  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.
