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.