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REVIEW 2 major objections 5 minor 53 references

Targeted maximum likelihood estimation for longitudinal two-stage designs with outcome subsampling

T0 review · 2 major / 5 minor · reviewed 2026-07-12 · grok-4.5

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Pith's one-line read Two new estimators recover large efficiency gains for survival parameters when outcomes are only measured on a second-stage subsample.

desk verdict Solid longitudinal TMLE toolkit for outcome-subsampling two-stage designs; efficiency and coverage claims hold under the reported DGP, with the single-DGP caveat already flagged by the authors. read the letter →

arxiv 2607.02702 v1 pith:CSBNWUVX submitted 2026-07-02 stat.ME

Kirsten E. Landsiedel , Maya L. Petersen , Mark J. van der Laan (University of California , Berkeley) This is my paper
classification stat.ME MSC 62N0262G0562P10
keywords two-stagedesignsoutcomesubsamplingtargetedmaximumlikelihoodestimationcross-fittingdoublesamplinglongitudinalsurvivalIPCW-LTMLEresampling
verification ladder T0 review T1 audit T2 compute T3 formal

The pith

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

The reading

In HIV mortality studies and related longitudinal settings, many people are lost to follow-up, so investigators resample only a fraction of them to learn their outcomes. The usual analysis reweights the complete cases with known sampling probabilities and throws away the rich longitudinal covariates collected on everyone. This paper shows that those designs are simply two-stage designs with outcome subsampling, and that modern sequential-regression tools can exploit the full covariate history. One estimator extends inverse-probability-weighted longitudinal targeted maximum likelihood estimation and further gains efficiency by estimating and targeting the known sampling weights; the other treats the second-stage sampling indicator itself as an intervention node, returning to pure plug-in estimation and never inverse-weighting. Simulations show variance reductions of 30–73 percent relative to the standard weighted Kaplan–Meier, while ordinary influence-curve variance estimates undercover badly; a simple cross-fitted variance estimator restores near-nominal coverage. The practical payoff is more precise survival curves from the same expensive tracing data.

What carries the argument

The LTMLE that inserts the second-stage sampling indicator Δ as an intervention node inside the usual sequential-regression recursion, so the target is the counterfactual mean under the joint intervention that both prevents censoring and sets Δ = 1 for everyone; this returns to plug-in estimation and never multiplies by inverse sampling weights.

What would settle it

Re-run the same Monte Carlo design at N = 500, 1 000 and 3 000 and check whether the LTMLE still shows 30–70 percent lower empirical variance than weighted Kaplan–Meier with known weights and whether the cross-fitted influence-curve intervals actually cover at the nominal 95 percent rate; a clear failure on either metric would refute the central claims.

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Extended reading notes

Core claim

In longitudinal two-stage designs with outcome subsampling, an LTMLE that treats the second-stage sampling indicator as an additional intervention node (and therefore never inverse-weights) achieves up to 73 percent lower empirical variance than weighted Kaplan–Meier with known sampling probabilities, with 30–50 percent reductions common; an IPCW-LTMLE that estimates and targets those known weights still yields consistent 20–35 percent gains. Cross-fitted influence-curve variance is required for valid 95 percent intervals when flexible nuisance estimators are used.

Load-bearing premise

The identification and efficiency arguments require that both the censoring process and the second-stage sampling decision are independent of the counterfactual outcome given the observed past (sequential randomization), plus positivity; the authors also work under a stronger model than pure coarsening-at-random for the bivariate censoring, so closed-form full efficiency is not claimed.

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Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, and a circularity audit.

Referee Report

2 major / 5 minor

Summary. The paper formalizes longitudinal resampling designs (HIV mortality with LTFU tracing) as two-stage designs with outcome subsampling, O = (V, Δ, ΔX). It develops two estimators for marginal means under joint intervention on right-censoring and stage-two sampling: (i) IPCW-LTMLE, a longitudinal extension of Rose & van der Laan IPCW-TMLE that applies full-data LTMLE among Δ = 1 subjects and shows that estimating and targeting known sampling weights Π_Δ yields further variance reductions; (ii) a plug-in LTMLE that treats Δ itself as an intervention node in sequential regression, avoiding inverse weighting. Identification is given under sequential randomization of censoring and of Δ (plus positivity). Simulations (N ∈ {500, 1000, 3000}, 1000 MC replications, ten time points) under a single HIV-like DGP report negligible bias, ordered efficiency gains (LTMLE up to 73 % lower variance than known-weight wKM; IPCW-LTMLE 20–35 %), and that standard influence-curve variance undercovers (coverage as low as ~76 %) while the proposed hybrid cross-fitted variance restores near-nominal coverage. The authors note that bivariate censoring by (τ, Δ) precludes closed-form full efficiency under CAR and that they work under the larger SRA model.

