Unmeasured confounding is a concern in studies relying on real-world data. Proximal causal inference (PCI) is a novel method that leverages proxy variables to validly estimate causal treatment effects with unmeasured confounding. However, when the unmeasured confounder is continuous, binary proxies will likely violate the PCI assumption of completeness.

To assess the robustness of PCI to completeness assumption violations with binary proxies for a continuous unmeasured confounder, we simulated 4000 cohorts of 1000 observations following a PCI structure with a binary exposure (A), a continuous outcome (Y), a continuous measured confounder (X), a continuous unmeasured confounder (U), and two proxy pairs (Z, W)—one continuous, one binary. The U-A and U-Y effects were both positive to create a confounding bias by U that overestimates a true positive effect of A on Y. Three U-Y confounding structures were specified: monotonic linear, restricted cubic splines, and step-wise (4 levels). We further varied U-A and proxy-confounder (U-Z, U-W) correlations. Both standard and PCI g-computation were applied to estimate the average causal effect using binary and continuous proxies separately. Performance metrics for the estimators included bias and 95% confidence interval coverage. Correctly specified bridge functions of observed proxy variables that reproduce the effect of an unmeasured confounder on the outcome were used for both proxy types.

Standard g-computation showed consistent residual confounding bias due to U in the expected positive direction (range: 0.072-0.244). When completeness was satisfied, PCI had biases near zero across all scenarios (range: -0.000 to 0.003) with nominal 95%CI coverage. When completeness was violated by use of binary proxies, PCI slightly over-corrected, producing bias opposite to the true confounding direction (range: -0.038 to -0.016) but with close to nominal 95%CI coverage (range: 93.7-95.7%). As U-A effect strengthened (OR: 1.22 to 3) and proxy-confounder correlations increased, PCI bias stabilized near zero while bias for standard g-computation persisted for both binary and continuous proxies.

These findings are based on an analysis of a causal effect on a continuous outcome, and results from this limited parameter space may not generalize to other scenarios. Further work on PCI robustness for binary outcomes and other mechanisms is underway.