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arxiv: 2510.22202 · v1 · submitted 2025-10-25 · 📊 stat.ME · stat.ML

Causal Effect Estimation with TMLE: Handling Missing Data and Near-Violations of Positivity

Christoph Wiederkehr (1) , Christian Heumann (2) , Michael Schomaker (1 , 2 , 3 , 4) ((1) Department of Statistics , Ludwig-Maximilians University Munich , (2) Centre for Integrated Data
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Epidemiological Research Cape Town (3) Institute of Public Health Medical Decision Making Health Technology Assessment UMIT - University for Health Sciences Medical Informatics Technology Hall in Tirol (4) Munich Center for Machine Learning (MCML) Ludwig-Maximilians University Munich)
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Pith reviewed 2026-05-18 04:31 UTC · model grok-4.3

classification 📊 stat.ME stat.ML
keywords causal inferencemissing dataTMLEpositivity violationmultiple imputationaverage treatment effectepidemiological studiescomplete case analysis
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The pith

Complete-case TMLE that models outcome missingness reduces bias more than multiple imputation when estimating causal effects under missing data and near-positivity violations.

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

The paper tests targeted maximum likelihood estimation for average treatment effect estimation when observations are missing according to five common mechanisms and when the positivity assumption is nearly violated. It pits eight missing-data strategies against one another, some that discard incomplete records and others that fill in values via multiple imputation using either parametric or tree-based models. Simulations include both abstract data-generating processes and a design-based setup that undersmooths highly adaptive lasso on the WASH Benefits Bangladesh dataset. The central result is that keeping only complete cases while embedding an explicit model for outcome missingness inside TMLE produces smaller bias and stays more stable when positivity is strained. This comparison matters for applied researchers who routinely face incomplete exposure, outcome, and covariate records in one-time epidemiological studies.

Core claim

When targeted maximum likelihood estimation is paired with complete-case analysis that incorporates an outcome-missingness model, the resulting average treatment effect estimates exhibit lower bias and greater robustness to near-positivity violations than estimates obtained from any of the multiple-imputation strategies examined, whether those imputations rely on parametric regressions or on classification and regression trees.

What carries the argument

Complete-case TMLE augmented by an explicit model for the probability that the outcome is observed, which uses the observed data to correct for selection while retaining the full sample structure for treatment and covariate relations.

If this is right

  • Non-multiple-imputation approaches, especially the outcome-missingness-adjusted complete-case version, are preferred when bias minimization is the primary goal.
  • Multiple imputation using classification and regression trees yields lower root mean squared error and maintains nominal coverage more reliably than other imputation variants.
  • Trade-offs between bias and interval coverage should guide method choice depending on whether point estimation or uncertainty quantification is prioritized.
  • The relative robustness of the recommended non-MI strategy persists across both model-based and design-based simulation settings that include not-at-random missingness.

Where Pith is reading between the lines

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

  • The same complete-case adjustment could be tested inside other doubly robust estimators such as augmented inverse-probability weighting to see whether the bias reduction generalizes beyond TMLE.
  • In studies with time-varying exposures, embedding missingness models for each time point might preserve the robustness property observed here.
  • When positivity violations are suspected, a preliminary check of the estimated propensity-score distribution could flag whether the outcome-missingness adjustment is likely to deliver the reported stability.
  • Applied analysts could embed the recommended procedure in sensitivity analyses that vary the assumed missingness mechanism to quantify how much the point estimate moves.

Load-bearing premise

The five missingness-directed acyclic graphs together with the undersmoothed highly adaptive lasso design-based simulation on the WASH Benefits Bangladesh data set accurately capture the missing-data patterns and positivity challenges that arise in typical one-point exposure epidemiological studies.

What would settle it

Apply the same eight methods to a randomized trial with known true average treatment effect, artificially introduce missingness according to one of the paper's DAGs, and check whether the complete-case TMLE with outcome-missingness model still shows the smallest bias.

