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R in logistic regression

In the case of logistic regression, usually fit by maximum likelihood, there are several choices of pseudo- R.

One is the generalized R originally proposed by Cox & Snell, and independently by Magee:

where L (0) {\displaystyle {\mathcal {L}}(0)} is the likelihood of the model with only the intercept, L (θ ^) {\displaystyle {{\mathcal {L}}({\widehat {\theta }})}} is the likelihood of the estimated model (i.e., the model with a given set of parameter estimates) and n is the sample size. It is easily rewritten to:

where D is the test statistic of the likelihood ratio test.

Nico Nagelkerke noted that it had the following properties:

However, in the case of a logistic model, where L (θ ^) {\displaystyle {\mathcal {L}}({\widehat {\theta }})} cannot be greater than 1, R is between 0 and R max 2 = 1 − (L (0)) 2 / n {\displaystyle R_{\max }^{2}=1-({\mathcal {L}}(0))^{2/n}}: thus, Nagelkerke suggested the possibility to define a scaled R as R / R.