# [R-meta] Freeman-Tukey double arcsine transformation and harmonic mean

Viechtbauer Wolfgang (SP) wolfgang.viechtbauer at maastrichtuniversity.nl
Fri Jul 7 15:41:31 CEST 2017

```Hi Naike,

The first linked got mangled up. It is: http://www.metafor-project.org/doku.php/analyses:miller1978

The exact back/inverse transformation of the Freeman-Tukey (double arcsine) transformation requires that we specify the sample size for the transformed value. So:

library(metafor)
dat <- escalc(measure="PFT", xi=4, ni=10)

> dat
yi     vi
1 0.6936 0.0238

transf.ipft(dat\$yi, ni=10)

yields a proportion of 0.4 as expected.

Now if you synthesize a whole bunch of transformed values and you want to back-transform that value to a proportion, you still need to specify some value for the sample size if you want to use the exact back-transformation. Miller (1978), who derived the back-transformation, suggested to use the harmonic mean of the sample sizes. That is what transf.ipft.hm() does. Using the harmonic mean of the sample sizes is a rather heuristic method that may or may not work so well. I would be interested in any published papers that show this to be a problem.

I don't know what MetaXL does for the back-transformation, but maybe it just pretends that the values are arcsine-square-root transformed proportions and then uses the back-transformation for that -- which does not require one to specify the sample size. The difference is typically negligible:

transf.iarcsin(dat\$yi)

yields 0.4086998. But then, one might as well just do the meta-analysis directly with the AS transformed proportions:

dat <- escalc(measure="PAS", xi=4, ni=10)
dat

> dat
yi     vi
1 0.6847 0.0250

transf.iarcsin(dat\$yi)

gives back 0.4 exactly.

Or one could go directly to a logistic mixed-effects model for the analysis. You can do that with rma.glmm().

Best,
Wolfgang

--
Wolfgang Viechtbauer, Ph.D., Statistician | Department of Psychiatry and
Neuropsychology | Maastricht University | P.O. Box 616 (VIJV1) | 6200 MD
Maastricht, The Netherlands | +31 (43) 388-4170 | http://www.wvbauer.com

>-----Original Message-----
>From: R-sig-meta-analysis [mailto:r-sig-meta-analysis-bounces at r-
>project.org] On Behalf Of Naike Wang
>Sent: Friday, July 07, 2017 15:25
>To: r-sig-meta-analysis at r-project.org
>Subject: [R-meta] Freeman-Tukey double arcsine transformation and harmonic
>mean
>
>Hi all,
>I have two questions.
><https://mail.jjay.cuny.edu/owa/redir.aspx?C=jnnID1xyBS33HM9BECtjC_Z23ilF5
>4mDEf0zdCS88qMPhZkvMsXUCA..&URL=http%3a%2f%2fwww.metafor-
>project.org%2fdoku.php%2fanalyses%3amiller1978>,
>Dr. Wolfgang Viechtbauer used the harmonic mean of the sample sizes to
>back-transform the estimated average transformed proportion (the pooled
>proportion). If I don't want to use the harmonic mean,  is it possible to
>use the *transf.ipft*, instead of the *transf**=**transf.ipft.hm
><http://transf.ipft.hm>*, to get the pooled proportion? If so, how do I do
>that?
>
>Meta-analysis
>of prevalence
>(MetaXL). MetaXL uses a different method to perform the double arcsine
>transformation. The differences are two-fold.
>First, MetaXL uses a different definition of  the Freeman-Tukey
>transformation. The PFT values (yi) are twice as large as the values
>computed by metafor and the variances (vi) are four times as large. The
>different definitions are also explained here
><http://www.metafor-project.org/doku.php/faq#how_is_the_freeman-
>tukey_trans>
>.
>Second, it does not use the harmonic mean to perform the
>back-transformation. According to the authors, it is better not to use the
>harmonic mean because their simulation studies suggest that the harmonic
>mean is not stable.
>Basically, I'm asking how to get metafor to get the same results as
>obtained in MetaXL? Do you agree with the MetaXL authors that it is better
>not to use the harmonic mean?
>
>I hope my questions make sense. Feel free to ask me anything if you don't
>understand.
>