In Theorem 11.2.5 we saw that the ANOVA null is equivalent to all contrasts being 0. We can also write the ANOVA null as the intersection over another set of hypotheses.
(a) Show that the hypotheses
H0: θ1 = θ2 = ... = θk versus H1: θi, ≠ θj for some i, j
and the hypotheses
H0: θi - θj = 0 for all i, j versus H1: θi - θj ≠ 0 for some i, j
are equivalent.
(b) Express H0 and H1 of the ANOVA test as unions and intersections of the sets
Θij = {θ = (θ1,..., θk) : θi - θj = O}.
Describe how these expressions can be used to construct another (different) union-intersection test of the ANOVA null hypothesis.