[R-meta] Sigma in Bayesian Multi-Level Meta-Analytic Models (brms)

[Yefeng Yang]

Dear Wolfgang,

Thank you for your confirmation and clarification. I have a further question. It is clear to me that for a dataset with dependent effect sizes, bf(yi | se(sei, sigma=TRUE) ~ X + (1 | study), sigma ~ X) will model location and estimate level scale parameter (or we call it effect size level in the context of meta-analysis) as a function of X.

But for a dataset with independent effect sizes (where between-study level is equal to estimate level), what sort of scale parameter will be modelled by bf(yi | se(sei, sigma=TRUE) ~ X + (1 | study), sigma ~ X)? (1 | study) already accounts for all random effects and there is nothing left for  'sigma ~ X'.

Another question is that a dataset with dependent effect sizes, if I want to model between-study variance (rather than estimate level) as a function of 'X', what the syntax will be?

Happy to see your comment on these.

Thanks,
Yefeng




________________________________
From: Viechtbauer, Wolfgang (NP) <wolfgang.viechtbauer using maastrichtuniversity.nl>
Sent: Friday, January 10, 2025 3:59
To: Yefeng Yang <yefeng.yang1 using unsw.edu.au>; R Special Interest Group for Meta-Analysis <r-sig-meta-analysis using r-project.org>
Subject: RE: [R-meta] Sigma in Bayesian Multi-Level Meta-Analytic Models (brms)

Dear Yefeng,

Yes, that is correct. To be precise, it should be:

bf(yi | se(sei, sigma=TRUE) ~ X, sigma ~ X)

Note that brms() parameterizes the scale part as log(tau) and not as log(tau^2) as is done in rma(), so the coefficients will be about half of the size (and of course can differ due to the priors).

Best,
Wolfgang

> -----Original Message-----
> From: Yefeng Yang <yefeng.yang1 using unsw.edu.au>
> Sent: Thursday, January 9, 2025 14:50
> To: R Special Interest Group for Meta-Analysis <r-sig-meta-analysis using r-
> project.org>
> Cc: Viechtbauer, Wolfgang (NP) <wolfgang.viechtbauer using maastrichtuniversity.nl>
> Subject: Re: [R-meta] Sigma in Bayesian Multi-Level Meta-Analytic Models (brms)
>
> Dear Hanna and Wolfgang,
>
> Just jump in. I have a following-up question regarding Wolfgang's comment about
> 'yi | se(sei, sigma=TRUE) ~ ...'
>
> If setting 'sigma=TRUE' is equivalent to add an estimate level random effect and
> if in the random part of the model I do not have (1 | estimate) (which means
> there is no redundant effect level random effect), whether "bf(y ~ X + (1 |
> study), sigma ~ 1 + X)" is equivalent to a location-scale meta-analysis?
>
> All the best,
> Yefeng
>
> ________________________________________
> From: R-sig-meta-analysis <mailto:r-sig-meta-analysis-bounces using r-project.org> on
> behalf of Viechtbauer, Wolfgang (NP) via R-sig-meta-analysis <mailto:r-sig-meta-
> analysis using r-project.org>
> Sent: Thursday, January 9, 2025 23:43
> To: R Special Interest Group for Meta-Analysis <mailto:r-sig-meta-analysis using r-
> project.org>
> Cc: Viechtbauer, Wolfgang (NP)
> <mailto:wolfgang.viechtbauer using maastrichtuniversity.nl>
> Subject: Re: [R-meta] Sigma in Bayesian Multi-Level Meta-Analytic Models (brms)
>
> Dear Hanna,
>
> Correct, when you use something like 'yi | se(sei) ~ ...' in brms, then the
> known standard errors replace the sigma parameter, which is just set to 0 (and
> given as such in the output). So sigma here is not an estimate but simply a
> fixed value. See help(brmsformula).
>
> It is possible to use 'yi | se(sei, sigma=TRUE) ~ ...', where sigma will be
> estimated. However, this just adds a regular error term to the model, which
> would be redundant with an estimate level random effect (i.e., (1 | estimate)),
> which should be part of the model anyway.
>
> Best,
> Wolfgang
>
> > -----Original Message-----
> > From: R-sig-meta-analysis <mailto:r-sig-meta-analysis-bounces using r-project.org>
> On Behalf
> > Of Hanna M��tze via R-sig-meta-analysis
> > Sent: Thursday, January 9, 2025 11:19
> > To: mailto:r-sig-meta-analysis using r-project.org
> > Cc: Hanna M��tze <mailto:muetzeh using uni-bremen.de>
> > Subject: [R-meta] Sigma in Bayesian Multi-Level Meta-Analytic Models (brms)
> >
> > Dear all,
> >
> > I conducted a Bayesian 3-level meta-analysis with |brms| and noticed
> > that �� is consistently estimated as 0, even in 2-level and 4-level
> > models. I observed the same behavior in this tutorial:
> > https://mvuorre.github.io/posts/2016-09-29-bayesian-meta-analysis/#ref-
> > mcelreathStatisticalRethinkingBayesian2020.
> > How should I interpret this result?
> >
> > How should I interpret this result? Am I correct in assuming that
> > Bayesian models do not estimate the sampling error because it is assumed
> > to be known based on the sample size? Unfortunately, I could not find
> > references supporting this interpretation and would appreciate any
> > clarification or guidance.
> >
> > Thank you for your time and help!
> >
> > Best regards,
> > Hanna M��tze

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