Quadratic Form based Multiple Contrast Tests for Comparison of Group Means

TL;DR

This paper introduces a novel quadratic form based multiple contrast test that combines the strengths of quadratic form tests and multiple contrast tests, with theoretical analysis and simulation validation.

Contribution

It develops a new combined testing approach that leverages quadratic forms and multiple contrast tests, with theoretical properties and improved small sample performance.

Findings

The proposed test has favorable asymptotic properties.

Monte-Carlo and resampling improve small sample performance.

Simulation shows advantages over existing methods.

Abstract

Comparing the mean vectors across different groups is a cornerstone in the realm of multivariate statistics, with quadratic forms commonly serving as test statistics. However, when the overall hypothesis is rejected, identifying specific vector components or determining the groups among which differences exist requires additional investigations. Conversely, employing multiple contrast tests (MCT) allows conclusions about which components or groups contribute to these differences. However, they come with a trade-off, as MCT lose some benefits inherent to quadratic forms. In this paper, we combine both approaches to get a quadratic form based multiple contrast test that leverages the advantages of both. To understand its theoretical properties, we investigate its asymptotic distribution in a semiparametric model. We thereby focus on two common quadratic forms - the Wald-type statistic and the Anova-type statistic - although our findings are applicable to any quadratic form. Furthermore, we employ Monte-Carlo and resampling techniques to enhance the test's performance in small sample scenarios. Through an extensive simulation study, we assess the performance of our proposed tests against existing alternatives, highlighting their advantages.