Two-stage Estimation for Causal Inference Involving a Semi-continuous Exposure

1. Problem Statement

In observational public health research, many exposures are semi-continuous: they have a substantial mass at zero (representing non-exposure) and a continuous, often right-skewed distribution among those who are exposed (e.g., alcohol consumption, smoking, drug use).

• Limitations of Existing Methods: Standard propensity score (PS) methods are well-developed for binary or purely continuous exposures. However, applying generalized PS methods (inverse density weighting) to semi-continuous data often leads to highly variable estimators due to extreme weights. Furthermore, existing two-part modeling approaches often lack a unified causal framework that explicitly defines and identifies separate causal estimands for exposure status (drinking vs. non-drinking) and exposure dose (intensity among drinkers).
• Motivating Example: The authors focus on Prenatal Alcohol Exposure (PAE) and its effect on child cognition. The data exhibits a point mass at zero (mothers who abstain) and a continuous distribution for drinkers. The research goal is to disentangle the causal effect of any drinking versus no drinking from the dose-response effect among those who drink.

2. Methodology

The authors propose a general causal framework utilizing a two-stage estimation strategy based on a two-part propensity score (PS) structure.

A. Causal Framework and Estimands

The exposure $T$ is decomposed into a binary indicator $A = I(T > 0)$ and a continuous dose $D = \log(T)$ (for $A=1$). The framework defines two distinct causal estimands within a Marginal Structural Model (MSM):

1. $\Delta_D$ (Dose-Response Effect): The causal effect of increasing the dose by one unit among the exposed population ($A=1$).
   $$\Delta_D = E\{Y(1, d+1) - Y(1, d) \mid A=1\}$$
2. $\Delta_0(c)$ (Status Effect): The causal effect of being exposed at a specific reference dose $c$ versus being unexposed ($A=0$).
   $$\Delta_0(c) = E\{Y(1, c) - Y(0, 0)\}$$

B. Two-Part Propensity Score

To adjust for confounding, the authors introduce a two-part PS vector $S(X; \alpha) = (S_1, S_2)'$:

• $S_1$ (Dose PS): The conditional expectation of the dose given covariates among the exposed: $S_1 = E(D \mid A=1, Z_1)$.
• $S_2$ (Status PS): The conditional probability of exposure given covariates: $S_2 = P(A=1 \mid Z_2)$.
• Note: The sets of confounders $Z_1$ and $Z_2$ can differ, allowing for distinct mechanisms driving exposure status vs. exposure intensity.

C. The Two-Stage Estimation Procedure

The estimation is sequential to separate the identification of the dose effect from the status effect:

• Stage I: Estimating the Dose-Response Effect ($\Delta_D$)

  • Target: Individuals with $A=1$.
  • Method: Propensity Score Regression Adjustment.
  • Model: Regress the outcome $Y$ on the observed dose $D$ and the dose PS $S_1$.
  • Estimator: The coefficient of $D$ provides a consistent estimate of the causal dose-response effect ($\psi_{11}$), provided the model for $S_1$ is correctly specified.

• Stage II: Estimating the Status Effect ($\Delta_0(c)$)

  • Target: Comparison between $A=1$ (at reference dose $c$) and $A=0$.
  • Strategy: The estimated dose-response effect from Stage I ($\hat{\gamma}_{11}$) is treated as a fixed offset in the Stage II model. This removes the variation in the outcome attributable to deviations from the reference dose, isolating the "status" effect.
  • Methods: The authors propose three approaches for Stage II, all utilizing the status PS ($S_2$):
    1. PS Regression Adjustment: Regress $Y$ on $A$ and $S_2$ (with the offset).
    2. Inverse Probability Weighting (IPW): Weight observations by $1/S_2$ (or $1/(1-S_2)$).
    3. Augmented Inverse Probability Weighting (AIPW): Combines IPW with an outcome imputation model. This estimator is doubly robust: it remains consistent if either the status PS model ($S_2$) or the outcome imputation model is correctly specified.

D. Theoretical Properties

• Consistency & Asymptotic Normality: The authors prove that the resulting estimators are consistent and asymptotically normal under standard regularity conditions and correct model specification.
• Misspecification Analysis: They derive the limiting values of the estimators when PS models are misspecified.
  • Misspecification in Stage I ($S_1$) biases the dose estimate and propagates to bias the Stage II status estimate.
  • Misspecification in Stage II ($S_2$) biases the status estimate unless the AIPW approach is used and the imputation model is correct.

3. Key Contributions

1. Unified Causal Framework: The paper formally defines two distinct causal estimands for semi-continuous exposures within a potential outcomes framework, addressing a gap in existing literature where such exposures were often treated as a single construct or discretized.
2. Two-Stage Estimation Strategy: By separating the estimation of dose-response and status effects, the method allows for flexible modeling (e.g., different confounders for status vs. dose) and enables the use of the offset technique to isolate the status effect.
3. Doubly Robust Option: The integration of AIPW in Stage II provides a safeguard against model misspecification for the binary exposure component, a feature previously lacking in two-part PS analyses.
4. Theoretical Characterization of Bias: The authors provide explicit mathematical derivations for the asymptotic bias under various misspecification scenarios, clarifying how errors in the continuous dose model affect the binary status estimate.

4. Results

Simulation Studies

• Performance: Under correct model specification, all proposed methods (PS regression, IPW, AIPW) showed negligible bias and coverage probabilities close to the nominal 95% level.
• Robustness:
  • Stage I Misspecification: If the dose PS ($S_1$) is misspecified, both Stage I and Stage II estimates become biased, highlighting the critical importance of correctly modeling the continuous exposure component.
  • Stage II Misspecification: The AIPW estimator demonstrated double robustness. When the status PS ($S_2$) was misspecified, the estimator remained consistent provided the outcome imputation model was correct. Conversely, if the imputation model was misspecified, consistency was maintained if $S_2$ was correct.
• Variance: Weighting-based methods (IPW/AIPW) generally yielded larger standard errors compared to regression adjustment, a known trade-off for robustness.

Application: Prenatal Alcohol Exposure (Detroit Cohort)

• Data: 377 children from the Detroit Longitudinal Cohort, with Full Scale IQ (WISC-III) as the outcome.
• Findings:
  • Conventional Covariate Adjustment: Suggested a small positive effect of drinking status (likely due to residual confounding).
  • Propensity Score Methods: All PS-based methods (Two-Part, Two-Stage PS, IPW, AIPW) yielded negative estimates for both the dose effect and the status effect, suggesting a harmful impact of PAE on cognition.
  • Significance: While the point estimates indicated harm, none reached statistical significance at the 5% level in this specific sample. However, the PS-based estimates were more consistent with prior epidemiological evidence than the naive covariate adjustment.
  • Handling Missing Data: The authors successfully applied multiple imputation (MICE) combined with the two-stage framework, estimating propensity scores within each imputed dataset to maintain validity.

5. Significance

This paper provides a rigorous methodological solution for a common but challenging data structure in epidemiology and social sciences. By decomposing semi-continuous exposures into status and dose components, the proposed framework allows researchers to:

• Answer distinct scientific questions (e.g., "Is any drinking harmful?" vs. "Does more drinking cause more harm?").
• Avoid the pitfalls of arbitrary discretization or unstable inverse density weighting.
• Leverage doubly robust estimation to protect against model misspecification in the binary exposure component.

The methodology is broadly applicable to any study involving semi-continuous treatments, such as environmental toxicology (pollutant levels), behavioral health (substance use), and healthcare utilization (costs or frequency of visits).