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

TL;DR

This paper bridges model-based and model-free methods in linear regression, showing that p-values have a data-analytic interpretation and introducing equivalence regions as a reinterpretation of confidence regions.

Contribution

It demonstrates the connection between classical statistical p-values and data-analytic measures, and introduces equivalence regions for model-free inference in linear regression.

Findings

P-values can be interpreted in a model-free, data-analytic way.

Equivalence regions serve as a reinterpretation of confidence regions.

The approach applies in a general regression context.

Abstract

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.