Correction for restriction of range when both X and Y are truncated

Abstract

The effect of range restriction on one variable in a bivariate normal distribution on the X-Y correlation and the problem of estimating unrestricted from restricted correlations has been widely studied for more than half a century. The behavior of correction formulas under truncation of both X and Y, however, remains largely unresearched. The performance of the correction formula for unidimensional truncation (Thorndike, 1947, Case 2) and an approximation procedure for correcting for bidimensional truncation proposed by Wells and Fruchter (1970) were investigated. The Thorndike correction formula undercorrects in most circumstances. The Wells and Fruchter procedure performs quite well under most conditions but often results in a slight overcorrection. The performance of the Wells and Fruchter and Thorndike formulas are also compared under truncation on X or Y alone. In these circumstances the Wells and Fruchter correction is either equal or markedly superior to the traditional correction. Based on overall performance in recapturing the unbiased population values under both unidimensional and bidimensional truncation, the Wells and Fruchter correction is recommended as the preferred procedure in many practical settings.