Coarsening Bias from Variable Discretization in Causal Functionals

TL;DR

Discretizing continuous variables in causal effect estimation introduces bias, which can be reduced by a simple bias-corrected functional, improving accuracy and confidence interval coverage.

Contribution

The paper identifies the first-order bias caused by variable discretization in causal functionals and proposes a bias-reduction method with estimators, enhancing estimation accuracy.

Findings

Discretization bias is proportional to bin width under smoothness conditions.

The proposed bias-reduced functional eliminates the leading bias term.

Simulations show improved bias and confidence interval coverage with the new method.

Abstract

A class of causal effect functionals requires integration over conditional densities of continuous variables, as in mediation effects and nonparametric identification in causal graphical models. Estimating such densities and evaluating the resulting integrals can be statistically and computationally demanding. A common workaround is to discretize the variable and replace integrals with finite sums. Although convenient, discretization alters the population-level functional and can induce non-negligible approximation bias, even under correct identification. Under smoothness conditions, we show that this coarsening bias is first order in the bin width and arises at the level of the target functional, distinct from statistical estimation error. We propose a simple bias-reduced functional that evaluates the outcome regression at within-bin conditional means, eliminating the leading term and yielding a second-order approximation error. We derive plug-in and one-step estimators for the bias-reduced functional. Simulations demonstrate substantial bias reduction and near-nominal confidence interval coverage, even under coarse binning. Our results provide a simple framework for controlling the impact of variable discretization on parameter approximation and estimation.