Controlling power in a heteroscedastic ANOVA procedure

The advantage of the Bishop–Dudewicz ANOVA procedure is that, without assuming equal variances, the experimenter can guarantee that the Type I and Type II error probabilities are exactly equal to α and β respectively. Because unequal variances are known to affect both Type I and Type II errors, their procedure can be important in practice. However, to use their procedure, the experimenter must determine the constant, say d , such that Pr[Σ( T j –T̄+( μ j –μ)/√ d ) 2 ≥ c ] = 1 – β, where the T j s are J independent Student's t random variables, each having ν degrees of freedom, c is the critical value used in the Bishop–Dudewicz procedure, μ = Σ μ j / J , and the μ j s are the means of J independent normal random variables. Bishop & Dudewicz proposed a method of approximating d , but for many researchers and students, the procedure is inconvenient to the point that few would apply it. This brief note proposes and examines two simple approximations to d . One of these, which is based on a new approximation of the non‐null distribution, appears to be especially accurate.