Independence in multi-way contingency tables: S. N. Roy's breakthroughs and later developments

In the mid-1950s S.N. Roy and his students contributed two landmark articles to the contingency table literature [Roy, S.N., Kastenbaum, M.A., 1956. On the hypothesis of no “interaction” in a multiway contingency table. Ann. Math. Statist. 27, 749–757; Roy, S.N., Mitra, S.K., 1956. An introduction to some nonparametric generalizations of analysis of variance and multivariate analysis. Biometrika 43, 361–376]. The first article generalized concepts of interaction from 2 × 2 × 2 contingency tables to threeway tables of arbitrary size and to larger tables. In the second article, which is the source of our primary focus, various notions of independence were clarified for three-way contingency tables, Roy’s union–intersection test was applied to construct chi-squared tests of hypotheses about the structure of such tables, and the chi-squared statistics were shown not to depend on the distinction between response and explanatory variables. This work pre-dates by many years later developments that expressed such results in the context of loglinear models. It pre-dates by a quarter century the development of graphical models. We summarize the main results in these key articles and discuss the connection between them and the later developments of loglinear modeling and of graphical modeling. We also mention ways in which these later developments have themselves been further generalized.