Dear Will,
Please see my responses below.
Best,
Wolfgang
-----Original Message-----
From: R-sig-meta-analysis <r-sig-meta-analysis-bounces at r-project.org> On Behalf
Of Will Hopkins via R-sig-meta-analysis
Sent: Sunday, March 3, 2024 22:15
To: 'R Special Interest Group for Meta-Analysis' <r-sig-meta-analysis at r-
project.org>
Cc: Will Hopkins <willthekiwi at gmail.com>
Subject: [R-meta] Standard error of heterogeneity with rma.uni
After a year of denial, a colleague and I are starting to use metafor
instead of SAS's Proc Mixed, since finding recently that selection models to
adjust for publication bias are not implementable in SAS (not by us, anyway,
and no-one else seems to be using SAS for meta-analyses). So I'm starting by
importing data into R that I have simulated and analyzed in SAS, to make
sure I am using metafor correctly. I have found that the standard error (SE)
for the heterogeneity variance provided by rma.uni is considerably smaller
than that provided by SAS, and the SE for a fixed effect can be smaller in
metafor, too. Here are two examples. Along the way, I have several questions
I hope you or someone can answer, Wolf.
In the first example, there were 20 studies (10 female, 10 male), with
sample size ranging from 10 to 30, with small and moderate mean changes for
males and females respectively, small heterogeneity within each group, and
standard errors of measurement such that the mean changes in most of the
male studies were non-significant and most of the females were significant.
("Small" is not trivial, "small" and "moderate" are not defined by
standardization. In the following examples, small and moderate mean changes
are 1 and 3, and small heterogeneity is and SD of 0.5.)
The code for metafor was:
rma.uni(yi=Ydelta, vi=YdeltaSEsq, mods=Female01, data=simraw).
(Female01 is a dummy variable.) The solution for fixed effects was the same
in metafor...
estimate se
intrcpt 0.9546 0.3819
mods 2.1568 0.5162
...as in SAS...
Effect Estimate StdErr
Mean 0.9546 0.3819
Female01 2.1568 0.5162
But for the heterogeneity, metafor produced...
tau^2 (estimated amount of residual heterogeneity): 0.2755 (SE = 0.3810)
...while SAS produced...
CovParm Estimate StdErr
StudyID 0.2755 0.5256
As you can see, the point estimates are the same, but the SE in metafor 0.72
that in SAS. I presume SAS is correct, because I can use the simulation in
SAS to generate thousands of meta-analyses for given population values of
everything, and the coverage of the confidence intervals for the estimates
of heterogeneity is exactly the level of confidence of the intervals (I use
90%, for various good reasons). Is this a small-sample issue in the way you
produce the SE in rma.uni, Wolf, one that is not a problem in SAS? Is there
a solution in metafor?
SAS uses by default the inverse of the Hessian (i.e., the observed Fisher information matrix) to calculate the SE of tau^2, while rma() uses the expected Fisher information matrix. If you use 'scoring=100' (should be sufficiently large), then SAS will also use the latter and the SEs are the same. Or one can use:
simraw$id <- 1:nrow(simraw)
res2 <- rma.mv(Ydelta, YdeltaSEsq, random = ~ 1 | id, cvvc=TRUE)
round(sqrt(res2$vvc), 4)
to get the SE based on the observed Fisher information matrix.
Which is to be preferred (the observed or expected information) is an open issue. Neither is particularly useful though in the present context, because better CIs for tau^2 can be constructed (e.g., using the Q-profile method, the generalized Q-statistic method, or the profile likelihood method) that do not make use of any SE of tau^2.
Will this matter, when I come to use selmodel?
No.
And will selmodel work with negative values of heterogeneity variance?
No.
In the second example, I used simulated data that produced negative
heterogeneity variance in SAS. I am "brave enough to step into risky
territory", to quote you from the documentation, Wolf, because negative
variance for point estimates and confidence limits are necessary to get
realistic unbiased estimates and correct coverage of random effects, when
the uncertainty in heterogeneity is large enough relative to its true point
estimate, so I wanted to make sure rma.uni produced correct negative
variance. When I asked this mailing list about getting negative variance a
year ago, James Pustejovsky provided the code (which is also in the metafor
documentation):
rma(yi = yi, vi = vi, data=dat, control=list(tau2.min=-min(vi))).
Unfortunately the above code doesn't quite work for me. Here's the line of
code for my data:
rma.uni(yi=Ydelta, vi=YdeltaSEsq, mods=Female01, data=simrawnv,
control=list(tau2.min=-min(vi)))
I got this error:
Error in min(vi) : invalid 'type' (closure) of argument.
Blindly copy-pasting code without the necessary adjustments to variable names isn't going to work.
And when I tried this...
rma.uni(yi=Ydelta, vi=YdeltaSEsq, mods=Female01, data=simrawnv,
control=list(tau2.min=-min(YdeltaSEsq)))
...I got this...
Error: object 'YdeltaSEsq' not found.
You can use control=list(tau2.min=-min(simrawnv$YdeltaSEsq)). The tau2.min argument does not make use of non-standard evaluation.
But I got it to work with this...
rma.uni(yi=Ydelta, vi=YdeltaSEsq, mods=Female01, data=simrawnv,
control=list(tau2.min=-99))
...which gave this message...
Warning message:
Value of 'tau2.min' constrained to -min(vi) = -0.1640.
So there's a bug, but it's easy to bypass it meantime.
This is not a bug. If we allow tau^2 to go below -min(vi), then the marginal variance becomes negative. Neither the all holy SAS(r) nor metafor can magically make this work.
The point estimates of the fixed effects in metafor and SAS were identical,
but the SE for the intercept (males) in metafor...
estimate se
intrcpt 1.3013 0.0013
mods 1.9245 0.2063
...was 0.67 times than in SAS...
Effect Estimate StdErr
Mean 1.3013 0.001930
Female01 1.9245 0.2063
This seems quite peculiar. Please provide a fully reproducible example replicating this issue.
The SE for heterogeneity in metafor...
tau^2 (estimated amount of residual heterogeneity): -0.1640 (SE =
0.0735)
... was less than half that in SAS...
CovParm Estimate StdErr
StudyID -0.1640 0.1636
Should I be disappointed that both SAS and metafor set the point estimate of
negative variance to minus the smallest variance of the study estimates?
No, because it is the only sensible thing that can be done.
It seems a pretty arbitrary and clunky thing to do, "to ensure that the
marginal variances are always non-negative", to quote Wolf in the
documentation.
That 'Wolf' guy seems to know what he is talking about here.
Perhaps someone can explain that. It obviously works as far
as coverage is concerned, in SAS anyway.
These problems presumably go away with large-enough numbers of studies
and/or sample sizes and/or study-estimate SEs relative to effect magnitudes
and/or heterogeneity, but mainly I am working with small numbers of studies,
small sample sizes, often large(-ish) study-estimate SEs relative to effect
magnitudes, and small heterogeneity. What to do? Bootstrap in metafor?
The main issue for me is whether I will be able to use selmodel to adjust
for publication bias, when the study estimates are such that negative
heterogeneity could arise purely from sampling uncertainty or from
publication bias (which results in underestimation of heterogeneity, in our
simulations), and will surely arise in simulations aimed at estimating bias
and coverage.
As mentioned above, selmodel() does not allow for negative tau^2 values.