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

Viechtbauer, Wolfgang (NP) wo||g@ng@v|echtb@uer @end|ng |rom m@@@tr|chtun|ver@|ty@n|
Mon Apr 24 13:06:44 CEST 2023

```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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