We propose a procedure for testing the linearity of a scalar-on-functionregression relationship. To do so, we use the functional generalized additivemodel (FGAM), a recently developed extension of the functional linear model.For a functional covariate X(t), the FGAM models the mean response as theintegral with respect to t of F{X(t),t} where F is an unknown bivariatefunction. The FGAM can be viewed as the natural functional extension ofgeneralized additive models. We show how the functional linear model can berepresented as a simple mixed model nested within the FGAM. Using thisrepresentation, we then consider restricted likelihood ratio tests for zerovariance components in mixed models to test the null hypothesis that thefunctional linear model holds. The methods are general and can also be appliedto testing for interactions in a multivariate additive model or for testing forno effect in the functional linear model. The performance of the proposed testsis assessed on simulated data and in an application to measuring diesel truckemissions, where strong evidence of nonlinearities in the relationship betweenthe functional predictor and the response are found.
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