Regression Analysis of Unmeasured Confounding
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When studying the causal effect of x on y, researchers may conduct regression and report a confidence interval for the slope coefficient β_x. This common confidence interval provides an assessment of uncertainty from sampling error, but it does not assess uncertainty from confounding. An intervention on x may produce a response in y that is unexpected, and our misinterpretation of the slope happens when there are confounding factors w. When w are measured we may conduct multiple regression, but when w are unmeasured it is common practice to include a precautionary statement when reporting the confidence interval, warning against unwarranted causal interpretation. If the goal is robust causal interpretation then we can do something more informative. Uncertainty in the specification of three confounding parameters can be propagated through an equation to produce a confounding interval. Here we develop supporting mathematical theory and describe an example application. Our proposed methodology applies well to studies of a continuous response or rare outcome. It is a general method for quantifying error from model uncertainty. Whereas confidence intervals are used to assess uncertainty from unmeasured individuals, confounding intervals can be used to assess uncertainty from unmeasured attributes.
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