In applied settings, it is typical to find a correlation between a predictor and some criterion. Unfortunately, if the predictor is used to choose the subjects, the range of the predictor is seriously reduced. This restricts the observed correlation to be less than would be observed in the full range of the predictor. A correction for this problem is well known as Thorndike Case 2:
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)The observed correlation
The unrestricted standard deviation)
The restricted standard deviation
Unrestricted standard deviation for case 4
Restricted standard deviation for case 4
Which of the four Thurstone/Stauffer cases to use
The corrected correlation.
When participants in a study are selected on one variable, that will reduce the variance of that variable and the resulting correlation. Thorndike (1949) considered four cases of range restriction. Others have continued this discussion but have changed the case numbers.
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 https://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.
cRRr in the psychometric package.
# NOT RUN {
rangeCorrection(.33,100.32,48.19) #example from Revelle (in prep) Chapter 4.
# }
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