[R-meta] Metafor Fixed Effects giving I2- why?

[Aman Dheri]

Hi all,

My question was exactly as Wolfgang interpreted, but Gerta the answer to
your question was great to know, too!

Thank you both so much!
Arthur

On Tue, Jul 30, 2019 at 10:39 Gerta Ruecker <ruecker using imbi.uni-freiburg.de>
wrote:

> Thanks again - this is exactly what I wanted to know (and thought the
> original post referred to).
>
> Gerta
>
> Am 30.07.2019 um 16:31 schrieb Viechtbauer, Wolfgang (SP):
> > And yes, one can construct examples where the estimate of tau^2 = 0,
> while I^2 > 0 (when I^2 is computed with max(0, (Q - (df)) / Q)). As you
> pointed out, this can't happen when estimating tau^2 wit the DL estimator,
> but it can happen when using other estimators. For example:
> >
> > yi <- c(.2, .3, .4, .5)
> > vi <- rep(.0125, 4)
> >
> > res <- rma(yi, vi, method="FE")
> > res$I2
> >
> > rma(yi, vi, method="ML")$tau2 # this is essentially 0
> > var(yi)*3/4 - .0125 # for this case, ML reduces to this
> >
> > (same goes for the HS estimator, which is identical to ML when the
> sampling variances are homoscedastic; all other estimators should give an
> estimate > 0)
> >
> > Best,
> > Wolfgang
> >
> > -----Original Message-----
> > From: Gerta Ruecker [mailto:ruecker using imbi.uni-freiburg.de]
> > Sent: Tuesday, 30 July, 2019 16:16
> > To: Viechtbauer, Wolfgang (SP); Aman Dheri;
> r-sig-meta-analysis using r-project.org
> > Subject: Re: [R-meta] Metafor Fixed Effects giving I2- why?
> >
> > Thank you, Wolfgang!
> >
> > The difference is between the *parameter* tau^2, which is zero if we
> > believe the fixed effect model, and the estimate \hat tau^2, which in
> > your example is likewise positive (all methods), though (of course)
> small.
> >
> > Best,
> >
> > Gerta
> >
> > Am 30.07.2019 um 15:28 schrieb Viechtbauer, Wolfgang (SP):
> >> My understanding of Arthur's question was that he is wondering why the
> following code yields a value of I^2 = 40% given that this fits a
> fixed-effects model where tau^2 = 0 by definition.
> >>
> >> library(metafor)
> >>
> >> yi <- c(.2, .3, .4, .5)
> >> vi <- rep(.01, 4)
> >>
> >> res <- rma(yi, vi, method="FE")
> >> res$I2
> >>
> >> (and in the 'devel' version, the value of I^2 is shown explicitly in
> the output also for a FE model).
> >>
> >> Best,
> >> Wolfgang
> >>
> >> -----Original Message-----
> >> From: Gerta Ruecker [mailto:ruecker using imbi.uni-freiburg.de]
> >> Sent: Tuesday, 30 July, 2019 10:37
> >> To: Viechtbauer, Wolfgang (SP); Aman Dheri;
> r-sig-meta-analysis using r-project.org
> >> Subject: Re: [R-meta] Metafor Fixed Effects giving I2- why?
> >>
> >> Hi all,
> >>
> >> The question whether it is possible in the fixed effect model that tau^2
> >> = 0 though I^2 > 0 was not answered in Wolfgang's reply. Following his
> >> response, in this case metafor calculates
> >>
> >> I^2 = (Q - (df)) / Q,
> >>
> >> which probably means
> >>
> >> I^2 = max(0, (Q - (df)) / Q).
> >>
> >> Now it depends on the way tau^2 is calculated. For the DerSimonian-Laird
> >> estimate, \hat tau^2 has the same numerator as I^2, that is, if \hat
> >> tau^2 = 0, then I^2 = 0.
> >>
> >> Thus the case Arthur describes can only, if at all, occur for other
> >> tau^2 estimates. Thus, I repeat his question: Is it possible and under
> >> which circumstances?
> >>
> >> Best,
> >>
> >> Gerta
> >>
> >> Am 29.07.2019 um 19:56 schrieb Viechtbauer, Wolfgang (SP):
> >>> Hi Arthur,
> >>>
> >>> Indeed, for random/mixed-effects models, I^2 is computed with tau^2 /
> (tau^2 + v), where tau^2 is the estimate of (residual) heterogeneity and v
> is the 'typical' sampling variance. So, if tau^2 = 0, then I^2 = 0.
> >>>
> >>> Since I^2 is a useful statistic also for fixed-effects models, I^2 is
> also computed then, but using (Q - (df)) / Q, where Q is the test statistic
> for (residual) heterogeneity and df = degrees of freedom of the test (k-1
> for a model without moderators). But this has no effect on the rest of the
> computations (such as the weights).
> >>>
> >>> Best,
> >>> Wolfgang
> >>>
> >>> -----Original Message-----
> >>> From: R-sig-meta-analysis [mailto:
> r-sig-meta-analysis-bounces using r-project.org] On Behalf Of Aman Dheri
> >>> Sent: Monday, 29 July, 2019 19:11
> >>> To: r-sig-meta-analysis using r-project.org
> >>> Subject: [R-meta] Metafor Fixed Effects giving I2- why?
> >>>
> >>> Hello,
> >>>
> >>> I am doing a fixed effects meta-analysis in Metafor and I see that it
> returns a non-zero I^2 value when tau^2 is always zero. According to
> Metafor documentation the fixed effects model in rma.uni should be 0
> because of its calculation from the tau^2. Does any one know why this is
> happening, and if it is changing the weighting during the meta-analysis vs
> if the I^2 was 0?
> >>>
> >>> I assumed my code wasn’t needed for the answer to this but please let
> me know if I’m wrong!
> >>>
> >>> Thank you!
> >>> Arthur
>
> --
>
> Dr. rer. nat. Gerta Rücker, Dipl.-Math.
>
> Institute of Medical Biometry and Statistics,
> Faculty of Medicine and Medical Center - University of Freiburg
>
> Stefan-Meier-Str. 26, D-79104 Freiburg, Germany
>
> Phone:    +49/761/203-6673
> Fax:      +49/761/203-6680
> Mail:     ruecker using imbi.uni-freiburg.de
> Homepage: https://www.imbi.uni-freiburg.de/persons/ruecker/person_view
>
>

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