A semiparametric risk score for physical activity.

We develop a generalized partially additive model to build a single semiparametric risk scoring system for physical activity across multiple populations. A score comprised of distinct and objective physical activity measures is a new concept that offers challenges due to the nonlinear relationship between physical behaviors and various health outcomes. We overcome these challenges by modeling each score component as a smooth term, an extension of generalized partially linear single-index models. We use penalized splines and propose two inferential methods, one using profile likelihood and a nonparametric bootstrap, the other using a full Bayesian model, to solve additional computational problems. Both methods exhibit similar and accurate performance in simulations. These models are applied to the National Health and Nutrition Examination Survey and quantify nonlinear and interpretable shapes of score components for all-cause mortality.