# Confidence of Effect Size Differences

Hedges and Olkin (1985) suggest the following procedure for computing confidence intervals for Hedges' g resp. d $$\text{CI} = g \pm 1.96 \sqrt{(n_2+n_1)\big/(n_2n_1) + g^2 \times .5\big/(n_1+n_2)}$$ (with 1.96 as the normal distribution cumulative density value for the confidence coefficient 95%)

When using pre-posttest designs with experimental and control groups, Klauer (1993) defined a corrected effect size d(corr) by subtracting the pre-posttest effect size of the experimental and the control group in order to measure intervention effects: $$\text{d(corr) = d(exp) - d(control)}$$ I could not find literature on computing confidence intervals for effect size differences, though. Does anybody know how to compute confidence intervals of effect size differences? Thanks!

You can compute Cohen’s $d$ as
$$d = (ME - MC)/SQRT [(SD2E - SD2C )2],$$ where ME and SDE are the posttest mean and SD within the experimental group and MC and SDC are the posttest mean and SD within the control group. This is for equal group sizes.

If the group sizes are unequal, you can use Glass’ $∆$: $$∆ = (ME - MC)/SDC.$$

A formula for calculating the confidence interval for an effect size is given by Hedges and Olkin (1985). If the effect size estimate from the sample is $d$, then it is Normally distributed, with standard deviation: $$\sigma(d) = \sqrt{ \frac{N_E+N_C}{N_E\times N_C}+\frac{d^2}{2(N_E+N_C)} }$$

Hence a 95% confidence interval for $d$ would be from $$(d – 1.96 σ[d],\; d + 1.96 σ[d])$$

(Where NE and NC are the numbers in the experimental and control groups, respectively.)

You can find more information in the article Effect Sizes, Confidence Intervals, and Confidence Intervals for Effect Sizes by Bruce Thompson.