Sensitivity Analysis with the R^2-Calculus

In observational studies, inference for causal effects necessarily relies on unverifiable assumptions, such as the absence of unmeasured confounders. For this reason, it is crucial to investigate how robust estimators and confidence intervals are. Sensitivity analysis accomplishes this by introducing additional parameters which quantify the degree of violation of model assumptions. Yet, this is rarely done in practice due to the lack of flexible and interpretable methods.

This project provides a new perspective on sensitivity analysis and casts it as a constrained optimisation problem. Practitioners can express their domain knowledge as constraints on the sensitivity parameters. These bounds are added to the optimisation problem and we can compute a range of plausible estimates by solving it.

We focus on linear models and the family of k-class estimators, which includes the commonly used ordinary least squares and instrumental variable estimator. In this set-up we can use a system of algebraic rules concerning R^2-values and partial correlations, which we call R^2-calculus. In the presence of an unmeasured confounder, we identify the causal effect of interest in terms of estimable quantities and two sensitivity parameters. Subsequently, we use the R^2-calculus to develop several options for practitioners to specify their domain knowledge about the unmeasured confounder and its relation to other variables. The ensuing bounds and equality constraints are added to the optimisation problem which is solved with an adapted grid search algorithm. Moreover, we introduce sensitivity frontier plots to illustrate how the specified bounds affect the stability of the estimate and confidence interval.

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