Dimensional Analysis in Statistical Modelling

Building on recent work in statistical science, the paper presents a theory for modelling natural phenomena that unifies physical and statistical paradigms based on the underlying principle that a model must be nondimensionalizable. After all, such phenomena cannot depend on how the experimenter chooses to assess them. Yet the model itself must be comprised of quantities that can be determined theoretically or empirically. Hence, the underlying principle requires that the model represents these natural processes correctly no matter what scales and units of measurement are selected. This goal was realized for physical modelling through the celebrated theories of Buckingham and Bridgman and for statistical modellers through the invariance principle of Hunt and Stein. Building on recent research in statistical science, the paper shows how the latter can embrace and extend the former. The invariance principle is extended to encompass the Bayesian paradigm, thereby enabling an assessment of model uncertainty. The paper covers topics not ordinarily seen in statistical science regarding dimensions, scales, and units of quantities in statistical modelling. It shows the special difficulties that can arise when models involve transcendental functions, such as the logarithm which is used e.g. in likelihood analysis and is a singularity in the family of Box-Cox family of transformations. Further, it demonstrates the importance of the scale of measurement, in particular how differently modellers must handle ratio- and interval-scales