Significance. If the reported efficiency hierarchy and the necessity of cross-fitted variance hold more generally, the paper supplies practically useful closed-form alternatives to the dominant weighted Kaplan–Meier analyses of resampling designs and, more broadly, to inverse-weighted estimators for longitudinal two-stage designs with outcome subsampling. Strengths include a clear identification argument, explicit algorithms for targeting Π_Δ and for hybrid cross-fitted variance, careful handling of deterministic outcomes, and transparent Monte Carlo evidence that standard IC variance fails under flexible Super Learner nuisance estimation. The work also correctly situates itself relative to bivariate-censoring theory (no claim of nonparametric efficiency under CAR). These contributions expand the methodological toolkit for a design class that arises in HIV cascade studies, EHR long-term outcomes, and related settings.

major comments (2)
  1. All headline numerical claims (LTMLE 30–73 % variance reduction vs known-weight wKM; IPCW-LTMLE 20–35 %; cross-fit restoring coverage from ~76 % to nominal) rest on a single carefully constructed DGP (Section 3.1: fixed visit/death/reporting rates, τ ∈ {5,7,9,10}, resampling probability 0.2 among LTFU, Super Learner library matched to the generative process, substantial deterministic information from V(t) and Id(t)). The Discussion itself flags this limitation. Without at least one qualitatively different regime (e.g., weaker positivity, lower resampling fraction, reduced determinism, or deliberate outcome-model misspecification), it remains unclear whether the efficiency hierarchy and the necessity of cross-fitting are general properties or artifacts of this sparsity/determinism pattern. A second simulation regime is needed to support the central empirical claims.
  2. Sections 2.3.2 and Discussion correctly note that the authors work under sequential randomization (SRA) treating censoring like a treatment node, whereas the honest missing-data model is coarsening-at-random (CAR) for bivariate censoring by (τ, Δ); estimators efficient for the larger SRA model need not be efficient for the smaller CAR model, and closed-form full efficiency is not claimed. This is an important caveat for applied readers who may interpret “highly efficient closed-form alternatives” as near-optimal. The manuscript should state more prominently (abstract or early methods) that the efficiency gains are relative to wKM and to untargeted IPCW, not relative to the nonparametric CAR bound, and should clarify what practical efficiency loss relative to a fully efficient (non-closed-form) estimator might be expected.
minor comments (5)
  1. Tables 1–4 report variance to six decimals and coverage to three; a compact relative-efficiency column (vs known-weight wKM) would make the hierarchy easier to read.
  2. Notation for the full-data structure X = (W, τ, L-bar(τ), Y-bar(τ)) deliberately includes right-censoring; a short remark early in Section 2.2 that this is a convenience definition (not the usual complete-data X) would reduce confusion for readers coming from the two-stage literature.
  3. The hybrid cross-fitting design (full-data point estimate, cross-fitted variance only) is well motivated by deterministic sparsity, but the fallback rule to non-cross-fitted variance when Super Learner fails inside a fold should be stated more precisely (how often it triggers at N = 500, t = 1).
  4. Several self-citations to the authors’ related hazard-TMLE preprint and TMLE monographs are appropriate background; a brief sentence distinguishing the present plug-in LTMLE from that hazard-based estimator would help readers who encounter both papers.
  5. Minor typographical issues: “substatially” (p. 17), “crosffit” (p. 17), and occasional missing spaces around mathematical operators.

Circularity Check

0 steps flagged · score 0.0 of 10

No significant circularity: standard TMLE targeting and Monte-Carlo evaluation against a known DGP, with self-citations only as background method references.

full rationale

The paper constructs two estimators (IPCW-LTMLE with optional targeting of known sampling weights, and LTMLE treating the stage-two indicator as an intervention node) from the longitudinal g-computation formula under sequential randomization, then evaluates finite-sample bias/variance/coverage by simulation under an explicitly stated data-generating process whose true survival curves are known. Targeting steps enforce the mean-zero efficient-influence-curve equation by design (the defining property of TMLE), which is not a circular prediction of an independent quantity. Variance reductions versus weighted Kaplan–Meier and the necessity of cross-fitted influence-curve variance are empirical Monte-Carlo findings, not algebraic identities forced by the inputs. Self-citations (to the authors’ related hazard-TMLE preprint, TMLE monographs, and Rose & van der Laan IPCW-TMLE) supply background algorithms and software; none is invoked as a uniqueness theorem that forces the reported efficiency hierarchy. Identification assumptions (SRA plus positivity) and the acknowledged SRA-versus-CAR efficiency gap are stated openly and do not reduce the simulation claims to tautologies. The derivation chain is therefore self-contained against external benchmarks.

