# modelling rare events with small sample size

I have a dataset of 600 observations and the target variable is binary(risky/ non-risky). The constraint I have while modelling is that the target variable is very imbalanced .i.e only 12 (only 2% incidence rate) of these observations are risky and others are non risky . I have gone through literature on rare events and have found out that logistic regression for rare events might produce biased estimates if the number of observations with incidence is low (in my case its 12)

To overcome this bias in the estimating process, people started to use penalized regrssion like firth regression (if incidence rate is low but number of observations with incidence is higher, like 2% of 100000 observations is 2000) . And in the case of the sample size and incidence rate both being low(like 3-6%), literature suggests to use something called as "Exact Logistic Regression" which use MCMC sampling.

Since I do my modelling generally in R, I had found tutorials wherein they had used "elrm" package in R to fit logistic regression. But unfortunately "elrm" package has been removed from the R Cran - Repository. Hence that made me question the credibility of the Exact Logistic Regression for "Small Sample Size Rare Events(incidence rate around 3-8%)". It would be of great help to me , if you folks could provide me some insights on the working mechanism of Exact Logistic Regression and whether should I be using regular Logistic Regression (non-penalized estimation) or Exact Logistic Regression (penalized estimation) for my dataset of 600 observations and 2% incidence rate.

It would also be of great help to me if you could provide me insights on the methods ( apart from firth regression and Exact Logistic Regression)which are generally used to model rare events for small sample size?

• Could you please give us some more context? What kind of events? How many covariables? with 12 cases you could fit about one predictor, more will be difficult. Commented Jul 2, 2018 at 13:39
• Hey Halvorsen , The data is in the context of loan default.I have around 100 covariables. By event i mean an observation. so 12 events in here means that out of 600 observations I have only 12 of them are risky and and 588 are non risky. Since I have a lot of covariables, does it make sense to use LASSO to predict the outcome variable. Or should i use the exact logistc regression like Bjorn had suggested below? Commented Jul 3, 2018 at 11:02
• About 100 covariables is way to much! You should start thinking about which might be really important, and which not. There must be some having experience in the field. Otherwise, data reduction is necessary, and that would be lasso or ridge. I am not sure which would be best here! But before anything, try to get more data. And investigate if the covariables are colinear, do a principal component analysis. Commented Jul 3, 2018 at 12:01
• Commented Jul 6, 2018 at 5:40