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Coarsening Bias from Variable Discretization in Causal Functionals

Xiaxian Ou, Razieh Nabi

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.

Coarsening Bias from Variable Discretization in Causal Functionals

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.
Paper Structure (21 sections, 4 theorems, 99 equations, 7 figures, 2 tables)

This paper contains 21 sections, 4 theorems, 99 equations, 7 figures, 2 tables.

Table of Contents

  1. Introduction
  2. Target estimands
  3. Coarsened estimands
  4. Debiased coarsened estimands
  5. Smoothed debiased coarsened estimands
  6. Smoothed estimands and plug-ins
  7. Influence function based estimation
  8. Simulation studies
  9. A real data application
  10. Discussion
  11. Extensions to multiple mediators
  12. Proofs
  13. Identification proofs
  14. Lemma \ref{['lem:coarsening_error']}
  15. Remark \ref{['remark:covariance']}
  16. ...and 6 more sections

Key Result

Lemma 3.1

For each $c\in\mathcal{C}$, the coarsening error $\Delta_h(Q)(c)=\theta_h(Q)(c)-\theta(Q)(c)$ satisfies If $m \mapsto \mu(m,a_1,c)$ is continuously differentiable on each ${\cal B}_k$ and for each fixed $c$ there exists a finite constant $L(c) < \infty$ such that $\sup_{m\in{\cal B}_k}|\mu'_m(m,a_1,c)|\le L(c)$ for all $k\in\{1, \ldots, K\}$ and $L(C)$ is square-integrable under $P_C$, then $|\De

Figures (7)

  • Figure 1: Within-bin differences in conditional mediator means, $m_k(a_1,c)-m_k(a_0,c)$, under discretizations with $K=2$ (red) and $K=6$ (blue). Each horizontal segment represents one mediator bin, with bin indices shown on the upper axis for $K=2$ and on the lower axis for $K=6$. Panels correspond to different values of the covariate $C$.
  • Figure 2: Within-bin covariance between $\mu(M,a_1,C)$ and $r_k(M \,|\, C)$ at $C=0$, as formalized in Remark \ref{['remark:covariance']}. Finer discretization reduces the magnitude of this covariance.
  • Figure 3: Bias of the coarsened plug-in estimator $\psi_h(\widehat{Q})$ and the debiased plug-in estimator $\widetilde{\psi}_h(\widehat{Q})$ as functions of sample size for several discretization levels $K$. For fixed $K$, the bias of $\psi_h(\widehat{Q})$ persists as $n$ increases, reflecting population coarsening error, whereas $\widetilde{\psi}_h(\widehat{Q})$ exhibits substantially reduced bias across sample sizes.
  • Figure 4: The performance of the coarsened plug-in estimator $\psi_h(\widehat{Q})$ and the debiased estimator $\widetilde{\psi}_h(\widehat{Q})$ as functions of the number of mediator bins $K$ for sample size $n=5{,}000$.
  • Figure 5: Comparison of plug-in and one-step estimators under nuisance-model misspecification for $n=5{,}000$. "Correct" indicates all nuisance models are correctly specified, whereas "False" indicates all nuisance models are misspecified. Conditions 1–3 correspond to misspecification of $\{\mu, \mu_k\}$, $\{g, g_k\}$, and $\pi$, respectively. Performance is summarized in terms of bias, variance, MSE, and 95% confidence interval coverage.
  • ...and 2 more figures

Theorems & Definitions (5)

  • Lemma 3.1
  • Remark 3.2
  • Lemma 4.1
  • Lemma 5.1
  • Theorem 5.2