Figures

Figures reproduced from arXiv: 2510.22202 by 2, (2) Centre for Integrated Data, 3, (3) Institute of Public Health, 4) ((1) Department of Statistics, (4) Munich Center for Machine Learning (MCML), Cape Town, Christian Heumann (2), Christoph Wiederkehr (1), Epidemiological Research, Hall in Tirol, Health Technology Assessment, Ludwig-Maximilians University Munich, Ludwig-Maximilians University Munich), Medical Decision Making, Medical Informatics, Michael Schomaker (1, Technology, UMIT - University for Health Sciences.

Figure 2
Figure 2. Figure 2: Workflow illustration: The study starts with different DGPs (e.g., DGP 1), explores three positivity levels for each DGP, and incorporates m-DAGs to represent different missingness mechanisms. Finally, various missing data handling methods are applied, alongside TMLE, to estimate causal effects. 3.3 Evaluation Criteria We assesed performance of the approaches for handling missing data, following Morris et … view at source ↗
Figure 3
Figure 3. Figure 3: Model-based simulation: Relative bias (%) in ATE estimation using different missing data [PITH_FULL_IMAGE:figures/full_fig_p016_3.png] view at source ↗
Figure 4
Figure 4. Figure 4: Model-based simulation: Coverage in ATE estimation using different missing data methods [PITH_FULL_IMAGE:figures/full_fig_p025_4.png] view at source ↗
Figure 5
Figure 5. Figure 5: Model-based simulation: RMSE in ATE estimation using different missing data methods [PITH_FULL_IMAGE:figures/full_fig_p026_5.png] view at source ↗
Figure 6
Figure 6. Figure 6: Results for the design-based simulation: Performance evaluation in ATE estimation using [PITH_FULL_IMAGE:figures/full_fig_p027_6.png] view at source ↗
read the original abstract

We evaluate the performance of targeted maximum likelihood estimation (TMLE) for estimating the average treatment effect in missing data scenarios under varying levels of positivity violations. We employ model- and design-based simulations, with the latter using undersmoothed highly adaptive lasso on the 'WASH Benefits Bangladesh' dataset to mimic real-world complexities. Five missingness-directed acyclic graphs are considered, capturing common missing data mechanisms in epidemiological research, particularly in one-point exposure studies. These mechanisms include also not-at-random missingness in the exposure, outcome, and confounders. We compare eight missing data methods in conjunction with TMLE as the analysis method, distinguishing between non-multiple imputation (non-MI) and multiple imputation (MI) approaches. The MI approaches use both parametric and machine-learning models. Results show that non-MI methods, particularly complete cases with TMLE incorporating an outcome-missingness model, exhibit lower bias compared to all other evaluated missing data methods and greater robustness against positivity violations across. In Comparison MI with classification and regression trees (CART) achieve lower root mean squared error, while often maintaining nominal coverage rates. Our findings highlight the trade-offs between bias and coverage, and we recommend using complete cases with TMLE incorporating an outcome-missingness model for bias reduction and MI CART when accurate confidence intervals are the priority.

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

2 major / 3 minor

Summary. The manuscript evaluates targeted maximum likelihood estimation (TMLE) for average treatment effect estimation under missing data and near-positivity violations. It employs both model-based and design-based simulations, the latter using undersmoothed highly adaptive lasso on the WASH Benefits Bangladesh dataset, along with five missingness DAGs that include MNAR mechanisms for exposure, outcome, and confounders. Eight missing-data methods (non-MI and MI, with parametric and ML variants) are compared when paired with TMLE; the central claim is that non-MI complete-case analysis with an explicit outcome-missingness model yields the lowest bias and greatest robustness to positivity violations, while MI with CART achieves lower RMSE and maintains nominal coverage.

