Calculator for the confidence interval of R squared (coefficient of determination).
Entirely free download of software


The correlation coefficient (R) is often represented by R squared, the coefficient of determination or explantion. This coefficient (between 0 and 1) is a measure for the goodness of fit of a regression line or curve. R=1 means a perfect fit while R=0 means zero fit.


The calculator program gives the range over which the calculated R or R squared value may deviate from the true value. The range is determined by the assumed probability (confidence) interval.

        Screenprint of the R-squared calculator:

        here comes an image    

Download:

R-squared
calculator.





Go to:


software
& models


articles
& manuals


reports
case studies


FAQ's
& papers


home
page
 

Thus the confidence limits for the R squared values found in segmented regression and in fitted probability distributions can be computed wirh this calculator.

For improvement, I am interested to learn about your experiences with the calculator. For this there is a contact form.
 

MATHEMATICS

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.

here comes an image


here comes an image


here comes an image


    The flowers are here
    to make the mathematics
    less boring.