= A suggestion for the standard error for Cohen's d =


Since from Taylor's series the ratio of two estimates a and b is given as

V(a/b) = [ a^2 ^ V(b) - 2ab Cov(a,b) + b^2 ^ V(a)]/ b^4 ^

with, for Cohen's d,  a = difference in two group means, xbar(1) and xbar(2) with respective sizes n1 and n2 and standard deviations, s1 and s2 and b the pooled standard deviation, sd. 

then V(a) = [s1^2 ^]/[n1] + [s2^2 ^]/[n2] and
V(b) = the square of [sqrt(2)sd]/sqrt(n1+n2) (from inverting and multiplying by (n1+n2) the diagonal term from the Fisher information matrix based upon the Normal distribution stated  [[attachment:fishinf.pdf|here]]). If Cov(a,b) is considered equal to zero assuming the variance within the groups is unrelated to their means

we have

s.e. (Cohen's d) = 

[ (xbar(1)-xbar(2))^2 ^ [(0.71sd)/sqrt(n1+n2)]^2 ^ + sd^2 ^ ([s1^2 ^]/[n1]+[s2^2 ^]/[n2] ) ]/[sd^4 ^]

In the example of the hospital waiting times xbar(1)=12.08, s1=2.89, n1=480, xbar(2)=16.07, s2=2.98, n2=493 and the (pooled) sd=3.55. Inserting these into the above equation we then get Cohen's d = -1.36, s.e.(Cohen's d) = 0.059 giving 95% CI of (-1.24, -1.48) compared to (-1.22, -1.50) using Smithson's approach and (-1.21, -1.50) with the bootstrap.

Howell (2013, p.309) gives a formula for the standard error of Cohen's d and uses this to evaluate a 95% confidence interval for d. 

* [[https://stats.stackexchange.com/questions/8487/how-do-you-calculate-confidence-intervals-for-cohens-d | Formulae for standard errors of Cohen's d using group sizes are given here]]
and reproduced [[http://imaging.mrc-cbu.cam.ac.uk/statswiki/FAQ/cohendse | here if this link is broken.]] 

__Reference__

Howell DC (2013) Statistical methods for psychologists. 8th Edition. International Edition. Wadsworth:Belmont,CA.