Sensitivity analyses informed by tests for bias in observational studies

In an observational study, the treatment received and the outcome exhibited may be associated in the absence of an effect caused by the treatment, even after controlling for observed covariates. Two tactics are common: (i) a test for unmeasured bias may be obtained using a secondary outcome for which the effect is known and (ii) a sensitivity analysis may explore the magnitude of unmeasured bias that would need to be present to explain the observed association as something other than an effect caused by the treatment. Can such a test for unmeasured bias inform the sensitivity analysis? If the test for bias does not discover evidence of unmeasured bias, then ask: Are conclusions therefore insensitive to larger unmeasured biases? Conversely, if the test for bias does find evidence of bias, then ask: What does that imply about sensitivity to biases? This problem is formulated in a new way as a convex quadratically constrained quadratic program and solved on a large scale using interior point methods by a modern solver. That is, a convex quadratic function of N variables is minimized subject to constraints on linear and convex quadratic functions of these variables. The quadratic function that is minimized is a statistic for the primary outcome that is a function of the unknown treatment assignment probabilities. The quadratic function that constrains this minimization is a statistic for subsidiary outcome that is also a function of these same unknown treatment assignment probabilities. In effect, the first statistic is minimized over a confidence set for the unknown treatment assignment probabilities supplied by the unaffected outcome. This process avoids the mistake of interpreting the failure to reject a hypothesis as support for the truth of that hypothesis. The method is illustrated by a study of the effects of light daily alcohol consumption on high-density lipoprotein (HDL) cholesterol levels. In this study, the method quickly optimizes a nonlinear function of $N=800$ variables subject to linear and quadratic constraints. In the example, strong evidence of unmeasured bias is found using the subsidiary outcome, but, perhaps surprisingly, this finding makes the primary comparison insensitive to larger biases.