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

In a regression setting with response vector $\mathbf{y} \in \mathbb{R}^n$ and given regressors $\mathbf{x}_1,\ldots,\mathbf{x}_p \in \mathbb{R}^n$, a typical question is to what extent $\mathbf{y}$ is related to these regressors, specifically, how well can $\mathbf{y}$ be approximated by a linear combination of them. Classical methods for this question are based on statistical models for the conditional distribution of $\mathbf{y}$, given the regressors $\mathbf{x}_j$. 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.