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We implemented the method by Algina, J., & Keselman, H. J. (2003). Approximate confidence intervals for effect sizes. Educational and Psychological Measurement, 63, 537-553.
The sample effect size is calculated as
d=ˉy1−ˉy2√(s21+s22)/2
where ˉy1 and ˉy1 are sample means, and s21 and s22 are sample variances of under two different conditions, respectively.
Algina and Keselman (2003) constructed a confidence interval for the population effect size based on a non-central t-distribution. In the method, one first gets the lower and upper bounds of the non-centrality parameter as λL and λU with
λL=t−1ncp(t,df=n−1,1−α/2)
and
λU=t−1ncp(t,df=n−1,α/2)
where
t=ˉy1−ˉy2√(s21+s22−2rs1s2)/n
with r being the correlation.
Now the confidence interval is given by
[λL√2(s21+s22−2rs1s2)n(s21+s22),λU√2(s21+s22−2rs1s2)n(s21+s22)]