Analysis of Variance in Nonparametric Regression Models

Abstract

In a nonparametric regression model with a multivariate explanatory variable we consider the problem of testing the hypothesis that specific interactions in a canonical decomposition of the model vanish. A simple consistent test is developed which is based on the difference between the corresponding sum of squares of the ordinary ANOVA for the multi-factor case and a nonparametric variance estimator. This presents an analogue of the classical ANOVA in the nonparametric regression setup. Asymptotic normality of the introduced test statistic is derived under the null hypothesis and under fixed alternatives. The finite sample behaviour of the proposed procedures is illustrated by a small simulation study and a data example. 2000Academic Press AMS 1991 subject classifications: 62G010, 62G020.

Optimal testing for additivity in multiple nonparametric regression

“…(1,2) n (with order l = 1) test of Dette and Derbort (2001), the M 2,α (with order l = 2) test of Derbort et al (2002), and the T 1n , T 2n , T 3n and T 4n tests of Dette et al (2005). The V d and V f tests are based on truncated Fourier series estimation of the interactive term and its further testing for the significant difference from zero.…”

“…Although early work dates back to Tukey (1949), it is only recently that the problem of testing for additivity has been of real interest. We refer to Barry (1993), Eubank et al (1995), Gozalo and Linton (2001), Dette and Derbort (2001), Dette and Wilkau (2001), Derbort et al (2002), Sperlich et al (2002), Li et al (2003), De Canditiis and Sapatinas (2004) and Dette et al (2005) for various approaches to testing for additivity in the nonparametric regression models. Despite a growing number of works, almost none of them investigated the optimality of the proposed additivity tests.…”

SiZer Map for inference with additive models

“…Testing for interactions has been studied only recently in papers such as Sperlich et al (2002) by considering marginal integration and Roca-Pardiñas et al (2005) for GAMs. Also, in a very general context, testing higher order interactions was considered by Dette and Derbort (2001).…”

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

“…The underlying idea is based on the sparsity of the underlying signal's representation in the Fourier or wavelet domains that allows significant reduction in dimensionality. Somewhat similar approaches were considered in Eubank 21 and Dette & Derbort 15 . Recently, Guo 28 suggested a maximum likelihood ratio based test for functional variance components in mixed-effects FANOVA models.…”