Dealing with positivity violations in mediation analysis via weighted controlled effects, with application to assessing immune correlates of protection in antigen-experienced participants

Imagine you are a coach trying to figure out which training drill makes your athletes run the fastest. You want to know: "If we force everyone to run at a specific speed, say 10 miles per hour, how likely are they to win the race?"

In the world of vaccines, scientists play the role of the coach. They want to know: "If we could magically set everyone's immune system to a specific strength (like a high level of antibodies), how well would that protect them from getting sick?"

This is called Mediation Analysis. It's a way of measuring how much of a vaccine's success is due to the specific immune response it creates.

The Problem: The "Backwards" Reality

For a long time, scientists studied people who had never seen the virus before (like a brand-new athlete). For these people, their immune system starts at zero. It's easy to imagine a scenario where we "set" their immune level to 10, 20, or 30. It's a clean, logical experiment.

But recently, the world changed. We started studying people who had already been infected or vaccinated before (like athletes who have already trained for years).

The Issue: These people already have a "baseline" immune level. Maybe they are already at level 20.
The Violation: The old math tried to ask, "What if we set their immune level to 10?"
The Reality Check: You can't magically lower a person's immune system from 20 down to 10 just by giving them a vaccine. The vaccine usually boosts them up, not down.

In statistics, this is called a Positivity Violation. It's like asking a chef, "What if we made this soup with less salt than the chef already put in?" The chef can't do that; the soup is already salty. The math breaks because the scenario is impossible.

The Solution: The "Relevant Group" Filter

The authors of this paper, Qijia He and Bo Zhang, came up with a clever new way to handle this. Instead of trying to force the impossible (lowering immune levels), they changed the question.

The Old Question: "What happens if we set everyone's immune level to X?" (Impossible for people who already have high levels).

The New Question: "What happens if we look only at the people who could realistically reach immune level X?"

They call this the Weighted Controlled Effect.

The Analogy: The "High Jump" Filter

Imagine you are testing a new high-jump bar.

The Naive Group: People who have never jumped before. You can set the bar at 1 foot, 2 feet, or 3 feet. Everyone can attempt it.
The Experienced Group: People who are already professional jumpers. They can clear 6 feet.
- If you ask, "What if we set the bar at 2 feet?" for the pros, it's a weird question. They will always clear it. It's not a fair test of their potential.
- The New Approach: Instead of forcing the pros to jump at 2 feet, you say, "Let's only look at the pros who have a chance of clearing 6 feet." You ignore the people who are too weak to ever reach 6 feet, and you ignore the people who are so strong that 6 feet is a joke. You focus on the "relevant" group where the target level is actually achievable.

How They Did It (The "Magic Weight")

To make this work mathematically, the authors invented a Weighting System.

They calculated the probability of each person reaching a certain immune level.
If a person had a very low chance of reaching that level (because their baseline was too high or too low), they gave that person a weight of zero (they are excluded from that specific calculation).
If a person had a good chance, they gave them a weight to include them in the average.

They also used a "smoothing" technique. Instead of drawing a hard line at "6 feet," they made the line fuzzy. This helps the math work better when dealing with continuous numbers like antibody levels.

The Real-World Test: The COVAIL Trial

The authors tested their new method on real data from the COVAIL trial (a study on COVID-19 booster shots).

The Participants: People who had already been vaccinated or infected before (the "Experienced Group").
The Goal: To see if higher antibody levels meant better protection against the Omicron variant.
The Result: Their new method worked! It showed that for the people who could realistically reach higher antibody levels, those higher levels did indeed lead to better protection.

They also checked if the vaccine worked directly (without just relying on antibodies). They found that for this specific group, the vaccine's main superpower was indeed boosting those antibody levels.

Why This Matters

This paper is a toolkit for scientists.

Before: If you studied people with prior immunity, your math might break or give you nonsense answers because you were trying to imagine impossible scenarios.
Now: You can use this "Weighted" method to get accurate answers. You can safely ask, "How much protection does a specific immune level provide?" even if your study group is a mix of newbies and veterans.

In short: They fixed the math so we can stop asking "What if we turned back the clock?" and start asking "What can we realistically achieve for the people we are studying?" This helps us design better vaccines and understand exactly how they protect us.

1. Problem Statement

Causal mediation analysis is a standard framework for evaluating immune correlates of protection (CoP) in vaccine research. The controlled effects approach (Gilbert et al., 2023) estimates the risk of a clinical outcome under a counterfactual scenario where all individuals are vaccinated and their post-vaccination immune markers ( $S$ ) are set to a fixed level ( $s$ ).

However, this framework relies on the positivity assumption: conditional on baseline covariates, every individual must have a non-zero probability of attaining any level of the mediator $S$ .

In Antigen-Naïve Populations: This assumption holds because baseline immunity ( $B$ ) is effectively zero (below the limit of detection), allowing $S$ to take any value.
In Antigen-Experienced Populations: This assumption is frequently violated. In populations with prior infection or vaccination (e.g., booster trials), the post-vaccination immune response ( $S$ ) is structurally constrained to be $\ge$ the baseline level ( $B$ ). It is biologically implausible to conceive of an intervention that sets a participant's immune marker to a level below their existing baseline.
Consequence: Standard estimators for controlled risk and controlled vaccine efficacy become undefined or biased when applied to non-naïve populations because the target counterfactuals involve subpopulations with zero probability of occurrence.

