Restricted likelihood ratio tests for linearity in scalar-on-function regression

Abstract

We propose a procedure for testing the linearity of a scalar-on-function regression relationship. To do so, we use the functional generalized additive model (FGAM), a recently developed extension of the functional linear model. For a functional covariate X(t), the FGAM models the mean response as the integral with respect to t of F {X(t), t} where F (·, ·) is an unknown bivariate function. The FGAM can be viewed as the natural functional extension of generalized additive models. We show how the functional linear model can be represented as a simple mixed model nested within the FGAM. Using this representation, we then consider restricted likelihood ratio tests for zero variance components in mixed models to test the null hypothesis that the functional linear model holds. The methods are general and can also be applied to testing for interactions in a multivariate additive model or for testing for no effect in the functional linear model. The performance of the proposed tests is assessed on simulated data and in an application to measuring diesel truck emissions, where strong evidence of nonlinearities in the relationship between the functional predictor and the response are found.

Methods for Scalar‐on‐Function Regression

“…McLean et al. () treat the linear model ( as the null, to be tested versus the additive model , while García‐Portugués et al. ) consider testing against a more general alternative.…”

Classical testing in functional linear models

“…Recently McLean, Hooker, and Ruppert (2014) proposed a restricted likelihood ratio test for testing for linear dependence between a scalar response and a functional covariate, in the class of functional generalized additive models (Mclean, Hooker, Staicu, Scheipl, and Ruppert 2014; Müller, Wu, and Yao 2013). In what follows, we write ∫ X i ( t ) β ( t ) dt instead of ∫ 𝒯 X i ( t ) β ( t ) dt for notational convenience.…”

A Smoothing-Based Goodness-of-Fit Test of Covariance for Functional Data

“…Hardle and Mammen () propose a smoothing‐based goodness‐of‐fit statistic for regression functions, derive the asymptotic normal distribution, and develop a “wild” bootstrap algorithm for finite samples. Comparisons have also been applied to functional regression for model diagnostics and evaluating assumptions (Chiou and Muller, ; Bucher et al, ) and testing functional coefficients (Swihart et al, ; McLean et al, ; Kong et al, ). The proposed method is an extension of smoothing‐based methods to test the form of the covariance function.…”