On the correlation of a naturally and an artificially dichotomized variable

Source abstract

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 ν is contrasted with the phi correlation and the biserial η. The biserial η was proposed by Karl Pearson and is conceptually related to the new correlation coefficient. However, in addition, Pearson's biserial η 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 ν and its standard error.