Diagnostics for the Scale of Functional Predictors in Generalized Linear Models

We develop diagnostics for fixed-effects generalized linear models (GLMs) with functional predictors. As a first step, we extend the GLM to allow a functional predictor to depend on a vector of parameters. A power transformation and a cubic spline transformation are the two primary models considered. We discuss methods for fitting the models, then derive the score test and a graphical tool for assessing the need for a transformation in models with functional predictors. These methods are natural extensions of the Box–Tidwell score test and constructed residual plots used to examine nonlinearities in classical GLMs. We consider both penalized and nonpenalized maximum likelihood methods and use two data sets to illustrate the methods.

[1]  D. Ruppert,et al.  Penalized Spline Estimation for Partially Linear Single-Index Models , 2002 .

[2]  Calyampudi R. Rao.  Large sample tests of statistical hypotheses concerning several parameters with applications to problems of estimation , 1948, Mathematical Proceedings of the Cambridge Philosophical Society.

[3]  D. Boos.  On Generalized Score Tests , 1992 .

[4]  B. Silverman,et al.  Functional Data Analysis , 1997 .

[5]  B. Silverman,et al.  Nonparametric Regression and Generalized Linear Models: A roughness penalty approach , 1993 .

[6]  B. Silverman,et al.  Nonparametric Regression and Generalized Linear Models: A roughness penalty approach , 1993 .

[7]  N. Altman,et al.  Variation in height acceleration in the Fels growth data , 1994 .

[8]  Franklin A. Graybill,et al.  Extensions of the General Linear Hypothesis Model , 1970 .

[9]  R. C. V. Nostrand,et al.  Transformations and Weighting in Regression , 1990 .

[10]  R. Todd Ogden,et al.  Functional regression in crop lodging assessment with digital images , 2002 .

[11]  William N. Venables,et al.  Modern Applied Statistics with S-Plus. , 1996 .

[12]  David J. Ostry,et al.  Functional data analyses of lip motion , 1995 .

[14]  T. Hassard,et al.  Applied Linear Regression , 2005 .

[15]  C. R. Rao,et al.  Linear Statistical Inference and its Applications , 1968 .

[16]  Gareth M. James.  Generalized linear models with functional predictors , 2002 .

[17]  P. McCullagh,et al.  Generalized Linear Models , 1992 .

[18]  Efstathia Bura,et al.  Two approaches to mutation detection based on functional data. , 2002, Statistics in medicine.

[19]  Gillian Z Heller,et al.  Functional data analysis with application to periodically stimulated foetal heart rate data. II: Functional logistic regression , 2002, Statistics in medicine.

[20]  Gillian Z Heller,et al.  Functional data analysis with application to periodically stimulated foetal heart rate data. I: Functional regression , 2002, Statistics in medicine.

[21]  Malik Beshir Malik,et al.  Applied Linear Regression , 2005, Technometrics.

[22]  S. R. Searle,et al.  Generalized, Linear, and Mixed Models , 2005 .