The linear stochastic order and directed inference for multivariate ordered distributions

TL;DR

This paper introduces the linear stochastic order, a new framework for testing ordered hypotheses in multivariate data, generalizing classical tests and applicable to mixed data types, with practical application in toxicology studies.

Contribution

It develops a novel nonparametric methodology for estimating and testing the order between multivariate distributions, extending classical methods to more general settings.

Findings

Generalizes Roy's largest root test to nonparametric multivariate context

Applicable to discrete and continuous data components

Demonstrated on NTP rodent bioassay data

Abstract

Researchers are often interested in drawing inferences regarding the order between two experimental groups on the basis of multivariate response data. Since standard multivariate methods are designed for two-sided alternatives, they may not be ideal for testing for order between two groups. In this article we introduce the notion of the linear stochastic order and investigate its properties. Statistical theory and methodology are developed to both estimate the direction which best separates two arbitrary ordered distributions and to test for order between the two groups. The new methodology generalizes Roy's classical largest root test to the nonparametric setting and is applicable to random vectors with discrete and/or continuous components. The proposed methodology is illustrated using data obtained from a 90-day pre-chronic rodent cancer bioassay study conducted by the National Toxicology Program (NTP).