Assessing Model Fit in Logistic Regression#

Stage 6 · 🎯 Classification & Logistic Regression · Lesson 40 of 56 · advanced

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No R² to lean on#

In ordinary regression, \(R^2\) gives a quick read of fit — the fraction of variance explained. Logistic regression has no such natural measure, because it predicts probabilities, not a continuous quantity with variance to partition. Assessing a logistic model’s fit therefore takes a different toolkit, built on the likelihood the model was fitted to maximise.

Likelihood-based fit#

The foundation is the log-likelihood of the fitted model — higher means the model assigns greater probability to what was actually observed. A closely related quantity is the deviance, defined as \(-2\) times the log-likelihood; lower deviance means better fit. On their own these numbers are not interpretable, but compared — fitted model against a baseline — they become the basis of every fit measure below.

Pseudo-R²#

To mimic the familiar 0-to-1 feel of \(R^2\), several pseudo-\(R^2\) measures compare the fitted model’s likelihood to that of the null (intercept-only) model. The most widely recommended is McFadden’s:

\[R^2_{\text{McF}} = 1 - \frac{\ln L_{\text{model}}}{\ln L_{\text{null}}}.\]

It is 0 when the predictors add nothing and approaches 1 as the fitted model explains the data far better. It is not the proportion of variance explained — McFadden values of 0.2–0.4 already indicate a very good fit — so it is read as a relative measure among models, not on the OLS scale. (Cox–Snell, Nagelkerke and Efron’s are common alternatives.)

Comparing to the null#

That comparison can be made a formal test. The likelihood-ratio test asks whether the fitted model fits significantly better than the null, using the drop in deviance as a chi-square statistic; a significant result means the predictors, taken together, genuinely contribute. And to balance fit against complexity across candidate models, AIC and BIC combine the log-likelihood with a penalty for the number of predictors — the same parsimony principle from the selection lessons. The next lesson turns this deviance-comparison idea into a tool for choosing features.

Hint