• Functional data;
  • Mixed effects model;
  • Reproducing kernel Hilbert space;
  • Smoothing spline analysis of variance;
  • Tensor product

Summary. Smoothing spline analysis of variance decomposes a multivariate function into additive components. This decomposition not only provides an efficient way to model a multivariate function but also leads to meaningful inference by testing whether a certain component equals 0. No formal procedure is yet available to test such a hypothesis. We propose an asymptotic method based on the likelihood ratio to test whether a functional component is 0. This test allows us to choose an optimal model and to compare groups of curves. We first develop the general theory by exploiting the connection between mixed effects models and smoothing splines. We then apply this to compare two groups of curves and to select an optimal model in a two-dimensional problem. A small simulation is used to assess the finite sample performance of the likelihood ratio test.