Investigating Targeting Strategies and Truncation in TMLE for the Average Treatment Effect under Practical Positivity Violations

1. Problem Statement

The paper addresses the challenge of estimating the Average Treatment Effect (ATE) from observational data when practical positivity violations occur.

• Context: In observational studies, treatment assignment is not random. The positivity assumption requires that every subject has a non-zero probability of receiving any treatment ($0 < P(A=1|W) < 1$).
• Practical Violation: While theoretical violations (probability = 0) are fatal, practical violations occur when probabilities are non-zero but extremely close to 0 or 1 in the observed data. This leads to extreme Inverse Probability Weighting (IPW), causing:
  • Inflated variance in estimators.
  • Systematic bias if not handled correctly.
  • Unreliable confidence intervals.
• Gap: While Targeted Maximum Likelihood Estimation (TMLE) is a state-of-the-art method known for double robustness and efficiency, its performance under practical positivity violations is sensitive to:
  1. The choice of targeting strategy (how the fluctuation step is formulated).
  2. The choice of truncation level (how extreme weights are capped).
  3. The method of variance estimation.
     Existing literature lacks systematic comparisons of these factors, particularly regarding data-adaptive truncation strategies.

2. Methodology

A. TMLE Framework and Targeting Strategies

The authors compare two specific strategies for the TMLE fluctuation step, which updates the initial outcome regression estimator ($\bar{Q}$):

1. Loss-Weighted Targeting ($g_{Wt}$): Inverse probability weights are applied directly to the loss function. This results in a globally regularized update that does not depend heavily on the magnitude of individual weights.
2. Clever-Covariate-Scaled Targeting ($g_{H}$): The loss function remains unweighted; instead, inverse probability factors are incorporated into the "clever covariate" ($H$) within the fluctuation submodel. This allows for stronger local corrections in strata with high weights.

B. Truncation Mechanisms

To stabilize estimation, estimated propensity scores ($\hat{g}$) are truncated at a lower bound $b$:
$$\hat{g} \leftarrow \max(\hat{g}, b)$$
where $b = \frac{c}{\sqrt{n} \ln n}$. The paper investigates:

• Fixed Truncation: Using pre-specified constants (e.g., $c=5, 6$).
• Adaptive Truncation (Lepski with Brake): A new procedure proposed by the authors.
  • It traverses a grid of truncation constants $c_1 < c_2 < \dots < c_K$ (from strong to weak truncation).
  • It moves to a weaker truncation level only if the shift in the point estimate is statistically significant relative to the increase in uncertainty (confidence interval width).
  • Brake Mechanism: To prevent instability when variance estimates are unreliable (common in positivity violations), the procedure defines a "safe zone" (localization region) around the most-truncated (stablest) estimate. It halts movement if the variance estimate becomes too unstable, ensuring the selected truncation level does not induce excessive variance.

C. Variance Estimation

The paper evaluates three variance estimators for inference:

1. EIF-based: Based on the sample variance of the Efficient Influence Function. Known to underestimate variance under positivity violations.
2. Plug-in: Decomposes variance into residual and regression terms, incorporating conditional outcome variance.
3. Targeted Bootstrap (TB): Re-estimates only the fluctuation parameter on bootstrap samples while keeping initial nuisance parameters fixed.

3. Simulation Design

The authors conducted extensive Monte Carlo simulations with the following variables:

• Sample Sizes: $n \in \{500, 1000, 2000\}$.
• Positivity Severity: Controlled by a scaling constant $\kappa_{pos} \in \{1, 2, 3\}$, where higher values induce more extreme propensity scores (severe violations).
• Outcome Misspecification: Ranging from "nearly correct" to "highly misspecified" (omitting key covariates in the working model).
• Truncation Grid: Constants $c \in \{1, \dots, 10\}$.

4. Key Results

A. Targeting Strategy Performance

• Loss-Weighted ($g_{Wt}$): Under practical positivity violations, this strategy exhibits substantial systematic bias that persists regardless of truncation level. Increasing truncation reduces variance but fails to correct the bias, leading to poor coverage probabilities (often near 0%).
• Clever-Covariate-Scaled ($g_{H}$): This strategy is significantly more robust. It adapts corrections to high-weight regions. While weak truncation leads to high variance, and aggressive truncation leads to bias, there exists a "sweet spot" where both bias and variance are minimized, yielding near-nominal coverage (95%).

B. Truncation Level Selection

• Fixed Rules: Common choices like $c=5$ or $c=6$ are robust defaults but are not optimal across all sample sizes.
  • Sample Size Dependence: The optimal truncation level increases with sample size. Smaller samples ($n=500$) require more aggressive truncation (lower $c$) to control variance, while larger samples ($n=2000$) benefit from less aggressive truncation (higher $c$) to reduce bias.
• Adaptive Lepski with Brake: The proposed adaptive procedure successfully selects truncation levels that balance bias and variance.
  • It performs competitively with or better than fixed rules ($c=5, 6$) across all sample sizes.
  • It effectively avoids the high-variance regime (too little truncation) and the bias-dominated regime (too much truncation).
  • The "brake" mechanism is crucial for stability, preventing the algorithm from selecting unstable, low-truncation levels when variance estimates are erratic.

C. Variance Estimation

• EIF Estimator: Consistently underestimates variance under positivity violations, leading to confidence intervals that are too narrow and under-cover the true parameter.
• Plug-in Estimator: Improves upon EIF by accounting for conditional variance but can still be unstable.
• Targeted Bootstrap (TB): Provides the most stable and conservative variance estimates. It closely tracks the Monte Carlo (true) variance across truncation levels. While it may slightly overestimate variance in mild settings, this conservativeness ensures valid inference under severe positivity violations.

5. Significance and Contributions

1. Clarification of Targeting Strategies: The paper provides empirical evidence that clever-covariate-scaled targeting is superior to loss-weighted targeting in the presence of practical positivity violations, particularly when combined with appropriate truncation.
2. Data-Adaptive Truncation: It introduces a Lepski-type adaptive truncation procedure with a brake mechanism. This solves the practical problem of choosing a truncation constant without prior knowledge of the sample size or violation severity, offering a robust, data-driven alternative to fixed rules.
3. Variance Estimation Guidance: It validates the Targeted Bootstrap as a reliable method for variance estimation in this context, addressing the critical failure of standard EIF-based estimators under positivity violations.
4. Practical Recommendations:
   • Use clever-covariate-scaled targeting ($g_H$).
   • Use Targeted Bootstrap for variance estimation.
   • If a fixed rule is necessary, $c=5$ or $c=6$ are reasonable defaults, but the adaptive Lepski-with-brake procedure is recommended for optimal performance across varying sample sizes and violation severities.

This work significantly advances the practical application of TMLE in real-world observational studies where data sparsity and extreme propensity scores are common, ensuring more reliable causal inference.