Impact of a confounding variable on a regression coefficient

Regression coefficients cannot be interpreted as causal if the relationship can be attributed to an alternate mechanism. One may control for the alternate cause through an experiment(e.g., with random assignment to treatment and control) or by measuring a corresponding confounding variable and including it in the model. Unfortunately, there are some circumstances under which it is not possible to measure or control for the potentially confounding variable. Under these circumstances, it is helpful to assess the robustness of a statistical inference to the inclusion of a potentially confounding variable. In this article, an index is derived for quantifying the impact of a confounding variable on the inference of a regression coefficient. The index is developed for the bivariate care and then generalized to the multivariate case, and the distribution of the index is discussed. The index also is compared with existing indexes and procedures. An example is presented for the relationship between socioeconomic background and educational attainment, and a reference distribution for the index is obtained. The potential for the index to inform causal inferences is discussed as are extensions.