Connecting model-based and model-free approaches to linear least squares regression

In a regression setting with a response vector and given regressor vectors, a typical question is to what extent the response is related to these regressors, specifically, how well it can be approximated by a linear combination of the latter. Classical methods for this question are based on statistical models for the conditional distribution of the response, given the regressors. In the present paper it is shown that various p-values resulting from this model-based approach have also a purely data-analytic, model-free interpretation. This finding is derived in a rather general context. In addition, we introduce equivalence regions, a reinterpretation of confidence regions in the model-free context.