) and (6) from Pustejovsky (2014) because in many of the extreme group studies that I am extracting data from, the researchers did not recruit participants of all ages (e.g., between 18-80), they’ve only recruited people that could be categorized into either a young or older age group, so in this case m* = n_younger + n_older (*total sample cases, p.96)
I am aware that combining effect sizes based on different designs has limitations, and I am unsure how to (or if it is possible/correct to do at all) combine these effect size metrics into one analysis. A possibility would be that I combine Pearson’s r coefficients, point-biserial correlation coefficients and phi coefficients (although I am aware that these are based on different assumptions) into a single meta-analysis and I would then conduct a meta-regression to see if the effect size metric has any effect. But I also have the following concern: can I draw *sound/proper* conclusions from this pooled estimate?
 Therefore, given the above information do you think that this would be a sensible approach? If not, other suggestions would be very much appreciated :-)

Thank you in advance for your help and any clarification
Best regards
Alexandra Bagaini

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
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

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Alexandra Bagaini | PhD Student
University of Basel | Faculty of Psychology | Cognitive and Decision Sciences
Missionsstrasse 64a | 4055 Basel | Switzerland
alexandra.bagaini using unibas.ch<mailto:alexandra.bagaini using unibas.ch> | psychologie.unibas.ch<http://psychologie.unibas.ch>


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