Hypothesis Testing in Functional Linear Models

Abstract

Summary Functional data arise frequently in biomedical studies, where it is often of interest to investigate the association between functional predictors and a scalar response variable. While functional linear models (FLM) are widely used to address these questions, hypothesis testing for the functional association in the FLM framework remains challenging. A popular approach to testing the functional effects is through dimension reduction by functional principal component (PC) analysis. However, its power performance depends on the choice of the number of PCs, and is not systematically studied. In this paper, we first investigate the power performance of the Wald-type test with varying thresholds in selecting the number of PCs for the functional covariates, and show that the power is sensitive to the choice of thresholds. To circumvent the issue, we propose a new method of ordering and selecting principal components to construct test statistics. The proposed method takes into account both the association with the response and the variation along each eigenfunction. We establish its theoretical properties and assess the finite sample properties through simulations. Our simulation results show that the proposed test is more robust against the choice of threshold while being as powerful as, and often more powerful than, the existing method. We then apply the proposed method to the cerebral white matter tracts data obtained from a diffusion tensor imaging tractography study.

Classical testing in functional linear models

“…It would be desirable to develop a method to select the number of basis functions, and it is an interesting topic for future research. As one referee pointed out, Su and Hsu (2016) developed a method to select the number of basis functions when studying the same testing problem presented in our paper.…”

Unified Tests for Nonparametric Functions in RKHS With Kernel Selection and Regularization

“…However, these methods were designed for a linear model and inapplicable to a general nonparametric function. When X i is considered as functional data, extensive studies have been done for hypothesis testing under various model settings, for example, under the functional linear model (e.g., Kong et al (2016); Su et al (2017)), under the generalized functional linear models (e.g., Shang and Cheng (2015); Li and Zhu (2020)) and under the consideration of nonparametric functions of functional covariates (e.g., Delsol et al (2011); Delsol (2012)). See Tekbudak et al (2019) for a recent review.…”

Nonparametric testing of lack of dependence in functional linear models

“…In parametric tests, the test statistics are usually established by first estimating the functional regression coefficient through dimension reduction, such as functional PCA [5,[6][7][8]. Methods for real-valued responses include [6], [7] and [8]. [6] used a test statistic based on the L 2 norm of the empirical crosscovariance operator of (X, Y).…”

“…[6] used a test statistic based on the L 2 norm of the empirical crosscovariance operator of (X, Y). [8] proposed a Wald-type test with varying thresholds in selecting the number of principal components. [7] developed four test statistics based on the functional principal component (FPC) scores.…”