Is there a way to increase the number of predictor in a logistic regression when sample size is small? One of the dependent variables in my dataset is a binomial variable indicating whether an invasive species has been eradicated (with $n_{1}$ = 23) or not (with $n_{0}$ = 75) after the use of a certain control method.
In order to highlight which factors affect the species eradication with this method, I would like to build a set of a priori logistic regression models with various combinations of explanatory variables for which I have biologically relevant hypotheses, and to select the "best" model(s) on the basis of something like an AIC (cf. Burnham & Anderson, 2002).
From what I understood from this post and this one, with such a low frequency of success (i.e. number of 1 in my variable), I should not build models including more than 1 or 2 explanatory variables to avoid overfitting. I don't want to fit that many predictors at once, but having models with at least 3-4 explanatory variables seem to me mandatory to properly grasp the phenomenon I'm studying.
My question is thus: is there a way to reasonably increase the number of parameters to be estimated in my models without affecting inferences and/or goodness of fit too badly? I wondered if perhaps approaches such as bootstraps, lasso or ridge-regression were not meant for this?
Any alternative, suggestion, documentation and constructive criticism is welcome.

I asked another quite related question here.
 A: This is less of an answer and more of an extended comment. There is no way to generate additional data in a reliable way in order to meet the (very liberal) 1-in-10 EPV rule, which is what I think you are working off. Here are some of my thoughts:

*

*Penalized methods are not what you want if your goal is inference.  You lose the ability to do inference since the models are by design biased.  New techniques for doing inference from these models exist (I think they go by the name of post selection inference or selective inference). I'm not intimately familiar with them and so won't comment on if they are good or bad.  I will leave that to you to decide.


*If you have good prior information, you might consider a Bayesian analysis.  The prior information acts as regularization (indeed the LASSO and Ridge regression have Bayesian interpretations).  That might be more work to do right than you are willing to commit.


*The bootstrap is a good idea, but might prove problematic in some cases.  The problem with too many predictors is that you risk complete or quasi-complete separation of the data.  If that happens you get incredibly large estimates of the standard error even from bootstrapping because the coefficients can be enormous.  You can try and get around that by using Firth's logistic regression.
A: Ridge regression would be a useful approach here. It allows you to include all predictors in the model while penalizing their coefficient values to minimize overfitting. A standard approach is to use cross validation to choose the level of penalty that provides the optimal performance (for example, gauged by deviance in logistic regression).
You could get into trouble, however, if you apply this approach to setting up and comparing a set of multiple different models to identify the "best" one. As the answer from Demetri Pananos emphasizes, inference is difficult at best with penalized methods, and bootstrapping won't get around the problem of too few events.
At this stage of your study, use your knowledge of the subject matter to identify potentially important predictors and use them all in a single penalized ridge-regression model. That should help you start "to properly grasp the phenomenon" you're studying and guide you toward studies that can provide more data efficiently.
