  Observational healthcare data offer the potential to estimate causal effects
of medical products on a large scale. However, the confidence intervals and
p-values produced by observational studies only account for random error and
fail to account for systematic error. As a consequence, operating
characteristics such as confidence interval coverage and Type I error rates
often deviate sharply from their nominal values and render interpretation
impossible. While there is longstanding awareness of systematic error in
observational studies, analytic approaches to empirically account for
systematic error are relatively new. Several authors have proposed approaches
using negative controls (also known as "falsification hypotheses") and positive
controls. The basic idea is to adjust confidence intervals and p-values in
light of the bias (if any) detected in the analyses of the negative and
positive control. In this work, we propose a Bayesian statistical procedure for
posterior interval calibration that uses negative and positive controls. We
show that the posterior interval calibration procedure restores nominal
characteristics, such as 95% coverage of the true effect size by the 95%
posterior interval.
