Rare events defined as binary outcomes, which have tens of thousands to hundreds of thousands of non-events (zeroes) compared to the outcome of interest (ones), can be a challenge in observational studies or clinical trials. Logistic regression methods for independent data have binary or ordinal outcomes but can produce predicted probabilities that grossly underestimate the true probability of a rare event [6]. At present, very few methods are available for modeling and analyzing longitudinal rare event data. The methods currently available are models based upon the Poisson distribution and are appropriate when the dependent variable involves count data. In the rare event situations, with dependent data, the variance matrix for the regression coefficients of the standard logistic regression model is biased; the estimated variances are smaller than the true variances. Furthermore, Carroll and colleagues discovered that under rare event conditions the use of the sandwich estimator with the logistic regression model produced undercoverage of Wald-type tests. In the case of logistic regression using the sandwich estimator, “an important part of sample size considerations is the number of events” [2]. In other words, decreasing sample sizes with rare outcomes can worsen the bias of the sandwich estimator.
