An F-type test for detecting departure from monotonicity in a functional linear model

Abstract

When studying associations between a functional covariate and scalar response using a functional linear model (FLM), scientific knowledge may indicate possible monotonicity of the unknown parameter curve. In this context, we propose an F-type test of monotonicity, based on a full versus reduced nested model structure, where the reduced model with monotonically constrained parameter curve is nested within an unconstrained FLM. For estimation under the unconstrained FLM, we consider two approaches: penalised least-squares and linear mixed model effects estimation. We use a smooth then monotonise approach to estimate the reduced model, within the null space of monotone parameter curves. A bootstrap procedure is used to simulate the null distribution of the test statistic. We present a simulation study of the power of the proposed test, and illustrate the test using data from a head and neck cancer study.

Keywords:

• monotonicity
• functional data analysis
• functional linear model
• hypothesis testing

AMS Subject Classifications:

• 62G05
• 62G08
• 62G10

Acknowledgments

We would like to thank the three referees, the Associate Editor, and the Editor for their valuable comments and suggestions that significantly improved the paper. The authors express their gratitude to Matthew Schipper, Jeremy M. G. Taylor, and Xihong Lin for sharing the head and neck cancer data.

Disclosure statement

No potential conflict of interest was reported by the authors.

Supplemental data

Supplemental data for this article can be accessed doi:10.1080/10485252.2016.1163352.

ORCID

Eduardo L. Montoya https://orcid/org/0000-0001-5040-5063

Wendy Meiring https://orcid/org/0000-0002-5843-6662

Additional information

Funding

This research was supported in part by NSF [grant number HRD-1137774].