A powerful test for comparing multiple regression functions

Abstract

In this article, we address the important problem of comparison of two or more population regression functions. Recently, Pardo-Fernández, Van Keilegom and González-Manteiga [2007, ‘Testing for Equality of k Regression Curves’, Statistica Sinica, 17, 1115–1137] developed test statistics for simple nonparametric regression models: Y _ij =θ _j (Z _ij )+σ _j (Z _ij )ε _ij , based on empirical distributions of the errors in each population j=1, …, J. In this article, we propose a test for equality of the θ _j (·) based on the concept of generalised likelihood ratio type statistics. We also generalise our test for other nonparametric regression set-ups, for example, nonparametric logistic regression, where the log-likelihood for population j is any general smooth function ℒ{Y _j , θ _j (Z _j )}. We describe a resampling procedure to obtain the critical values of the test. In addition, we present a simulation study to evaluate the performance of the proposed test and compare our results to those in Pardo-Fernández et al. [2007, ‘Testing for Equality of k Regression Curves’, Statistica Sinica, 17, 1115–1137].

Keywords:

• bootstrap
• comparison of regression functions
• generalised likelihood ratio
• kernel regression
• local likelihood
• nonparametric regression

Acknowledgements

This research was partly supported by Award Number R00ES017744 from the National Institute of Environmental Health Sciences. The content is solely the responsibility of the authors and does not necessarily represent the official views of the National Institute of Environmental Health Sciences or the National Institutes of Health. I am also grateful to an anonymous referee for their careful evaluation of the article and constructive comments that lead to a significantly improved version of the article.