Biserial and point-biserial correlation with correction for nonoptimal dichotomies

Abstract

Usingh/(pq)V2, one can convertthe biserial rback to the point-biserial r that would have resulted from any other piq split. Because thesecomputations use population equations , the program may occasionally producebizarre outcomes , such as rh or corrected rph greater than one. This shouldhappenonly rarely if reasonable sample sizes are used. Also, the equations assumebivariate normality and linearity of regression; thus they can be expected to be accurateonly when the standardcorrelationassumptions obtain. The Program. Three forms of initial input are permitted: a previously computed point-biserial r, togetherwith respective sample sizes,can be entereddirectly; the point-biserial correlation can be computed from raw data; and the point-biserial correlation can be computed from group means, standard deviations, and sample sizes. The program outputs the point-biserial correlation, the corresponding biserial correlation, and the point-biserial correlation corrected to a .51.5 split. The program then inquires whether the user wishes to correct rph for other possibleplq splits in the dichotomous variable. The only the same, the correlationwill be maximal forp=0.5, and will be increasingly attenuated as a function of the disparity betweenp and q. Because (pq)'h is the standard deviationof the dichotomized variable, this is analogous to the problem of range restrictionattenuating the correlation between continuous variables. In order to correct rph for a nonoptimumplq split, one must think of the dichotomous variable as having arisen from an arbitrary cut through an underlying continuous distribution. Take, for example, the problem of predicting employeeturnover. If the tendency to quit or change jobs has an underlying continuous distribution, then reexamining a cohort after a given elapsed time period represents just such an arbitrary cut; if one had waited a longer period, more of the group would have quit. If it can further be assumed that this underlying distribution is normal, then it is possible to estimate the correlation that would have existed if the underlying distribu-tionwere directly measured andcorrelated. Thisestimated underlying correlation is called the biserial correlation, rs. and is related to rph by r, = rph(pq) 'hlh, (2) where h is the height of the standard normal distribution at the point where the cut was made, p the area to the left, and q the area to the right. To convert from r, back to rph, the conversion factoris obviously h/(pq) 'h. Therefore , to estimate what rph would have been had the cutting point been at 0.5, one first converts to rh, then back-converts to rph withp =q=0.5. The resultingconversion factor is In a recent paper on the cost of dichotomizing variables , Cohen (1983) addressed the impact of having unequal proportions in eachcategory of a dichotomous variable. When a continuous bivariate normal variable is dichotomized at the mean of the underlying distribution (i.e., p=q=.5), the upper limit of the observed correla-tionwillbe equal to 0.798 times the population correlation. In instances in whicha variable has been dichotomized at either 0.5, 1.0, or 1.5 standard deviations from the mean, the observed correlation will be reduced by a factor of 0.762,0.662, and 0.519, respectively. In other words, if a variable wasdichotomized at 1.5 standard deviations from the mean (i.e., p=.841 and q=.159) and the underlying population product-moment correlation was 0.7, the observed correlation would be .3633 (i.e., .519x.7). Kemery, Dunlap, and Griffeth (in press) derived and empirically testeda procedure for correcting point-biserial correlationsbased on a particularpiq split in the dichoto-mous variable, to estimate what that correlation would have been had an optimum .51.5 split occurred for the dichotomous variable. This procedure first involves calculating the biserial correlation, which is an estimate of the correlationbetweentwo continuous normalvariables, assuming that the dichotomized variable was originally normal and had been arbitrarily dichotomized at the pointp. The biserial r can thenbe converted backto show what the point-biserial r wouldhavebeen withan optimal .51.5 split, or any other plq split. Mathematical Development. Letp equal the proportion in the first category and q = 1-p be the proportion in the second category of the dichotomized variable. When the second variable is continuous, the correlation coefficient is called the point-biserial correlation, r.; and the Pearson equation simplifies to rph = [(Yp-Yq)(pq)'h]/Sy , (1) where Y p and Y q are the respectivemeans of the continuous variable for the two categories of the binomial variable , and Sy is the standard deviation of the continuous variable. Because this equationcontains the crossproduct term, pq, if the Y means and standard deviation remain