How does a Logistic regression model converge if most variables are not linear with the log odds of the dependent variable?

Question

I have a dataset (unfortunately cannot disclose any part of it) which has a binary response variable. For each independent variable, I calculate the log odds of the positive cases given each value of the IV and plot them to check linearity, i.e., x-axis is the IV and y-axis is the $logodds(DV=1|IV)$. I find that at least 50% of my variables (which includes interaction effects) are not linear in log odds with the DV. How does my model converge then? Is the linearity assumption not a very strong one or any model can converge but in such cases simply cannot be trusted? If any additional information is required, just let me know and I will try to my best to provide to clarify my question even further.

As a side question, I am wondering if my approach of plotting logodds against each IV to check linearity is correct because everywhere else, people just advice to use the Box-Tidwell approach.

Answer 1

Maximum likelihood will give you the "best" parameters given the model, but "best" does not necessarily mean "good enough". Especially when your model is not very appropriate, the best given the bad model can be quite poor. However, that does not necessarily preclude the model from converging.