  Linear regression is a frequently used tool in statistics, however, its
validity and interpretability relies on strong model assumptions. While robust
estimates of the coefficients' covariance extend the validity of hypothesis
tests and confidence intervals, a clear interpretation of the coefficients is
lacking if the mean structure of the model is miss-specified. We therefore
suggest a new intuitive and mathematical rigorous interpretation of the
coefficients that is independent from specific model assumptions. It relies on
a new population based measure of association. The idea is to quantify how much
the population mean of the dependent variable Y can be changed by changing the
distribution of the independent variable X. Restriction to linear functions for
the distributional changes in X provides the link to linear regression. It
leads to a conservative approximation of the newly defined and generally
non-linear measure of association. The conservative linear approximation can
then be estimated by linear regression. We show how this interpretation can be
extended to multiple regression and how far and in which sense it leads to an
adjustment for confounding. We point to perspectives for new analysis
strategies and illustrate the utility and limitations of the new interpretation
and strategies by examples and simulations.
