In a situation in which the observations are frequencies in a multi-way contingency table such that the observations are supposed to be independent and it is only the total number that is supposed to be fixed from sample to sample, a hypothesis on the structure of the probabilities in the different cells or categories is put forward. This hypothesis, by a certain analogy with the customary terminology of analysis of variance, is defined to be the hypothesis of "no interaction" and a large sample test of this hypothesis in terms of $\chi^2$ is offered. Bartlett's results  for the case of a $2 \times 2 \times 2$ table and Norton's results  for the case of a $2 \times 2 \times t$ table formally turn out to be special cases of the results given here with these differences; (i) Bartlett's and Norton's results refer to "analysis of variance" situations, with marginal frequencies along at least two ways of the table being fixed, while in this paper, for reasons explained elsewhere , it is only the total $n$ that is held fixed. (ii) Bartlett's and Norton's papers do not give any indication of the mechanism behind the formulae for the hypothesis of "no interaction," while this paper attempts to give a definite mathematical (and perhaps also physical) mechanism behind the formulae.
"On the Hypothesis of No "Interaction" In a Multi-way Contingency Table." Ann. Math. Statist. 27 (3) 749 - 757, September, 1956. https://doi.org/10.1214/aoms/1177728181