On a Class of Aligned Rank Order Tests in Two-way Layouts

Abstract

The present investigation is concerned with the formulation of a multivariate approach for the construction of a class of aligned rank order tests for the analysis of variance (ANOVA) problem relating to two-way layouts. The problems of simultaneous testing and testing for ordered alternatives based on aligned rank order statistics are also considered. Various efficiency results pertaining to the proposed tests are studied. Let us consider a two factor experiment comprising $n$ blocks, each block containing $p(\geqq 2)$ plots receiving $p$ different treatments. In accordance with the two-way ANOVA model, we express the yield $X_{ij}$ of the plot receiving the $j$th treatment in the $i$th block as \begin{equation*}\tag{1.1}X_{ij} = \mu + \alpha_i + \tau_j + \epsilon_{ij}, \quad i = 1, \cdots, n, j = 1, \cdots p;\end{equation*} where $\mu$ stands for the mean effect, $\alpha_1, \cdots, \alpha_n$ for the block effects (may or may not be stochastic), $\tau_1, \cdots, \tau_p$ for the treatment effects (assumed to be non-stochastic), and $\epsilon_{ij}$'s are the residual error components. It is assumed that $\mathscr{\epsilon}_i = (\epsilon_{i1}, \cdots, \epsilon_{ip}), i = 1, \cdots, n$ are independent and identically distributed stochastic vectors having a continuous (joint) cumulative distribution function (cdf) $G(x_1, \cdots, x_p)$ which is symmetric in its $p$ arguments; (this includes the conventional situation of independence and identity of distributions of all the $np$ error components as a special case). We may set without any loss of generality $\sum^p_1 \tau_j = 0$, and frame the null hypothesis of no treatment effect as \begin{equation*}\tag{1.2}H_0 : \tau_1 = \cdots = \tau_p = 0.\end{equation*} The usual ANOVA test based on the variance-ratio criterion is valid only when $G$ is a $p$-variate (totally symmetric) multinormal cdf. For arbitrarily continuous cdf $G(x_1, \cdots, x_p)$, intra-block rank tests are due to Friedman [7], Brown and Mood [3], and Sen [21]; generalizations of these tests to incomplete layouts are due to Durbin [6], Benard and Elteren [1], and Bhapkar [2]. Hodges and Lehmann [9] have pointed out that intra-block rank tests do not utilize the information contained in the interblock comparisons, and hence, are comparatively less efficient. They have suggested the use of ranking after alignment and also considered the Wilcoxon's and Kruskal-Wallis tests based on aligned observations whose asymptotic efficiency have been studied very recently by Mehra and Sarangi [14]: (After the first draft of the paper was submitted, the author came to know through the editor about the paper by Mehra and Sarangi [14], submitted earlier to the Annals. In view of this, the overlapping part is not considered here.) The object of the present investigation is to formulate briefly the theory of rank order tests based on observations after alignment and through a multivariate approach to justify the validity and efficacy of the proposed tests (which include the earlier works as special cases).