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

Gerta Ruecker ruecker @end|ng |rom |mb|@un|-|re|burg@de
Tue Jul 30 16:39:35 CEST 2019

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
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Mail:     ruecker using imbi.uni-freiburg.de
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