Parametrization and penalties in spline models with an application to survival analysis
Costa, M. J. and Shaw, J. Ewart H. (2008) Parametrization and penalties in spline models with an application to survival analysis. Working Paper. Coventry: University of Warwick. Centre for Research in Statistical Methodology . (Working papers).
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In this paper we show how a simple parametrization, built from the definition of cubic splines, can aid in the implementation and interpretation of penalized spline models, whatever configuration of knots we choose to use. We call this parametrization value-first derivative parametrization. We perform Bayesian inference by exploring the natural link between quadratic penalties and Gaussian priors. However, a full Bayesian analysis seems feasible only for some penalty functionals. Alternatives include empirical Bayes methods involving model selection type criteria. The proposed methodology is illustrated by an application to survival analysis where the usual Cox model is extended to allow for time-varying regression coefficients.
|Item Type:||Working or Discussion Paper (Working Paper)|
|Subjects:||Q Science > QA Mathematics|
|Divisions:||Faculty of Science > Statistics|
|Library of Congress Subject Headings (LCSH):||Spline theory, Knot theory|
|Series Name:||Working papers|
|Publisher:||University of Warwick. Centre for Research in Statistical Methodology|
|Place of Publication:||Coventry|
|Number of Pages:||19|
|Status:||Not Peer Reviewed|
|Access rights to Published version:||Open Access|
|Funder:||Fundação para a Ciência e a Tecnologia (FCT)|
|Grant number:||SFRH/BD/16955/2004 (FCT)|
|References:||Abrahamowicz, M., MacKenzie, T. & Esdaile, J. M. (1996). Time-dependent hazard ratio: modelling and hypothesis testing with application in lupus nephritis. Journal of the American Statistical Association 91 1432–1439. Aldrin, M. (2006). Improved predictions penalizing both slope and curvature in additive models. Computational Statistics and Data Analysis 50 267–284. Biller, C. & Fahrmeir, L. (1997). Bayesian spline-type smoothing in generalized regression models. Computational Statistics 12 135–151. Breslow, N. E. (1972). Discussion of the paper by D. R. Cox. Journal of the Royal Statistical Society, Series B 34 216–217. Brezger, A. & Lang, S. (2006). Generalized structured additive regression based on Bayesian P-splines. Computational Statistics and Data Analysis 50 957–991. Cox, D. R. (1972). Regression models and life-tables. Journal of the Royal Statistical Society, Series B 34 187–220. de Boor, C. (1978). A Practical Guide to Splines. Springer. Denison, D. G. T., Mallick, B. K. & Smith, A. F. M. (1998a). Automatic Bayesian curve fitting. Journal of the Royal Statistical Society, Series B 60 333–350. Denison, D. G. T., Mallick, B. K. & Smith, A. F. M. (1998b). Bayesian MARS. Statistics and Computing 8 337–346. Eilers, P. H. C. & Goeman, J. J. (2004). Enhancing scatterplots with smoothed densities. Bioinformatics 20 623–628. Eilers, P. H. C. & Marx, B. D. (1996). Flexible smoothing with B-splines and penalties. Statistical Science 11 89–121. Eilers, P. H. C. & Marx, B. D. (2003). Multivariate calibration with temperature interaction using two-dimensional penalized signal regression. Chemometrics and Intelligent Laboratory Systems 66 159–174. Eilers, P. H. C. & Marx, B. D. (2004). Splines, knots, and penalties. URL http://www.stat. lsu.edu/faculty/marx/splines_knots_penalties.pdf. Fahrmeir, L. & Lang, S. (2001). Bayesian inference for generalized additive mixed models based on Markov random field priors. Applied Statistics 50 201–220. Fleming, T. R. & Harrington, D. P. (1991). Counting Processes and Survival Analysis. Wiley- Interscience, New York. Friedman, J. H. (1991). Multivariate adaptive regression splines. The Annals of Statistics 19 1–67. Gamerman, D. (1997). Sampling from the posterior distribution in generalized linear mixed models. Statistics and Computing 7 57–68. Gray, R. J. (1994). Spline based tests in survival analysis. Biometrics 50 640–652. Green, P. J. & Silverman, B. W. (1994). Nonparametric Regression and Generalized Linear Models. Chapman & Hall. Hastie, T. & Tibshirani, R. (1990a). Exploring the nature of covariate effects in the proportional hazards model. Biometrics 46 1005–1016. Hastie, T. & Tibshirani, R. (1990b). Generalized Additive Models. Chapman & Hall. Hastie, T. & Tibshirani, R. (1993). Varying-coefficients models. Journal of the Royal Statistical Society, Series B 55 757–796. Hastie, T. & Tibshirani, R. (2000). Bayesian backfitting. Statistical Science 15 196–213. Hennerfeind, A., Brezger, A. & Fahrmeir, L. (2006). Geoadditive survival models. Journal of the American Statistical Association 101 1065–1075. Kauermann, G. (2005). Penalized spline smoothing in multivariable survival models with varying coefficients. Computational Statistics and Data Analysis 49 169–186. Lambert, P. & Eilers, P. H. C. (2005). Bayesian proportional hazards model with time-varying regression coefficients: A penalized Poisson regression approach. Statistics in Medicine 24 3977– 3989. Lang, S. & Brezger, A. (2004). Bayesian P-splines. Journal of Computational and Graphical Statistics 13 183–212. Martinussen, T. & Scheike, T. H. (2006). Dynamic Regression Models for Survival Data. Springer. Ruppert, D. & Carroll, R. J. (2000). Spatially-adaptive penalties for spline fitting. Australian and New Zealand Journal of Statistics 42 205–223. Shaw, J. E. H. (1987). Numerical Bayesian analysis of some flexible regression models. The Statistician 36 147–153. Sinha, D., Ibrahim, J. G. & Chen, M.-Hui. (2000). A Bayesian justification of Cox’s partial likelihood. Biometrika 90 629–641. Tian, L., Zucker, D. & Wei, L. J. (2005). On the Cox model with time-varying regression coefficients. Journal of the American Statistical Association 100 172–183. Wahba, G. (1990). Spline Models for Observational Data. CBMS-NSF 59, Regional Conference Series in Applied Mathematics. Wood, S. N. (2003). Thin plate regression splines. Journal of the Royal Statistical Society, Series B 65 95–114.|
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