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Targeted Learning Estimation of Sampling Variance for Improved Inference

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Pith's one-line read The paper proposes a one-step targeted variance estimator whose confidence intervals for the log causal risk ratio reach nominal 0.95 coverage where the standard influence-function variance underestimates.

desk verdict Plausible new EIF-based variance estimator for log(CRR), but the paper overclaims asymptotic efficiency for the algorithm it actually implements, and the formulas need a close check before the method can be trusted. read the letter →

arxiv 2505.10624 v1 pith:BZJ7X3OQ submitted 2025-05-15 stat.ME

Yunwen Ji , Mark van der Laan , Alan Hubbard This is my paper · ORCID
classification stat.ME MSC 62G0562G2062P10
keywords targetedmaximumlikelihoodestimationinfluencefunctionvariancecausalriskratiopositivityviolationsone-stepTMLEuniversalleastfavorablesubmodelconfidenceintervalcoverage
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

Standard errors for causal-effect estimates are usually obtained as the sample variance of the estimated influence function, but these standard errors can be too small in small samples or when some treatment probabilities are near zero, producing confidence intervals with coverage far below the claimed 0.95. This paper treats the variance of the influence function itself as the target parameter and derives its efficient influence function, then estimates that variance with a one-step targeted maximum likelihood estimator (TMLE), a plug-in method that makes a small likelihood-based update to remove bias for a chosen target. The result is a plug-in variance estimator whose confidence intervals for the log causal risk ratio hold nominal coverage in simulations, and whose Type I error stays near 0.05 where the usual estimator's error inflates under positivity violations. A reader should care because reliable standard errors are what make causal conclusions trustworthy in the finite samples typical of observational studies.

What carries the argument

The load-bearing object is the efficient influence function of the variance parameter $\Sigma^2(P) = P\{D^*_{\Psi,P}\}^2$, decomposed orthogonally into components in the tangent spaces of the covariate distribution, the outcome regression, and the propensity score (Theorem 1, Eq. 9). It is used twice: its clever covariates $H_{\bar{Q}}$ and $H_g$ define a one-dimensional universal least favorable parametric submodel along which both the outcome regression and propensity score are fluctuated in small logistic steps, and solving its estimating equation is what makes the resulting variance plug-in estimator asymptotically efficient. The one-step TMLE is the engine that carries the argument: one targeted update, rather than iterative refitting, corrects bias for the variance without the additional variance inflation that the paper attributes to iterative TMLE.

What would settle it

Numerically verify Theorem 1 by finite-difference Gateaux differentiation of $\Sigma^2(P)$ along one-parameter submodels for the outcome regression and propensity score on a small simulated dataset; if the numerical derivative disagrees with Eq. 9, especially in the $Q_W$ component, the targeting step is misaligned. Independently run the paper's $n=100$, $\beta_p=1$, $\beta_{\psi_0}=0$ simulation 1000 times: if Wald intervals using the one-step variance estimator cover the true log(CRR) less than 90% of the time, the claimed restoration of nominal coverage fails.

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Extended reading notes

Core claim

The paper claims that the anti-conservatism of the standard influence-function variance estimator is not an inevitable small-sample artifact but a correctable bias: by defining the variance of the efficient influence function of log(CRR) as a statistical parameter and estimating it with a one-step TMLE, the variance estimator tracks the true sampling variance even when propensity scores approach 0 or 1. The central object is the analytic variance formula, which averages terms with 1/g(1|W) and 1/g(0|W) in the denominators, so rare treatment and outcome patterns are explicitly integrated rather than being missed through indicator functions. The one-step TMLE, built on the derived EIF, targets this variance directly using the universal least favorable submodel, avoiding the overfitting the authors attribute to iterative TMLE. Simulations across effect sizes 0, 0.5, and 2, sample sizes 100 to 1000, and six positivity levels show coverage near 0.95 and Type I error near 0.05, with the empirical estimator dropping below 0.85 coverage and inflating Type I error in extreme cases.

Load-bearing premise

The derived efficient influence function in Theorem 1 must be correct; the appendix obtains component influence functions from a book chapter and the printed equation contains apparent typographical slips (for example, a $\psi_1$ where $\psi_0$ is expected), so an error here would move the targeting update in the wrong direction and the claimed efficiency would collapse.

