Testing linear combinations of multiple variance components

TL;DR

This paper introduces a new, efficient statistical testing method for linear combinations of multiple variance components in Gaussian models, applicable to complex designs.

Contribution

It offers a novel, general approach for testing multiple variance components simultaneously, including a likelihood ratio test for zero variance components.

Findings

Provides a computationally efficient decomposition of the residual log-likelihood.

Develops a modified Newton method for minimization.

Enables testing of multiple variance components simultaneously, including cases not covered by existing tests.

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.