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is there like a diagnostic checking(like those in the time series) done for logistic regression and complementary loglog model?

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Yes, there are many diagnostic checks done in these situations. Any textbook on generalized linear models should introduce them to you. A lot of them are adaptions of those used in the linear regression case eg plotting residuals to see if they have the distribution expected in the model; plotting residuals against explanatory variables or candidate explanatory variables to see if there is structure that is not explained by the model; etc. Checking for autocorrelation in the residuals would certainly be one thing that should be done if they are a time series.

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