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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Knafl, G.J., Ding, K. (2016). Adaptive Regression Modeling of Univariate Continuous Outcomes. In: Adaptive Regression for Modeling Nonlinear Relationships. Statistics for Biology and Health. Springer, Cham. https://doi.org/10.1007/978-3-319-33946-7_2
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DOI: https://doi.org/10.1007/978-3-319-33946-7_2
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