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 logistic regressions do exist.  The most "famous" example is that of a species occupancy model where the probability of presence of a species follows a logistic model dependent on various predictors AND the probability of detection on a single visit is less than 1.
If only a single visit is observed, then as @MartinModrak has mentioned only the product of the occupancy probability and the detection probability can be estimated.  However, if multiple independent observations are taken on the same site under the same conditions (with the true but unknown presence status constant), then the two probabilities can be separated.
The R package unmarked deals with such models (logistic regressions for both the probability of occupancy and the probability of detection).  A reference for this is

MacKenzie, D. I., J. D. Nichols, G. B. Lachman, S. Droege, J. Andrew
Royle, and C. A. Langtimm. 2002. Estimating Site Occupancy Rates When
Detection Probabilities Are Less Than One. Ecology 83: 2248-2255.

For your problem with a rare disease such models might be appropriate if the test for having the rare disease does not always detect the disease when the patient does have the disease.  You would need to have patients tested multiple times (with an appropriate model describing the probability of detection).
These methods are certainly data hungry and having a "rare" disease (i.e., a small probability of having the disease) raises the need for lots of patients.
But my main point is that there are models that deal with excess zeros in a logistic regression but a secondary model (and subject matter rationale) that describes the generation of excess zeros is required.
