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Direct Doubly Robust Estimation of Conditional Quantile Contrasts

This paper proposes a novel direct, doubly robust estimator for the conditional quantile comparator (CQC) that enables explicit modeling and interpretation while achieving superior estimation accuracy compared to existing inversion-based methods, as demonstrated through theoretical analysis, simulations, and a real-world employment study.

Original authors: Josh Givens, Song Liu, Henry W J Reeve, Katarzyna Reluga

Published 2026-01-28

📖 5 min read🧠 Deep dive

CC BY 4.0

Original authors: Josh Givens, Song Liu, Henry W J Reeve, Katarzyna Reluga

Original paper licensed under CC BY 4.0 (http://creativecommons.org/licenses/by/4.0/). ✨ This is an AI-generated explanation of the paper below. It is not written or endorsed by the authors. For technical accuracy, refer to the original paper. Read full disclaimer

The Big Picture: Measuring "What If?"

Imagine you are a doctor trying to figure out if a new medicine works. You have two groups of patients: those who took the medicine (Treatment) and those who didn't (Control).

Usually, statisticians ask two main questions:

The Average Question (CATE): "On average, how much better did the treated group do?" (e.g., "The medicine added 2 years to life expectancy.")
The Percentile Question (CQTE): "If you were in the bottom 10% of health outcomes, how much did the medicine help you? What about the top 10%?"

The authors introduce a third, newer question called the Conditional Quantile Comparator (CQC). Instead of asking about averages or abstract percentiles, the CQC asks:

"If a specific person without the medicine would have earned $50,000, how much would they have earned with the medicine, given their age and background?"

It connects a specific starting point (untreated outcome) directly to a specific ending point (treated outcome).

The Problem: The "Roundabout" Route

Before this paper, calculating this specific connection (the CQC) was like trying to find a destination by first driving to a different city, looking at a map, and then driving back.

The old method worked in two confusing steps:

Step 1: Estimate the probability of earning less than $50k for both groups (this is the "Cumulative Distribution Function").
Step 2: Do a complex mathematical "inversion" to figure out the final dollar amount.

Why was this bad?

It was a black box: You couldn't easily see the formula. It was hard to explain to a boss or a patient.
It was fragile: If your Step 1 estimate was slightly wrong, the final answer could be wildly off.
It was slow: Calculating the "inversion" for every single person took a lot of computer power.

The Solution: A Direct Highway

The authors propose a Direct Estimator. Instead of taking the roundabout route, they built a highway that goes straight from "Untreated Outcome" to "Treated Outcome."

How does it work?
Think of the CQC as a machine with knobs (parameters). The authors created a new "loss function" (a scorecard) that tells the computer: "If you turn the knobs this way, you are closer to the truth."

They use a technique called Gradient Descent (like rolling a ball down a hill to find the lowest point) to adjust the knobs until the machine perfectly predicts the treated outcome based on the untreated one.

Why is this better? (The Metaphors)

1. The "Blueprint" vs. The "Photo"

Old Way: The old method was like taking a photo of a building and then trying to guess the blueprints by squinting at the picture. You could see the building, but you couldn't easily change the design or explain how the beams fit together.
New Way: This paper gives you the actual blueprint. Because the model is "explicitly parameterized," you can look at the numbers and say, "Ah, for every year of age, the earnings boost drops by 2%." It's interpretable and easy to tweak.

2. The "Double-Edged Sword" (Double Robustness)
In statistics, there are "nuisance parameters"—extra things you have to estimate to get the answer right (like how likely someone is to get the treatment based on their age).

The Magic: The authors prove their method is "Doubly Robust." Imagine you are trying to hit a target with two different guns. If either gun is accurate, you hit the target. Even if your estimates for the "nuisance" factors are messy or imperfect, as long as one part of the math is decent, the final result for the treatment effect remains accurate.

3. The "Complexity" Advantage
The old method's accuracy depended on how hard it was to estimate the complex "probability maps" (Step 1).
The new method's accuracy depends only on how complex the relationship itself is.

Analogy: If the relationship between untreated and treated income is a simple straight line, the new method learns it instantly. The old method struggled because it was busy trying to map the complex curves of the probability data first, even though the final answer was simple.

Real-World Test: The Employment Program

The authors tested this on real data from a job training program.

The Finding: They looked at how the program affected earnings for people of different ages.
The Insight: They found that for younger participants, the program seemed to act like a "multiplier" (if you were earning a bit, you earned a lot more). For older participants, the boost was more like a "flat shift" (everyone got a similar bump, regardless of where they started).
Because their method was direct and interpretable, they could see this pattern clearly. The old method would have made this pattern much harder to spot.

Summary

This paper replaces a convoluted, two-step "guess and invert" process with a direct, transparent, and efficient way to calculate how a treatment changes a specific outcome. It allows researchers to build models that are easier to understand, faster to compute, and more accurate, even when some of the background data is a bit noisy.

Technical Summary: Direct Doubly Robust Estimation of Conditional Quantile Contrasts

Problem Formulation

In heterogeneous treatment effect (HTE) analysis, researchers aim to quantify the effect of a treatment $A$ on an outcome $Y$ conditional on covariates $X$ . While the Conditional Average Treatment Effect (CATE) and Conditional Quantile Treatment Effect (CQTE) are standard estimands, they possess distinct limitations. CATE offers interpretability but lacks granularity regarding distributional shifts, while CQTE provides granular quantile-level summaries but often lacks direct interpretability in the response space (e.g., relating a treatment effect to a specific untreated outcome value rather than a quantile level).

