The analysis of contingency tables under inequality constraints

Abstract

This article surveys ways of taking order restrictions into account in the analysis of contingency tables. The methods are best suited for tables in which at least one classification is ordinal. Primary topics include (1) inequality constraints for parameters for a set of independent binomial random variables, (2) inequality constraints for odds ratios, (3) inequality constraints for parameters in loglinear or logit models, and (4) structured terms such as linear trends in loglinear and logit models. We discuss methods for contingency tables with two rows or two columns that result from comparing several binomial parameters or two multinomial distributions, two-way contingency tables with arbitrary numbers of rows and columns, multi-way contingency tables in which two-way analyses are relevant within levels of a stratification factor, and square contingency tables having the same categories in each dimension. We discuss the advantages and disadvantages of various order-restricted approaches and highlight possible problems for future research.