Upper and lower bounds for correlations in 2 X 2 tables--revisited

The correlation coefficient (r) between two dichotomous variables requires a different interpretation from that of the customary correlation between normally distributed continuous variables, since perfect correlation is usually not +1 or -1. The maximum and minimum possible correlations between two dichotomous variables depend on the marginal distributions. An example using data on smoking and lung cancer illustrates that what appears to be a small correlation in the usual sense may in fact be quite large in relation to the maximum possible. Similarly, the interpretation of R2 as the proportion of variance of the dependent variable that is explained by the independent variable(s) is subject to the same consideration. This paper describes the calculation of the upper and lower limits of r for two dichotomous variables. The problem of interpreting R2 in linear regression and the use of R2 for variable selection in stepwise regression applied to dichotomous data are also discussed and illustrated.