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

[Viechtbauer, Wolfgang (NP)]

I am sorry, but I am still not following. The RE model with non-standard weights doesn't really use a multiplicative factor to adjust the SE of the overall effect. We can directly derive Var[mu-hat] = Var[sum(wi*yi)/sum(wi)] (with Var[yi] = vi + tau2) using standard variance-covariance algebra.

Best,
Wolfgang

>-----Original Message-----
>From: Yefeng Yang [mailto:yefeng.yang1 using unsw.edu.au]
>Sent: Monday, 24 April, 2023 13:24
>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,
>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.pd
>f
>>
>>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