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

While the answer by Stephan gives a good overview of the bigger picture, I think the answer in the narrow sense is IMHO that
No, zero-inflated logistic regression does not make much sense
Why? Assume the true data-generating process is indeed a mixture of a Bernoulli distribution and a constant zero, specifically:
$$
\begin{align}
y_i &= \begin{cases}
  0  & z_i = 0 \\
  \bar{y}_i & z_i = 1
\end{cases}\\
\bar{y}_i &\sim \text{Bernoulli}(p_i)\\
z_i &\sim \text{Bernoulli}(\theta_i)
\end{align}
$$
Where both $p_i$ and $\theta_i$ are some function of some predictors (usually a logit-transformed linear predictor term). We can quickly see that the outcome is 1 if and only if both $z_i$ and $\bar{y}_i$ are 1, so $P(y_i = 1) = \theta_i p_i$ and thus simply $y_i \sim \text{Bernoulli}(\theta_i p_i)$. This means $y_i$ can only give you information about the product $\theta_i p_i$ and cannot disentangle the individual contributions of the "logistic regression" and "zero inflation" components. unless you make strong restricting assumptions about the possible forms of predictors for $\theta_i$ and $p_i$. (In theory, there is a very tiny difference as this "zero-inflated" formulation implies a slightly different link function and thus different behavior of continuous predictor terms than the logistic regression, but I think this is highly unlikely to be relevant to any practical analysis task).
A similar line of reasoning applies to hurdle logistic regression.
So standard logistic regression model is likely sensible, but it is known that maximum likelihood estimators can be biased when there is little information in the data (small sample size and/or rare events) and bias corrected methods such as Firth's correction (e.g. via logistf) are thus likely to be preferred to R's glm or similar.
The case would be different if you had zero inflation to a binomial response with more than one trial - then you could in fact at least in some cases learn something about the zero-inflation/hurdle component separately from the success probability.
