I've been thinking about this problem. The usual logistic function for modeling binary data is: $$ \log\left(\frac{p}{1-p}\right)=\beta_0+\beta_1X_1+\beta_2X_2+\ldots $$ However is the logit function, which is an S-shaped curve, always the best for modeling the data? Maybe you have reason to believe your data does not follow the normal S-shaped curve but a different type of curve with domain $(0,1)$.
Is there any research into this? Maybe you can model it as a probit function or something similar, but what if it is something else entirely? Could this lead to better estimation of the effects? Just a thought I had, and I wonder if there is any research into this.