Lack-of-Fit Tests for Generalized Linear Models via Splines

Abstract

Cubic B-splines are used to estimate the nonparametric component of a semiparametric generalized linear model. A penalized log-likelihood ratio test statistic is constructed for the null hypothesis of the linearity of the nonparametric function. When the number of knots is fixed, its limiting null distribution is the distribution of a linear combination of independent chi-squared random variables, each with one df. The smoothing parameter is determined by giving a specified value for its asymptotically expected value under the null hypothesis. A simulation study is conducted to evaluate its power performance; a real-life dataset is used to illustrate its practical use.

Keywords:

• B-spline
• Generalized linear model
• Generalized partially linear model
• Penalized log-likelihood ratio test
• Semiparametric generalized linear model

Mathematics Subject Classification:

• 62G05
• 62G08
• 62G10
• 62G20

Acknowledgments

The author is very grateful to the Editor and a referee whose helpful comments improved the presentation of this article. The project described was supported by the National Center for Advancing Translational Sciences (NCATS) and the National Institutes of Health (NIH) through grant UL1, TR000002.