Qualitative variance,entropy, and correlation ratios for nominal dependent variables

In this paper we derive a measure of dispersion for a nominal variable having k ? 2 categories and compare it with ordinary quantitative variance and with entropy. We then develop two qualitative analogs to the R² statistic, one based on qualitative variance, the other on entropy. For concreteness, data from the study of The American Soldier by S. A. Stouffér, E. A. Suchman, L. C. Devinney, S. A. Star, and R. M. Williams, Jr. (1949, Princeton Univ. Press, Princeton, N.J.) are used to illustrate the computations and compare these two correlation ratios for a variety of logit models. It is shown that both measures are reasonable measures of the proportion of variation of one or more qualitative dependent variables explainable by any mixture of qualitative and quantitative predictor variables. They can be used with logit, probit, dummy variable regression models, or probability models of some other type.