Coarsening Bias from Variable Discretization in Causal Functionals

Technical Summary: Coarsening Bias from Variable Discretization in Causal Functionals

Problem Statement
Causal identification functionals, particularly in mediation analysis, path-specific effects, and front-door identification within hidden-variable Directed Acyclic Graphs (DAGs), often require integrating an outcome regression over the conditional density of a continuous mediator. When the mediator is continuous or high-dimensional, estimating these densities and computing the resulting integrals is statistically and computationally demanding. A common practical workaround is to discretize the continuous mediator and replace integrals with finite sums.

While convenient, this discretization fundamentally alters the population-level parameter being estimated. Even when causal identification assumptions hold and nuisance parameters are estimated consistently, the discretized functional represents a different target than the original continuous functional. The authors identify this discrepancy as coarsening bias (or coarsening error). The paper argues that this bias is distinct from statistical estimation error; it is an approximation error arising at the level of the target functional itself. Previous literature often overlooks this bias or treats it as secondary, yet it can induce non-negligible errors even with correct identification.

Methodology and Theoretical Framework
The authors formalize the difference between the original causal functional $\theta(Q)(c)$ and its discretized analogue $\theta_h(Q)(c)$. Let $M$ be a continuous mediator discretized into $K$ bins $B_k$ with width $w_k$.

1. Characterization of Coarsening Error:
   The paper demonstrates that the naive discretized functional incurs a first-order error proportional to the maximum bin width ($O(w_{\max, K})$). This error is driven by the shift in within-bin conditional means of the mediator across treatment levels ($m_k(a_1, c) - m_k(a_0, c)$) and the local slope of the outcome regression $\mu(m, a, c)$.

   • Specifically, the error term is proportional to $\mu'_m \times \Delta m_k$.
   • The error vanishes only if the outcome regression is locally constant within bins or if the conditional distribution of the mediator does not differ between treatment groups within each bin.

2. Debiased Coarsened Functional:
   To eliminate this leading error, the authors propose a debiased coarsened functional, $\tilde{\theta}_h(Q)(c)$. Instead of evaluating the outcome regression at the bin boundaries or using raw bin averages, this method evaluates the regression $\mu(m, a_1, c)$ at the within-bin conditional mean of the mediator under the reference treatment distribution ($m_k(a_0, c)$).

   • Using a Taylor expansion, the authors show that this modification cancels the first-order term.
   • The resulting approximation error becomes second-order in the bin width ($O(w_{\max, K}^2)$), significantly reducing bias even under coarse binning.

3. Statistical Estimation:
   The paper derives both plug-in and one-step estimators for the debiased functional.

   • Smoothed Estimation: Because point evaluation of the regression at estimated means can violate pathwise differentiability requirements for influence-function-based methods (like one-step estimators) when $M$ is continuous, the authors introduce a smoothed version. This replaces the point evaluation with a localized average using a kernel function and bandwidth $b$.
   • Influence Functions: The authors derive the efficient influence function (EIF) for the smoothed debiased functional, enabling the construction of asymptotically linear one-step estimators that are robust to nuisance model misspecification.

Key Results

• Theoretical Bounds: Under smoothness conditions, the naive discretized estimator has an asymptotic bias of $O(1/K)$ (for equal-width bins), whereas the proposed debiased estimator achieves $O(1/K^2)$.
• Simulation Studies:
  • Bias Reduction: Simulations confirm that the naive discretized estimator retains substantial bias that does not vanish as sample size $n$ increases (reflecting population-level error). In contrast, the debiased estimator exhibits near-zero bias across various sample sizes and bin counts.
  • Coverage: The debiased estimator achieves near-nominal confidence interval coverage (e.g., 95%) even with coarse binning, whereas the naive estimator suffers from poor coverage due to uncorrected bias.
  • Robustness: The debiased approach performs well under both correct specification and various forms of nuisance model misspecification, particularly when combined with one-step corrections.
  • Discretization Schemes: The study compares equal-frequency, empirical equal-width, and predefined-width discretizations. While the choice of binning affects the magnitude of the naive bias, the debiased estimator remains robust and accurate across all schemes.
• Real Data Application: In an analysis of the B_PROUD study (Mobile Stroke Unit dispatch), the debiased estimator produced results consistent with a sequential regression approach (which avoids density estimation entirely) and differed significantly from the naive coarsened estimator, particularly with coarse binning ($K=2$).

Significance and Claims
The paper claims to provide a formal foundation for controlling the impact of variable discretization on causal inference. Its primary contributions are:

1. Distinction of Errors: It clearly separates functional approximation error (coarsening bias) from statistical estimation error, noting that standard influence-function corrections (like one-step estimators) address the latter but fail to correct the former if the target functional itself is misspecified by discretization.
2. Simple Correction: It proposes a computationally simple correction (evaluating at within-bin means) that eliminates the leading bias term without requiring complex density estimation or abandoning the discretization framework.
3. Practical Utility: The results suggest that researchers can use discretization for computational convenience or interpretability (e.g., clinical thresholds) without sacrificing the validity of their causal estimates, provided they apply the proposed debiasing correction.

The authors conclude that while discretization is often necessary in practice, ignoring the resulting coarsening bias leads to invalid inference. The proposed framework allows for the control of this bias, enabling valid statistical estimation even when continuous variables are coarsened.