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← Revision 20 as of 2018-07-26 13:22:03 ⇥
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| V(a/b) = $$\frac{\mbox{a}^text{2} \mbox{V(b) - 2ab Cov(a,b) + } \mbox{b}^text{2} \mbox{V(a)}}{\mbox{b}^text{4}}$$ | V(a/b) = [ a^2 ^ V(b) - 2ab Cov(a,b) + b^2 ^ V(a)]/ b^4 ^ |
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| with, for Cohen's d, a = difference in two group means, $$\bar{x}_text{1}$$ and $$\bar{x}_text{2}$$ with respective sizes n1 and n2 and standard deviations, s1 and s2 and b the pooled standard deviation, sd. | 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. |
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| then V(a) = $$ \frac{s1^text{2}}{n1} + \frac{s2^text{2}}{n2}$$ and V(b) = the square of $$\frac{\sqrt{2}sd}{sqrt{n1+n2}}$$ (from inverting and multiplying by (n1+n2) the diagonla term from the Fisher information matrix based upon the Normal distribution stated [http://en.wikipedia.org/wiki/Normal_distribution here]) using the Fi. If Cov(a,b) is equal to zero as the groups are independent |
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 |
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| s.e. (Cohen's d) = $$\frac{ (\bar{x}_text{1}-\bar{x}_text{2})^text{2} \(\frac{0.71sd}{sqrt{n1+n2}})^text{2} + sd^text{2} (\frac{s1^text{2}}{n1}+\frac{s2^text{2}}{n2}) }{sd^text{4}}$$ | s.e. (Cohen's d) = |
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| In the example of the hospital waiting times $$\bar{x}_text{1}$$=12.08, s1=2.89, n1=480, $$\bar{x}_text{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. | [ (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. |
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) = [ a2 V(b) - 2ab Cov(a,b) + b2 V(a)]/ b4
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) = [s12 ]/[n1] + [s22 ]/[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 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 + sd2 ([s12 ]/[n1]+[s22 ]/[n2] ) ]/[sd4 ]
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.
* Formulae for standard errors of Cohen's d using group sizes are given here and reproduced here if this link is broken.
Reference
Howell DC (2013) Statistical methods for psychologists. 8th Edition. International Edition. Wadsworth:Belmont,CA.