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

Yefeng Yang ye|eng@y@ng1 @end|ng |rom un@w@edu@@u
Mon Apr 24 12:56:08 CEST 2023

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.

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:



>-----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
>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!
>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.
>Yefeng Yang PhD
>UNSW, Sydney

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