[R-meta] Meta-analysis of linear and non-linear associations: R2 as the “effect size”?

Michael Dewey ||@t@ @end|ng |rom dewey@myzen@co@uk
Sun Mar 14 14:30:38 CET 2021

Some comments in-line

On 12/03/2021 09:02, Werner, M.A. (Marlene) wrote:
> Dear all,
> I have a question regarding meta-analysis of effect sizes of associations.
> I was wondering whether it is possible to somehow quantitatively summarize results across studies that modelled the association between two variables "linearly" (e.g. using simple linear regression) vs. others that modelled the association "non-linearly" (e.g. polynomials or splines). One variable is continuous (hormone concentration) and the other is ordinal (self-report of well-being).

Note that if you include, say, a quadratic for hormone concentration it 
is still a linear model. I suspect you know that as you put 
"non-linearly" in quotes.

> For instance, would it be possible to meta-analyze the (adjusted) R2 as the "effect size"? If so, how would one go about the meta-analytic steps?
> I figured that meta-analysis of proportions might be applicable, but then I would need the counts on which the proportion is based, which is not directly available for this �proportion�, especially not in the non-linear case.
> I started thinking about meta-analyzing "explained variance" in the first place because I cannot summarize the "regression coefficients" across these models; �non-linear models� include several coefficients, whereas �linear models� only provide one.
> Then again, R2 is not really applicable as a goodness of fit measure for non-linear models.

I do not think that is true for the reason stated above.

  So which other "summary statistic" to use that is available and 
comparable for these models, even when comparing them in a 
non-quantitative way? Would the Standard Error of the Regression be an 
option? But how to standardize it to make it comparable across studies?
> I would be very grateful for any ideas or pointers! Thanks in advance!

The main problem I see here is that R^2 is something which could arise 
in many ways in the polynomial models. It just tells you how much 
variance in well being is explained by knowing hormone concentration but 
that might be either a strong linear effect, a strong quadratic, both, 
or something else.

> (Note that I have posted the same q on stackexchange: https://stats.stackexchange.com/questions/513164/meta-analysis-of-linear-and-non-linear-associations-r2-as-the-effect-size?; I hope that is ok!)

As long as you tell us cross-posting is OK although there is a danger 
that someone might answer on CV not knowing about this post.


> All the best,
> Marlene Werner
> Drs. M.A. (Marlene) Werner  |  PhD candidate
> Department of Gynaecology and Obstetrics / Sexology and Psychosomatic Gynaecology
> Location AMC | H4-236 | Meibergdreef 9, 1105 AZ Amsterdam
> E: m.a.werner using amsterdamumc.nl<mailto:m.a.werner using amsterdamumc.nl>
> www.amsterdamumc.nl<http://www.amsterdamumc.nl/>
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