Assumptions & free parameters 5 free parameters · 5 assumptions · 0 invented entities

The paper is a semiparametric estimation methods paper. Load-bearing content is identification under sequential randomization/positivity, the TMLE sequential-regression construction, and simulation truth under one DGP. No new physical entities. Free parameters are simulation design choices and nuisance-library choices, not constants fitted to claim a universal law.

free parameters (5)
  • Stage-two resampling probability among LTFU = 0.2
    Simulation sets P(resample | LTFU)=0.2; efficiency rankings can depend on this rate.
  • Natural death-reporting probability Id(t) = 0.2
    Simulation reports death to clinic with probability 0.2 among newly deceased; affects how often Δ=1 without resampling.
  • Administrative censoring time distribution τ∈{5,7,9,10} = discrete support {5,7,9,10}
    Participant-specific end-of-study times with stated unequal probabilities shape bivariate censoring and effective sample size by t.
  • Super Learner library and screening choices
    Nuisance fits use a fixed library (linear, lasso, MARS ± screening) and time-varying fallbacks to non-CV lasso/GLM under sparsity; variance and coverage depend on these choices.
  • Cross-fitting fold count V and convergence tolerance for Π targeting
    Variance procedure and targeting loop use implementer-chosen folds and a mean-IC ≤ sd/√n log n stopping rule.
assumptions (5)
  • domain assumption Sequential randomization of censoring: Y(t0)_{C̄(t0)=1} ⊥ C(t) | history for t≤t0.
    Used for g-computation identification of counterfactual survival under no censoring (§2.1.2–2.1.3).
  • domain assumption Randomization of stage-two sampling: Y(t0)_{Δ=1,C̄(t0)=1} ⊥ Δ | stage-one history (by design in resampling).
    Identifies the joint intervention that sets Δ=1 (§2.1.3); holds by design when sampling probabilities depend only on V.
  • domain assumption Positivity of censoring and of stage-two sampling conditional on history.
    Required for identification and for stable inverse weights / intervention support.
  • ad hoc to paper Working under sequential randomization (SRA) for censoring treated like a treatment node, rather than coarsening-at-random only.
    Authors explicitly note SRA is larger than CAR and that IPCW-LTMLE is therefore not guaranteed efficient for bivariate censored data (§2.3.2).
  • standard math Standard TMLE regularity / Donsker or cross-fit conditions for asymptotic linearity of sequential regression estimators.
    Background for influence-curve inference; paper mitigates overfitting via cross-fitted variance (§2.4).

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Pith. "Pith review of Targeted maximum likelihood estimation for longitudinal two-stage designs with outcome subsampling." pith.science (2026). https://pith.science/paper/CSBNWUVX

@misc{pith2026260702702,
  author       = {Pith},
  title        = {Pith review of: Targeted maximum likelihood estimation for longitudinal two-stage designs with outcome subsampling},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/CSBNWUVX}},
  note         = {Machine review of arXiv:2607.02702}
}
read the original abstract

We consider efficient estimation of causal parameters in longitudinal two-stage designs with outcome subsampling, motivated by resampling designs in HIV-related mortality studies. In these studies, many participants become lost to follow-up; resampling designs address this by tracing a subset of lost individuals to ascertain their outcomes. Analyses often use inverse-probability-weighted Kaplan-Meier (wKM) estimators that discard longitudinal covariate information and suffer from efficiency losses. We note that resampling designs are an instance of a broader class: two-stage designs with outcome subsampling, in which a first stage collects some data on all participants and a second stage collects outcome information on a selected subset. This connection motivates two estimators. First, drawing on inverse probability of censoring weighted targeted maximum likelihood estimation (IPCW-TMLE) for two-stage designs, we develop its longitudinal extension, IPCW longitudinal TMLE (IPCW-LTMLE) and show that estimating and targeting the known second-stage sampling weights yields variance reductions of up to 36% over the use of known sampling probabilities. Second, given that inverse weighting sacrifices efficiency, we propose an LTMLE that incorporates the second-stage sampling indicator as an intervention node in the sequential regression framework, returning to plug-in estimation and avoiding inverse weighting entirely. Simulations across sample sizes show that LTMLE achieves up to 73% lower variance than wKM with known sampling weights, with reductions of 30-50% common across settings, while IPCW-LTMLE achieves consistent gains of 20-35%. We further demonstrate that cross-fitted variance estimation is essential for valid inference: standard variance estimators yield confidence interval coverage as low as 76%, while our cross-fitted variants consistently restore coverage to nominal levels.

