Sign Changes When Correcting for Range Restriction: A Note on Pearson's and Lawley's Selection Formulas

K. Pearson (1903) recognized that the correlation coefficient was subject to distortion when a sample was censored or preselected in some way. He proposed 3 univariate correction formulas for better estimates in these circumstances. These have become well known from the work of R. L. Thorndike (1949). D. N. Lawley (1943) proposed a general solution usually called the multivariate correction for range restriction. Both Pearson's and Lawley's corrections are discussed and examples are presented. Of particular interest are the opportunities for the corrected correlations to change sign as a result of the correction. Numerical examples are presented that show that correlations can change signs in the Pearson-Thorndike Case 3 and in Lawley's general solution