Identifying Which of J Independent Binomial Distributions Has the Largest Probability of Success

Abstract

Let p₁,…, p_J denote the probability of a success for J independent random variables having a binomial distribution and let p₍₁₎ ≤ … ≤ p_(J) denote these probabilities written in ascending order. The goal is to make a decision about which group has the largest probability of a success, p_(J). Let p̂₁,…, p̂_J denote estimates of p₁,…,p_J, respectively. The strategy is to test J − 1 hypotheses comparing the group with the largest estimate to each of the J − 1 remaining groups. For each of these J − 1 hypotheses that are rejected, decide that the group corresponding to the largest estimate has the larger probability of success. This approach has a power advantage over simply performing all pairwise comparisons. However, the more obvious methods for controlling the probability of one more Type I errors perform poorly for the situation at hand. A method for dealing with this is described and illustrated.