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The model including one binary outcome (0/1; incident rate ~1.2%), one main exposure, and 13 covariates. The whole model is significant and the goodness-of-fit is OK. However, model diagnostic is completely questionable (See Fig). Almost all the observations fall into the category of y=1 had residuals larger than 3. I wonder whether it is due to the low event rate of the outcome, and can anyone give me some advice as to how to deal with this problem? If the purpose of the model is inference rather than prediction, can I just ignore the model diagnostic results?

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Logistic Regressions require different diagnostics: http://www.r-bloggers.com/residuals-from-a-logistic-regression/

You can used an offset to help with a label imbalance: Using offset in binomial model to account for increased numbers of patients

Additionally, when you evaluate the quality of your prediction, I would recommend a Mathews Correlation Coefficient rather than AUC. AUC tends to overestimate given a label imbalance while MCC is normalizes for the label imbalance and will give you a more appropriate answer.

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