# Tjur R-Squared¶

This fit statistic applies only to logistic regression.

Also known as Tjur’s $$D$$ or Tjur’s coefficient of discrimination, the Tjur pseudo $$R^2$$ value compares the average fitted probability $$\bar{\pi}$$ of the two response outcomes. In particular it is the difference between the average fitted probability for the binary outcome coded to 1 (success level) and the average fitted probability for the binary outcome coded to 0 (the failure level).

If the coded response $$y$$ has $$n_1$$ 1s and $$n_0$$ 0s then,

$R_{Tjur}^2 = \frac{1}{n_1}\sum \hat{\pi}(y=1) - \frac{1}{n_0} \sum \hat{\pi}(y=0)$

Note that $$0 \leq R_{Tjur}^2 \leq 1$$. If the model has no discriminating power, then $$R_{Tjur}^2 = 0$$. If the model has perfect discriminating power, then $$R_{Tjur}^2 = 1$$.

References

• Tue Tjur. Coefficients of determination in logistic regression models–a new proposal: the coefficient of discrimination. The American Statistician, 63(4):366–372, November 2009.