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Abstract:
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Many modern applications seek to understand the relationship between an outcome variable Y and a covariate X in the presence of a confounding variable Z. Although much attention has been paid to testing whether Y depends on X given Z, in this paper we seek to go beyond testing by inferring the strength of that dependence. We first define our estimand, the minimum mean squared error (mMSE) gap, which quantifies the conditional relationship between Y and X in a way that is deterministic, model-free, interpretable, and sensitive to nonlinearities and interactions. We then propose a new approach called floodgate that can leverage any working regression function chosen by the user to construct asymptotic confidence bounds, and we apply it to the mMSE gap. In addition to proving floodgate's asymptotic validity, we rigorously quantify its accuracy and robustness. We then show we can apply the same floodgate principle to a different measure of variable importance when Y is binary. Finally, we demonstrate floodgate's performance in a series of simulations and apply it to data from the UK Biobank to infer the strengths of dependence of platelet count on various groups of genetic mutations.
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