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2016 | OriginalPaper | Chapter

2. Adaptive Regression Modeling of Univariate Continuous Outcomes

Authors : George J. Knafl, Kai Ding

Published in: Adaptive Regression for Modeling Nonlinear Relationships

Publisher: Springer International Publishing

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Abstract

This chapter presents a series of analyses of data on death rates per 100,000 for 60 metropolitan statistical areas, addressing how these death rates depend on the primary predictors: the nitric oxide pollution index, the sulfur dioxide pollution index, and the average annual precipitation in inches. These analyses demonstrate adaptive regression modeling of univariate continuous outcomes using fractional polynomials, including how to set the number k of folds for computing k-fold likelihood cross-validation (LCV) scores, how to compare alternative models using LCV ratio tests analogous to likelihood ratio tests, and how to adaptively model variances as well as means. The analyses demonstrate the benefits of considering nonlinearity over linearity in primary predictors and of fractional polynomial modeling over standard polynomial modeling.
The chapter also provides a formulation for multiple regression models of univariate continuous outcomes and for k-fold LCV scores. Other alternatives for conducting cross-validation are defined as well. Penalized likelihood criteria (PLCs), including the Akaike information criterion (AIC), Bayesian information criterion (BIC) and Takeuchi information criterion (TIC), are defined and their use in model selection compared to the use of LCV.

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Literature
go back to reference Ahlberg, J. H., Nilson, E. N., & Walsh, J. L. (1967). The theory of splines and their applications. New York: Academic.MATH Ahlberg, J. H., Nilson, E. N., & Walsh, J. L. (1967). The theory of splines and their applications. New York: Academic.MATH
go back to reference Allen, D. M. (1974). The relationship between variable selection and data augmentation and a method of prediction. Technometrics, 16, 125–127.MathSciNetCrossRefMATH Allen, D. M. (1974). The relationship between variable selection and data augmentation and a method of prediction. Technometrics, 16, 125–127.MathSciNetCrossRefMATH
go back to reference Burman, P. (1989). A comparative study of ordinary cross-validation, ν-fold cross-validation and the repeated learning-testing methods. Biometrika, 76, 503–514.MathSciNetCrossRefMATH Burman, P. (1989). A comparative study of ordinary cross-validation, ν-fold cross-validation and the repeated learning-testing methods. Biometrika, 76, 503–514.MathSciNetCrossRefMATH
go back to reference Claeskens, G., & Hjort, N. L. (2009). Model selection and model averaging. Cambridge: Cambridge University Press.MATH Claeskens, G., & Hjort, N. L. (2009). Model selection and model averaging. Cambridge: Cambridge University Press.MATH
go back to reference Geisser, S., & Eddy, W. F. (1979). A predictive approach to model selection. Journal of the American Statistical Association, 74, 153–160.MathSciNetCrossRefMATH Geisser, S., & Eddy, W. F. (1979). A predictive approach to model selection. Journal of the American Statistical Association, 74, 153–160.MathSciNetCrossRefMATH
go back to reference Gunst, R. F., & Mason, R. L. (1980). Regression analysis and its applications: A data-oriented approach. New York: Dekker.MATH Gunst, R. F., & Mason, R. L. (1980). Regression analysis and its applications: A data-oriented approach. New York: Dekker.MATH
go back to reference Hastie, T., Tibshirani, R., & Friedman, J. (2009). The elements of statistical learning (2nd ed.). New York: Springer.CrossRefMATH Hastie, T., Tibshirani, R., & Friedman, J. (2009). The elements of statistical learning (2nd ed.). New York: Springer.CrossRefMATH
go back to reference Knafl, G. J., Dixon, J. K., O’Malley, J. P., Grey, M., Deatrick, J. A., Gallo, A., et al. (2012). Scale development based on likelihood cross-validation. Statistical Methods in Medical Research, 21, 599–619.MathSciNetCrossRef Knafl, G. J., Dixon, J. K., O’Malley, J. P., Grey, M., Deatrick, J. A., Gallo, A., et al. (2012). Scale development based on likelihood cross-validation. Statistical Methods in Medical Research, 21, 599–619.MathSciNetCrossRef
go back to reference Knafl, G. J., & Grey, M. (2007). Factor analysis model evaluation through likelihood cross-validation. Statistical Methods in Medical Research, 16, 77–102.MathSciNetCrossRefMATH Knafl, G. J., & Grey, M. (2007). Factor analysis model evaluation through likelihood cross-validation. Statistical Methods in Medical Research, 16, 77–102.MathSciNetCrossRefMATH
go back to reference McDonald, G. C., & Ayers, J. A. (1978). Some applications of the “Chernoff Faces”: A technique for graphically representing multivariate data. In P. C. C. Wang (Ed.), Graphical representation of multivariate data (pp. 183–197). New York: Academic.CrossRef McDonald, G. C., & Ayers, J. A. (1978). Some applications of the “Chernoff Faces”: A technique for graphically representing multivariate data. In P. C. C. Wang (Ed.), Graphical representation of multivariate data (pp. 183–197). New York: Academic.CrossRef
go back to reference McDonald, G. C., & Schwing, R. C. (1973). Instabilities of regression estimates relating air pollution to mortality. Technometrics, 15, 463–482.CrossRef McDonald, G. C., & Schwing, R. C. (1973). Instabilities of regression estimates relating air pollution to mortality. Technometrics, 15, 463–482.CrossRef
go back to reference Royston, P., & Altman, D. G. (1994). Regression using fractional polynomials of continuous covariates: Parsimonious parametric modeling. Applied Statistics, 43, 429–467.CrossRef Royston, P., & Altman, D. G. (1994). Regression using fractional polynomials of continuous covariates: Parsimonious parametric modeling. Applied Statistics, 43, 429–467.CrossRef
go back to reference Sclove, S. L. (1987). Application of model-selection criteria to some problems in multivariate analysis. Psychometrika, 52, 333–343.CrossRef Sclove, S. L. (1987). Application of model-selection criteria to some problems in multivariate analysis. Psychometrika, 52, 333–343.CrossRef
go back to reference Stone, M. (1977). An asymptotic equivalence of choice of model by cross-validation and Akaike’s criterion. Journal of the Royal Statistical Society, Series B, 39, 44–47.MathSciNetMATH Stone, M. (1977). An asymptotic equivalence of choice of model by cross-validation and Akaike’s criterion. Journal of the Royal Statistical Society, Series B, 39, 44–47.MathSciNetMATH
go back to reference Takeuchi, K. (1976). Distribution of informational statistics and a criterion of model fitting. Suri-Kagaku (Mathematical Sciences), 153, 12–18. In Japanese. Takeuchi, K. (1976). Distribution of informational statistics and a criterion of model fitting. Suri-Kagaku (Mathematical Sciences), 153, 12–18. In Japanese.
Metadata
Title
Adaptive Regression Modeling of Univariate Continuous Outcomes
Authors
George J. Knafl
Kai Ding
Copyright Year
2016
DOI
https://doi.org/10.1007/978-3-319-33946-7_2

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