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arxiv: 2605.21793 · v1 · pith:WQJGDTFLnew · submitted 2026-05-20 · 📊 stat.ME · stat.AP· stat.ML

Targeted maximum likelihood estimation of vaccine effectiveness and immune correlates in test-negative design studies with missing data

Leah I. B. Andrews , Lars van der Laan , Peter B. Gilbert This is my paper

Pith reviewed 2026-05-22 08:10 UTC · model grok-4.3

classification 📊 stat.ME stat.APstat.ML
keywords test-negative designtargeted maximum likelihood estimationvaccine effectivenessmissing datacausal inferenceimmune correlatesCOVID-19 vaccine
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The pith

A targeted maximum likelihood estimator provides valid causal inference for vaccine effectiveness in test-negative designs with missing exposure data.

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The paper introduces a statistical method for analyzing test-negative design studies that assess how well vaccines prevent symptomatic disease. These studies recruit people seeking tests for symptoms and compare vaccination or immune marker status between those who test positive and negative. Missing data on the exposure like vaccination status often occurs due to incomplete records or sampling designs. The new approach uses targeted maximum likelihood estimation to produce an efficient estimator that adjusts for confounding in a flexible way and gives correct causal estimates under missing at random conditions. This matters because it lets researchers use real-world data more reliably to inform vaccination strategies without needing complete information or randomized trials.

Core claim

Under causal and missing at random assumptions, the targeted maximum likelihood estimation approach involving a semiparametric logistic regression model produces an efficient, asymptotically linear estimator that provides flexible, data-driven confounding control and valid causal inference when analyzing TND studies with missing exposure variable data.

What carries the argument

Targeted maximum likelihood estimation with a semiparametric logistic regression model that targets the causal conditional risk ratio of symptomatic disease in the healthcare-seeking population.

If this is right

  • The method yields efficient and asymptotically linear estimators for the targeted causal parameter.
  • Flexible, data-driven control for confounding is achieved beyond basic adjustments for healthcare-seeking behavior.
  • Valid causal inference holds for TND studies using two-phase sampling designs with missing exposure data.
  • Finite sample performance is demonstrated via plasmode simulations based on immune correlates studies.

Where Pith is reading between the lines

These are editorial extensions of the paper, not claims the author makes directly.

  • Analysts could apply this estimator to other observational vaccine studies with similar missing data patterns to improve inference accuracy.
  • Policy decisions on vaccine recommendations might incorporate more data from incomplete records if this method is adopted widely.
  • Future work could test the method's robustness when the missing at random assumption is mildly violated through sensitivity analyses.

Load-bearing premise

The method requires that missing exposure data occur at random given the observed variables and that the causal assumptions needed to identify the risk ratio are satisfied.

What would settle it

Generating data from a TND study where missingness depends on unobserved factors and checking whether the estimator shows bias or incorrect coverage would falsify the validity claim if systematic errors appear.

Figures

Figures reproduced from arXiv: 2605.21793 by Lars van der Laan, Leah I. B. Andrews, Peter B. Gilbert.

Figure 1
Figure 1. Figure 1: Directed acyclic graph (DAG) of causal relationships in the general population ( [PITH_FULL_IMAGE:figures/full_fig_p004_1.png] view at source ↗
Figure 2
Figure 2. Figure 2: Bias of seven estimators from 1000 simulated two-phase test-negative design (TND) study [PITH_FULL_IMAGE:figures/full_fig_p012_2.png] view at source ↗
Figure 3
Figure 3. Figure 3: 95% confidence interval coverage of seven estimators from 1000 simulated two-phase [PITH_FULL_IMAGE:figures/full_fig_p013_3.png] view at source ↗
Figure 4
Figure 4. Figure 4: Moderna COVE primary COVID-19 vaccine efficacy and vaccine effectiveness estimates. [PITH_FULL_IMAGE:figures/full_fig_p016_4.png] view at source ↗
Figure 5
Figure 5. Figure 5: Causal conditional risk ratios of CDC COVID-19 by 50% inhibitory dilution neutralizing [PITH_FULL_IMAGE:figures/full_fig_p019_5.png] view at source ↗
Figure 6
Figure 6. Figure 6: Causal conditional risk ratios of CDC COVID-19 by anti-spike immunoglobulin G binding [PITH_FULL_IMAGE:figures/full_fig_p020_6.png] view at source ↗
Figure 7
Figure 7. Figure 7: Causal conditional risk ratios of CDC COVID-19 by anti-receptor-binding domain im [PITH_FULL_IMAGE:figures/full_fig_p021_7.png] view at source ↗
read the original abstract

