Discussion of the Paper "Connecting Model-Based and Model-Free Approaches to Linear Least Squares Regression" by Lutz Dümbgen and Laurie Davies (2024)

We congratulate Lutz Dmbgen & Laurie Davies for their new approach to measuring the relevance of covariates in linear regression (Dmbgen and Davies, 2024).The approach is based on a randomization scheme that replaces observed covariates with Gaussian covariates -i.e.covariates that are known to be irrelevant -and leads, inter alia, to the formulation of Gaussian p-values to measure the observed covariate's relevance without assuming a stochastic model for the given data.That means, their randomization scheme avoids any (implicit) reference to a "true" representation of data which is usually assumed to hold in statistical estimation problems.Yet, the obtained Gaussian p-values are still formally equivalent to model-based ones.This is achieved without resorting to asymptotic reasoning, distinguishing the approach from permutation tests, which are mathematically equivalent to exact tests only as the sample size goes to infinity, see Good (1994).We think this is a truly genuine and remarkable piece of work deserving wide recognition.Moreover, it needs emphasis that their approach to linear regression is based on a general approximation framework developed by Davies (2014) that avoids the assumption of "assumed (revealed?)truth" (Tukey, 1993, p. 2).This framework has not gained much attention yet and furthering it is important in our opinion as it appears to be universally applicable and to be a better account for the complexity of data-based inference compared to conventional statistical parlance.In the following, we want to comment on a connection to a fundamental issue not only of statistical research.The Big Picture: