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

Zero inflated models are using a distribution (like Poisson or something else) that is mixed with a point mass at zero.

Logistic regression relates to binary data which is modelled with a Bernoulli distribution. When you mix a Bernoulli distribution with a point mass at zero then you get another Bernoulli distribution. So zero inflation does not make much sense.
An exception could be when your response is a Binomial distribution that is modelled with a logistic function for a parameter $p$. E.g. instead of whether a patient has a disease or not, the response could be something like during how many years $x$ out of $n$ a patient has had some rare symptom.