The Conditional Quantile Comparator (CQC), denoted $g^*(y_0|x)$ , was introduced to bridge this gap. It maps an untreated response $y_0$ and covariate $x$ to the treated response value that shares the same conditional quantile. This allows for a natural interpretation of treatment effects as additive or multiplicative shifts on the response itself. However, the existing estimation method for CQC (Givens et al., 2024) relies on an indirect two-step process:

Estimating an intermediate "CCDF contrasting function" $h(y_1, y_0, x) = F_1(y_1|x) - F_0(y_0|x)$ .
Inverting this function to find $y_1$ such that $h=0$ .

This inversion approach suffers from three primary shortcomings:

Lack of Explicit Form: The resulting estimate $\hat{g}$ is not explicitly parameterized, hindering model constraints, regularization, and interpretability.
Error Propagation: Estimation accuracy depends on the complexity of the intermediate function $h$ rather than the target function $g^*$ .
Computational Cost: Evaluating the estimate requires searching over a grid of $y_1$ values for each query, which is computationally expensive.

Methodology

The paper proposes the first direct estimator for the CQC, bypassing the inversion step by formulating the problem as an M-estimation task.

Loss Function Construction

The authors define a loss function $\bar{\ell}(y_1, y_0, x)$ based on the integral of the contrasting function $h$ :
$\bar{\ell}(y_1, y_0, x) = \int_{g^*(y_0|x)}^{y_1} h(t, y_0, x) \, dt$
where $h(t, y_0, x) = F_1(t|x) - F_0(y_0|x)$ . By the Fundamental Theorem of Calculus, this loss is minimized when $y_1 = g^*(y_0|x)$ . The authors parameterize the CQC as $g_\theta(y_0|x)$ and define the population loss $L(\theta) = \mathbb{E}[\ell(\theta, Y_0, X)]$ .

Doubly Robust Gradient Estimator

Since the true conditional distributions $F_a$ and propensity score $\pi$ are unknown, the authors construct a doubly robust (DR) gradient estimator $\zeta_{dr}$ to perform gradient descent on the empirical loss. The gradient of the loss with respect to parameters $\theta$ is estimated using:
$\zeta_{dr}(\theta, y_0, z) = \nabla_\theta g_\theta(y_0|x) \left[ \frac{A}{\pi(x)}\{1\{Y \le g_\theta(y_0|x)\} - F_1(g_\theta(y_0|x)|x)\} - \frac{1-A}{1-\pi(x)}\{1\{Y \le y_0\} - F_0(y_0|x)\} + F_1(y_1|x) - F_0(y_0|x) \right]$
This estimator utilizes sample splitting (or cross-fitting) to ensure independence between the nuisance parameter estimates ( $\hat{\pi}, \hat{F}_0, \hat{F}_1$ ) and the data used for fitting the CQC parameters.

Optimization

The estimation procedure (Algorithm 1) involves:

Splitting data into two sets to estimate nuisance parameters and fit the CQC.
Using the estimated nuisance parameters to compute the Monte Carlo gradient estimate $\hat{J}_{dr}$ .
Updating parameters $\theta$ via gradient descent (e.g., Adam) to minimize the loss.

Key Contributions

First Direct Estimator: The paper provides the first method to directly estimate the CQC without inverting an intermediate function, allowing for explicit parameterization (e.g., linear models, neural networks).
Theoretical Guarantees: The authors derive finite-sample bounds on the estimation error. They prove that the method retains the double robustness property: the estimator achieves optimal convergence rates even if the nuisance parameters (propensity score and CCDFs) are estimated at slower rates, provided their product of errors converges sufficiently fast.
Error Dependency: Theoretical analysis shows that the estimation error depends directly on the complexity of the CQC itself, rather than the complexity of the intermediate CCDF contrast function.
Enhanced Interpretability and Control: By explicitly parameterizing $g_\theta$ , the method allows for enforcing structural constraints (e.g., smoothness, monotonicity) and facilitates direct model inspection.

Experimental Results

The authors evaluate their method (labeled "Est. DR") against the existing inversion-based method ("Est. Inv. DR") and an inverse probability weighting (IPW) variant across simulated and real-world datasets.

Simulation (10-dimensional):
- Slope Sensitivity: As the steepness of the true CQC increases, the direct estimator maintains low error, whereas the inversion-based method's error increases significantly.
- Nuisance Error: The direct estimator performs comparably or better than the inversion method under increasing levels of noise in nuisance parameter estimates.
- Sample Size: The direct estimator consistently achieves lower mean absolute error across varying sample sizes.
Real-World Applications:
- Employment Scheme: Applied to data on employment interventions, the model revealed that while the intervention generally improves earnings, the range of potential improvement decreases as participant age increases. The linear parameterization allowed for clear interpretation of these age-dependent scaling effects.
- Colon Cancer Trial: Using a Neural Network parameterization, the method captured a nonlinear treatment effect where younger patients showed significant improvement, while the effect was more binary for older patients. The direct estimate provided a smoother, more interpretable surface compared to the noisy output of the inversion method.

Significance and Claims

The paper claims that this direct estimation procedure represents a significant improvement over existing CQC methods. By removing the inversion step, the authors argue that the method:

Improves Accuracy: Reduces estimation error by decoupling the target complexity from the nuisance complexity.
Enhances Interpretability: Provides a function $g_\theta(y_0|x)$ that can be directly analyzed, constrained, and visualized, making it more suitable for real-world decision-making where understanding the treatment effect on specific response values is crucial.
Maintains Robustness: Preserves the desirable double robustness property essential for causal inference in the presence of model misspecification.

The authors acknowledge that while the direct estimator is theoretically robust, it may be practically more sensitive to nuisance parameter errors than the inversion method in certain regimes, a trade-off they suggest warrants further investigation. They also note that future work could extend these convergence results to more complex non-linear architectures like Bayesian additive regression trees.