Multivariate Analysis of Variance (MANOVA)

Abstract

Theoretical results, mostly in the areas of tests of significance and estimation, have been available for some time now for certain problems of MANOVA. [e.g. 6, 9, 10, 11, 14, 16, 17, 19, 26]. It must, of course, be recognized that most of these theoretical results on estimation and tests of significance are useful in applications only as tools for considering rather specific questions. For the more important problems of general evaluation of what is shown in the data, techniques of MANOVA similar to the one discussed in [25] are important. Situations in which the interest of the experimenter is in specific evaluation of fairly specific questions do arise and tools, such as tests of significance and confidence inter-val estimation, are useful in these situations although they should be recognized as being very limited in their scope. The application of these techniques to real life problems has, however, lagged behind perhaps due, among other things, to the computational complexities involved in applying them. In any problem, the relevant design of the experiment determines a systematic rule or scheme by which the various factors are assigned to the experimental units in the problem. Now, for any given design, the problem is a uni-response or a multi-response one, according as one or more than one response is measured on each of the experimental units under the desighn. For unii-response problems we have the ussual techniques of ANOVA for analysing and interpreting the data. For multi-response problems the current practice, if not universal at least usual, seems to be to analyse the results for each of the responses separately thus ignoring and failing to exploit the real correlations that may exist between the responses. The present paper, on the other hand, using a bio-chemical problem from the literature [2] illustrates some of the techniques of simultaneously analysing and interpreting multi-response data under the Model I of MANOVA [22]. The paper deals mainly with tests of significance and only one type of confidence interval estimation. As such the paper is concerned primarily witlh