Robust Estimation in Incomplete Blocks Designs

Abstract

Robust estimates of contrasts in treatment effects for experiments with one observation per cell were proposed by Lehmann [6] for complete (randomized) blocks designs. The model for the observations $X_{i\alpha} (i = 1, \cdots, c; \alpha = 1, \cdots, n)$ is in this case \begin{equation*}\tag{1}X_{i\alpha} = \nu + \xi_i + \mu_\alpha + U_{i\alpha}\quad (\sum \xi_i = \sum \mu_\alpha = 0)\end{equation*} where the $\xi$'s are the treatment effects, the $\mu$'s are the block effects, and the $U$'s are independent with a common continuous distribution. Here we shall generalize these estimates to experiments in which the block size is smaller than the number of treatments to be compared, and we shall obtain their asymptotic efficiencies relative to the classical estimates. Since we are concerned with large sample theory, we shall be interested in designs in which the blocks are replicated a large number (at least 4) times. Such designs could be applied to situations in which only a few different treatment combinations are practicable but each could be replicated several times. For example, in an experiment to compare various diets for pigs, the natural block is the litter. One may wish to compare $c$ diets and have available a number of litters of size $b < c$. An incomplete blocks design using some $J$ litters could first be selected and then the whole design replicated several times using the remaining litters or (e.g. if some comparisons were of greater interest than others) some groups of $b$ diets could be given to more litters than others. Thus the situation to be considered is that in which $c$ treatments are to be compared and the blocks of experimental units are all of size $b < c$. An incomplete blocks design $D$ consisting of $J$ blocks of size $b$ is selected, the number $n_j$ of replications of the $j$th block is decided upon $(j = 1, \cdots, J; n_j = \rho_jn), \sum n_j$ blocks of experimental units are selected and numbered, and $\sum n_j$ sets of $b$ treatments are assigned to the selected blocks as specified by $D$ and the $n_j$. The set of blocks receiving the same treatments will be called a replication set. After the assignment of treatments to blocks, the order of application within the blocks is randomized. Assuming fixed effects and no interaction between treatment and block effects, the model for $D$ is \begin{equation*}\tag{2}X_{ij} = \nu' + \xi_i + \mu_j + U_{ij} (j = 1, \cdots, J; i \varepsilon S_j)\end{equation*} $\sum^c_{i = 1} \xi_i = \sum^J_{j = 1} \mu_j = 0$ where $S_j$ consists of the numbers of the $b$ treatments applied in the $j$th block, the $\xi$'s are the treatment effects, the $\mu$'s are the block effects, and the $U$'s are independent and identically distributed according to a continuous distribution $F$ with mean zero and variance $\sigma^2$ (not necessarily finite). Under the same assumptions, the model for the whole design is \begin{equation*}\tag{3}X_{ij\alpha} = \nu + \xi_i + \mu_j + \beta_{j\alpha} + U_{ij\alpha},\end{equation*} $(j = 1, \cdots, J; i \varepsilon S_j; \alpha = 1, \cdots, n_j)$ $\sum^c_{i = 1} \xi_i = \sum^J_{j = 1} \mu_j = 0; \sum^{n_j}_{\alpha = 1} \beta_{j\alpha} = 0\quad \text{for each} j$ where $S_j$ is defined as before (the treatments applied to the $j$th block of $D$ now being applied in the $n_j$ blocks of the $j$th replication set), $\xi_i$ is the effect of the $i$th treatment, $\mu_j$ is the effect of the $j$th replication set, $\beta_{j\alpha}$ is the effect of the $\alpha$th block in the $j$th replication set, and the $U$'s are distributed as in the model (2). (Although it might appear that the models (2) and (3) are valid only for a fixed order of application of the assigned treatments to the units within a block, the models remain valid under randomization. For, consider a model in which any of the $b$ treatments may be assigned to each unit within the block. If the corresponding $U$'s are independent, identically distributed random variables, then any selection according to specified probabilities will again be independent and identically distributed, which justifies the assumptions of the above models.)