Calculation of Chi-Square to Test the No Three-Factor Interaction Hypothesis

Koch’s Contributions to Statistical Practice, 1965–1985

“…Roy had carefully distinguished between probability models for multidimensional contingency tables based on which dimensions are considered fixed factors versus random responses; in a regression setting, these are, respectively, independent and dependent variables. Roy and Kastenbaum had worked extensively on hypotheses and tests of no interaction for three-way tables (Roy andKastenbaum 1955, 1956;Roy 1957;Kastenbaum and Lamphiear 1959).…”

“…The technology was applied to varied examples collectively encompassed by no prior analytic framework: testing second-order loglinear interaction in Bartlett's (1935) 2 3 data, and marginal homogeneity in Cochran's (1955) 2 3 drug comparison datasets and a 4 × 4 opthalmological comparison of paired left and right eyes; a two-factor ANOVA analog for an ordinal surgical sequel scored 0, 1, or 2; and a two-factor loglinear model for three-category litter depletion response data of Kastenbaum and Lamphiear (1959).…”

Simple Methods for Analyzing Three-Factor Interaction in Contingency Tables

“…More generally, for the 2 XJ XK tables we note that S2 given by (2.2.9), which is equal to gkM(k)gk given in (2.2.4), can be used to test the hypothesis that the J measures fjk (j= 1, 2, ... , J) do not differ significantly from each other (see Appendix A2). To apply Plackett's analysis [23 ], the user would invert K+ 1 (J -1) X (J -1) matrices, and to apply the BFNRKLD test he would solve the set of (J -1) (K-1) simultaneous third-degree equations given in Roy and Kastenbaum [24] and Kastenbaum and Lamphiear [19] or he would solve the JK+2(J+K) nonlinear equations given in Darroch [8]. If this hypothesis is rejected, the subtraction of g'Qg from R2 k=l [as in (2.2.8) ] will then test the hypothesis that the difference between the J measures flk (j= 1, 2, * * *, J) can be explained simply in terms of the differences between the J means fj*-=Ek Z jk/K (j= 1, 2, * , J); i.e., the hypothesis Ho that with respect to the J X K measures (Pjk the interaction in the J X K twoway layout is zero.…”

“…For the more general I X J X K three-way table, Roy and Kastenbaum [24 ] have extended the definition of zero three-factor interaction, and they have constructed a test of the hypothesis that this interaction is zero, a test which is identical to the tests given by Bartlett [2] and Norton [22] for the special case of the 2 X 2 X K table. To solve this system of equations, Kastenbaum and Lamphiear [19 ] have generalized Norton's iterative procedure and have adapted it for a high-speed computer. (When min [I, J, K] = 2, these equations are of the third-degree.)…”

“…The user applying Plackett's method of analysis will be required to invert K+ 1 matrices, each (J -1) X (J -1), while the user applying Goodman's method [14] will be required to invert only one such matrix. (For the 2 XJ XK table, the set of (J -1) (K-1) simultaneous thirddegree equations given by Roy and Kastenbaum [24] and Kastenbaum and Lamphiear [19], and the set of JK+2(J+K) non-linear equations given by Darroch [8], can be avoided; and so can the inversion of K of the K+1 matrices given by Plackett [23].) In the present article, we shall introduce a new and still simpler "approximate" test of Ho, which is an approximation to the method of analysis given earlier by Goodman [14], and we shall indicate here the conditions under which this approximation will be adequate.…”