A formal framework for generalized reporting methods in parametric settings

Effect size measures and visualization techniques aimed at maximizing the interpretability and comparability of results from statistical models have long been of great importance and are recently again receiving increased attention in the literature. However, since the methods proposed in this context originate from a wide variety of disciplines and are more often than not practically motivated, they lack a common theoretical framework and many quantities are narrowly or heuristically defined. In this work, we put forward a common mathematical setting for effect size measures and visualization techniques aimed at the results of parametric regression and define a formal framework for the consistent derivation of both existing and new variants of such quantities. Throughout the presented theory, we utilize probability measures to derive weighted means over areas of interest. While we take a Bayesian approach to quantifying uncertainty in order to derive consistent results for every defined quantity, all proposed methods apply to the results of both frequentist and Bayesian inference. We apply selected specifications derived from the proposed framework to data from a clinical trial and a multi-analyst study to illustrate its versatility and relevance.