[R-meta] Metafor results tau^2 and R^2

Dr. Gerta Rücker ruecker @end|ng |rom |mb|@un|-|re|burg@de
Sun Aug 9 16:19:46 CEST 2020

Dear Wolfgang,

Thank you for clarifying this. I really thought it was the Higgins R^2, 
as it stands in the neighborhood of I^2 and H^2 and also as in the given 
case also its value 1 is plausible (however, in fact , Higgins's R^2 
would not be expressed in percent).

I confused these two R^2s, and I might not be the only person confusing 
these. Do you see a way to avoid this misconception, for example by 
mentioning Raudenbush in the output text?



Am 09.08.2020 um 12:57 schrieb Viechtbauer, Wolfgang (SP):
> Hi All,
> R^2 in the output of metafor is *not* R^2 from Higgins et al. (2002). It is in fact a (pseudo) coefficient of determination that goes back to Raudenbush (1994). It estimates how much of the (total) heterogeneity is accounted for by the moderator(s) included in the model. If the *residual* amount of heterogeneity (i.e., the unaccounted for heterogeneity) is 0 after including the moderator(s) in the model, then R^2 is going to be 100% (i.e., all of the heterogeneity has been accounted for). One would in fact expect then that the moderator (or set of moderators) is significant -- it would actually be a bit odd if a moderator accounts for all of the heterogeneity, but fails to be significant (although one could probably construct an example where this is the case). And reporting R^2 is definitely useful, although should be cautiously interpreted given that R^2 can be rather inaccurate when k is small (as discussed in López‐López et al., 2014).
> Best,
> Wolfgang
>> -----Original Message-----
>> From: R-sig-meta-analysis [mailto:r-sig-meta-analysis-bounces using r-project.org]
>> On Behalf Of Dr. Gerta Rücker
>> Sent: Saturday, 08 August, 2020 23:09
>> To: Dustin Lee; r-sig-meta-analysis using r-project.org
>> Subject: Re: [R-meta] Metafor results tau^2 and R^2
>> Dear Dustin,
>> The results you report show that in this analysis there was no
>> between-study heterogeneity found at all. As explained in the message,
>> all measures given are measures of heterogeneity, also R^2. You find all
>> definitions in Higgins JP, Thompson SG. Quantifying heterogeneity in a
>> meta-analysis. Stat Med. 2002;21(11):1539-1558. doi:10.1002/sim.1186.
>> R^2 should not be confused with the coefficient of determination (which
>> is also often denoted R^2). It is unusual to report the heterogeneity
>> measure R^2 in a study report; most authors would report tau, tau^2 or I^2.
>> See also Rücker G, Schwarzer G, Carpenter JR, Schumacher M. Undue
>> reliance on I(2) in assessing heterogeneity may mislead. /BMC Med Res
>> Methodol/. 2008;8:79. Published 2008 Nov 27. doi:10.1186/1471-2288-8-79.
>> Best,
>> Gerta
>> Am 08.08.2020 um 22:13 schrieb Dustin Lee:
>>> Dear all,
>>> I am currently conducting a meta regression in which we are examining the
>>> role of temporal effects (year of study) in the relationship between
>>> organizational attitudes and job performance. Using a mixed-effects model
>>> using ML estimation, our analyses have thus far produced results that do
>>> not appear to be irregular.
>>> Our problem: With one relationship the analysis is showing the following:
>>> tau^2 (estimated amount of residual heterogeneity):     0 (SE = 0.0152)
>>> tau (square root of estimated tau^2 value):             0
>>> I^2 (residual heterogeneity / unaccounted variability): 0.00%
>>> H^2 (unaccounted variability / sampling variability):   1.00
>>> R^2 (amount of heterogeneity accounted for):            100.00%
>>> However, the significance of the effect of 'year of study' is significant
>>> along with the omnibus Q_M statistic. While I inherently understand this
>> is
>>> due to the way in which these values (R^2, tau^2, I^2, etc.) are
>> calculated
>>> and that it may be due to the smaller than ideal sample size (k =32) as
>>> suggested by López‐López and colleagues (2014). I am unsure on how these
>>> findings should be reported, particularly the 100% R^2 with the
>> significant
>>> predictor 'year of study' result.
>>> Thank you for any assistance you may be able to provide.
>>> All the best,
>>> Dustin
>>> Reference:
>>> López‐López, J. A., Marín‐Martínez, F., Sánchez‐Meca, J., Van den
>>> Noortgate, W., & Viechtbauer, W. (2014). Estimation of the predictive
>> power
>>> of the model in mixed‐effects meta‐regression: A simulation study.
>> *British
>>> Journal of Mathematical and Statistical Psychology*, *67*(1), 30-48.

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