Figures

Figures reproduced from arXiv: 2607.02702 by the authors.

Figure 1
Figure 1. Variance of the point estimate for IPCW-LTMLE with known weights, estimated weights, and [PITH_FULL_IMAGE:figures/full_fig_p018_1.png] view at source ↗
Figure 2
Figure 2. Variance of the point estimate for each estimator across 1,000 simulations of sample size N=500. [PITH_FULL_IMAGE:figures/full_fig_p020_2.png] view at source ↗
Figure 3
Figure 3. Variance of the point estimate for each estimator across 1,000 simulations of sample size N=1,000. [PITH_FULL_IMAGE:figures/full_fig_p022_3.png] view at source ↗
Figures from the paper (14 more)
Figure 4
Figure 4. Figure 4: Variance of the point estimate for each estimator across 1,000 simulations of sample size N=3,000. [PITH_FULL_IMAGE:figures/full_fig_p024_4.png]
Figure 5
Figure 5. Figure 5: Confidence interval coverage for targeted IPCW-LTMLE across 1,000 simulations of sample size [PITH_FULL_IMAGE:figures/full_fig_p026_5.png]
Figure 6
Figure 6. Figure 6: Confidence interval coverage for targeted LTMLE across 1,000 simulations of sample size [PITH_FULL_IMAGE:figures/full_fig_p026_6.png]
Figure 7
Figure 7. Figure 7: Confidence interval coverage for hazard TMLE across 1,000 simulations of sample size [PITH_FULL_IMAGE:figures/full_fig_p027_7.png]
Figure 8
Figure 8. Figure 8: Coverage, mean standard error (mean se), 95% confidence interval width (width), and standard error ratio (se ratio) relative to the true (empirical se) for the four variance estimators for IPCW-LTMLE at sample size N = 500. A.1.2 IPCW-LTMLE, N = 1,000 [PITH_FULL_IMAGE…
Figure 9
Figure 9. Figure 9: Coverage, mean standard error (mean se), 95% confidence interval width (width), and standard error ratio (se ratio) relative to the true (empirical se) for the four variance estimators for IPCW-LTMLE at sample size N = 1,000. 32 [PITH_FULL_IMAGE:figures/full_fig_p032_9.png]
Figure 10
Figure 10. Figure 10: Coverage, mean standard error (mean se), 95% confidence interval width (width), and standard error ratio (se ratio) relative to the true (empirical se) for the four variance estimators for IPCW-LTMLE at sample size N = 3,000. A.1.4 LTMLE, N = 500 [PITH_FULL_IMAGE:fig…
Figure 11
Figure 11. Figure 11: Coverage, mean standard error (mean se), 95% confidence interval width (width), and standard error ratio (se ratio) relative to the true (empirical se) for the regular (non-cross-fit) versus cross-fit (CF) variance estimators for LTMLE at sample size N = 500. 33 [PIT…
Figure 12
Figure 12. Figure 12: Coverage, mean standard error (mean se), 95% confidence interval width (width), and standard error ratio (se ratio) relative to the true (empirical se) for the regular (non-cross-fit) versus cross-fit (CF) variance estimators for LTMLE at sample size N = 1,000. A.1.6 …
Figure 13
Figure 13. Figure 13: Coverage, mean standard error (mean se), 95% confidence interval width (width), and standard error ratio (se ratio) relative to the true (empirical se) for the regular (non-cross-fit) versus cross-fit (CF) variance estimators for LTMLE at sample size N = 3,000. A.1.7 …
Figure 14
Figure 14. Figure 14: Coverage, mean standard error (mean se), 95% confidence interval width (width), and standard error ratio (se ratio) relative to the true (empirical se) for the regular (non-cross-fit) versus cross-fit (CF) variance estimators for Hazard TMLE at sample size N = 500. 34…
Figure 15
Figure 15. Figure 15: Coverage, mean standard error (mean se), 95% confidence interval width (width), and standard error ratio (se ratio) relative to the true (empirical se) for the regular (non-cross-fit) versus cross-fit (CF) variance estimators for Hazard TMLE at sample size N = 1,000. …
Figure 16
Figure 16. Figure 16: Coverage, mean standard error (mean se), 95% confidence interval width (width), and standard error ratio (se ratio) relative to the true (empirical se) for the regular (non-cross-fit) versus cross-fit (CF) variance estimators for Hazard TMLE at sample size N = 3,000. …
Figure 17
Figure 17. Figure 17: Sampling distribution of the point estimate for IPCW-LTMLE, LTMLE, and Hazard TMLE at [PITH_FULL_IMAGE:figures/full_fig_p035_17.png]

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Pith tools

Reviewed July 12, 2026 · model on record in the stance chip above.