Positive Quadrant Dependence and Marginal Modeling in Two-Way Tables With Ordered Margins

Abstract

For a collection of two-way tables, where subjects are cross-classified according to the same pair of ordinal categorical variables conditionally on the value of one or more discrete explanatory variables, we propose a general approach to likelihood inference that combines marginal modeling with fitting and testing of inequality constraints such as those implied by the assumption that one marginal distribution is stochastically larger than the other, positive dependence and stronger positive dependence. The approach is based on parameterizing bivariate conditional distributions with global logits and global log-odds ratios, and we provide a general framework for handling models defined by equality and inequality constraints on these parameters. In this way, such models as marginal homogeneity, proportional odds among row or columns margins, and Plackett distribution may be treated together with various models defined by inequality constraints on the same parameters, such as, for instance, those implied by the positive quadrant dependence. For this class of models, we define a Fisher scoring algorithm for computing maximum likelihood estimates and derive the asymptotic distribution of the likelihood ratio tests that turn out to be of the chi-bar squared type when inequalities are involved. When the main interest is on positive dependence, we derive tight bounds on the asymptotic distribution of these statistics that are independent of the marginal logits. These also may be removed by conditioning on one or both observed margins in each table, and we describe in detail how the approach may be adapted to these sampling schemes. Three applications to real datasets are discussed.

KEY WORDS:

• Chi-bar squared distribution
• Conditional inference
• Constrained Fisher scoring algorithm
• Marginal homogeneity
• Positive quadrant dependence
• Proportional odds model
• Stochastic ordering