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When researching effect size for proportions, in particular the paper Effect-Size Indices for Dichotomized Outcomes in Meta-Analysis, that at least two of the usual effect sizes are realy variance stabilizing transformations on the binomial distribution (from which the proportions are samples). The log odds ratio and the arcsin effect sizes (as discussed in the paper above) are just the difference between the transformed data.

The log odds ratio is (without the multiplicative correction)

$$ \log \frac{P_E(1-P_C)}{P_C(1-P_E)} $$

is just

$$ \log \frac{P_E}{1-P_E} - \log \frac{P_C}{1-P_C} $$
and logit is a variance stabilizing transformation for binomial. The more exotic arcsin effect size (again without the multiplicative correction) $$ \arcsin \sqrt{P_E} - \arcsin \sqrt{P_C} $$ is obviously related to the arcsin square root transform (for example Transformations Related to the Angular and the Square Root)

This relation between effect size and variance seems to make some intuitive sense, since effect size should measure of difference and one would like for the difference to "mean the same" regardless of the value of the "factors" in the difference. For example a chance of 0.01 in a proportion is a "big thing" if one of the factors is 0.99 (that is the difference is 1.00-0.99), but that same difference does not "mean the same" if the difference is 0.61-0.60! The two differences do not mean the same because the variance is different around 0.99 and 0.60.

a) Any idea or pointers in this direction?

b) If this relation between variance stabilizing and effect size is correct, there should be many more effect size indexes based on more precise transforms (for example http://www.jstor.org/discover/10.2307/2282340?sid=21106186616973&uid=4&uid=3737664&uid=70&uid=2129&uid=2)

Jacques Wainer
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    logit is not the variance-stabilizing transformation for the binomial. – Glen_b Mar 21 '15 at 07:13
  • https://en.wikipedia.org/wiki/Variance-stabilizing_transformation – Glen_b Mar 21 '15 at 11:19
  • @Glen_b - You may be right. I did not find a paper explicitly mentioning logit as variance stabilizing for binomial. But I did find some mentions, such as http://strata.uga.edu/6370/rtips/proportions.html I end up assuming it was a known fact (also because of the link function in GLMs). – Jacques Wainer Mar 22 '15 at 02:26
  • It's used as a link function because it's the natural link for the binomial (as well as having a convenient interpretation for many applications). In terms of ratios of variances for large $n$ as $p$ gets small (that is, if you look at the ratio of the larger variance to the smaller for two different small $p$), logit and identity link are about equally "unstable" but move in opposite directions, while arcsin-square-root is nearly flat. – Glen_b Mar 22 '15 at 03:07

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