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Logistic regression models (generalised linear models with a binary response) are odd things. I am in the habitat of using randomised quantile residuals (R package statmod) for GLM diagnostics. I have read elsewhere on this site that:

if the model contains only categorical [predictor] variables and interactions among the variables are not needed, the model must fit the data and no calibration assessment is needed

This statement seems correct because the response variable can take on only two values (overdispersion can't happen) and there are no continuous predictors. Can someone please provide a citeable reference in support of this statement? The closest I can get is p. 595 of Crawley 2007 (The R Book), where it was noted that overdispersion is impossible for a binary response.

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  • $\begingroup$ I'm using the Crawley reference although it only concerns overdispersion and it's not a statistical text. $\endgroup$
    – stweb
    Jul 31, 2021 at 10:28

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For binary Y on independent observations there are no distributional assumptions (including overdispersion) in need of checking. So think about what can possibly go wrong on the right hand side of the model. And realize that there are direct assessments for the right hand side without resorting to residuals.

  • all predictors categorical: the only error can be omitting interaction terms; test all pre-specified interactions together using a likelihood ratio $\chi^2$ "chunk test"
  • some predictors continuous: problems can include nonlinearity of main effects on the logit scale or interactions between predictors; for linearity expand continuous predictors not known to act linearly on the logit scale and test all the nonlinear terms together in a chunk test

But note that testing for lack of fit involves dichotomous thinking and results in model uncertainty, inflation of type I assertion probabilities $\alpha$, and incorrect standard errors. So the strategy in RMS is aimed at formulating flexible models and sticking with them no matter what "significance" tests show. The degree of complexity in the pre-specified with reference to the available information in the data (when using frequentist methods) and if there is lack of fit (more complexity than you specify) there's not much you can do about it anyway (unless using a Bayesian model).

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