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Increasing the usefulness of additive spline models by knot removal

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  • Binder, Harald
  • Sauerbrei, Willi
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    Abstract

    Modern techniques for fitting generalized additive models mostly rely on basis expansions of covariates using a large number of basis functions and penalized estimation of parameters. For example, a mixed model approach is used to fit a model for children's lung function that allows for non-linear influence of several covariates available in a substantial data set. While the resulting model is expected to have good prediction performance, its handling beyond simple visual presentation is problematic. It is shown how the number basis functions of the underlying B-spline representation can be reduced by knot removal techniques without refitting, while preserving the shape of the fitted functions. The condition for exact knot removal is extended towards approximate knot removal by incorporating the covariance matrix of the initial parameter estimates, resulting in considerable simplification of the model. Covariance matrices for the transformed parameter estimates are provided. It is demonstrated that enforcing the knot removal condition during estimation leads to the difference penalties employed in the P-spline approach for estimation of B-spline coefficients, and therefore provides a further justification for this type of penalty. A final transform to a truncated power basis provides a simple equation for the model. This increases transportability, while retaining properties of the initial fit such as good prediction performance.

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    File URL: http://www.sciencedirect.com/science/article/B6V8V-4SK07N9-3/2/30499c53a57dd33c05c0e3c61eec1b12
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    Bibliographic Info

    Article provided by Elsevier in its journal Computational Statistics & Data Analysis.

    Volume (Year): 52 (2008)
    Issue (Month): 12 (August)
    Pages: 5305-5318

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    Handle: RePEc:eee:csdana:v:52:y:2008:i:12:p:5305-5318

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    Web page: http://www.elsevier.com/locate/csda

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    1. W. Sauerbrei & P. Royston, 1999. "Building multivariable prognostic and diagnostic models: transformation of the predictors by using fractional polynomials," Journal of the Royal Statistical Society Series A, Royal Statistical Society, vol. 162(1), pages 71-94.
    2. Bradley Efron, 2004. "The Estimation of Prediction Error: Covariance Penalties and Cross-Validation," Journal of the American Statistical Association, American Statistical Association, vol. 99, pages 619-632, January.
    3. Simon N. Wood, 2004. "Stable and Efficient Multiple Smoothing Parameter Estimation for Generalized Additive Models," Journal of the American Statistical Association, American Statistical Association, vol. 99, pages 673-686, January.
    4. Wenxin Mao & Linda H. Zhao, 2003. "Free-knot polynomial splines with confidence intervals," Journal of the Royal Statistical Society Series B, Royal Statistical Society, vol. 65(4), pages 901-919.
    5. Ruppert,David & Wand,M. P. & Carroll,R. J., 2003. "Semiparametric Regression," Cambridge Books, Cambridge University Press, number 9780521785167, October.
    6. Daniel Gervini, 2006. "Free-knot spline smoothing for functional data," Journal of the Royal Statistical Society Series B, Royal Statistical Society, vol. 68(4), pages 671-687.
    7. Zhou S. & Shen X., 2001. "Spatially Adaptive Regression Splines and Accurate Knot Selection Schemes," Journal of the American Statistical Association, American Statistical Association, vol. 96, pages 247-259, March.
    8. Molinari, Nicolas & Durand, Jean-Francois & Sabatier, Robert, 2004. "Bounded optimal knots for regression splines," Computational Statistics & Data Analysis, Elsevier, vol. 45(2), pages 159-178, March.
    9. Breiman, Leo, 1993. "Fitting additive models to regression data : Diagnostics and alternative views," Computational Statistics & Data Analysis, Elsevier, vol. 15(1), pages 13-46, January.
    10. Ruppert,David & Wand,M. P. & Carroll,R. J., 2003. "Semiparametric Regression," Cambridge Books, Cambridge University Press, number 9780521780506, October.
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