On a General Class of Designs for Multiresponse Experiments

Abstract

The purpose of this paper is to present a class of designs suitable for experiments where several responses or characteristics are under study, but all characteristics are not measured on each unit. Usually, while choosing the design, not much consideration has been given to the multiresponse aspect of the experiment, the choice being often made as if only a single response had been under study. Furthermore, while planning multiresponse experiments, we merely take over such designs and then assume that each experimental unit is studied on all variates or characteristics, and carry out the analysis accordingly. However, in a large number of cases, it is either physically impossible, uneconomic, or inadvisable on account of unequal importance or measuring costs of the various characteristics of interest, to study each of them on each experimental unit. Such situations arise in diverse areas of research in the humanities and the natural sciences. An interesting example will be found in [15] and many others in [14]. These communications bring out the important fact that in many situations we need to have experiments where observations on some of the responses are missing not by accident (as, for example, in [7] or [8]), but by design. For the sake of illustration, we include an example here too. Often the process of taking measurements is quite time-consuming. Suppose a biologist has a number of growing organisms of a similar kind, on each of which he could observe (say) $p$ responses provided that the process of taking measurements on any given unit were fast enough to enable him to finish in the limited amount of time, during which the experimental conditions remain unchanged. However, if the process is necessarily slow he may have to content himself with fewer (than $p$) responses on each unit. The above indicates that every design has two aspects: one relative to the responses and experimental units, and the other relative to treatments and blocks. The first decides for each unit the responses to be studied therein, while the latter tells us for each block the treatments (and units) allocated to it. However, the first aspect may also influence the second in the sense that relative to each response, we may have a different system of blocks. Designs which possess this last property are called multiresponse designs with $p$ block systems. These have been studied in [11] and will not be considered here. In [11], we have also studied another class called hierarchical designs, which are suitable for those situations in which the various responses could be graded in a descending order of importance, say $(V_1, V_2, \cdots)$, such that the response $V_i$ is supposed to be more important than $V_j$, if $i < j$. The designs are defined essentially by requiring that if $V_i$ is more important than $V_j$, then $V_i$ should also be observed on each experimental unit on which $V_j$ is observed. However, in many cases such a grading may not exist, or it may be otherwise inadvisable to impose such a hierarchical structure on the experiment. More general classes of designs are then called for. In this paper we introduce a class of designs (which may be called `regular incomplete multiresponse designs') that are response-wise incomplete, i.e., in which there are (at least some) experimental units on which all responses are not measured. However, in addition, they may or may not be treatment-wise incomplete (in the sense of an ordinary incomplete block design). As will be evident from the definition in the next section, these designs will be useful in a large variety of multiresponse experiments. The approach to the analysis of response-wise incomplete designs, by Trawinski (unpublished dissertation, [14]) and Trawinski and Bargmann [15], is through the use of maximum likelihood estimates and the likelihood ratio tests. However, as mentioned by the author in [13], apart from large sample approximations, etc., (inherent in the likelihood ratio tests), the formulae for the estimates of the parameters and for the test statistic that one obtains this way are somewhat cumbersome even for an electronic computer. In this paper therefore, a different approach which works for regular designs, is adopted. The attempt is to transform the data back into the framework of linear estimation and multivariate analysis of variance. Once this is done, the usual techniques of analysis become available. It may be stressed that the purpose of the regular designs is not merely to permit the estimation of treatment effects by linear analysis methods (which in itself is important), but also to make available valid (free of nuisance parameters) and exact test regions for testing linear hypotheses on the treatments. As usual, we require normality assumptions for the latter but not the former. It will also be noted that the approach in this paper possesses in a sense a symmetry with respect to the responses, unlike, for example, the one used in the analysis of hierarchical designs in [11] where the responses are arranged in an order of importance, say $V_1, V_2, \cdots, V_p$, and then the analysis for $V_{i+1} (i = 0, \cdots, p - 1)$ is performed conditional to the data observed for $V_1, V_2, \cdots, V_i$. Finally, we remark that though the derivations of the condition for regular designs may seem a little complex, their actual use is simple relative particularly to the other existing techniques.