Bias testing, bias correction, and confounder selection using an instrumental variable model

Instrumental variable (IV) analysis can be used to address bias due to unobserved confounding when estimating the causal effect of a treatment on an outcome of interest. However, if a proposed IV is correlated with unmeasured confounders and/or weakly correlated with the treatment, the standard IV estimator may be more biased than an ordinary least squares (OLS) estimator. Several methods have been proposed that compare the bias of the IV and OLS estimators relying on the belief that measured covariates can be used as proxies for the unmeasured confounder. Despite these developments, there is lack of discussion about approaches that can be used to formally test whether the IV estimator may be less biased than the OLS estimator. Thus, we have developed a testing framework to compare the bias and a criterion to select informative measured covariates for bias comparison and regression adjustment. We also have developed a bias-correction method, which allows one to use an invalid IV to correct the bias of the OLS or IV estimator. Numerical studies demonstrate that the proposed methods perform well with realistic sample sizes.