We propose a simple and fast approach to testing polynomial regression versus a general nonparametric alternative modeled by penalized splines. For the construction of the test we exploit novel results on simultaneous confidence bands using the approximation to the tail probability of maxima of Gaussian processes by the volume-of-tube formula (see Krivobokova et al., 2010, and Sun, 1993).
In simulations we show that the proposed test not only performs competitively compared to restricted likelihood ratio tests (RLRT, see Crainiceanu et al., 2005), but also allows to incorporate smooth curves that enter an additive model, are spatially heterogeneous (see Krivobokova et al., 2008) and are estimated from heteroscedastic data. Moreover, the test can also be used for investigating the statistical significance of certain features in a curve, such as dips and bumps. Further advantages include very good small sample properties and the analytical availability, i.e. no computationally intensive procedures such as bootstrapping are necessary.
We apply the method to analyze determinants of child undernutrition in Kenya with special interest in the investigation of catch-up growth as well as in possible nonlinearities in the effects of the mother's BMI and height. The method is implemented in the R package AdaptFitOS, making it readily available for practitioners.
Crainiceanu, C., Ruppert, D., Claeskens, G., and Wand, M. (2005). Exact likelihood ratio tests for penalised splines. Biometrika, 92(1):91.
Krivobokova, T., Crainiceanu, C., and Kauermann, G. (2008). Fast adaptive penalized splines. Journal of Computational and Graphical Statistics, 17(1):1-20.
Krivobokova, T., Kneib, T., and Claeskens, G. (2010). Simultaneous confidence bands for penalized spline estimators. Journal of the American Statistical Association, 105(490):852-863.
Sun, J. (1993). Tail probabilities of the maxima of Gaussian random fields. The Annals of Probability, 21(1):34-71.
Keywords: Simultaneous confidence band; Lack-of-fit test; Mixed model representation of penalized splines
Biography: Manuel Wiesenfarth is a PhD student at the Courant research Centre “Poverty, Equity and Growth in Developing and Transition countries” in Göttingen, Germany. Before he started his PhD he studied statistics in Munich. The focus of his thesis are inferential methods for nonparametric regression via penalized splines and their application to topics in development economics.