Interaction Tests for 2 × s × t Contingency Tables

SUMMARY The interaction test for 2 x 2 x t contingency tables of Zelen (1971) is extended to 2 x 8 x t tables. This test is related to the tests of partial association, or 'main effect', of Birch (1965) and Armitage (1966). These tests are frequency analogues of the log-frequency tests con- sidered by Plackett (1962) and Goodman (1964). The various analyses are illustrated and compared in several numerical examples. Bartlett (1935) considered a model and statistical analysis for interaction in 2 x 2 x 2 tables. Norton (1945) generalized these results to 2 x 2 x t tables and Roy & Kastenbaum (1956) gave the general r x s x t results. These authors used tests based on Pearson's chi- squared statistic with parameters fitted by iterative maximum likelihood methods. Woolf (1955), using the same concept of interaction, gave a much simpler, asymptotically equiva- lent test for 2 x 2 x t tables based on the logit transformation of the data. Plackett (1962) extended Woolf's result to the general three-dimensional table and Goodman (1964) showed how Plackett's test statistics might be more easily computed, particularly for 2 x s x t tables. For further discussion and references on the various models and analyses of three-dimensional contingency tables, see Lancaster (1969, Chapter XII) and Plackett (1969). More recently, various authors have discussed simplified iterative maximum likeli- hood estimation techniques in three-dimensional tables and Goodman (1970) has extended these results to m-dimensional tables. Zelen (1971) derived a conditional test for interaction in 2 x 2 x t tables based on the reference set of all fixed marginal totals. The asymptotic version of this test may be con- sidered the analogue, in the untransformed space, of Woolf's logit test. This paper extends Zelen's test to 2 x s x t tables; the resulting test is the close analogue, in the untransformed space, of Goodman's logit analyses for such tables. The structure of the test statistic is analogous to a sum of squares in an analysis of variance. The summed terms are related to the usual homogeneity chi-squares for the s 2 x t tables and the 'correction term' is Birch's (1965) test statistic for partial association, or 'main effect'.