[R-meta] Standard error of heterogeneity with rma.uni

[Viechtbauer, Wolfgang (NP)]

Dear Will,

Please see my responses below.

Best,
Wolfgang

> -----Original Message-----
> From: R-sig-meta-analysis <r-sig-meta-analysis-bounces using 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 using r-
> project.org>
> Cc: Will Hopkins <willthekiwi using 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?

If you think this is going to fix whatever problem you perceive, then here is some code to get you started:

https://www.metafor-project.org/doku.php/tips:bootstrapping_with_ma

> 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.