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

Naike Wang wangnaike1989 at gmail.com
Sat Jul 8 01:18:44 CEST 2017

```Thank you for your answer, Dr. Viechtbauer.
I think you were right. MetaXL seems to apply the inverse transformation of
the arcsine transformation to double arcsine transformed proportions in
order to pool an average proportion. I tested this assumption with
the example dataset "SchizophreniaPrev" built in MetaXL and yielded very
similar results.
Here's my code:

individual transformed proportions using the double arcsine transformation
transf.pes=rma(yi,vi,data=transf.ies,method="DL",weighted=TRUE) #pooling
transformed proportions under the random effect size model
pes=predict(transf.pes,transf=transf.iarcsin) #back-transforming
with inverse of the arcsine transformation
print(pes,digits=4)

>My results:
pred         ci.lb       ci.ub
0.5856    0.5089   0.6603

>MetaXL results:
pred         ci.lb       ci.ub
0.5875 0.5098   0.6632

Thank you again for your time.

Cheers,
Naike

2017-07-07 9:41 GMT-04:00 Viechtbauer Wolfgang (SP) <
wolfgang.viechtbauer at maastrichtuniversity.nl>:

> 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.
> >
> >P.S. Dowload MetaXL here: http://www.epigear.com/index_files/metaxl.html
> >P.S.S. After you install MetaXL, open example "SchizophreniaPrev" to get a
> >sense of how it performs meta-analysis of proportions.
> >
> >Cheers,
> >Naike
>
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