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 $F(\cdot ,\cdot )$ 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.

Appendix

1.1 Form of the one variance component RLRT statistic distribution

For the linear mixed model $\varvec{y}=\mathbb {X}{\varvec{\beta }}+\mathbb {Z}\varvec{b},\ \varvec{b}\sim N(\mathbf {0},\sigma _1^2\mathbb {I}_{q_1})$,

${\varvec{\varepsilon }}\sim N(\mathbf {0},\sigma ^2\mathbb {I})$, the restricted likelihood test statistic has distribution

$$\begin{aligned}&\text {RLRT}=\mathop {2\sup \ell _R(\varvec{y})}\limits _{H_1}-\mathop {2\sup \ell _R(\varvec{y})}\limits _{H_0}\nonumber \\&\qquad \!=\!\mathop {\sup }\limits _{\lambda }\left[ (N\!-\!q_0\!-\!1)\log \{1\!+\!U_n(\lambda )\}\!-\!\sum \limits _{k\!=\!1}^{q_1}\log (1\!+\!\lambda \mu _{k})\right] ;\nonumber \\ \end{aligned}$$

(7)

where $\lambda =\sigma _1^2/\sigma ^2$ and $U(\lambda )=N(\lambda )/D(\lambda )$ with

$$\begin{aligned} N(\lambda )&= \sum \limits _{k=1}^{q_1}\frac{\lambda \mu _k}{1+\lambda \mu _k}w _k^2,\\ D(\lambda )&= \sum \limits _{k=1}^{q_1}\log (1+\lambda \mu _k)+\sum \limits _{k=q_1+1}^{N-q_0}w _k^2; \end{aligned}$$

for $w _k\mathop {\sim }\limits ^{{\text{ i.i.d. }}}N(0,1);\ k=1,\ldots ,N-q_0-1$ (Crainiceanu and Ruppert 2004). The $\mu _k$ are the eigenvalues of the matrix $\mathbb {Z}^T(\mathbb {I}_N-\mathbb {X}(\mathbb {X}^T\mathbb {X})\mathbb {X}^T)\mathbb {Z}$. The eigendecompostion of the $q_1\times q_1$ matrix need only be computed once, and then all that is required to obtain a draw from the RLRT distribution is simulation of $q_1$ $\chi ^2_1$ random variables and one $\chi ^2_{N-q_0-q_1-1}$ random variable.