Ordered Regressions

Abstract

There are often situations where two or more regression functions are ordered over a range of covariate values. In this paper, we develop efficient constrained estimation and testing procedures for such models. Specifically, necessary and sufficient conditions for ordering generalized linear regressions are given and shown to unify previous results obtained for simple linear regression, for polynomial regression and in the analysis of covariance models. We show that estimating the parameters of ordered linear regressions requires either quadratic programming or semi-infinite programming, depending on the shape of the covariate space. A distance-type test for order is proposed. Simulations demonstrate that the proposed methodology improves the mean square error and power compared with the usual, unconstrained, estimation and testing procedures. Improvements are often substantial. The methodology is extended to order generalized linear models where convex semi-infinite programming plays a role. The methodology is motivated by, and applied to, a hearing loss study.Key words: constrained inference, quadratic programming, semi-infinite programming Theoretical foundationThe development of estimation and testing procedures for ordered regressions requires the reformulation of the hypotheses (2) in terms of the parameters of the model (1). It turns © 2017 Board of the Foundation of the Scandinavian Journal of Statistics. Scand J Statist 44Ordered regressions 819 out that the shape of the covariate space X plays an important role in determining the constraints required for ordering regressions. In the following, necessary and sufficient conditions for ordering regressions are provided. These are followed by a discussion of implications, applications and some specific examples.

On the Design of Experiments with Ordered Treatments

“…(), Davidov et al . () and Rosen and Davidov (). A general proof of the superiority of the restricted likelihood ratio test (RLRT) compared with the unrestricted LRT (ULRT) can be found in Praestgaard () and Davidov and Iliopoulos ().…”

“…Researchers specializing in ordered inference have long known that incorporating constraints on the model parameters improves the precision of the estimators, as measured by their mean-squared errors (MSEs), and increases the power of the associated tests. For some examples see Davidov and Herman (2012), Farnan et al (2014), Davidov et al (2014) and Rosen and Davidov (2017). A general proof of the superiority of the restricted likelihood ratio test (RLRT) compared with the unrestricted LRT (ULRT) can be found in Praestgaard (2012) and Davidov and Iliopoulos (2019).…”