2. Methodology

The authors propose a Weighted Controlled Risk approach that redefines the target estimand to focus on a "relevant" subpopulation where the counterfactual intervention is plausible.

A. Target Population and Weighting

Instead of estimating effects for the entire population, the method targets a subpopulation of non-naïve participants who have a prespecified probability ( $> t$ ) of achieving the specific post-vaccination immune marker level $S=s$ given their baseline covariates ( $B, X$ ).

Trimmed Weighted Controlled Risk (TWCR): The authors define a weighted risk parameter where individuals are weighted by an indicator function:
$\omega_s^{trim}(B, X) = \mathbb{1}\{ P(S=s \mid A=a, B, X) > t \}$
This effectively "trims" the population, excluding individuals for whom the mediator level $s$ is impossible or highly unlikely given their baseline $B$ .

B. Smoothing for Differentiability

The indicator function in the weighting scheme and the continuous nature of the mediator $S$ create non-differentiability issues, preventing the derivation of Efficient Influence Functions (EIFs) required for asymptotic inference. The authors introduce two smoothing techniques:

Smoothing the Indicator: Replace the hard threshold $\mathbb{1}\{\cdot > t\}$ with a smooth function $\phi(\cdot)$ , typically a Normal CDF with variance $\epsilon^2$ .
Kernel Smoothing: Apply a kernel function $K_h(\cdot)$ to smooth across the continuous mediator levels $S=s$ .

This leads to the Smoothed Trimmed Weighted Controlled Risk (STWCR) and Smoothed Trimmed Weighted Controlled Relative Vaccine Efficacy (STWCRVE).

C. Estimation and Inference

Estimators: The authors derive the Efficient Influence Functions (EIFs) for the numerator and denominator of the STWCR and STWCRVE estimands.
Implementation: They construct cross-fitted one-step estimators using the EIFs. This allows for the use of flexible machine learning methods (e.g., Super Learner) to estimate nuisance parameters (propensity scores, outcome regression) while maintaining $\sqrt{n}$ -consistency and asymptotic normality.
Inference: Confidence intervals are constructed based on the asymptotic normal distribution of the estimators. For relative efficacy, log-transformation is used to ensure positivity of the interval.

3. Key Contributions

Theoretical Extension: Generalizes propensity score trimming methods (previously used in point-exposure settings) to causal mediation analysis with continuous mediators.
Handling Positivity Violation: Provides a rigorous framework to estimate controlled effects in antigen-experienced populations where the standard positivity assumption fails due to the unidirectional relationship between baseline and post-vaccination immunity.
New Estimands: Introduces STWCR and STWCRVE, which are well-defined causal contrasts for subpopulations where the intervention is biologically plausible.
Statistical Machinery: Derives EIFs and proves asymptotic normality for these complex, smoothed, trimmed estimands, enabling valid statistical inference.

4. Results

Simulation Studies

The authors conducted simulations mimicking the COVAIL trial population with varying sample sizes ( $n=1000, 2000, 5000$ ) and baseline distributions (discrete, continuous, and mixed).

Bias and Coverage: The proposed estimators exhibited low bias and empirical coverage close to the nominal 95% level across most scenarios.
Boundary Effects: Performance degraded slightly when the target mediator level $s$ was near the boundary of the feasible range (e.g., very high $s$ values where few individuals can reach them). However, increasing the sample size mitigated this issue.
Robustness: The method performed well under both discrete and continuous baseline immune marker distributions.

Application: COVAIL Trial

The method was applied to reanalyze data from the Coronavirus Variant Immunologic Landscape (COVAIL) trial, focusing on neutralizing antibody titers ($nAb$-ID50) against Omicron BA.4/BA.5.

Context: Participants were non-naïve (previously vaccinated/infected).
Findings:
- Controlled Risk CoP: Higher peak neutralizing antibody titers were associated with lower controlled risk of infection within the relevant subpopulations.
- Controlled Direct Effects: When comparing Omicron-containing vaccines to Prototype vaccines while fixing the immune marker level ( $s_1 = s_0 = s$ ), no significant controlled direct effects were observed. This suggests that the superior efficacy of Omicron vaccines is largely mediated through the induction of higher antibody titers, rather than other unmeasured mechanisms.
- Subpopulation Specificity: The analysis highlighted that the "relevant" population changes depending on the target antibody level (e.g., those capable of reaching high titers vs. low titers), providing nuanced insights into immune mechanisms.

5. Significance

Practical Relevance: As vaccine research shifts from naïve populations (primary series) to experienced populations (boosters, seasonal updates), standard mediation methods are becoming invalid. This paper provides the necessary statistical tools to analyze these critical modern datasets.
Interpretability: The method clarifies that controlled effects in experienced populations are inherently subpopulation-specific. It prevents researchers from making invalid causal claims by forcing comparisons on individuals who could never biologically achieve the counterfactual mediator levels.
Generalizability: The framework is applicable beyond vaccine research to any biomedical setting where a mediator is constrained by baseline characteristics (e.g., biomarkers constrained by disease severity or prior treatment history).

In summary, He and Zhang resolve a critical identification problem in modern vaccine immunology by introducing a weighted, smoothed, and trimmed estimation framework that respects the biological constraints of antigen-experienced populations.