Abstract
In parametric regression models the sign of a coefficient often plays an important role in its interpretation. One possible approach to model selection in these situations is to consider a loss function that formulates prediction of the sign of a coefficient as a decision problem. Taking a Bayesian approach, we extend this idea of a sign based loss for selection to more complex situations. In generalized additive models we consider prediction of the sign of the derivative of an additive term at a set of predictors. Being able to predict the sign of the derivative at some point (that is, whether a term is increasing or decreasing) is one approach to selection of terms in additive modelling when interpretation is the main goal. For models with interactions, prediction of the sign of a higher order derivative can be used similarly. There are many advantages to our sign-based strategy for selection: one can work in a full or encompassing model without the need to specify priors on a model space and without needing to specify priors on parameters in submodels. Also, avoiding a search over a large model space can simplify computation. We consider shrinkage prior specifications on smoothing parameters that allow for good predictive performance in models with large numbers of terms without the need for selection, and a frequentist calibration of the parameter in our sign-based loss function when it is desired to control a false selection rate for interpretation.
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References
Abramovich, F., Steinberg, D.: Improved inference in nonparametric regression using L k -smoothing splines. J. Stat. Plan. Inference 49, 327–341 (1996)
Biller, C.: Adaptive Bayesian regression splines in semiparametric generalized linear models. J. Comput. Graph. Stat. 12, 122–140 (2000)
Chan, D., Kohn, R., Nott, D., Kirby, C.: Locally adaptive semiparametric estimation of the mean and variance functions in regression models. J. Comput. Graph. Stat. 15, 915–936 (2006)
Chaudhuri, P., Marron, S.: SiZer for exploration of structure in curves. J. Am. Stat. Assoc. 94, 807–823 (1999)
Cottet, R., Kohn, R., Nott, D.: Variable selection and model averaging in semiparametric overdispersed generalized linear models. J. Am. Stat. Assoc. 103, 661–671 (2008)
De Boor, C.: A Practical Guide to Splines. Springer, New York (1978)
Denison, D.G.T., Mallick, B.K., Smith, A.F.M.: Automatic Bayesian curve fitting. J. R. Stat. Soc. B 60, 333–350 (1998)
Dias, R., Gamerman, D.: A Bayesian approach to hybrid splines non-parametric regression. J. Stat. Comput. Simul. 72, 285–297 (2002)
DiMatteo, I., Genovese, C.R., Kass, R.E.: Bayesian curve-fitting with free-knot splines. Biometrika 88, 1055–1071 (2001)
Eilers, P.H.C., Marx, B.D.: Flexible smoothing using B-splines and penalties (with discussion). Stat. Sci. 11, 89–121 (1996)
Gamerman, D.: Sampling from the posterior distribution in generalized linear mixed models. Stat. Comput. 7, 57–68 (1997)
Ganguli, B., Wand, M.P.: Feature significance in generalized additive models. Stat. Comput. 17, 179–192 (2007)
Gelman, A.: Prior distributions for variance parameters in hierarchical models. Bayesian Anal. 1, 515–533 (2006)
Gelman, A., Tuerlinckx, F.: Type S error rates for classical and Bayesian single and multiple comparison procedures. Comput. Stat. 15, 373–390 (2000)
Gelman, A., Carlin, J.B., Stern, H.S., Rubin, D.B.: Bayesian Data Analysis, 2nd edn. CRC Press, Boca Raton (2003)
Gilley, O.W., Pace, R.K.: On the Harrison and Rubinfeld data. J. Environ. Econ. Manage. 31, 403–405 (1996)
Green, P.J.: Reversible jump Markov chain Monte Carlo computation and Bayesian model determination. Biometrika 82, 711–732 (1995)
González-Manteiga, W., Martínez-Miranda, M.D., Raya-Miranda, R.: SiZer map for inference with additive models. Stat. Comput. 18, 297–312 (2008)
Gu, C., Wahba, G.: Smoothing spline ANOVA with component-wise Bayesian “confidence intervals”. J. Comput. Graph. Stat. 2, 97–117 (1993)
Gustafson, P.: Bayesian regression modeling with interactions and smooth effects. J. Am. Stat. Assoc. 95, 795–806 (2000)
Harrison, D., Rubinfeld, D.L.: Hedonic prices and the demand for clean air. J. Environ. Econ. Manage. 5, 81–102 (1978)
