[R-meta] Why total variation differs in two rma.mv models?

[Viechtbauer, Wolfgang (NP)]

And to add to this:

When I intially saw Yuhang's mail, I actually expected that the question was going to phrased the other way around -- that m2 had a higher amount of heterogeneity than m1 -- which indeed is counterintuitive and in which case I would have pointed Yuhang to this:

https://www.metafor-project.org/doku.php/tips:increasing_tau2_when_adding_moderators

(which only looks at the standard random/mixed-effects model, but the same issue can of course arise in more complex multilevel models).

Best,
Wolfgang

>-----Original Message-----
>From: Yefeng Yang [mailto:yefeng.yang1 using unsw.edu.au]
>Sent: Tuesday, 10 January, 2023 12:34
>To: Viechtbauer, Wolfgang (NP); r-sig-meta-analysis using r-project.org
>Cc: yh342 using nau.edu; Viechtbauer, Wolfgang (NP)
>Subject: Re: [R-meta] Why total variation differs in two rma.mv models?
>
>Dear Yuhang,
>
>Just add to Wolfgang's explanation. If you have basic knowledge of regression
>analysis, you will understand that it is expected that m2 would have less total
>variance than m1. This is basically the aim of why you added the a priori
>moderators to a meta-analytic model - explain the heterogeneity. However, as
>Wolfgang pointed out, if the moderators you added to the model do not account for
>any of the heterogeneity, the residual heterogeneity might be even larger than
>that from a null model - this is somewhat counterintuitive, but it can happen in
>the context of meta-analysis.
>
>Best,
>Yefeng
>________________________________________
>From: R-sig-meta-analysis <r-sig-meta-analysis-bounces using r-project.org> on behalf
>of Viechtbauer, Wolfgang (NP) <wolfgang.viechtbauer using maastrichtuniversity.nl>
>Sent: Tuesday, 10 January 2023 18:12
>To: r-sig-meta-analysis using r-project.org <r-sig-meta-analysis using r-project.org>
>Cc: yh342 using nau.edu <yh342 using nau.edu>
>Subject: Re: [R-meta] Why total variation differs in two rma.mv models?
>
>Dear Yuhang,
>
>If the moderators account for (at least some of the) heterogeneity, then this is
>exactly what should happen (in m2, the variance components reflect heterogeneity
>not accounted for by the moderators).
>
>Best,
>Wolfgang
>
>>Dear Colleagues,
>>
>>I have fit an intercept-only model like:
>>
>>m1 = rma.mv(yi ~ 1, V=V, random = ~1|study/effect)
>>
>>And then the same model with some moderators:
>>
>>m2 = rma.mv(yi ~ mod1*mod2 + X1 + X2, V=V, random = ~1|study/effect)
>>
>>When I compare the **total variation** (sum of the variance components)
>>across the two models, "m1" has a much larger estimate than "m2".
>>
>>I wonder how that could be when both models use the same set of effect
>>sizes?
>>
>>Thank you so much!
>>Yuhang Hu