On the bias and variance of odds ratio, relative risk and false discovery proportion

This paper develops a method to calculate the moments of statistical ratios as functionals of Bernoulli random variables via inverse moments of binomial distributions. We derive exact expressions for the mean, bias and variance formulas of several statistics, including the "modified" versions of the odds ratio and relative risk as well as the (positive) false discovery proportion (FDP), in which the case of zero cells is handled via indicator functions. Similar to the "positive" FDP, we introduce the "positive" odds ratio and relative risk statistics. We provide numerical examples to illustrate the biases and variances of these statistical ratios.