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When examining the difference between two proportions I typically use Cohen's h (i.e, the difference between two arcsin-transformed proportions) for the effect size.

Does anyone know how I could calculate 95% confidence intervals for Cohen's h?

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You can calculate 95% confidence intervals of arcsine difference, using the function "metabin" on the package "meta" in R. Cohen's h is the double of arcsine difference. You can calculate 95% confidence intervals of Cohen's h, the double of the lower and upper limit of confidence intervals of arcsine difference.

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You can use this function:

cohens_h = function( prop_1, prop_2, n1, n2, ci = 0.95 ){
    x1 = asin(sign(prop_1) * sqrt(abs(prop_1)))
    x2 = asin(sign(prop_2) * sqrt(abs(prop_2)))
    es = x1 - x2
    se = sqrt(0.25 * (1 / n1 + 1 / n2 ))
    ci_diff = qnorm(1 - (1-ci) / 2) * se
    return( c( h = es*2, h_low = (es-ci_diff)*2, h_upp = (es+ci_diff)*2 ) )
}

E.g.:

acc1 = .70
acc2 = .90
N1 = 200
N2 = 300
cohens_h(acc1, acc2, N1, N2)

Returns:

         h      h_low      h_upp 
-0.5157784 -0.6946978 -0.3368590 
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