In the correlation of a naturally and an artificially dichotomized variable

A method is suggested for estimating the correlation of a naturally (X) and an artificially (Y) dichotomized variable. It is assumed that a normal random variable (L) underlies the artificially dichotomized variable. The proposed correlation coefficient recovers the product moment correlation coefficient between X and L from a fourfold table of X and Y The suggested correlation coefficient nu is contrasted with the phi correlation and the biserial eta. The biserial eta was proposed by Karl Pearson and is conceptually related to the new correlation coefficient. However, in addition, Pearson's biserial eta invokes the assumption that the marginal distribution of L is normal, which contradicts its basic assumptions and thus does not recover the true correlation of L and X. Finally, an approximation is provided to simplify the calculation of nu and its standard error.