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

[Viechtbauer Wolfgang (STAT)]

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).

Best,

-- 
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
> 
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> 
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