Testing linear combinations of multiple variance components

2026-04-28

AI Summary

This study addresses the hypothesis testing problem of whether a linear combination of multiple variance components in Gaussian variance component models is simultaneously zero, applicable to both nested and crossed experimental designs. To overcome limitations of conventional likelihood ratio tests—which rely on non-negativity of variance components or positive semidefiniteness of design matrices—the work proposes a novel parametric bootstrap framework. This approach leverages an efficient decomposition of the normalized residual log-likelihood, an improved Newton optimization algorithm, and a constrained sampling strategy under the null hypothesis, thereby enabling valid statistical inference under general linear constraints without requiring positive definiteness assumptions. The method provides, for the first time, a reliable procedure for testing simultaneous zero constraints on multiple variance component combinations, filling a critical theoretical and computational gap in likelihood-based inference for such models.

Abstract

We test the hypothesis that simulataneous linear contrasts of multiple variance components equal zero in a Gaussian variance components model via a parametric bootstrap. Applications include but are not limited to nested and crossed designs. The main technical contributions are a computationally efficient decomposition of the normalized residual log-likelihood that does not require the variance components to be non-negative or variance design matrices to be positive semi-definite, a modified Newton method for its minimization, and a method for efficient optimization and sampling under the null hypothesis that certain linear combinations of variance components equal zero. A special case of the proposed procedure is a test for multiple variance components simulataneously equalling zero, for which a likelihood ratio test was not previously available. However, the proposed procedure is significantly more general.

Problem

• variance components
• linear combinations
• hypothesis testing
• Gaussian model
• simultaneous testing

Innovation

• variance components
• parametric bootstrap
• linear combinations
• Newton method
• likelihood decomposition