Regression diagnostics for categorical variables
Some people feel a little anxious expressing correlations between dichotomous variables and a continuous variable in a regression, for example, as input for multicollinearity diagnostics.
When we have have a dichotomous variable (or dummy variable) in a simple regression the correlation with the outcome measure is termed a point-biserial correlation. Rosenthal, R. (1994) shows that this correlation is related both to the F and t statistics and also to the difference in group means expressed in terms of the pooled group standard deviation.
In particular, for the former two,
$$r_text{pb} = \sqrt{ ttext{2} / (ttext{2} + df) $$
and
$$F_text{1,df} = df(Residual)r_text{pb}$$ $$r_text{pb} / (1-r_text{pb}r_text{pb}) $$
For the more general case of a categorical predictor, representing k groups, say, Rsq, the square of the semi-partial correlation for the categorical predictor with outcome is related to the F value by
$$F_text{k-1,df} = [df(Residual)/(k-1)]$$ $$Rsq /(1-Rsq) $$
Semi-partial R-squared for group, Rsq(group), is defined as
Rsq(group) = Rsq(all predictors) - Rsq(removing group)
Semi-partial R-squareds and F ratios are routinely used as indicators of predictive strength in simple and multiple regressions. Cohen, J. Cohen, P. (1983), for example, give an example of semi-partial correlations in a four predictor multiple regression involving sex.
References
Cohen, J. Cohen, P. (1983) Applied multiple regression/correlation analysis for the behavioral sciences. Second edition. Lawrence Erlbaum:London.
Rosenthal, R. (1994) Parametric measures of effect size. In H.Cooper amd L.V. Hedges (Eds) The handbook of research synthesis. New York: Russell Sage Foundation.