Direct generalized additive modeling with penalized likelihood. (English) Zbl 1042.62580

Summary: Generalized additive models (GAMs) have become an elegant and practical option in model building. Estimation of a smooth GAM component traditionally requires an algorithm that cycles through and updates each smooth, while holding other components at their current estimated fit, until specified convergence. We aim to fit all the smooth components simultaneously. This can be achieved using penalized B-spline or P-spline smoothers for every smooth component, thus transforming GAMs into the generalized linear model framework. Using a large number of equally spaced knots, P-splines purposely overfit each B-spline component. To reduce flexibility, a difference penalty on adjacent B-spline coefficients is incorporated into a penalized version of the Fisher scoring algorithm. Each component has a separate smoothing parameter, and the penalty is optimally regulated through extensions of cross validation or information criterion. An example using logistic additive models provides illustrations of the developments.


62J12 Generalized linear models (logistic models)
65C60 Computational problems in statistics (MSC2010)


Fahrmeir; FITPACK
Full Text: DOI


[1] De Boor, C.: A practical guide to splines. (1978) · Zbl 0406.41003
[2] Buja, A.; Hastie, T.; Tibshirani, R.: Linear smoothers and additive models. Ann statist. 17, 453-555 (1989) · Zbl 0689.62029
[3] Dierckx, P.: Curve and surface Fitting with splines. (1993) · Zbl 0782.41016
[4] Dobson, A. J.: An introduction to generalized linear models. (1990) · Zbl 0727.62074
[5] Eilers, P. H. C.; Max, B. D.: Flexible smoothing using B-splines and penalized likelihood (with comments and rejoinder). Statist. sci. 11, No. 2, 89-121 (1996) · Zbl 0955.62562
[6] Fahrmeir, L.; Tutz, G.: Multivariate statistical modelling based on generalized linear models. (1994) · Zbl 0809.62064
[7] Green, P. J.; Silverman, B. W.: Nonparametric regression and generalized linear models. (1994) · Zbl 0832.62032
[8] Hastie, T.; Tibshirani, R.: Generalized additive models. Statist. sci. 1, 297-318 (1986) · Zbl 0645.62068
[9] Hastie, T.; Tibshirani, R.: Generalized additive models. (1990) · Zbl 0747.62061
[10] Mccullagh, P.; Nelder, J. A.: Generalized linear models. (1989) · Zbl 0744.62098
[11] Hosmer, D. W.; Lemeshow, S.: Applied logistic regression. (1989) · Zbl 0715.62125
[12] Nelder, J. A.; Wedderburn, R. W. M.: Generalized linear models. J. roy. Statist. soc. A 135, 370-384 (1972)
[13] O’sullivan, F.: A statistical perspective on ill-posed inverse problems (with discussion). Statist. sci. 1, 505-527 (1986) · Zbl 0625.62110
[14] O’sullivan, F.: Fast computation of fully automated log-density and log-hazard estimators. SIAM J. Sci. statist. Comput. 9, 363-379 (1988) · Zbl 0688.65083
[15] Pregibon, D.: Logistic regression diagnostics. Ann. statist. 9, 705-724 (1981) · Zbl 0478.62053
[16] Stone, C. J.: The use of polynomial splines and their tensor products in multivariate function estimation. Ann. statist. 22, 118-184 (1994) · Zbl 0827.62038
[17] Stone, C. J.; Koo, K. Y.: Additive splines in statistics. Proc. statistical computing section of the American statistical association, 45-47 (1985)
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.