[R-meta] formula of variance of mean effect from a RE model with 1/vi with weights

[Yefeng Yang]

Dear Wolfgang,
Because RE model with a weight of 1/vi has the same overall effect of the fixed-effect model (FE model), which assumes no heterogeneity. But RE model with a weight of 1/vi  uses a multiplicative factor or scaling factor to adjust SE of the overall effect, which in essence is to account for heterogeneity. When the multiplicative factor or scaling factor is equal to 1, this model is exactly the FE model. This multiplicative factor or scaling factor is related to vi + tau^2, as indicated by the formula in p7 in your slide and (Eq. 7 in Bakbergenuly et
al. 2020), but I do not know the exact formula of this factor. This is my understanding. I could be wrong.

Best,
Yefeng
________________________________
From: Viechtbauer, Wolfgang (NP) <wolfgang.viechtbauer using maastrichtuniversity.nl>
Sent: Monday, 24 April 2023 21:06
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: formula of variance of mean effect from a RE model with 1/vi with weights

I wouldn't call this a multiplicative model.

And I don't understand what you are after when trying to "derive the formula for this multiplicative factor".

Best,
Wolfgang

>-----Original Message-----
>From: Yefeng Yang [mailto:yefeng.yang1 using unsw.edu.au]
>Sent: Monday, 24 April, 2023 12:56
>To: Viechtbauer, Wolfgang (NP); R Special Interest Group for Meta-Analysis
>Subject: Re: formula of variance of mean effect from a RE model with 1/vi with
>weights
>
>Dear Wolfgang,
>
>Amazing - you almost touch everything. Eq. 7 you pointed out in Bakbergenuly et
>al. 2020 is pretty clear. But your p7 is more straightforward.
>
>I would be grateful if you would like to answer one more question. So, RE model
>with weights of 1/vi has the same weights as the fixed-effect model (I mean the
>weights do not account for heterogeneity if we specify user-defined weights in
>rma). But RE model with weights of 1/vi will adjust SE of the model coefficient
>to account for the variance of the observed effect (in this case tau^2 + vi). In
>this sense, can we say this is a sort of multiplicative heterogeneity because it
>does not assume variance is exactly known? Rather, they are up to some values (in
>this case, the values are changing rather than a constant). If I am correct, can
>you help derive the formula for this multiplicative factor (I tried but I
>failed).
>
>Bakbergenuly I, Hoaglin D C, Kulinskaya E. Estimation in meta-analyses of
>response ratios[J]. BMC medical research methodology, 2020, 20: 1-24.
>
>Best,
>Yefeng
>________________________________________
>From: Viechtbauer, Wolfgang (NP) <wolfgang.viechtbauer using maastrichtuniversity.nl>
>Sent: Monday, 24 April 2023 19:50
>To: R Special Interest Group for Meta-Analysis <r-sig-meta-analysis using r-
>project.org>
>Cc: Yefeng Yang <yefeng.yang1 using unsw.edu.au>
>Subject: RE: formula of variance of mean effect from a RE model with 1/vi with
>weights
>
>See slide 7 in this presentation:
>
>https://www.wvbauer.com/lib/exe/fetch.php/talks:2016_viechtbauer_srsm_weights.pdf
>
>Best,
>Wolfgang
>
>>-----Original Message-----
>>From: R-sig-meta-analysis [mailto:r-sig-meta-analysis-bounces using r-project.org] On
>>Behalf Of Yefeng Yang via R-sig-meta-analysis
>>Sent: Monday, 24 April, 2023 11:01
>>To: r-sig-meta-analysis using r-project.org
>>Cc: Yefeng Yang
>>Subject: Re: [R-meta] formula of variance of mean effect from a RE model with
>>1/vi with weights
>>
>>Dear Wolfgang,
>>Brilliant! The general form looks quite complex: Var[b] = (X'WX)^-1 X'W Var[y]
>WX
>>(X'WX)^-1.
>>
>>If we do not consider predictor analysis (i.e., intercept-only MA model), can I
>>bother you to reformulate it using non-matrix notation?  I appreciate it!
>>
>>Best,
>>Yefeng
>>________________________________
>>From: R-sig-meta-analysis <r-sig-meta-analysis-bounces using r-project.org> on behalf
>>of Yefeng Yang via R-sig-meta-analysis <r-sig-meta-analysis using r-project.org>
>>Sent: Monday, 24 April 2023 18:38
>>To: r-sig-meta-analysis using r-project.org <r-sig-meta-analysis using r-project.org>
>>Cc: Yefeng Yang <yefeng.yang1 using unsw.edu.au>
>>Subject: [R-meta] formula of variance of mean effect from a RE model with 1/vi
>>with weights
>>
>>Dear experts,
>>I see the rma() in metafor package can run a random-effects meta-analysis with
>>inverse sampling variance (say vi) as weights (setting weights = 1/vi). I wonder
>>about the formula used to calculate the variance (or standard error) of the
>>overall mean in this model. I would be grateful if someone can provide the
>>corresponding estimator.
>>
>>Best
>>Yefeng Yang PhD
>>UNSW, Sydney

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