[R-meta] sd of blups vs tau in RE model

[Viechtbauer, Wolfgang (NP)]

Dear Yefeng,

This was actually recently discussed on this mailing list:

https://stat.ethz.ch/pipermail/r-sig-meta-analysis/2023-May/004627.html

It is NOT true that the variance of the BLUPs will be equal to or approximate tau^2 when k is large. As I explain in that post, one can decompose tau^2 into two parts by the law of total variance. The variance of the BLUPs is only one part of this. To demonstrate:

library(metafor)

tau2 <- .02
k <- 2500
vi <- runif(k, .002, .05)
yi <- rnorm(k, 0, sqrt(vi + tau2))

res <- rma(yi, vi)
res$tau2
blups <- ranef(res)

# variance of the BLUPs (way too small as this is essentially only var(E(u_i|y_i)))
var(blups$pred)

# by adding what is essentially E(var(u_i|y_i)), we get (approximately) tau^2
var(blups$pred) + mean(blups$se^2)

The larger tau^2 is relative to the sampling variances, the less relevant mean(blups$se^2) will be (try running the code above with tau2 <- .2). But still, the variance of the BLUPs will underestimate tau^2.

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: Friday, 30 June, 2023 6:57
>To: James Pustejovsky
>Cc: Yefeng Yang; R Special Interest Group for Meta-Analysis
>Subject: Re: [R-meta] sd of blups vs tau in RE model
>
>Exactly. I was meant to the number of studies. I have no idea why I typed within-
>study replicates. Sorry for the confusion.
>________________________________
>From: James Pustejovsky <jepusto using gmail.com>
>Sent: Friday, 30 June 2023 14:41
>To: Yefeng Yang <yefeng.yang1 using unsw.edu.au>
>Cc: R Special Interest Group for Meta-Analysis <r-sig-meta-analysis using r-
>project.org>
>Subject: Re: [R-meta] sd of blups vs tau in RE model
>
>The approximations there are predicated on k (the number of studies) being large
>enough that the estimated heterogeneity (tau-hat) converges to the true
>heterogeneity parameter.
>
>On Thu, Jun 29, 2023 at 11:29 PM Yefeng Yang
><yefeng.yang1 using unsw.edu.au<mailto:yefeng.yang1 using unsw.edu.au>> wrote:
>Hi both,
>
>I happen to come across a paper, which can answer both of your comments.
>
>Eq. 1 and the following Eqs. show the derivation of the equivalence mentioned by
>my earlier email.
>
>Wang C C, Lee W C. A simple method to estimate prediction intervals and
>predictive distributions: summarizing meta‐analyses beyond means and confidence
>intervals[J]. Research Synthesis Methods, 2019, 10(2): 255-266.
>
>Best,
>Yefeng
>________________________________
>From: James Pustejovsky <jepusto using gmail.com<mailto:jepusto using gmail.com>>
>Sent: Friday, 30 June 2023 13:08
>To: Yefeng Yang <yefeng.yang1 using unsw.edu.au<mailto:yefeng.yang1 using unsw.edu.au>>
>Cc: R Special Interest Group for Meta-Analysis <r-sig-meta-analysis using r-
>project.org<mailto:r-sig-meta-analysis using r-project.org>>
>Subject: Re: [R-meta] sd of blups vs tau in RE model
>
> Thanks for your clarification. Your explanations are very clear. Actually, the
>SD of BLUPs and tau will converge when the within-study replicates are getting
>large.
>
>Can you say more about this? Is this claim based on simulations or something? I
>see the intuition, but it also seems like this property might depend not only on
>the within-study replicates all being large, but also on their _relative_ sizes.