On the Wald-Optimality of Rank-Order Tests for Paired Comparisons

Abstract

Consider $p(\geqq 2)$ treatments in an experiment involving paired comparisons. The $(i, j)$th pair yields $n_{ij}$ observations $X_{ijm}$ with distribution functions (df's) $F_{ij}(x) = F(x - \beta_i + \beta_j), (1 \leqq m \leqq n_{ij}; 1 \leqq i < j \leqq p), \mathbf{\beta} = (\beta_1, \cdots, \beta_p)'$ being the vector involving treatment effects. We assume that $F$ is symmetric about zero, i.e. $F(x) + F(-x) = 1$, for all real $x$. This is, for example, the situation, when one considers a replicated balanced incomplete block design with each block of size two under the usual assumption of additivity in an analysis of variance model. The null hypothesis to be tested is as follows: (1.1) $H_0': F_{ij}(x) = F(x)$, for all real $x$, and $1 \leqq i < j \leqq p,$ against all alternatives, which is equivalent to (1.2) $H_0: \mathbf{\beta} = \mathbf{0},$ against $\beta \neq \mathbf{0, 0}$ being a $p$-component column vector. A class of rank-order tests for the above problem was considered in Mehra and Puri (1967) and Puri and Sen (1969b), and asymptotic distributions of test statistics were obtained both under the null hypothesis, and under a sequence of alternatives converging to the null hypothesis at a suitable rate. However, any asymptotic optimality properties of the test procedures were not considered in either of the two papers. For subsequent notational convenience, we first pool the $n = \sum\sum_{i\leqq i<j\leqq p} n_{ij}$ observations, and label the $m$th observation for the $(i, j)$th pair by $\alpha = \sum^p_{\lambda=k+1} \sum^{i-1}_{k=0} n_{k\lambda} + \sum^{j-1}_{\lambda=i} n_{i\lambda} + m(1 \leqq m \leqq n_{ij})$, where we define $n_{0\lambda} = 0, n_{ii} = 0 (1 \leqq \lambda \leqq p, 1 \leqq i < j \leqq p, 1 \leqq \alpha \leqq n)$. Also, set $c_{in\alpha} = 1, -1$ or 0 according as the $\alpha$th observation is from a block where the treatment $i$ is paired with some treatment, the index of which is $> i, <i$, or the treatment $i$ is not involved in the block at all $(1 \leqq \alpha \leqq n; 1 \leqq i \leqq p)$. Then one can write \begin{equation*}\tag{1.3}P(X_{n\alpha} \leqq x) = F_{n\alpha}(x) \text{(say)} = F(x - \mathbf{\beta'c}_{n\alpha}),\end{equation*} $\mathbf{c}_{n\alpha} = (c_{1n\alpha}, \cdots, c_{pn\alpha})', 1 \leqq \alpha \leqq n.$ We may remark at this point that the paired comparison problem can be regarded as a particular case of the more general regression problem considered in Puri and Sen (1969a). The object of the present note is to show that, while tests considered in Puri and Sen (1969a), are, under certain conditions asymptotically optimal for the regression problem (and, hence, for the paired comparison case if the same conditions are satisfied), tests considered in Mehra and Puri (1967) or Puri and Sen (1969b) are, in general, not so. A counter-example is given in Section 3 to illustrate this fact.