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

[James Pustejovsky]

Yes, a known property of empirical Bayes (BLUP) estimates is that their SD
is not equal to the (estimated) SD of the random effects. See the following
articles and references therein:

Louis, T. A. (1984). Estimating a population of parameter values using
Bayes and Empirical Bayes methods. Journal of the American Statistical
Association, 79, 393–398.

Howard S. Bloom, Stephen W. Raudenbush, Michael J. Weiss & Kristin Porter
(2016): Using Multisite Experiments to Study Cross-Site Variation in
Treatment Effects: A Hybrid Approach With Fixed Intercepts and a Random
Treatment Coefficient, Journal of Research on Educational Effectiveness.
https://doi.org/10.1080/19345747.2016.1264518

Personally, I don't think this is a big problem because the SD of the BLUPs
is not the same thing as the SD of the population distribution. If it's a
concern, you could try putting a kernel density smooth over the BLUPs. The
resulting density estimate will have larger variance than the BLUPs. (I've
wondered for a while whether there's a way to choose the smoothing
bandwidth to make it match the random effects SD, but haven't had time to
look into it.)

James

On Thu, Jun 29, 2023 at 7:10 AM Yefeng Yang via R-sig-meta-analysis <
r-sig-meta-analysis using r-project.org> wrote:

> Dear experts,
>
>
> I kindly request your guidance in resolving my matter.
>
> To simplify my query, I would like to focus solely on the random-effects
> model for now. My question pertains to the conceptual equivalence between
> the tau (standard deviation of the mean/overall effect) and the standard
> deviation of the study-specific effects within the studies included in a
> meta-analysis. Some researchers refer to these study-specific effects as
> BLUPs.
>
> To explore this further, I conducted an analysis using a specific dataset
> in metafor. However, my findings seem to suggest otherwise. I present a
> reproducible example below for your reference:
>
> # calculate es and var
> dat <- escalc(measure="RR", ai=tpos, bi=tneg, ci=cpos, di=cneg,
> data=dat.bcg)
>
> # RE model
> res <- rma(yi, vi, data=dat)
>
> # calculate blups
> blup_value <- blup(res)
>
> # test whether the overall effect from res is equal to the simple mean of
> blup_value
> res$beta[1] # -0.7145323
> mean(blup_value$pred) # -0.7145323
>
> We see the two values are equal.
>
> # test whether tau from res is equal to the var of blup_value
> sqrt(res$tau2) # 0.5596815
> sd(blup_value$pred) # 0.4816293
>
> We see the two values are not equal and seem to have a big gap.
> Any comments and insights?
>
> Best,
> Yefeng
>
>
>
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