The confidence interval for R or R squared is based on the normal probability distribution.
To use that distribution Fisher's transformation needs to be applied to R:
Z = 0.5 * ln [ (1-R) / (1+R) ]
where Z is the transformed R value. Z has been proved to follow a normal distribtion with standard deviation (S) defined by:
S squared = 1/(N-3)
where N is the number of data sets.
With S, confidence intervals for Z can be found as follows:
ZL = Z - F * S
ZU = Z + F * S
where ZL is the lower confidence limit of Z, ZU is the upper confidence limit, and F is a factor dependeing on the degree of confidence desired:
For 50% confidence F = 0.674
For 75% confidence F = 1.150
For 90% confidence F = 1.65
For 95% confidence F = 1.96
For 97.5% confidence F = 2.24
For 99% confidence F = 2.58
For 99.9% confidence F = 3.29
After finding ZL and ZU, these values must be transformed back to RL and RU, RL being the lower confidence limit of R and RU the upper confidence limit. We have:
RL = [exp(2*ZL)-1] / [exp(2*ZL) +1]; RU = [exp(2*ZU)-1] / [exp(2*ZU) +1];
The upper and lower confidence limits of R squared can now be found using the squared values of RL and RU.
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