Inference in contingency tables with ordered categories using Plackett's coefficient of association for bivariate distributions

Abstract

SUMMARY By using a one-parameter class of bivariate distributions proposed by Plackett (1965), methods are derived for analysing the pattern of association in contingency tables with ordered categories. In an r x c table (r - 1) (c - 1) estimates of this parameter can be con- sidered and all of them, or subsets of them, can be tested for equality. This can answer the question whether the pattern of association is consistently the same over the whole table or not. Two well-known examples are considered using this method. The problems arising in the analysis of contingency tables with ordered categories usually go further than mere tests for independence. Generally, a specific analysis of the pattern of association between the two categories is required. For this purpose, contingency tables with ordered categories can be considered as samples from bivariate distributions. Models for symmetry, quasisymmetry, marginal homogeneity or palindromic symmetry can be applied (Bishop, Fienberg & Holland, 1975, p. 281 ff.; McCullagh, 1978) in order to analyse such tables. It is the major aim of these analyses to evaluate differences between the marginal variables by taking into account the association between them. In the present paper the central emphasis is put on the association itself. We use a class of bivariate distributions proposed by Plackett (1965) which seems to be very useful in this context. This class of bivariate distributions, introduced in detail in ? 2, is characterized by one real-valued parameter representing a measure of association. This parameter is closely related to the cross-product ratio in 2 x 2 tables. In an r x c table this parameter can be estimated in (r - 1) (c - 1) different ways. We derive the asymptotic distribution for this set of estimates. This enables us to construct tests for the equality of all parameters estimated or certain subsets of them. By this means we can determine whether the structure of association between the two marginal random variables is consistently the same over the whole table. This would mean that the bivariate distribu- tion, incorporating the interaction of the two marginal variables, can be explained by a simple one-parameter model. The interpretation of the model depends on the numerical value of the parameter and ranges from negative association over independence to positive association. Regardless of the numerical values of the estimates, a rejection of the hypothesis of equality of the para- meters would indicate that there is an interaction between the marginal variables which