The test-negative design (TND) is a resource-efficient observational study design that can assess vaccine effectiveness and exposure-proximal immune correlates of disease. The TND enrolls symptomatic individuals seeking diagnostic testing and compares case status by an exposure variable, such as vaccination status or immune marker level, that is measured at testing. While the TND reduces confounding by healthcare-seeking behavior, other sources of confounding may remain. TND studies may also have missing data in the exposure variable due to incomplete records or two-phase sampling designs. We present a targeted maximum likelihood estimation approach involving a semiparametric logistic regression model that targets a causal conditional risk ratio of symptomatic disease in the healthcare-seeking population. Under causal and missing at random assumptions, our method produces an efficient, asymptotically linear estimator that provides flexible, data-driven confounding control and valid causal inference when analyzing TND studies with missing exposure variable data. We evaluate our method's finite sample properties using plasmode simulations of a two-phase TND immune correlates study. We also apply our method to assess COVID-19 vaccine effectiveness and antibody marker correlates of COVID-19 from TND study cohorts derived from the Moderna Coronavirus Efficacy phase 3 trial.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Referee Report

0 major / 3 minor

Summary. The manuscript develops a targeted maximum likelihood estimator (TMLE) for vaccine effectiveness and immune correlates in test-negative design (TND) studies that feature missing exposure data. It specifies a semiparametric logistic regression model that targets the causal conditional risk ratio of symptomatic disease in the healthcare-seeking population, incorporates a missingness indicator into the efficient influence function, and claims that the resulting estimator is asymptotically linear and efficient under standard causal and missing-at-random assumptions. Finite-sample behavior is examined via plasmode simulations of a two-phase TND immune-correlates study, and the method is illustrated on COVID-19 vaccine effectiveness and antibody-marker data derived from the Moderna phase-3 trial.

Significance. If the derivations hold, the work supplies a practical, semiparametric tool for valid causal inference in TND studies that routinely encounter incomplete exposure records or two-phase sampling. The explicit use of the efficient influence function to handle missingness, together with the plasmode simulation design and the Moderna application, provides reproducible evidence of finite-sample performance and real-data utility that is uncommon in this literature.

minor comments (3)
  1. [theoretical development section] The abstract and introduction state that the estimator is asymptotically linear and efficient under the stated assumptions, yet the manuscript does not display the explicit form of the efficient influence function that incorporates the missingness indicator (presumably in the theoretical development section). Adding this expression would allow readers to verify the rate conditions on the nuisance estimators directly.
  2. [simulation section] In the plasmode simulation description, the number of Monte Carlo replicates, the specific sample sizes for the two-phase design, and the exact missingness mechanism should be stated more precisely so that the reported coverage and bias results can be reproduced.
  3. [application section] The application to the Moderna trial data would benefit from a brief sensitivity analysis that varies the missing-at-random assumption or the choice of nuisance estimators, even if only as a supplementary table.

Simulated Author's Rebuttal

0 responses · 0 unresolved

We thank the referee for the positive and constructive review, which highlights the practical value of the TMLE approach for TND studies with missing exposure data. We appreciate the recommendation for minor revision and will address all points raised.

Circularity Check

0 steps flagged

No significant circularity

full rationale

The paper applies standard TMLE to a TND study with missing exposure data by specifying a semiparametric logistic regression that targets the conditional risk ratio and augments the efficient influence function with the missingness mechanism. This construction follows directly from existing TMLE theory once the causal and MAR assumptions are stated; the resulting estimator is asymptotically linear and efficient under standard rate conditions on the nuisance functions. No step reduces a claimed prediction or uniqueness result to a fitted parameter or to a self-citation whose content is itself defined by the present work. The plasmode simulations and Moderna application are external checks rather than definitional tautologies, leaving the derivation self-contained.

Axiom & Free-Parameter Ledger

1 free parameters · 2 axioms · 0 invented entities

The central claim rests on standard causal identification assumptions plus the missing-at-random condition for the exposure variable; the semiparametric logistic model is used for flexible adjustment but its exact specification is not detailed in the abstract.

free parameters (1)
  • nuisance parameters in semiparametric logistic regression
    Estimated from data within the targeted likelihood procedure
axioms (2)
  • domain assumption Missing at random (MAR) for the exposure variable
    Required for valid inference with incomplete records or two-phase sampling
  • domain assumption Causal assumptions sufficient to identify the conditional risk ratio
    Needed to interpret the target parameter as a causal quantity in the healthcare-seeking population

pith-pipeline@v0.9.0 · 5754 in / 1278 out tokens · 51673 ms · 2026-05-22T08:10:49.208518+00:00 · methodology

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Reference graph

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