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Using collinearity diagnostics on dummy 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

$$ F_{1,df} = df(Residual) \frac{r_{pb}*r_{pb}}{1-r_{pb}*r_{pb}} $$

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

$$ F_{k-1,df} = \frac{df(Residual)}{(k-1)} \frac{Rsq}{1-Rsq} $$

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.