Double Negative Control Inference With Some Invalid Negative Control Exposures for Continuous Outcome

Negative controls have been increasingly used for causal inference when unmeasured confounding exist. Valid negative control exposures (NCEs) could not causally affect outcome, and valid negative control outcomes (NCOs) are not to be causally affected by exposure. In most observational studies, it is easy to find a valid NCO but NCEs are harder to verify due to the current limited knowledge. Invalid NCEs associated with outcome result in biased estimate of causal effects. However, previous work considering invalid negative controls is very limited. In this paper, we develop a double negative control framework for continuous outcomes in the presence of some invalid NCEs. First, we prove that it is possible to identify causal effects with a known pre-defined valid NCO and a pre-defined set of NCEs without knowing exactly their validity. Furthermore, as long as more than 50% of NCEs are valid, the average causal effect could be consistently estimated. Then we design an ${\ell }_1$ procedure to select valid NCEs. Finally, we give two kinds of double negative control estimators (sisvNCE and naiveNCE-Median) with a guarantee of theoretical estimation performance. Simulation results show that the performance of our method is robust when the number of invalid NCEs does not exceed a certain threshold. Application results indicate that our method has a promising role in public health.