Testing order restrictions in contingency tables. (English) Zbl 1349.62231
Summary: Though several interesting models for contingency tables are defined by a system of inequality constraints on a suitable set of marginal log-linear parameters, the specific features of the corresponding testing problems and the related procedures are not widely well known. After reviewing the most common difficulties which are intrinsic to inequality restricted testing problems, the paper concentrates on the problem of testing a set of equalities against the hypothesis that these are violated in the positive direction and also on testing the corresponding inequalities against the saturated model; we argue that valid procedures should consider these two testing problems simultaneously. By reformulating and adapting procedures appeared in the econometric literature, we propose a likelihood ratio and a multiple comparison procedure which are both based on the joint distribution of two relevant statistics; these statistics are used to divide the sample space into three regions: acceptance of the assumed equality constraints, rejection towards inequalities in the positive direction and rejection towards the unrestricted model. A simulation study indicates that the likelihood ratio based procedure perform substantially better. Our procedures are applied to the analysis of two real data sets to clarify how they work in practice.
MSC:
| 62H17 | Contingency tables |
| 62H15 | Hypothesis testing in multivariate analysis |
| 62E20 | Asymptotic distribution theory in statistics |
| 62F30 | Parametric inference under constraints |
| 60E15 | Inequalities; stochastic orderings |
| 62P20 | Applications of statistics to economics |
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