  Negative controls are increasingly used to evaluate the presence of potential
unmeasured confounding in observational studies. Beyond the use of negative
controls to detect the presence of residual confounding, proximal causal
inference (PCI) was recently proposed to de-bias confounded causal effect
estimates, by leveraging a pair of treatment and outcome negative control or
confounding proxy variables. While formal methods for statistical inference
have been developed for PCI, these methods can be challenging to implement as
they involve solving complex integral equations that are typically ill-posed.
We develop a regression-based PCI approach, employing two-stage generalized
linear regression models (GLMs) to implement PCI, which obviates the need to
solve difficult integral equations. The proposed approach has merit in that (i)
it is applicable to continuous, count, and binary outcomes cases, making it
relevant to a wide range of real-world applications, and (ii) it is easy to
implement using off-the-shelf software for GLMs. We establish the statistical
properties of regression-based PCI and illustrate their performance in both
synthetic and real-world empirical applications.
