Bayesian Indexes of Superiority and Equivalence and the p-value of the F-test for the Variances of Normal Distributions

Abstract

We here consider the problem of comparing the variances of two normal populations. To make a more efficient decision than that made with the conventional F-test, we propose using the Bayesian index of the superiority of the variance of one group to the other θ=Pr(σ₁² > σ₂² | x₁, x₂). We express this index according to the hypergeometric series and the cumulative distribution functions of well-known distributions. Furthermore, we investigate the relationship between the Bayesian index and the p-value of the F-test. In addition, we propose another index, the Bayesian index of equivalence of two groups, θ_e(Δ) = Pr(Δ < σ₁/σ₂ < 1/Δ | x₁, x₂) for 0 < Δ < 1, which is also expressed according to the hypergeometric series and the cumulative distribution functions of well-known distributions. Finally, we evaluate the properties of the Bayesian index of equivalence using simulation, and illustrate the application of the Bayesian indexes with data from actual clinical trials.