Log-Linear Models for Frequency Tables with Ordered Classifications

Contingency tables with ordered classifications frequently arise in statistical practice. Various methods for examining such tables are explored by Yates [1948], Armitage [1955], Mantel [1963], Kendall and Stuart ([1967] p.p. 562-78), and Williams and Grizzle [1972]. These techniques generally involve attempts to assign scores to the categories of the table and then analyze the data as if they were continuous. An alternate approach which exploits the actual distributional structure of these tables may be made by use of the general loglinear models discussed by Haberman [1973]. The present paper considers a class of log-linear models appropriate for the analysis of r X c contingency tables with ordered row and column categories. Methods are presented for selection of models, computation of MVL estimates, performance of likelihood ratio or Pearson chi-square tests, and calculation of asymptotic variances. Although the discussion in this paper is confined to two-way contingency tables, no difficulty exists in applying the methods advocated here to m-way contingency tables.