Statistica Sinica 15(2005), 235-256

TESTING GENERALIZED LINEAR MODELS USING

SMOOTHING SPLINE METHODS

Anna Liu$^1$, Wendy Meiring$^2$ and Yuedong Wang$^2$

$^1$University of Massachusetts, Amherst and

$^2$University of California, Santa Barbara

Abstract: This article considers testing the hypothesis of Generalized Linear Models (GLM) versus general smoothing spline models for data from exponential families. The tests developed are based on the connection between smoothing spline models and Bayesian models (Gu (1992)). They are extensions of the locally most powerful (LMP) test of Cox, Koh, Wahba and Yandell (1988), the generalized maximum likelihood ratio (GML) test and the generalized cross validation (GCV) test of Wahba (1990) for Gaussian data. Null distribution approximations are considered and simulations are done to evaluate these approximations. Simulations show that the LMP and GML tests are more powerful for low frequency functions while the GCV test is more powerful for high frequency functions, which is also true for Gaussian data (Liu and Wang (2004)). The tests are applied to data from the Wisconsin Epidemiology Study of Diabetic Retinopathy, the results of which confirm and provide more definite analysis than those of previous studies. The good performances of the tests make them useful tools for diagnosis of GLM.

Key words and phrases: Diagnosis, generalized cross validation, generalized maximum likelihood, locally most powerful test, hypothesis test, reproducing kernel hilbert space.