Bridging Binarization: Causal Inference with Dichotomized Continuous Exposures

Technical Summary: Bridging Binarization: Causal Inference with Dichotomized Continuous Exposures

Problem Statement
In causal inference, the Average Treatment Effect (ATE) is a standard parameter defined for binary exposures. However, many scientific disciplines seek to quantify the causal effects of continuous exposures (e.g., income inequality, alcohol consumption, pesticide levels). A common practice to address this is "binarization" (or dichotomization), where a continuous exposure variable $A$ is converted into a binary dummy variable $T$ based on a specific cut-off (e.g., $T=1$ if $A \geq c$, else $0$). Researchers then estimate the ATE using $T$, termed the "binarized ATE" (BATE).

Despite its prevalence, binarization faces significant methodological criticism. Critics argue it violates the Stable Unit Treatment Value Assumption (SUTVA) by implying multiple versions of treatment exist within a single binary category. Furthermore, it is often criticized for assuming homogeneity of risk within categories and for obscuring the specific mechanism of intervention. Consequently, many researchers are dissuaded from using this approach, yet it remains widely employed in public health and social sciences.

Methodology
The authors propose a framework that validates binarization by reinterpreting the BATE through the lens of Modified Treatment Policies (MTPs).

1. Equivalence to Modified Treatment Policies:
   The paper demonstrates that the BATE is mathematically equivalent to the difference in average outcomes under two specific MTPs. These policies do not simply assign a fixed value; rather, they alter the underlying distribution of the continuous exposure $A$ conditional on confounders $W$.

   • Policy 1 ($\tilde{A}_1$): Restricts the exposure to the region where $T=1$ (e.g., $A \geq c$).
   • Policy 0 ($\tilde{A}_0$): Restricts the exposure to the region where $T=0$ (e.g., $A < c$).
   • Key Assumption: Both policies assume the preservation of relative self-selection. This means that within the allowed region, the ratio of the probability density of any two exposure values remains unchanged relative to the observed world. Formally, for any $a, a' \in \text{allowed region}$, $\frac{p(\tilde{A}|W=a)}{p(\tilde{A}|W=a')} = \frac{p(A|W=a)}{p(A|W=a')}$. This is analogous to Luce's Independence of Irrelevant Alternatives (IIA).

2. Identification:
   Under standard causal assumptions (Positivity and Conditional Independence/No Unmeasured Confounding), the BATE is identified as the difference in expected outcomes under these two MTPs: $\Psi_{BATE} = E[Y^{\tilde{A}_1}] - E[Y^{\tilde{A}_0}]$. This formulation provides a transparent causal mechanism, negating SUTVA violations by explicitly defining how the exposure distribution changes.

3. New Estimand: Causal Attributable Effect of Binarization (CAB):
   The authors introduce a novel parameter, the CAB ($\Psi_{CAB}$), defined as the difference between the expected outcome under one MTP (e.g., the "treated" policy) and the expected outcome in the observed world (status quo): $\Psi_{CAB} = E[Y^{\tilde{A}_t}] - E[Y]$.

   • The authors argue this is often more relevant than the BATE because it compares a specific policy intervention against the current reality, rather than comparing two hypothetical counterfactual worlds.
   • Identification of CAB requires weaker assumptions (one-way positivity) compared to BATE (two-way positivity).

4. Estimation:
   The paper details four estimators for both BATE and CAB: Regression Adjustment, Inverse Probability Weighting (IPW), Augmented Inverse Probability Weighting (AIPW), and Targeted Maximum Likelihood Estimation (TMLE).

   • A critical technical finding is that estimating these parameters does not require estimating the full density of the continuous exposure $A$. Due to the equivalence with the binarized ATE, researchers only need to estimate the propensity score of the binary variable $T$ (i.e., $P(T=1|W)$) and the outcome regression $E[Y|T, W]$. This simplifies implementation significantly compared to other MTPs that require density estimation.

Results

• Simulation Study: The authors conducted a simulation study with 5,000 replicates using sample sizes of 150, 300, and 500. The results showed that the regression, IPW, and AIPW estimators for both BATE and CAB were consistent with negligible bias. The standard errors were comparable across methods, and the estimators performed well even with smaller sample sizes. Crucially, the simulation confirmed that consistent estimation is possible using only the binarized variable $T$ and covariates $W$, without needing the full distribution of $A$.
• Application (California Oil/Gas Wells): The method was applied to evaluate a proposed California law creating 1km Health Protection Zones around oil and gas wells. The exposure was the distance to the nearest well, binarized at 1km.
  • The estimated BATE suggested a 0.205% increase in the probability of low birth weight if all wells were moved outside 1km compared to a world where all were inside 1km.
  • The estimated CAB suggested only a 0.0080% increase in low birth weight probability if the policy were implemented compared to the current status quo.
  • The authors note that the BATE may overstate the policy effect because it compares two extreme hypothetical worlds, whereas the CAB compares the policy to the observed reality. They also discuss that the "preservation of relative self-selection" assumption may not hold perfectly if removing wells changes the spatial configuration of the next nearest well, but the method provides a structured way to evaluate such policies.

Significance and Claims
The paper claims to provide a rigorous theoretical justification for the widespread practice of binarizing continuous exposures. By establishing the equivalence between the BATE and MTPs that preserve relative self-selection, the authors argue that binarization is a statistically valid method for answering specific causal questions, provided the underlying assumptions are plausible for the specific policy context.

Key contributions include:

1. Theoretical Validation: Resolving the debate on binarization by defining the precise causal mechanism (MTPs) it represents.
2. New Parameter (CAB): Proposing the CAB as a more policy-relevant estimand that compares intervention to status quo, with weaker identification requirements.
3. Practical Estimation: Demonstrating that these complex MTPs can be estimated using standard ATE techniques (regression, IPW, AIPW, TMLE) without the need for non-parametric density estimation of the continuous exposure, making the approach accessible to practitioners.
4. Contextual Guidance: Emphasizing that the validity of binarization depends on domain-specific knowledge regarding how individuals would behave under the policy (e.g., whether they would relocate or adjust behavior to maintain relative preferences). The paper does not claim binarization is universally correct, but rather that it is a valid tool when the "preservation of relative self-selection" assumption aligns with the real-world policy mechanism.