Omitted variable bias sensitivity analysis with clustered treatment assignment

2026-07-14

AI Summary

This study addresses the inconsistency in sensitivity analyses for unmeasured confounding that arises when observational studies with clustered treatment assignment are analyzed at different levels—individual versus cluster. Focusing on linear regression models under clustered treatment, the authors propose a correction method based on Pearson’s partial eta-squared. By applying the Mundlak transformation to incorporate cluster means of covariates and parameterizing unmeasured confounding bias through partial R², the approach ensures equivalence between individual- and cluster-level sensitivity analyses. The method explicitly accounts for between-cluster variation in driving bias, thereby reconciling cross-level discrepancies and substantially enhancing the robustness and reliability of causal inference in clustered data settings.

Abstract

Cinelli and Hazlett (2020) develops a sensitivity analysis method for the linear regression model that parameterizes omitted variable bias in terms of two partial $R^2$ parameters capturing the residual variation explained by an omitted confounder in the treatment and outcome respectively. This method is often applied to regressions fit to unit-level data when treatment is assigned at a higher level of aggregation -- as in clustered observational designs. This paper shows that despite the numerical equivalence of the unit-level regression and an appropriately weighted cluster-aggregated regression for estimating the treatment effect, the sensitivity analysis procedure yields different conclusions depending on the chosen level of analysis. The outcome-confounder partial $R^2$ reflects both between- and within- group variation but the latter is irrelevant to omitted variable bias as it by construction cannot be explained by a group-level confounder. Straightforward corrections to the robustness value and the extreme scenario analysis from the unit-level regression using Pearson's partial-$η$ recover equivalence between these two approaches. The paper concludes with a point of caution when benchmarking against unit-level covariates and recommends always including cluster-level averages of these covariates as regressors (Mundlak, 1978).

Problem

• omitted variable bias
• sensitivity analysis
• clustered treatment assignment
• partial R-squared
• ecological confounding

Innovation

• omitted variable bias
• sensitivity analysis
• clustered treatment assignment
• partial R²
• Mundlak correction