Hypothesis testing for functional linear models

Functional data arise frequently in many biomedical studies, where it is often of interest to investigate the dynamic 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 the theoretical properties of the proposed test 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 apply the proposed method to the cerebral white matter tracts data obtained from a diffusion tensor imaging tractography study. In this application, classical tests with varying thresholds yield inconsistent conclusions, while the same conclusion can be drawn by applying the proposed approach with different thresholds. This demonstrates the robustness of the proposed approach, which makes it easy for practitioners to interpret the findings.