In this report (i) right from the beginning we abandon the conditional probability approach and start either from a single multinomial distribution or a product of an appropriate number of different multinomial distributions according as, with a multi-way frequency data, all marginals are variable or some are variable and some are fixed, (ii) Also right from the beginning the hypotheses that are posed are of different kinds altogether according as we have all marginals variable which we would call a "multivariate analysis" situation or all marginals fixed exact in one direction which we would call 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 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. The hypetheses discussed in the two different situations are the exact analogues (for categorical data) of the usual hypotheses of multivariate analysis and analysis of variance for normal variate data.
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