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

Yefeng Yang ye|eng@y@ng1 @end|ng |rom un@w@edu@@u
Fri Jun 30 10:42:28 CEST 2023

```Dear Wolfgang,

Excellent!  Sorry that I did not realize this post was on the mailing list (the title of this post really hard let me relate it to my question - anyway sorry for repeating the question)

I am arguing a bit. As pointed out by Wolfgang, the total variance of the population can be decomposed into two parts:
tau^2 = var(u_i) = E(var(u_i|y_i)) + var(E(u_i|y_i))

If k is larger enough, the expectation of the variance of the random effects should be zero or trivial, assuming the random effects are normally distributed. So, the first part of the above formula E(var(u_i|y_i)) is decreasing to 0 as K -> infinite. This is my understanding. I might be silly.

A relevant question is do you think it is meaningful to use the distribution of BLUPs to calculate something like "the proportion of the true effects above a certain value" to represent the heterogeneity?

Let's use your numerical example to show my point:
Say we want to know the proportion of the true effects (which are denoted by BLUPs) above 0,  we get a proportion of 0.5046695. Can we say the data is moderately heterogeneous?
BLUP = 0
Z = (BLUP - res\$beta[1]) / sqrt(res\$tau2)
1 - pnorm(Z)

Best,
Yefeng

________________________________
From: Viechtbauer, Wolfgang (NP) <wolfgang.viechtbauer using maastrichtuniversity.nl>
Sent: Friday, 30 June 2023 17:38
To: R Special Interest Group for Meta-Analysis <r-sig-meta-analysis using r-project.org>; James Pustejovsky <jepusto using gmail.com>
Cc: Yefeng Yang <yefeng.yang1 using unsw.edu.au>
Subject: RE: [R-meta] sd of blups vs tau in RE model

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,
>
>
>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.
>