[R-meta] H2 in the context of rma.mv

[Werner, M.A. (Marlene)]

Thank you Wolfgang!

The reference and rewritten calculation are very helpful.

All the best,
Marlene

-----Oorspronkelijk bericht-----
Van: Viechtbauer, Wolfgang (SP) <wolfgang.viechtbauer using maastrichtuniversity.nl>
Verzonden: vrijdag 18 maart 2022 10:58
Aan: Werner, M.A. (Marlene) <m.a.werner using amsterdamumc.nl>; r-sig-meta-analysis using r-project.org
Onderwerp: RE: H2 in the context of rma.mv

Dear Marlene,

It's not there, but you can calculate it manually. You can find a discussion on computing I^2 for more complex models here:

https://eur04.safelinks.protection.outlook.com/?url=https%3A%2F%2Fwww.metafor-project.org%2Fdoku.php%2Ftips%3Ai2_multilevel_multivariate&data=04%7C01%7Cm.a.werner%40amsterdamumc.nl%7Ce44e912f5cbd4865757008da08c5e292%7C68dfab1a11bb4cc6beb528d756984fb6%7C0%7C0%7C637831943240660937%7CUnknown%7CTWFpbGZsb3d8eyJWIjoiMC4wLjAwMDAiLCJQIjoiV2luMzIiLCJBTiI6Ik1haWwiLCJXVCI6Mn0%3D%7C3000&sdata=eXYGwfpD5lzaehVitYQjgFMIi7nAGEOyvvYUhOPnkMg%3D&reserved=0

Using the same principles discussed there, one can also compute H^2. And yes, your calculation below is doing just that. Using the notation in the page above, H^2 for the Konstantopoulos example would be:

W <- diag(1/res$vi)
X <- model.matrix(res)
P <- W - W %*% X %*% solve(t(X) %*% W %*% X) %*% t(X) %*% W
(sum(res$sigma2) + (res$k-res$p)/sum(diag(P))) / ((res$k-res$p)/sum(diag(P)))

Note that I put sum(res$sigma2) in the numerator instead of just res$sigma2[1]. I think for H^2, that is more sensible, since we are asking how much larger the *total variance* is compared to the amount of variance expected based on sampling variability alone. Just putting part of the variance components in the numerator leads to a rather convoluted interpretation of what the H^2 value represents.

Best,
Wolfgang

>-----Original Message-----
>From: R-sig-meta-analysis
>[mailto:r-sig-meta-analysis-bounces using r-project.org] On Behalf Of Werner,
>M.A. (Marlene)
>Sent: Thursday, 17 March, 2022 18:49
>To: r-sig-meta-analysis using r-project.org
>Subject: [R-meta] H2 in the context of rma.mv
>
>Dear all,
>
>I have a question about retrieving or calculating the H2 statistic in
>the context of fitting a multilevel (random effects) model with rma.mv.
>Can the H2 statistic be calculated or retrieved based on rma.mv output?
>I cannot seem to find it in the rma.mv output. My apologies if I am
>just plainly overseeing it.
>Or does it actually not make sense to calculate it in the context of a
>multilevel model, considering it is not straightforward which heterogeneity variance to use?
>
>To provide a reproducible example:
>
>library(metafor)
>dat <- dat.konstantopoulos2011
>
>res2 <- rma.mv(yi, vi, random = ~ 1 | district/study, data=dat)
>
>vs <- ((res2$k-1)* sum(1/res2$vi, na.rm = TRUE))/((sum(1/res2$vi, na.rm
>= TRUE))^2-sum((1/res2$vi)^2, na.rm = TRUE))
>H2 <- (res2$sigma2[1] + vs)/vs
>
>Does the above make sense (I followed the equation on p. 250 of the
>metafor reference manual)? Or is the H2 easily retrievable in another way?
>
>Thank you and all the best,
>
>Marlene
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