Estimation and inference concerning ordered means in analysis of covariance models with interactions

When interactions are identified in analysis of covariance models it becomes important to identify values of the covariates for which there are significant differences or, more generally, significant contrasts among the group mean responses. Inferential procedures that incorporate a priori order restrictions among the group mean responses would be expected to be superior to those that ignore this information. In this paper we focus on analysis of covariance models with pre-specified order restrictions on the mean response across the levels of a grouping variable when the grouping variable may interact with model covariates. In order for the restrictions to hold in the presence of interactions, it is necessary to impose the requirement that the restrictions hold over all levels of interacting categorical covariates and across pre-specified ranges of interacting continuous covariates. The parameter estimation procedure involves solving a quadratic programming minimization problem with a carefully specified constraint matrix. Simultaneous confidence intervals for treatment group contrasts and tests for equality of the ordered group mean responses are determined by exploiting previously unconnected literature. The proposed methods are motivated by a clinical trial of the dopamine agonist pramipexole for the treatment of early-stage Parkinson's disease.