Order-Restricted Score Parameters in Association Models for Contingency Tables

原文摘要

The row effects and column effects models for two-way contingency tables have parameters for the row and column categories pertaining to the association between the variables. For classifications having ordered categories, it is often reasonable to assume that the association parameters have a corresponding ordering. This article proposes order-restricted estimates of the association parameters in these models. The maximum likelihood solution can be determined by the solution of a simple isotonic regression of some of the model sufficient statistics. The models are primarily log-linear in form and can be expressed in terms of odds ratios for 2×2 subtables consisting of adjacent rows and adjacent columns. For the order-restricted solution, these local log-odds ratios have uniform sign. Goodness-of-fit statistics for this solution are related to corresponding statistics for collapsed tables and to statistics for testing equality of sets of the parameters. The row effects model discussed in this article has been proposed by Haberman (1974), Simon (1974), and Goodman (1979), among others. This model contains parameters for the rows in the contingency table that describe the structure of the association and can be used to describe dependence in corresponding logit models. This article deals with applications of the model in which there is a monotonic relationship between the variables, in the sense that the population values of the local log-odds ratios are uniformly nonnegative or uniformly nonpositive. For instance, one might expect a nonnegative relationship for the data analyzed by Haberman (1974) and by Goodman (1979) on mental health and socioeconomic status, and a nonpositive relationship for the data analyzed in Section 2 of this article on age and severity of disturbances in dreams. By using the methods described in this article, one can obtain monotone estimates of the association parameters, which imply a monotone relationship between the variables. With this approach, one obtains a simpler description of the relationship, and better estimates, when the parameter scores truly are ordered.

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