Zero Inflated Logistic Regression - Does This Exist? I have a dataset with information belonging to medical patients (e.g. age, gender, height, etc.). Suppose that the response variable is whether or not the patient has a specific disease or not - thus, the goal would be to understand the effects and significance of the different explanatory
variables on having the disease.
Now, imagine that this is a rare disease and only 5% of the patients in your dataset has the disease. If you fit a Regression model to this data (e.g. Logistic Regression), the model might not have observed enough disease cases to effectively "learn" the difference between diseased cases and non-diseased cases - and therefore poorly generalize to new data.
I tried to read more about this online and came across "Zero Inflated Models" (https://en.wikipedia.org/wiki/Zero-inflated_model)  and "Gamma Hurdle Models" (https://en.wikipedia.org/wiki/Hurdle_model) - these models seem to be appropriate for instances where there are many "Zeros" within the response variable. However, it seems that these models are intended for "Count Data", whereas the problem I am working with has a "Binary Response".
I tried to read more online to see if there are extensions of these frameworks for Logistic Models (e.g. Zero Inflated Logistic Model, Logistic Hurdle Model) and if it would be possible to implement these in R - but there does not seem to be anything at first glance.
I was thinking of just "tricking" my model into believing that the binary response variable I have is actually "count data" and then using Zero Inflated Models/Hurdle Models - but I feel that this is disingenuous and will likely result in problems later on.
As such, the closest thing I could find was "Weighted Logistic Regression" - but again, there do not seem to be many references and R implementations for this approach. Initially, I had thought of using an "Oversampling Approach" to correct for class imbalance, but I was advised that this might not be suitable (Logistic Regression With Imbalanced Data?)
Can someone please comment if it is possible to adapt Zero Inflated Models/Hurdle Models to a Logistic Regression? What kind of strategies can I employ in such a problem?
Possible References:

*

*What does "weighted logistic regression" mean?


*Does an unbalanced sample matter when doing logistic regression?


*https://stackoverflow.com/questions/50078497/weighted-logistic-regression-in-r

Logistic regression will not "state that all future patients do not have the disease". Logistic regression yields probabilistic predictions, i.e., probabilities that a patient has the disease.
In the case of a rare disease, this probability may be extremely low (for a patient that is essentially healthy - no need for action), or very low (better to run another non-invasive test), or "merely" low. If the disease in question is rare but dangerous, it may make sense to run an invasive test, e.g., taking biopsies, even if the predicted probability your logistic regression yields is only $\hat{p}=0.2$. You need to adapt your decision thresholds (possibly multiple ones, as here!) to the costs of decisions.
Thus, while a "zero-inflated logistic regression" could in principle make sense (e.g., in the case where we suspect two data generating processes to be at work, one of which always yields a zero), that does not seem to be the case here. Logistic regression can deal quite well with rare instances of the target variable. If all goes well, it will simply output low probabilities. If these are well-calibrated, this is precisely what should happen.
And no, oversampling (or weighting, which is essentially the same as oversampling) won't address a non-problem.
