[R] Converting F-value from ANOVA to cohen's d in meta-analysis (metafor-package)

Viechtbauer Wolfgang (STAT) wolfgang.viechtbauer at maastrichtuniversity.nl
Wed Jul 27 15:35:28 CEST 2011

Dear Jokel,

Unfortunately, this won't work. The derivation of the equation given in the Handbook shows why this is so. First of all, note that

d = (m1 - m2) / sp,

where m1 and m2 are the means of the two groups and sp is the pooled SD. The two independent samples t-test (assuming homoscedastic variances in the two groups) uses the test statistic

t = d / sqrt((n1 + n2) / (n1*n2)),

which we can turn around to get:

d = t * sqrt((n1 + n2) / (n1*n2))

Since t^2 = F for the one-way ANOVA for two groups, we get the equation in the Handbook. For three groups, this won't work (the F-statistic then has 2 df (in the numerator) and does not reflect a simple contrast between two groups).

If you are willing to make the assumption that m_C = m_D in those 3 group studies, then you can reconstruct the F value for the one-way ANOVA with two groups (where C and D are collapsed into a single group) from the F value for the one-way ANOVA with three groups. In particular,

F_two_groups = F_three_groups * 2 * (N - 2) / (N - 3),

where N = total sample size of all three groups combined. Then you can use that F-value in the equation given in the Handbook, where n1 = n_E and n2 = n_C + n_D.

This will be exactly correct if m_C = m_D (i.e., in the sample). It will be approximately correct if you are willing to assume that any difference between m_C and m_D is only due to sampling error and does not reflect a true difference between those two groups (i.e., the population means must be the same).


Wolfgang Viechtbauer 
Department of Psychiatry and Neuropsychology 
School for Mental Health and Neuroscience 
Maastricht University, P.O. Box 616 
6200 MD Maastricht, The Netherlands 
Tel: +31 (43) 368-5248 
Fax: +31 (43) 368-8689 
Web: http://www.wvbauer.com 

> -----Original Message-----
> From: r-help-bounces at r-project.org [mailto:r-help-bounces at r-project.org]
> On Behalf Of Jokel Meyer
> Sent: Wednesday, July 27, 2011 13:29
> To: r-help at r-project.org
> Subject: [R] Converting F-value from ANOVA to cohen's d in meta-analysis
> (metafor-package)
> Dear R-experts!
> Running a meta-analysis (using the magnificent metafor-package), I use
> cohen's d as a main outcome measure in a random-effects model.
> For most of the samples cohen's d is derived form a comparison of two
> groups
> (A & B). However some studies report results from an ANOVA (one-factor
> with
> three levels: C,D,E) whereas two groups (C,D) correspond to one group in
> the
> other studies (B=C,D). Is there a way?
> The handbook of research synthesis and meta-analysis By Harris M. Cooper
> says that:
> d=sqrt((F*(n1+n2)/n1*n2)) for ANOVA (one-factor with two-levels)
> but does this also hold for ANOVA (one-factor with three levels)?
> Many thanks for your help!
> Jokel
> 	[[alternative HTML version deleted]]
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