Rank Tests for Randomized Blocks when the Alternatives have an a Priori Ordering

Let Xᵢⱼ, i = 1, ⋯, n, j = 1, ⋯, k, be independent with Xᵢⱼ having the continuous distribution function P(Xᵢⱼ x) = Fⱼ(x - bᵢ) where bᵢ is the nuisance parameter corresponding to block i. (These assumptions shall be called the HA assumptions.) This paper is concerned with procedures for testing the null hypothesis {equation*}{0.1}H_0 : F_j = F (unknown), j = 1, ⋯, k,{equation*} which are sensitive to the ordered alternatives {equation*}{0.2}H_a : F_1 F_2 ⋯ F_k,{equation*} where at least one of the inequalities is strict. In particular, we introduce a test statistic $(Y)$ based on a sum of Wilcoxon signed-rank statistics. In Section 2 we develop the asymptotic distribution of Y and find that, under H₀, Y is neither distribution-free for finite n, nor asymptotically distribution-free. However, a consistent estimate of the null variance of Y is used to define a procedure which is asymptotically distribution-free. In Section 3 we derive, under the HA assumptions, necessary and sufficient conditions for the consistency of Y and two of its nonparametric competitors, viz., (1) Jonckheere's τ test [11] based on Kendall's rank correlation coefficient between observed order and postulated order in each block; (2) Page's ρ test [17] based on Spearman's rank correlation coefficient between observed order and postulated order in each block. We find that (i) Y is consistent if and only if ∑u < v ∫ Hᵤ dHᵥ/k(k - 1) > 1/4 where Hᵤ = F^ᵤ Fᵤ, u = 1, ⋯, k, (ii) Jonckheere's test is consistent if and only if ∑u < v ∫ Fᵤ dFᵥ/k(k - 1) > 1/4, and (iii) Page's test is consistent if and only if ∑u < v (v - u) ∫ Fᵤ dFᵥ > k(k - 1) · (k + 1)/12. Section 4 is devoted to efficiency comparisons of the rank tests with respect to a normal theory t-test defined in Section 1. For a class of shift alternatives we show that the Pitman efficiency of Y with respect to $t(E(Y, t))$ is greater than .864 for every F and every k. When F is normal, $E(Y, t) = .963$ for $k = 3$ and → .989 as k → ∞. These values compare favorably with the corresponding ones of Page's test (.716, .955) and Jonckheere's procedure (.694, .955). For these shift alternatives we also show that .576 E(ρ, t) ∞ and .576 E(τ, t) ∞.