Causal Vaccine Effects on Post-infection Outcomes in the Naturally Infected

1. Problem Statement

Evaluating the full value of a vaccine requires quantifying its effects not only on preventing infection but also on reducing the severity of disease and sequelae (post-infection outcomes) among those who do become infected. However, defining appropriate causal estimands for these outcomes is challenging due to selection bias.

• The Doomed Stratum Limitation: Traditional approaches focus on the "Doomed" principal stratum (individuals who would be infected regardless of vaccination). The authors argue this underestimates the vaccine's true benefit because it excludes individuals whose adverse post-infection outcomes were prevented entirely because the vaccine stopped the primary infection.
• The Marginal Effect Limitation: Conversely, analyzing the marginal effect (average outcome across the whole population) is often underpowered. The "Immune" stratum (those who would never be infected) dilutes the signal, as they show no difference in post-infection outcomes regardless of vaccination.
• The Gap: There is a need for a causal estimand that captures the benefit in the "Naturally Infected" (individuals who would be infected in the absence of the vaccine, comprising both the Protected and Doomed strata) while addressing the fundamental identifiability issues caused by cross-world counterfactuals (e.g., outcomes for the Protected stratum under vaccination are unobservable).

2. Methodology

The paper develops a framework for identifying and estimating causal effects in the Naturally Infected principal stratum, defined as $S(0)=1$ (infected under placebo).

A. Identification Strategies

The authors derive identification results under varying assumptions:

1. Partial Identification (Bounds): Without strong assumptions, the effect in the Naturally Infected is not point-identified. The authors derive sharp bounds for the average post-infection outcome in the Protected stratum ($S(0)=1, S(1)=0$) by utilizing the observed distribution of outcomes in uninfected vaccinated individuals (a mixture of Immune and Protected).
2. Point Identification via Exclusion Restriction (ER): If the vaccine has no causal effect on post-infection outcomes in the absence of infection (i.e., $Y(1, 0) = Y(0, 0)$), the effect is point-identified. This is plausible for infection-necessary outcomes or when the only mechanism of vaccine benefit is via infection prevention.
3. Point Identification via Partial Principal Ignorability (PI): If, conditional on covariates $X$, the post-infection outcome in the Protected stratum is independent of the potential infection status under placebo ($Y(1) \perp S(0) \mid X, S(1)=0$), the effect is point-identified. This is a weaker assumption than full principal ignorability.
4. Exposure-Conditional Interpretation: To address concerns about untestable cross-world assumptions, the authors introduce an unmeasured variable $E$ representing a "sufficient infectious dose." They show that under specific assumptions (exposure is necessary/sufficient for infection), the principal stratum estimands align with effects in the subpopulation exposed to a sufficient infectious dose ($E=1$). This allows for empirical testing of assumptions if exposure data is collected.

B. Estimation

The authors propose efficient one-step estimators (based on influence functions) for the identified functionals:

• Robustness: The estimators are multiply robust, meaning they remain consistent if specific combinations of nuisance parameters (e.g., propensity scores, outcome regression models) are correctly specified.
• Efficiency: They leverage covariates to achieve semiparametric efficiency.
• Specific Estimators:
  • $\psi_0$: Effect of placebo in the Naturally Infected (identified without assumptions).
  • $\psi_{1,ER}$: Effect under vaccine in the Naturally Infected under the Exclusion Restriction.
  • $\psi_{1,PI}$: Effect under vaccine in the Naturally Infected under Partial Principal Ignorability.
  • $\psi_{1,\cdot}$: A semiparametric estimator when both ER and PI hold, utilizing data from uninfected individuals in both arms.

3. Key Contributions

1. New Estimand Definition: Formalizes the "Naturally Infected" stratum as a superior target for evaluating vaccine benefits on post-infection outcomes compared to the Doomed stratum or marginal effects.
2. Identification Theory: Provides a comprehensive set of identification results, including:
   • Derivation of sharp bounds without assumptions.
   • Point identification under ER and Partial PI.
   • A novel bridge between principal strata and exposure-conditional effects, offering a way to interpret these effects in terms of observable exposure events (e.g., infectious dose) rather than purely counterfactual worlds.
3. Efficient Estimation: Develops one-step estimators with double/multiple robustness properties, ensuring valid inference even if some nuisance models are misspecified.
4. Sensitivity Analysis: Proposes methods to assess sensitivity to the Partial Principal Ignorability assumption by parameterizing the deviation from the assumption.

4. Results

Simulation Studies

• Bounds: The derived bounds are valid but often wide, confirming that without assumptions, precise estimation is difficult.
• Point Estimators: When identifying assumptions (ER or PI) hold, the one-step estimators are unbiased and achieve nominal coverage. When assumptions are violated, estimators relying on the violated assumption exhibit high bias.
• Power: In realistic scenarios mimicking the PROVIDE rotavirus trial, Naturally Infected estimators (specifically those using Partial PI) demonstrated significantly higher statistical power to detect vaccine effects on post-infection outcomes compared to marginal effects or Doomed-stratum analyses. Marginal effects were often diluted by the Immune stratum, while Doomed analyses missed the benefit of prevented infections.

Data Analysis (PROVIDE Trial)

• Context: Re-analysis of a rotavirus vaccine trial in Bangladesh ($N=700$) assessing antibiotic use for diarrhea.
• Findings:
  • Marginal and Doomed Analyses: Showed no evidence of an effect on antibiotic use.
  • Naturally Infected (ER): Showed no significant effect (consistent with the null).
  • Naturally Infected (PI): Suggested a protective effect, estimating an 8% reduction in the absolute probability of antibiotic use among the Naturally Infected ($p \approx 0.07$).
  • Conclusion: The PI-based Naturally Infected analysis revealed a potential benefit that was masked in other analyses, highlighting the importance of the chosen estimand.

5. Significance

• Clinical Relevance: This work provides a rigorous statistical framework to capture the full value of vaccines, particularly for secondary outcomes like antibiotic resistance or long-term sequelae, which are often missed by traditional analyses.
• Methodological Advancement: It advances causal inference in infectious disease by moving beyond the restrictive "Doomed" stratum and offering a practical alternative (Naturally Infected) with robust estimation tools.
• Interpretability: By linking principal strata to exposure-conditional effects, the paper mitigates the criticism that principal strata effects rely on "fundamentally untestable" assumptions, offering a pathway for empirical validation in future trials with exposure monitoring.
• Policy Impact: The ability to detect protective effects in the Naturally Infected could influence vaccine licensing and public health recommendations, ensuring that vaccines are credited for preventing the consequences of infection, not just the infection itself.

The authors provide an R package (vaxstrat) to facilitate the implementation of these methods.