Optimal Testing in a Fixed-Effects Functional Analysis of Variance Model

Abstract

We consider the testing problem in a fixed-effects functional analysis of variance model. We test the null hypotheses that the functional main effects and the functional interactions are zeros against the composite nonparametric alternative hypotheses that they are separated away from zero in L 2 -norm and also possess some smoothness properties. We adapt the optimal (minimax) hypothesis testing procedures for testing a zero signal in a Gaussian "signal plus noise" model to derive optimal (minimax) non-adaptive and adaptive hypothesis testing procedures for the functional main effects and the functional interactions. The corresponding tests are based on the empirical wavelet coefficients of the data. Wavelet decompositions allow one to characterize different types of smoothness conditions assumed on the response function by means of its wavelet coefficients for a wide range of function classes. In order to shed some light on the theoretical results obtained, we carry out a simulation study to examine the finite sample performance of the proposed functional hypothesis testing procedures. As an illustration, we also apply these tests to a real-life data example arising from physiology. Concluding remarks and hints for possible extensions of the proposed methodology are also given.

The effect of the spatial domain in FANOVA models with ARH(1) error term

“…(see equation (2) in p.279 in [60]). Also, in general, the eigenvalues of the Dirichlet negative Laplacian…”

“…From classical ANOVA tests, an asymptotic approach is derived in [17], for studying the equality of the functional means from k independent samples of functional data. The testing problem for mixed-effect functional analysis of variance models is addressed in [1] and [2], developing asymptotically optimal (minimax) testing procedures for the significance of functional global trend, and the functional fixed effects. The wavelet transform of the data is again used in the implementation of this approach (see also [4]).…”

A comparison of tests for the one-way ANOVA problem for functional data

“…These data sets are commonly used to illustrate the use of statistical methods for real functional data (see Abramovich et al 2004;Ramsay and Silverman 2005;Zhang 2013;Zhang and Liang 2014). The Canadian temperature data are available in the R package fda, and the orthosis data can be downloaded from "http://www.stat.nus.edu.sg/~zhangjt/books/Chapman/FANOVA.…”

“…As reported by Abramovich et al (2004), the orthosis data were acquired and computed in an experiment by Dr. Amarantini David and Dr. Martin Luc (Laboratoire Sport et Performance Motrice, EA 597, UFRAPS, Grenoble University, France). They investigated how muscle redundancy could be appropriately used to cope with an external perturbation while complying with the mechanical requirements related either to balance control and/or minimum energy expenditure.…”

Functional regression analysis using an F test for longitudinal data with large numbers of repeated measures

“…Other important work addressing the functional testing problem was provided by Fan and Lin [22], Eubank [23], and Abramovich et al [24], but they only considered ANOVA-type models and their test statistics were formed by orthogonal (Fourier or Wavelets) expansion coefficients of response curves. Eubank [23] proved that among different ways of combining the coefficients into a test statistic, the L 2 norm, a simple sum of the squared coefficients, is asymptotically equivalent to the uniformly most powerful test when the grid size m goes to infinity.…”