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Model summaries based on the ratio of fitted and null likelihoods have beenproposed for generalised linear models, reducing to the familiar $R^2$coefficient of determination in the Gaussian model with identity link. In thisnote I show how to define the Cox--Snell and Nagelkerke summaries underarbitrary probability sampling designs, giving a design-consistent estimator ofthe population model summary. I also show that for logistic regression modelsunder case--control sampling the usual Cox--Snell and Nagelkerke $R^2$ are notdesign-consistent, but are systematically larger than would be obtained with across-sectional or cohort sample, even in settings where the weighted andunweighted logistic regression estimators are similar or identical.

Pseudo-$R^2$ statistics under complex sampling