A generalized correlated binomial distribution with application in multiple testing problems

A typical microarray experiment often involves comparisons of hundreds or thousands of genes. Since a large number of genes are compared, simple use of a significance test without adjustment for multiple comparison artifacts could lead to a large chance of false positive findings. In this context, Tsai et al. (Biometrics 59:1071–1081, 2003) have presented a model that studies the overall error rate when testing multiple hypotheses. This model involves the distribution of the sum of non-independent Bernoulli trials and this distribution is approximated by using a beta-binomial structure. Instead of using a beta-binomial model, in this paper, we derive the exact distribution of the sum of non-independent and non-identically distributed Bernoulli random variables. The distribution obtained is used to compute the conditional false discovery rates and the results are compared to those obtained, in Table 3, by Tsai et al. (Biometrics 59:1071–1081, 2003).