Adaptive Poisson Regression Modeling of Univariate Count Outcomes

Abstract

This chapter presents adaptive analyses of data on the incidence of non-melanoma skin cancer for women in St. Paul, Minnesota and Fort Worth, Texas, addressing how skin cancer rates for women of varying ages in these two locations depend on age and location. These analyses demonstrate adaptive Poisson regression modeling of univariate count outcomes using fractional polynomials, including modeling means of univariate count outcomes, possibly adjusted to rate outcomes through offsets, and modeling their dispersions as well as means. Formulations are also provided for these alternative regression models, for associated k-fold LCV scores for unit dispersions models, extended quasi-likelihood cross-validation (QLCV⁺) scores for non-unit dispersions models based on extended quasi-likelihoods, and for residuals and standardized or Pearson residuals. The example analyses demonstrate assessing whether the log of the means of an outcome is nonlinear in individual predictors, whether those relationships are better addressed with multiple predictors in combination compared to using singleton predictors, whether those relationships are additive in predictors, whether the predictors interact using geometric combinations, and whether there is a benefit to considering constant dispersions compared to unit dispersions and non-constant dispersions compared to constant dispersions.