Hastie, T.: Pseudosplines. J. R. Stat. Soc. B 58, 379–396 (1996)
Hastie, T., Tibshirani, R.: Generalized Additive Models. Chapman and Hall, London (1990)
Hastie, T., Tibshirani, R.: Bayesian backfitting (with discussion). Stat. Sci. 15, 196–223 (2000)
Jara, A., Hanson, T., Quintana, F., Mueller, P., Rosner, G.: The DPpackage package. Reference manual. Available at http://cran.r-project.org/web/packages/DPpackage/DPpackage.pdf (2008)
Jones, L.V., Tukey, J.W.: A sensible formulation of the significance test. Psychol. Methods 5, 411–414 (2000)
Kass, R.E., Wasserman, L.: A reference Bayesian test for nested hypotheses and its relationship to the Schwarz criterion. J. Am. Stat. Assoc. 90, 928–934 (1995)
Kohn, R., Smith, M., Chan, D.: Nonparametric regression using linear combinations of basis functions. Stat. Comput. 11, 313–322 (2001)
Lang, S., Brezger, A.: Bayesian P-splines. J. Comput. Graph. Stat. 13, 183–212 (2004)
Lenk, P.J., DeSarbo, W.S.: Bayesian inference for finite mixtures of generalized linear models with random effects. Psychometrika 65, 93–119 (2000)
Liu, J.S.: Monte Carlo Strategies in Scientific Computing. Springer, Berlin (2001)
Lunn, D.J., Best, N., Whittaker, J.C.: Generic reversible jump MCMC using graphical models. Stat. Comput. (2008, in press)
Nott, D.: Semiparametric estimation of mean and variance functions for non-Gaussian data. Comput. Stat. 21, 603–620 (2006)
Nott, D.J., Kuk, A.Y.C.: Coefficient sign prediction methods for model selection. J. R. Stat. Soc. B 69, 447–461 (2007)
Rigby, R.A., Stasinopoulos, D.M.: Generalized additive models for location, scale and shape. Appl. Stat. 54, 507–554 (2005)
Ruppert, D., Wand, M.P., Carroll, R.J.: Semiparametric Regression. Cambridge University Press, Cambridge (2003)
Shively, T.S., Kohn, R., Wood, S.: Variable selection and function estimation in additive nonparametric regression using a data-based prior (with discussion). J. Am. Stat. Assoc. 94, 777–806 (1999)
Smith, M., Kohn, R.: Nonparametric regression using Bayesian variable selection. J. Econom. 75, 317–344 (1996)
Smith, J.W., Everhart, J.E., Dickson, W.C., Knowler, W.C., Johannes, R.S.: Using the ADAP learning algorithm to forecast the onset of diabetes mellitus. In: Proceedings of the Symposium on Computer Applications and Medical Care, pp. 261–265. IEEE Comput. Soc., Los Alamitos (1988)
Spiegelhalter, D.J., Thomas, A., Best, N.: WinBUGSVersion 1.2 User Manual. MRC Biostatistics Unit. Software available at http://www.mrcbsu.cam.ac.uk/bugs/winbugs/contents.shtml (1999)
Wood, S.N.: Generalized Additive Models: An Introduction with R. Chapman and Hall/CRC Press, London/Boca Raton (2006)
Wood, S., Kohn, R.: A Bayesian approach to robust binary nonparametric regression. J. Am. Stat. Assoc. 93, 203–213 (1998)
Wood, S., Kohn, R., Shively, T., Jiang, W.: Model selection in spline nonparametric regression. J. R. Stat. Soc. B 64, 119–139 (2002)
Wu, Y., Boos, D., Stefanski, L.A.: Controlling variable selection by the addition of pseudo-variables. J. Am. Stat. Assoc. 102, 235–243 (2007)
Yau, P., Kohn, R.: Estimation and variable selection in nonparametric heteroscedastic regression. Stat. Comput. 13, 191–208 (2003)
Yau, P., Kohn, R., Wood, S.: Bayesian variable selection and model averaging in high-dimensional multinomial nonparametric regression. J. Comput. Graph. Stat. 12, 23–54 (2003)
Zaslavsky, A.M.: From ANOVA to variance components. Discussion of Gelman (2005) Analysis of variance—why it is more important than ever. Ann. Stat. 33, 1–53 (2005)
Zheng, X., Loh, W.: Consistent variable selection in linear models. J. Am. Stat. Assoc. 90, 151–156 (1995)
Zou, H., Hastie, T.: Regularization and variable selection via the elastic net. J. R. Stat. Soc. B 67, 301–320 (2005)
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This research was supported by an Australian Research Council grant.
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Nott, D.J., Jialiang, L. A sign based loss approach to model selection in nonparametric regression. Stat Comput 20, 485–498 (2010). https://doi.org/10.1007/s11222-009-9139-6
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DOI: https://doi.org/10.1007/s11222-009-9139-6