I would like to study rare events in a finite population. Since I am unsure about which strategy is best suited, I would appreciate tips and references related to this matter, although I am well-aware it has been largely covered. I just don't really know where to begin.
My problem is a political sciences one and I have a finite population comprising 515,843 records. They are associated to a binary dependent variable with 513,334 "0"s and 2,509 "1"s. I can coin my "1"s as rare events since they account for only 0.49% of the population.
I have a set of around 10 independent variables I would like to build a model with to explain the presence of "1"s. Like many of us, I read King & Zeng's 2001 article about rare events correction. Their approach was to use a case-control design to reduce the number of "0"s, then apply correction to the intercept.
However, this post says that King & Zeng's argument was not necessary if I already collected my data over the whole population, which is my case. Therefore, I have to use the classical logit model. Unfortunately for me, although I obtain good significant coefficients, my model is completely useless in terms of prediction (fails to predict 99.48% of my "1"s).
After reading King & Zeng's article, I wanted to try a case-control design and selected only 10% of the "0"s with all the "1"s. With almost the same coefficients, the model was able to predict almost one third of the "1"s when applied to the full population. Of course, there are a lot of false-positive.
I have thus three questions I would like to ask you:
1) If King & Zeng's approach is prejudiciable when you have full knowledge of the population, why do they use a situation where they know the population in their article to prove their point?
2) If I have good and siginificant coefficients in a logit regression, but very poor predictive power, does that mean that the variation explained by these variable is meaningless?
3) What is the best approach to deal with rare events? I read about King's relogit model, Firth's approach, the exact logit, etc. I must confess I am a lost among all these solutions.