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Re: st: using Freeman-Tukey arcsine transformation with metan command

From   David Hoaglin <>
Subject   Re: st: using Freeman-Tukey arcsine transformation with metan command
Date   Mon, 2 Apr 2012 21:44:04 -0400


To complicate matters further, I must point out that neither of the
papers that Cameron cited deals with meta-analysis.  In meta-analysis,
the arcsine-square-root transformation has the advantage that (to
first order) the weights depend only on the sample size(s), which are
known.  The customary methods for meta-analysis of risk difference,
risk ratio, and odds ratio use weights based on estimated variances,
but they pretend that those variances are known and, as a result, may
have substantial bias.  Since you plan to use random effects, you
should avoid using the DerSimonian-Laird method for risk difference,
risk ratio, or odds ratio.

It would be a good idea to examine the heterogeneity in your data
carefully, to see whether it involves clusters, outliers, or other
interesting structure.

For some background on the arcsine-square-root (a.k.a. angular)
transformation in meta-analysis, you may want to look at the following
paper (and some of its references):
Ruecker G, Schwarzer G, Carpenter J, Olkin I. Why add anything to
nothing? The arcsine difference as a measure of treatment effect in
meta-analysis with zero cells.  Statistics in Medicine 2009;

I also suggest the following book:
Kulinskaya E, Morgenthaler S, Staudte RG. Meta Analysis: A Guide to
Calibrating and Combining Statistical Evidence. Wiley, 2008.

The same authors discuss an alternative approach:
Kulinskaya E, Morgenthaler S, Staudte RG. Variance stabilizing the
difference of two binomial proportions. The American Statistician
2010; 64:350-356.

If you do the analysis in the angular scale, you can transform a
difference back to the proportion scale by choosing a particular
proportion as the "base rate" (a similar consideration applies to the
odds ratio and its log).

Another possible alternative would take a Bayesian approach, modeling
the observed proportions by binomial distributions and expressing the
difference between the true proportions on the logit scale.  See, for
Sutton AJ, Abrams KR. Bayesian methods in meta-analysis and evidence
synthesis. Statistical Methods in Medical Research 2001; 10:277-303.

David Hoaglin
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