Smoothing Parameter and Model Selection for General Smooth Models

ABSTRACT

This article discusses a general framework for smoothing parameter estimation for models with regular likelihoods constructed in terms of unknown smooth functions of covariates. Gaussian random effects and parametric terms may also be present. By construction the method is numerically stable and convergent, and enables smoothing parameter uncertainty to be quantified. The latter enables us to fix a well known problem with AIC for such models, thereby improving the range of model selection tools available. The smooth functions are represented by reduced rank spline like smoothers, with associated quadratic penalties measuring function smoothness. Model estimation is by penalized likelihood maximization, where the smoothing parameters controlling the extent of penalization are estimated by Laplace approximate marginal likelihood. The methods cover, for example, generalized additive models for nonexponential family responses (e.g., beta, ordered categorical, scaled t distribution, negative binomial and Tweedie distributions), generalized additive models for location scale and shape (e.g., two stage zero inflation models, and Gaussian location-scale models), Cox proportional hazards models and multivariate additive models. The framework reduces the implementation of new model classes to the coding of some standard derivatives of the log-likelihood. Supplementary materials for this article are available online.

KEYWORDS:

• Additive model
• AIC
• Distributional regression
• GAM
• Location scale and shape model
• Ordered categorical regression
• Penalized regression spline
• REML
• Smooth Cox model
• Smoothing parameter uncertainty
• Statistical algorithm
• Tweedie distribution.

Supplementary Materials

The online supplementary materials contain additional appendices for the article.

Acknowledgment

We thank the anonymous referees for a large number of very helpful comments that substantially improved the paper and Phil Reiss for spotting an embarrassing error in Supplementary Appendix A.

Funding

SNW and NP were funded by EPSRC grant EP/K005251/1 and NP was also funded by EPSRC grant EP/I000917/1. BS was funded by the German Research Association (DFG) Research Training Group “Scaling Problems in Statistics” (RTG 1644). SNW is grateful to Carsten Dorman and his research group at the University of Freiburg, where the extended GAM part of this work was carried out.