Asymptotic Efficiency of a Class of Aligned Rank Order Tests for Multiresponse Experiments in some Incomplete Block Designs

Abstract

Consider $n$ replications of an incomplete block design $D$ consisting of $b$ blocks of constant size $k(\geqq 2)$ to which $v(\geqq k)$ treatments are applied in such a way that (i) no treatment occurs more than once in any block, (ii) the $j$th treatment occurs in $r_j(\leqq b)$ blocks, and (iii) the $(j, j')$th treatments occur together in $r_{jj}'(> 0)$ blocks, for $j \neq j' = 1, \cdots, v$. Let then $S_i$ stand for the set of treatments occurring in the $i$th block, $i = 1, \cdots, b$. For the $\alpha$th replicate, the response of the plot in the $i$th block and receiving the $j$th treatment is a stochastic $p$-vector $\mathbf{X}_{\alpha ij}$ and is expressed as \begin{equation*}\tag{1.1} \mathbf{X}_{\alpha ij} = \mathbf{\mu}_\alpha + \mathbf{\beta}_{\alpha i} + \mathbf{\tau}_j + \mathbf{\varepsilon}_{\alpha ij},\quad j \in S_i, i = 1, \cdots, b, \alpha = 1, \cdots, n; \quad \sum \mathbf{\tau}_j = 0,\end{equation*} where the $\mathbf{\mu}_\alpha$ and the $\mathbf{\beta}_{\alpha i}$ are respectively the replicate and block effects (nuisance parameters in "fixed effect" model or spurious random variables in "mixed effect" model), $\mathbf{\tau}_1, \cdots, \mathbf{\tau}_v$ are the treatment effects (parameters of interest) and the $\mathbf{\varepsilon}_{\alpha ij}$ are the error vectors. It is assumed that $\{\mathbf{\varepsilon}_{\alpha ij}, j\in S_i\}$ have jointly a continuous cumulative distribution function ($\operatorname{cdf}$) $G(\mathbf{x}_1,\cdots, \mathbf{x}_k)$ which is symmetric in its $k$ vectors. This includes the conventional assumption of independence and identity of the $\operatorname{cdf's}$ of all the $N( = nbk)$ error vectors as a special case. We want to test the null hypothesis \begin{equation*}\tag{1.2} H_0:\mathbf{\tau}_1 = \cdots = \mathbf{\tau}_\nu = \mathbf{0} \text{vs} H_1:\mathbf{\tau}_j \neq \mathbf{0},\quad \text{for at least one} j(=1,\cdots, \nu)\end{equation*} In the univariate case (i.e., $p = 1$), intra-block rank tests for this problem are due to Durbin (1951), Benard and Elteren (1953), and Bhapkar (1961), among others. For some special balanced designs, the studies made by Elteren and Noether (1959) and Bhapkar (1963) reveal the low (Pitman-) efficiency of these tests, particularly when $k$ is small. In complete block designs, it is known [cf. Hodges and Lehmann (1962) and Sen (1968)] that the use of ranking after alignment increases the efficiency of the rank tests. The purpose of the present paper is to show that this merit of the ranking after alignment is preserved for a broad class of incomplete block designs. In fact, certain bounds for the efficiency are derived which do not depend on the design $D$, i.e., on the particular values of $b, \nu, r_j, r_{jj'}$ and $k$.