An introduction to some non-parametric generalizations of analysis of variance and multivariate analysis

Abstract

It is clear that a p-variate body of data arranged in a q-way classification will formally look llke a (p+q)-dimensional contingency table, but a distinction can be made between a 'variate' and a ‘way of classification’ in that along the direction of a 'variate' the marginal frequencies are supposed to be stochastic variates while along a 'way of classification' the marginal frequencies are supposed to be fixed. When, along certain directions, the marginals are fixed, an approach based on a conditional probability argument has been used. In the present paper (i) the conditional probability approach is abandoned and we start either from a single multinomial distribution or a product of an appropriate number of different multinomial distributions according as, with multi-way frequency data, all ways are 'variates' or some are 'variates' and some are 'ways of classification'. (ii) Also the hypotheses that are posed are of different kinds altogether according as we have a 'multivariate analysis' situation or an 'analysis of variance' situation. The hypotheses that are meaningful for one situation would not be too meaningful for the other and vice versa. Since the conditional probability approach is altogether abandoned, the mathematical theorems to which appeal is made are the two theorems as stated and proved by Cramer (1946, chapter 30) and a number of other such theorems which have been proved the same way and which, between them, take care of all the hypotheses discussed in this paper. When all ways are 'variates' the hypotheses are analogous to those in the usual multivariate analysis, and when some ways are 'variates' and some are 'ways of classification' the hypotheses are analogous to those in the analysis of variance. The general methods discussed in this paper arose out of an attempt to analyse a large mass of categorical data. The analysis has been carried out, and a few typical cases (together with the numerical analysis) illustrating different parts of the theoretical development will be presented in a subsequent paper. In this paper only the large-sample tests are considered. How 'large' the sample size has to be for the validity of the use of these asymptotic techniques, or in other words, some results on the nature of the approximation involved will also be discussed in a later paper.