Let R the unrestricted correlaton, r the restricted correlation, S the unrestricted standard deviation, s the restricted standard deviation, then
R = (rS/s)/ sqrt(1-r^2 + r^2(S^2/s^2)).
Several other cases of restriction were also considered by Thorndike and are implemented in rangeCorrection
.
rangeCorrection(r,sdu,sdr,sdxu=NULL,sdxr=NULL,case=2)
Can be used to find correlations in a restricted sample as well as the unrestricted sample. Not the same as the correction to reliability for restriction of range.
Revelle, William. (in prep) An introduction to psychometric theory with applications in R. Springer. Working draft available at http://personality-project.org/r/book/
Stauffer, Joseph and Mendoza, Jorge. (2001) The proper sequence for correcting correlation coefficients for range restriction and unreliability. Psychometrika, 66, 63-68.
rangeCorrection(.33,100.32,48.19) #example from Revelle (in prep) Chapter 4.
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