Permutation tests for the equality of covariance operators of functional data with applications to evolutionary biology

TL;DR

This paper extends permutation tests for covariance equality to multiple functional data samples, combining pairwise comparisons with non-parametric methods, and demonstrates its effectiveness through simulations and biological data applications.

Contribution

It introduces a new permutation test for multiple groups' covariance operators using non-parametric combination, with multiple testing adjustments and practical applications.

Findings

The new test effectively detects covariance differences in simulations.

It controls for multiple testing using step-down procedures.

Applied to mouse activity data, it identified significant covariance differences.

Abstract

In this paper, we generalize the metric-based permutation test for the equality of covariance operators proposed by Pigoli et al. (2014) to the case of multiple samples of functional data. To this end, the non-parametric combination methodology of Pesarin and Salmaso (2010) is used to combine all the pairwise comparisons between samples into a global test. Different combining functions and permutation strategies are reviewed and analyzed in detail. The resulting test allows to make inference on the equality of the covariance operators of multiple groups and, if there is evidence to reject the null hypothesis, to identify the pairs of groups having different covariances. It is shown that, for some combining functions, step-down adjusting procedures are available to control for the multiple testing problem in this setting. The empirical power of this new test is then explored via simulations and compared with those of existing alternative approaches in different scenarios. Finally, the proposed methodology is applied to data from wheel running activity experiments, that used selective breeding to study the evolution of locomotor behavior in mice.