[R-meta] Meta-analysis of intra class correlation coefficients

Wed, Oct 16, 2024 1:20 PM

I don't understand what you mean by 'related traits'. Also, if multiple estimates are extracted from the same cohort, you may need to account for this as well. But leaving this aside, you should fit a single bivariate model to the z(rMZ) and z(rDZ) values. Maybe something like this:

model <- rma.mv(
  zi,
  vzi,
  random = list(~ MZDZ_factor | Study_ID, ~ MZDZ_factor | ESID), struct="UN",
  data = data,
  mods = ~ 0 + MZDZ_factor + DZMZ_factor,
  method = "REML",
  tdist = TRUE)

although not sure whether this will converge.

coef(model) then gives you the two coefficients and vcov(model) the corresponding var-cov matrix thereof. The multivariate delta method is described here:

https://en.wikipedia.org/wiki/Delta_method#Multivariate_delta_method

This is implemented, for example in msm::deltamethod(). Double-check this, but:

deltamethod(~ 2*((exp(2*x1) - 1) / (exp(2*x1) + 1) - (exp(2*x2) - 1) / (exp(2*x2) + 1)), coef(model), vcov(model))

should then give you the SE of 2*(rMZ - rDZ).

Best,
Wolfgang

-----Original Message-----
From: St Pourcain, Beate <Beate.StPourcain at mpi.nl>
Sent: Wednesday, October 16, 2024 10:59
To: Viechtbauer, Wolfgang (NP) <wolfgang.viechtbauer at maastrichtuniversity.nl>; R
Special Interest Group for Meta-Analysis <r-sig-meta-analysis at r-project.org>
Subject: RE: Meta-analysis of intra class correlation coefficients

ATTACHMENT(S) REMOVED: Funnel_DZ_ICC_simple_withN_rZ.png |
Funnel_MZ_ICC_simple_withN_rZ.png

Dear Wolfgang,
Many thanks for the suggestions!
The r-Z transformation was easily applied, as suggested.

data$zi <- FisherZ(data$r)
data$vzi <- 1/(data$n-3/2)
egger_seMZ <- regtest(MZdata$zi, MZdata$vzi)
egger_seDZ <- regtest(DZdata$zi, DZdata$vzi)

...and improved the Egger regression
zrMZ_se_z       zrDZ_se_z
-3.968          -5.519

Also, the funnel plots look better (see attached). We subsequently carried out
the rma.mv as the ICC estimates partially represent related traits estimated
within the same study (i.e. the estimates are related within rMZ and within
rDZ), and of course rMZ and rDZ are nested within the same study and also
sometimes the same cohort (for code see below).

For the subsequent analysis, we back-transformed the Z-score intercept from
rma.mv with

rMZ <- FisherZInv(model2$b[1])
rDZ <- FisherZInv(model1$b[1])

However, we are unsure as to how to best backtransform model2$se[1] and
model1$se[1], i.e. the SEs of the rZ intercepts from rma.mv using the delta
method as you recommended. Could you please advise? This will create the input
for the subsequent heritability analyses.

Best wishes,
Beate

#################################################################
Study_ID - Cohort ID
ESID - actual Study ID
MZDZ_factor - (MZ=1, DZ=0)
DZMZ_factor - (DZ=1,MZ=0)

For completeness, I add the code below that was adapted from the Austerberry
study for z values:
model1 <- rma.mv(
  zi,
  vzi,
  random = list(~MZDZ_factor|Study_ID, ~ MZDZ_factor | ESID),
  data = data,
  mods = ~ MZDZ_factor,
  method = "REML",
  tdist = TRUE
)

model2 <- rma.mv(
  zi,
  vzi,
  random = list(~DZMZ_factor|Study_ID, ~ DZMZ_factor | ESID),
  data = data,
  mods = ~ DZMZ_factor,
  method = "REML",
  tdist = TRUE
)

Beate St Pourcain, PhD
Senior Investigator & Group Leader
Room A207
Max Planck Institute for Psycholinguistics | Wundtlaan 1 | 6525 XD Nijmegen |
The Netherlands

@bstpourcain
Tel: +31 24 3521964
Fax: +31 24 3521213
ORCID: https://orcid.org/0000-0002-4680-3517
Web: https://www.mpi.nl/departments/language-and-genetics/projects/population-
variation-and-human-communication/
Further affiliations with:
MRC Integrative Epidemiology Unit | University of Bristol | UK
Donders Institute for Brain, Cognition and Behaviour | Radboud University | The
Netherlands

-----Original Message-----
From: Viechtbauer, Wolfgang (NP)
<mailto:wolfgang.viechtbauer at maastrichtuniversity.nl>
Sent: Monday, October 14, 2024 3:32 PM
To: St Pourcain, Beate <mailto:Beate.StPourcain at mpi.nl>; R Special Interest
Group for Meta-Analysis <mailto:r-sig-meta-analysis at r-project.org>
Subject: RE: Meta-analysis of intra class correlation coefficients

It is not ideal to meta-analyze raw correlations (or raw ICC values) if they are
so large. In this case, I think the r-to-z transformation is highly advisable.

What you observe is the fact that the variance of a raw correlation coefficient
depends on the true correlation. In particular, the large-sample estimate of the
variance of a raw correlation coefficient (or ICC based on pairs) is (1-r^2)^2 /
(n-1), where r is the observed correlation and n the sample size. Therefore, as
r gets close to 1, the variance will get small, as you noted.

It is therefore no surprise that the Egger regression test is highly
significant. Of course, this then says nothing about potential publication bias.
There is an inherent link between the correlation and its varianace (and hence
standard error). The same issue also arises with other outcome measures / effect
sizes (e.g., standardized mean differences).

In the present case, the r-to-z transformation 'solves' this issue, since
Var[z_r] =~ 1/(n-3) for Pearson correlations and Var[z_ICC] =~ 1/(n-3/2) for
ICC(1) values.

I would then consider doing a bivariate meta-analysis of the z_ICC_mz and
z_ICC_dz values. Since they are based on independent samples, their sampling
errors are uncorrelated, but the random effects of the bivariate model then
account for potential correlation in the underlying true (transformed) ICC
values. This is analogous to what people do when pooling sensitivity and
specificity values in a diagnostic test meta-analysis and also directly relates
to the bivariate model discussed by van Houwelingen et al. (2002):

https://www.metafor-project.org/doku.php/analyses:vanhouwelingen2002

You can then estimate the heritability from this model by back-transforming the
pooled estimate for the MZ twins and the pooled estimate from the DZ twins and
taking twice the difference. The SE and hence CI for this can then be obtained
via the delta method.

This raises interesting questions about the difference between between
'pooled(x) - pooled(y)' versus 'pooled(x - y)' -- there are papers in the
literature that discuss this issue (not in the present context) -- but the
latter option doesn't appear sensible to me here anyway.

Best,
Wolfgang

[R-meta] Meta-analysis of intra class correlation coefficients

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