In the present article, we shall propose a definition of the $r$th order interactions in a $m$-dimensional $d_1 \times d_2 \times \cdots \times d_m$ contingency table ($r = 0, 1, 2, \cdots, m - 1)$, and we shall present methods for testing the hypothesis that any specified subset of these interactions is equal to zero. In addition, we shall present simple methods for obtaining simultaneous confidence intervals for these interactions or for any specified subset of them. In the special case where the $m$-dimensional contingency table is a $2 \times 2 \times \cdots \times 2$ table (i.e., where $d_i = 2$ for $i = 1, 2, \cdots, m$), the $r$th order interactions defined herein are the same as Good's interactions , but the tests proposed by Good are different even in this case from those presented herein. When $d_i > 2$ for some values of $i$, Good's interactions are complex valued, whereas the interactions presented here are real valued. We shall show herein that the hypothesis $H_r$ that all $r$th order and higher-order complex interactions (defined by Good) are equal to zero is equivalent to the hypothesis $H^\ast_r$ that all $r$th order and higher-order real interactions (defined herein) are equal to zero, and that the test of $H_r$ within $H_s (r < s)$ presented by Good is asymptotically equivalent (under $H_r$) to the test of $H^\ast_r$ within $H^\ast_s$ presented herein. The tests presented herein are, in some cases, easier to apply than Good's tests. In addition, the methods presented herein are applicable to a wider range of problems in the sense that Good's methods can be used to test the null hypothesis that all $r$th order and higher-order interactions are equal to zero, whereas the methods presented herein can be used to test the more general null hypothesis that any specified subset of these interactions is equal to zero. The tests presented herein are generalizations of methods proposed earlier by Plackett  and Goodman  for testing the null hypothesis $H^\ast_2$ in a three-dimensional table. The test proposed by Good  is a generalization of the methods proposed earlier by Bartlett , Roy and Kastenbaum , and Darroch  for testing $H^\ast_2$ in the three-dimensional table. All of these earlier papers were concerned mainly with the testing of null hypotheses. In the present article, in addition to our treatment of hypothesis testing, we shall also present two different methods for obtaining confidence intervals for the $r$th order real interactions in the $m$-dimensional contingency table $(r = 0, 1, 2, \cdots, m - 1)$.
"Interactions in Multidimensional Contingency Tables." Ann. Math. Statist. 35 (2) 632 - 646, June, 1964. https://doi.org/10.1214/aoms/1177703561