Significance. If the simulation results hold under broader conditions, the work supplies concrete, actionable guidance on bias-coverage trade-offs when applying TMLE to incomplete epidemiological data with positivity concerns, a setting where practitioners routinely face these issues.

major comments (2)
  1. [Design-based simulation] Design-based simulation (described in the methods for the WASH Benefits analysis): the decision to undersmooth HAL and then impose the five fixed missingness mechanisms does not include a sensitivity check on the tail quantiles of g(A|W); because near-positivity violations are driven precisely by those tails, the reported ordering of bias and robustness may be an artifact of the chosen DGP rather than a general property of the estimators.
  2. [Results] Results tables comparing bias across the eight methods: the claim that complete-case TMLE with outcome-missingness model exhibits 'lower bias' and 'greater robustness' is presented as the headline finding, yet no formal comparison (e.g., paired t-tests or bootstrap intervals on the Monte Carlo bias differences) is reported; without this, it is impossible to judge whether the observed advantage exceeds simulation noise.
minor comments (3)
  1. [Abstract] Abstract: the final sentence is truncated ('across.'); it should read 'across scenarios' or equivalent.
  2. [Methods] Notation: the manuscript introduces 'outcome-missingness model' without an explicit equation or diagram showing how this model enters the TMLE targeting step; a short display equation would remove ambiguity.
  3. [Figures] Figure captions: the DAGs in the five missingness scenarios would benefit from explicit node labels (A, Y, W, R) and a legend distinguishing observed versus missing arrows.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for their constructive comments. We address each major comment below and describe the changes we will make to the manuscript.

read point-by-point responses

  1. Referee: [Design-based simulation] Design-based simulation (described in the methods for the WASH Benefits analysis): the decision to undersmooth HAL and then impose the five fixed missingness mechanisms does not include a sensitivity check on the tail quantiles of g(A|W); because near-positivity violations are driven precisely by those tails, the reported ordering of bias and robustness may be an artifact of the chosen DGP rather than a general property of the estimators.

    Authors: We appreciate the referee's point that near-positivity violations are driven by the tails of g(A|W) and that a sensitivity check would strengthen the design-based results. Our choice of undersmoothed HAL was intended to retain the empirical distribution and tail behavior observed in the WASH Benefits data rather than impose an artificial DGP. The five missingness mechanisms are then applied to this fixed empirical structure. Nevertheless, we agree that additional checks are warranted. In the revision we will add a sensitivity analysis that varies the undersmoothing parameter of HAL and reports the resulting changes in the tail quantiles of the estimated propensity scores, together with the corresponding performance metrics for the leading methods. revision: yes

  2. Referee: [Results] Results tables comparing bias across the eight methods: the claim that complete-case TMLE with outcome-missingness model exhibits 'lower bias' and 'greater robustness' is presented as the headline finding, yet no formal comparison (e.g., paired t-tests or bootstrap intervals on the Monte Carlo bias differences) is reported; without this, it is impossible to judge whether the observed advantage exceeds simulation noise.

    Authors: We agree that formal assessment of whether the observed bias differences exceed Monte Carlo error would improve the credibility of the headline claim. In the revised manuscript we will report Monte Carlo standard errors for all bias estimates and add bootstrap intervals (or paired t-tests) on the differences in bias between the complete-case TMLE with outcome-missingness model and the other seven methods. These additions will allow readers to evaluate whether the reported advantages are statistically distinguishable from simulation noise. revision: yes

Circularity Check

0 steps flagged

No circularity: empirical simulation results are independent of fitted inputs

full rationale

The paper evaluates TMLE performance under missing data and positivity violations exclusively through model- and design-based simulations on specified DAGs and the WASH Benefits dataset. Performance metrics (bias, RMSE, coverage) are computed directly from applying the estimators to generated data; these quantities do not reduce by any equation or self-citation to previously fitted parameters. No self-definitional steps, fitted-input predictions, or load-bearing self-citations appear in the reported chain. The simulation design is externally specified and falsifiable, making the findings self-contained against the chosen benchmarks.

Axiom & Free-Parameter Ledger

0 free parameters · 1 axioms · 0 invented entities

The evaluation rests on standard causal-inference and missing-data assumptions plus the claim that the chosen DAGs and the real-data simulation design are representative; no new free parameters or invented entities are introduced.

axioms (1)
  • domain assumption Five missingness-directed acyclic graphs capture common missing data mechanisms in epidemiological research, particularly in one-point exposure studies.
    Stated directly in the abstract as the basis for the simulation scenarios.

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Reference graph

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