A Proof That Kramer\u27s Multiple Comparison Procedure for Differences Between Treatment Means Is Level-α for 3, 4, or 5 Treatments

Let Xij, i = 1, ..., p; j = 1, ... . N1 be independent normal variables with E(Xij) = μi, Var Xij = σ2. Let Xi = Ni-1 ∑j=1Ni Xij and S2 = (∑(Ni-1))-1 ∑(Xij-Xi)2. Then if p ≤ 5 we show Pr(|(μi-μj) - (Xi-Xj) | ≧ S(Ni-1 + Nj-1) qp(α), √21/2 for some i≠ j) ≤