Technical Summary: Bracketing Relationships of Weighted Average Treatment Effects

1. Problem Statement

In observational causal inference, researchers often estimate different weighted average treatment effects (ATEs) depending on the target population. Common estimands include the Average Treatment Effect on the Treated (ATT), the Average Treatment Effect on the Controls (ATC), and the Average Treatment Effect under the Overlap Weight (ATO). While the standard ATE is known to be a convex combination of ATT and ATC, the relationship between the overlap-weighted estimand (ATO) and these traditional estimands has not been generally characterized.

The ATO is defined by weighting units proportional to the conditional variance of the treatment given covariates, $e(X)\{1-e(X)\}$, where $e(X)$ is the propensity score. This weighting scheme is valued for its robustness to extreme propensity scores and its connection to regression-based estimators. However, without a theoretical ordering, it remains unclear whether ATO lies between ATT and ATC or if it can exceed them. This paper seeks to establish a "bracketing" relationship between these estimands under specific monotonicity conditions.

2. Methodology and Theoretical Framework

2.1 Observational Setting

The authors work within the standard potential outcomes framework under the assumption of unconfoundedness. They define the Conditional Average Treatment Effect (CATE) as $\tau(X) = E\{Y(1) - Y(0) | X\}$. The key theoretical object is the P-CATE function, $g(e) = E\{\tau(X) | e(X) = e\}$, which represents the conditional mean of the CATE given a specific propensity score value.

The core assumption driving the results is Assumption 1, which posits that $g(e)$ is either non-decreasing or non-increasing. This assumption implies that units with a higher (or lower) propensity to receive treatment tend to have systematically larger (or smaller) treatment effects on average. This is a weaker condition than requiring the pointwise relationship between $\tau(X)$ and $e(X)$ to be monotonic.

2.2 Instrumental Variable (IV) Setting

The framework is extended to the Local Average Treatment Effect (LATE) setting with a binary instrumental variable $Z$ and binary treatment $D$. The analysis focuses on the subpopulation of "compliers." The authors define local analogs of ATT, ATC, and ATO ($\tau^c_{ATT}, \tau^c_{ATC}, \tau^c_{ATO}$) and introduce a local P-CATE function, $g_c(e)$, defined over the complier stratum. Assumption 4 imposes the same monotonicity condition on $g_c(e)$.

2.3 Generalization to Beta Weights

The paper generalizes these findings to a family of beta weights defined as $\beta_{u,v}(x) = e(x)^{u-1}\{1-e(x)\}^{v-1}$ for $u, v \geq 1$. This family encompasses ATE ($u=1, v=1$), ATT ($u=2, v=1$), ATC ($u=1, v=2$), and ATO ($u=2, v=2$) as special cases.

2.4 Regression Interpretations

The authors connect these weighted estimands to regression coefficients:

• OLS: Under linearity of the propensity score, the coefficient of the treatment in a population OLS regression of $Y$ on $Z$ and $X$ equals $\tau_{ATO}$.
• TSLS: In the IV setting, under linearity of the IV propensity score, the population Two-Stage Least Squares (TSLS) estimator equals the overlap-weighted LATE ($\tau^c_{ATO}$).

3. Key Results

3.1 Bracketing Theorems

• Theorem 1 (Observational): Under the monotonicity assumption on $g(e)$, $\tau_{ATO}$ is bounded between $\tau_{ATT}$ and $\tau_{ATC}$. Specifically, if $g(e)$ is non-decreasing, $\tau_{ATC} \leq \tau_{ATO} \leq \tau_{ATT}$; if non-increasing, the inequality reverses.
• Theorem 2 (IV): Analogously, under the monotonicity assumption on $g_c(e)$, the overlap-weighted LATE ($\tau^c_{ATO}$) is bracketed by the local ATT and local ATC.
• Proposition 1: Under a linear specification of the conditional mean CATE, $\tau_{ATO}$ is explicitly shown to be a convex combination of $\tau_{ATT}$ and $\tau_{ATC}$, with a mixing weight $\lambda$ determined by the distribution of the propensity score.

3.2 Beta-Weight Ordering

• Theorem 4: For the general beta-weight family, the estimand $\tau_{u,v}$ is monotonic in its parameters $u$ and $v$. If $g(e)$ is non-decreasing, $\tau_{u,v}$ increases with $u$ and decreases with $v$. This establishes a lattice of ordering among all beta-weighted estimands.
• Proposition 3: Adjacent beta-weighted estimands satisfy a convex-combination relationship under the linear specification assumption.

3.3 General Weighting

• Theorem 6: The authors provide a sufficient condition for comparing any two weighted estimands based on a "one-crossing" property of the difference in their weighting functions relative to the monotonicity of the conditional mean effect.

4. Empirical Illustrations

The paper validates the theoretical bracketing through two sets of empirical analyses:

1. Observational Studies: Eight datasets (including Right Heart Catheterization, abortion legalization, and 401(k) eligibility) were analyzed. The authors constructed "CP-plots" (Conditional Propensity plots) of estimated CATE ($\hat{\tau}(X)$) against estimated propensity scores ($\hat{e}(X)$).

   • Findings: In all eight cases, the relationship between $\hat{\tau}(X)$ and $\hat{e}(X)$ was approximately monotonic over the region of well-supported data (the central overlap region).
   • Result: The estimated ATO consistently fell between the estimated ATT and ATC, with the direction of the ordering matching the visual trend (increasing or decreasing) in the CP-plots.

2. IV Study: The 401(k) dataset from Abadie (2003) was revisited.

   • Method: A complier-weighted smoother of the conditional Wald ratio was plotted against the IV propensity score.
   • Result: The relationship was mildly increasing, and the estimated overlap-weighted LATE fell between the local ATC and local ATT, consistent with the theoretical predictions.

5. Significance and Recommendations

The paper claims to fill a gap in the literature by characterizing the relationship between the increasingly popular overlap-weighted estimand and traditional estimands. The significance lies in:

1. Theoretical Unification: It demonstrates that ATO is not an isolated estimand but is structurally bounded by ATT and ATC under a plausible monotonicity condition.
2. Practical Diagnostic (CP-plot): The authors recommend plotting estimated CATE against estimated propensity scores as a diagnostic tool. If the plot shows monotonicity in the well-supported region, researchers can be confident that the ATO (and overlap-weighted LATE) lies between the ATT and ATC.
3. Robustness: The results hold even when the monotonicity assumption is violated in the tails of the propensity score distribution, provided the overlap weight downweights those regions (which it naturally does).
4. Reporting Strategy: The paper suggests that researchers should report ATT, ATO, and ATC (or their local IV analogs) alongside the CP-plot. This allows readers to assess the sensitivity of causal conclusions to the target population and verify if the bracketing relationship holds empirically.

The authors emphasize that these plots are descriptive diagnostics rather than formal statistical tests of monotonicity, and the bracketing conclusions should be interpreted with caution if the relationship is visibly non-monotonic in the central region of the data.