Exact, Nonparametric Sensitivity Analysis for Observational Studies of Contingency Tables

In observational studies, contingency tables provide a simple and intuitive approach to study associations between categorical variables. However, any test of association in contingency tables may be biased due to unmeasured confounders. Existing sensitivity analyses that assess the impact of unmeasured confounding on association tests typically assume a binary treatment variable or impose strong parametric assumptions on the non-binary treatment variable. We overcome these limitations with an exact (i.e., non-asymptotic) and nonparametric sensitivity analysis for unmeasured confounding in $I \times J$ or $I \times J \times K$ contingency tables, accommodating both non-binary treatment and non-binary outcome. Specifically, we extend Rosenbaum's sensitivity model for generic bias and develop a general method to calculate the exact worst-case null distribution for any permutation-invariant test, which includes the chi-square test and many likelihood-based or score-based tests of association. We also propose specialized results for sub-families of permutation-invariant tests that lead to more efficient computation of the worst-case null distribution. Finally, we show that sensitivity analyses based on tests that utilize all treatment and outcome levels typically have higher power than "naive" approaches that dichotomize the categorical variables. We conclude with a re-analysis of the effect of pre-kindergarten care on math achievement using data from the Early Childhood Longitudinal Study, Kindergarten Class of 1998-1999. An R package, sensitivityIxJ, implements the proposed methods.