On Plackett’s Test for Contingency Table Interactions

SUMMARY Based on a transformation matrix the rows of which are orthogonal to each other, a matrix that comes naturally to analysis of variance users, Plackett (1962) has proposed a test for zero second-order interaction in an r × s × t contingency table which requires that the user invert t +1 matrices, each having (r − 1) (s − 1) rows and (r −1) (s −1) columns. Based on a transformation matrix the rows of which are not orthogonal to each other, we present herein a modification of Plackett’s method of analysis which will require that the user invert only one matrix of side (r − 1) (s − 1) and t matrices each of side (u − 1), where u = min r, s. Thus, when u = 2, the user applying the method given here will compute the inverse of only one matrix of side (r — 1) (s − 1), while the user applying Plackett’s method will have to compute the inverse of t+ 1 such matrices (when max r, s 2). The test of zero second-order interaction presented herein will usually be easier to apply than the other valid tests of this null hypothesis appearing in the literature; its outcome is actually equivalent to Plackett’s test, which we note in turn is asymptotically equivalent (under the null hypothesis) to tests proposed by Bartlett (1935), Roy and Kastenbaum (1956), Darroch (1962), and Goodman (1963a).