Let Xijkl (i, j, k = 0, 1, l = 1,...,L) denote the entries in a 2 × 2 × 2 × L table with factors A, B, C, and D, and let

Γl = PABcCDlPABCD ˜ l PAB˜CD˜ l PAB˜ CD˜ l PABCDlPA˜BCD ˜ l PABCD˜ l PA˜B˜CD˜ l

.

Then

(i) under the assumption Γl = Γ there exists a UMP unbiased test of the hypothesis Γ ≤ Γ0 to for any fixed Γ0;

(ii) When l = 2, there exists a UMP unbiased test of the hypothesis Γ1 = Γ2

—in both cases regardless of whether 0, 1, 2 or 3 of the sets of margins are fixed.