An investigation of the accuracy of restriction of range correction formulas



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The major purpose of this study was to investigate the effects of violation of the linearity and homoscedasticity assumptions on the accuracy of restriction of range correction formulas. Both fallible empirical test score data and simulated test score data generated by Monte Carlo methods were utilized. Simulated bivariate distributions (N = 4,000) with correlations ranging from .10 to .90 were generated to study the effects of selected violations of the linearity and homoscedasticity assumptions. The empirical data included scores on a variety of psychological measures including cognitive abilities, biographical information, temperament, interests, academic achievement, and employee performance appraisals for three different samples. These samples consisted of (a) high school students from Project TALENT (N = 13,478), (b) applicants for hourly employment in an oil refinery (N = 2,250), and (c) white collar employees in a large international petroleum company (N = 5,900). Analysis of the empirical data involved (a) computing statistical tests for violation of the linearity and homoscedasticity assumptions, (b) generating comparison distributions to control for the effects of nonlinearity or heteroscedasticity or both by using a pseudorandom number generator to produce new Y distributions for the X arrays so that the mean of the array would fall on the regression line and the standard deviation would equal the standard error of estimate, and (c) truncating the distributions at successive 10% intervals from the lower end and computing the corresponding corrected correlations. Corrections for restriction of range were limited to correction for explicit selection for the two variable case. Major findings indicated (a) corrections are sensitive to even slight violations of the linearity assumption, (b) corrections are relatively insensitive to moderate violations of the homoscedasticity assumption, (c) corrections are least accurate for small correlations and severe degrees of restriction, (d) corrected correlations are usually better estimates of population correlations than restricted correlations, and (e) empirical data distributions may violate the linearity and homoscedasticity assumptions such as to produce substantial overcorrection or undercorrection.