Recent Trends of Research Work in Multivariate Analysis

Abstract

In the issue of Biometrics (Vol. 20, part 2) dedicated to the memory of Ronald Aylmer Fisher I reviewed the contributions by Fisher and some of the salient features of research work done in multivariate analysis up to 1964. Fisher's contributions as well as the related methodology developed by Wilks ([1932; 1946]-likelihood ratio criteria for testing multivariate hypotheses), Bartlett ([1947; 1951]-decomposition of Wilks's criteria for testing different aspects of a null hypothesis), Rao ([1946; 1948]-analysis of dispersion' as a generalization of the univariate analysis of variance, and tests for additional information supplied by a subset or functions of measurements), and Williams ([1952a; 19591-residual canonical correlations in testing goodness of fit of specified discriminants) may be described as a study of association between two sets of variables. One set is called predictor and another set, criterion variables. Some of the variables may be hypothetical (unobservable), and others may have values on a dummy or interval scale. I have indicated in Rao [1960] how various multivariate methods such as regression, canonical correlations, analysis of dispersion and canonical analysis, discriminant function, factor and latent structure analyses, treatment of contingency tables etc., can be classified by the nature of predictor and criterion variables. However, the classification provides only a suitable framework for the discussion of different problems but does not imply that the statistical methods appropriate for one problem can be simply deduced from those of another. Since 1964 considerable progress has been made in several directions of the work initiated by Fisher. Exact distributions have been found of several likelihood ratio criteria and of roots of determinantal equations involving random matrices which arise in multivariate statistical analysis and in some problems of physics. Further progress has been made in testing goodness of fit of assigned discriminant functions. The theory and construction of discriminant functions for deciding between composite hypotheses (involving nuisance parameters) has been developed. A satisfactory