Varying-coefficient models with categorical effect modifiers are considered within the framework of generalized linear models. We distinguish between nominal and ordinal effect modifiers, and propose adequate Lasso-type regularization techniques that allow for (1) selection of relevant covariates, and (2) identification of coefficient functions that are actually varying with the level of a potentially effect modifying factor. For computation, a penalized iteratively reweighted least squares algorithm is presented. We investigate large sample properties of the penalized estimates; in simulation studies, we show that the proposed approaches perform very well for finite samples, too. In addition, the presented methods are compared with alternative procedures, and applied to real-world data.
BondellHDReichBJ (2009) Simultaneous factor selection and collapsing levels in ANOVA. Biometrics, 65, 169–77.
2.
EfronBHastieTJohnstoneITibshiraniR (2004) Least angle regression. The Annals of Statistics, 32, 407–99.
3.
FahrmeirLKaufmannH (1985) Consistency and asymptotic normality of the maximum likelihood estimator in generalized liner models. The Annals of Statistics, 13, 342–68.
4.
FanJLiR (2001) Variable selection via nonconcave penalized likelihood and its oracle properties. Journal of the American Statistical Association, 96, 1348–60.
5.
FanJZhangW (1999) Statistical estimation in varying coefficient models. The Annals of Statistics, 27, 1491–518.
6.
GertheissJTutzG (2010) Sparse modelling of categorial explanatory variables. Annals of Applied Statistics, 4, 2150–80.
7.
GertheissJTutzG (2012) Regularization and model selection with categorial effect modifiers. Statistica Sinica, 22, 957–82.
8.
HastieTTibshiraniR (1993) Varying-coefficient models. Journal of the Royal Statistical Society. Series B, Statistical methodology, 55, 757–96.
HofnerBHothornTKneibT (2012) Variable selection and model choice in structured survival models. Computational Statistics, 28, 1079–101.
11.
HooverDRRiceJAWuCOYangL-P (1998) Nonparametric smoothing estimates of time-varying coefficient models with longitudinal data. Biometrika, 85, 809–22.
12.
KauermannGTutzG (2000) Local likelihood estimation in varying-coefficient models including additive bias correction. Journal of Nonparametric Statistics, 12, 343–71.
13.
KochI (1996) On the asymptotic performance of median smoothers in image analysis and nonparametric regression. The Annals of Statistics, 24, 1648–66.
14.
LengC (2009) A simple approach for varying-coefficient model selection. Journal of Statistical Planning and Inference, 139, 2138–46.
15.
LinYZhangHH (2006) Component selection and smoothing in multivariate nonparametric regression. The Annals of Statistics, 34, 2272–97.
16.
LuYZhangRZhuL (2008) Penalized spline estimation for varying-coefficient models. Communications in Statistics - Theory and Methods, 37, 2249–61.
17.
McCullaghP (1983) Quasilikelihood functions. The Annals of Statistics, 11, 59–67.
18.
Meier-DinkelLTrautmannJFriedenLTholenEKnorrCSharifiARBückingMWickeMMörleinD (2013) Consumer perception of boar meat as affected by labelling information, malodorous compounds and sensitivity to androstenone. Meat Science, 93, 248–56.
19.
MörleinDGraveASharifiARBückingMWickeM (2012) Different scalding techniques do not affect boar taint. Meat Science, 91, 435–40.
20.
OelkerM-R (2013) gvcm.cat: regularized categorial effects/categorial effect modifiers in GLMs. R package version 1.5.
21.
OelkerM-RTutzG (2013) A general family of penalties for combining differing types of penalties in generalized structured models. Department of Statistics: Technical Report 139, http://epub.ub.uni-muenchen.de/14735/.
22.
R Development Core Team (2012) R: a language and environment for statistical computing. Vienna, Austria, ISBN 3-900051-07–0.
23.
TibshiraniR (1996) Regression shrinkage and selection via the LASSO. Journal of the Royal Statistical Society. Series B, Statistical methodology, 58, 267–88.
24.
TibshiraniRSaundersMRossetSZhuJKnightK (2005) Sparsity and smoothness via the fused LASSO. Journal of the Royal Statistical Society. Series B, Statistical methodology, 67, 91–108.
25.
UlbrichtJ (2010) Variable selection in generalized linear models. Dissertation, Department of Statistics, Ludwig-Maximilians-Universität München: Verlag Dr. Hut.
26.
WangHXiaY (2009) Shrinkage estimation of the varying coefficient model. Journal of the American Statistical Association, 104, 747–57.
27.
WangLLiHHuangJZ (2008) Variable selection in nonparametric varying-coefficient models for analysis of repeated measurements. Journal of the American Statistical Association, 103, 1556–69.
28.
WuCOHooverDR (1998) Asymptotic confidence regions for kernel smoothing of a varying-coefficient model with longitudinal data. Journal of the American Statistical Association, 93, 1388–89.
29.
YuanMLinY (2006) Model selection and estimation in regression with grouped variables. Journal of the Royal Statistical Society. Series B, Statistical methodology, 68, 49–67.
30.
ZouH (2006) The adaptive LASSO and its oracle properties. Journal of the American Statistical Association, 101, 1418–29.
