The correction for restriction of range and nonlinear regressions: An analytic study

Abstract

The effect of a nonlinear regression function on the accuracy of the restriction of range correction formula was investigated using analytic methods. Expressions were derived for the expected mean square error (EMSE) of both the correction formula and the squared correlation computed in the selected group, with respect to their use as estimators of the population relationship. The relative accuracy of these two estimators was then studied as a function of the form of the regression, the form of the marginal distribution of x scores, the strength of the relationship, sample size, and the degree of selection. Although the relative accuracy of the correction formula was comparable for both linear and concave regression forms, the correction formula performed poorly when the regression form was convex. Further, even when the regression is linear or concave, it may not be advantageous to employ the correction formula unless the xy relationship is strong and sample size is large.