The sensitivity of linear regression coefficients confidence limits to the omission of a confounder

Omitted variable bias can affect treatment effect estimates obtained from observational data due to the lack of random assignment to treatment groups. Sensitivity analyses adjust these estimates to quantify the impact of potential omitted variables. This paper presents methods of sensitivity analysis to adjust interval estimates of treatment effect—both the point estimate and standard error—obtained using multiple linear regression. Central to our approach is what we term benchmarking, the use of data to establish reference points for speculation about omitted confounders. The method adapts to treatment effects that may differ by subgroup, to scenarios involving omission of multiple variables, and to combinations of covariance adjustment with propensity score stratification. We illustrate it using data from an influential study of health outcomes of patients admitted to critical care.

💡 Research Summary

The paper addresses a fundamental problem in observational research: omitted‑variable bias (OVB) that can distort treatment‑effect estimates because subjects are not randomly assigned to treatment groups. While many sensitivity‑analysis techniques exist, most focus only on the point estimate of the effect and ignore how OVB inflates the uncertainty around that estimate. The authors therefore develop a comprehensive framework that simultaneously adjusts both the regression coefficient and its standard error, thereby producing corrected confidence intervals that reflect the possible influence of unmeasured confounders.
The cornerstone of the approach is “benchmarking.” Instead of relying on arbitrary assumptions about the size of an omitted variable, the method uses the observed covariates to create reference points. By examining the covariance matrix of the included predictors, the analyst identifies one or more observed variables whose relationship with the treatment and outcome most closely resembles that which a hypothetical omitted variable might have. These benchmark variables provide empirical estimates of two key quantities: (1) the covariance between the omitted variable and the set of observed covariates (ΣxU) and (2) a plausible range for the omitted variable’s regression coefficient (βU).
With these quantities in hand, the ordinary‑least‑squares estimator is analytically corrected. The adjusted point estimate is
 β̂* = β̂ – (X′X)⁻¹ ΣxU βU,
and the adjusted standard error incorporates the extra variance contributed by the unmeasured factor:
 SE*(β̂) = √