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

Abstract

Let $X_{ij}, i = 1, \cdots, n, j = 1, \cdots, k$, be independent with $X_{ij}$ having the continuous distribution function $P(X_{ij} \leqq x) = F_j(x - b_i)$ where $b_i$ is the nuisance parameter corresponding to block $i$. (These assumptions shall be called the $H_A$ assumptions.) This paper is concerned with procedures for testing the null hypothesis \begin{equation*}\tag{0.1}H_0 : F_j = F \text{(unknown)}, \quad j = 1, \cdots, k,\end{equation*} which are sensitive to the ordered alternatives \begin{equation*}\tag{0.2}H_a : F_1 \geqq F_2 \geqq \cdots \geqq F_k,\end{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 $H_0, Y$ 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 $H_A$ assumptions, necessary and sufficient conditions for the consistency of $Y$ and two of its nonparametric competitors, viz., (1) Jonckheere's $\tau$ test [11] based on Kendall's rank correlation coefficient between observed order and postulated order in each block; (2) Page's $\rho$ 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 $\sum_{u < v} \int H_u dH_v/k(k - 1) > \frac{1}{4}$ where $H_u = F^\ast_u F_u, u = 1, \cdots, k$, (ii) Jonckheere's test is consistent if and only if $\sum_{u < v} \int F_u dF_v/k(k - 1) > \frac{1}{4}$, and (iii) Page's test is consistent if and only if $\sum_{u < v} (v - u) \int F_u dF_v > k(k - 1) \cdot (k + 1)/12$. 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 $\rightarrow .989$ as $k \rightarrow \infty$. 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 $.576 \leqq E(\rho, t) \leqq \infty$ and $.576 \leqq E(\tau, t) \leqq \infty$.