Editorial extensions

If this is right

  • Point estimates of log(CRR) from TMLE will come with standard errors that are not systematically too small when treatment probabilities approach zero.
  • Type I error rates for testing no causal effect will stay near 0.05 in small samples, instead of rising as positivity degrades.
  • The same derivation template—derive the EIF of the variance of the influence function and target it with a universal least favorable submodel—applies to other causal parameters such as treatment-specific means, risk differences, and odds ratios.
  • Because the estimator is a plug-in with an analytic form, it can be computed without iteratively refitting the data, avoiding the variance inflation associated with iterative TMLE.
  • The derived influence function of the variance also opens the possibility of constructing confidence intervals for the variance itself, which could be used in adaptive experiments to bound Type I error.

Reading between the lines

Editorial extensions of the paper, not claims the author makes directly.

  • Extension: the gap between the empirical influence-function variance and the targeted variance could serve as a data-driven positivity diagnostic—a large gap flags samples where standard inference is unreliable.
  • Extension: in rare-outcome longitudinal analyses where nonparametric bootstrap has been observed to beat the empirical influence-function variance, the one-step targeted variance may provide a computationally cheaper alternative without bootstrap resampling.
  • Extension: simultaneously targeting the causal estimand and its variance in a joint TMLE is a direct next step; if realized, it would give jointly efficient estimation and inference.
  • Extension: the ratio of the one-step TMLE variance to the empirical IF variance could be used to calibrate existing software outputs, giving applied users a finite-sample correction without changing their estimator.
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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

3 major / 6 minor

Summary. The paper proposes a one-step targeted maximum likelihood estimator (TMLE) for the variance of the efficient influence function of the log causal risk ratio (log(CRR)). Using a one-dimensional universally least favorable submodel, the authors update initial estimates of the outcome regression and propensity score, then plug the updated estimates into an analytic variance formula. The manuscript derives the efficient influence function of this variance parameter, states an asymptotic efficiency theorem, and compares the proposed estimator with the empirical influence-function variance, a simple substitution estimator, and an iterative TMLE in simulations across sample sizes, effect sizes, and positivity violations. The reported simulations show that confidence intervals using the one-step TMLE achieve coverage closer to the nominal 0.95 and lower Type-I error than the empirical influence-function variance estimator, especially in small samples and under near-positivity violations. A real-data-based simulation using the WASH Benefits Bangladesh study illustrates the method.

Significance. If the theoretical claims were fully established, the paper would provide a principled way to improve inference for causal risk ratios in finite samples and near-positivity settings, where the standard influence-function variance is known to be anti-conservative. The numerical results are plausible and the code is released, which is a strength. However, the central theoretical contribution is an asymptotic efficiency claim for the proposed one-step TMLE, and that claim is not currently supported: the algorithm as described does not solve the EIF estimating equation, the printed EIF formulas contain apparent typos, and the conditions of the efficiency theorem are not met by the simulation implementation. The finite-sample coverage improvements may well be real, but the manuscript does not yet substantiate the stated efficiency result, and the practical value depends on correcting the targeting step and verifying the formulas against the code.

major comments (3)
  1. [Section 3.2.4, Step 5; Theorem 2; Appendix C] The stopping criterion in Step 5, |P_n D^{(i)}_{Σ2,P_n}| > σ̂_n √n log n, is dimensionally inconsistent with the requirement P_n D*_{Σ2,P_n} = o_p(n^{-1/2}) that is used in the von Mises expansion in Appendix C. For a mean-zero function, the sample mean P_n D* has typical scale σ̂_n/√n, so the threshold is larger than the typical scale by a factor of order n log n. As a result, the loop will stop after one or very few dε = 0.001 updates, leaving P_n D* at O_p(n^{-1/2}) rather than o_p(n^{-1/2}). The implemented estimator therefore does not solve the EIF estimating equation, and Theorem 2 (Eq. 15) does not apply to the procedure as described. The authors should replace the stopping rule with one that targets |P_n D*| ≤ tol·n^{-1/2} and re-run the simulations and code checks.
  2. [Eq. (10) and Corollary 1 (Eqs. 12-14)] The EIF formulas as printed cannot serve as the validated basis for the targeting step. In Eq. (10), the last term of D*_{Σ2,QW,P} contains (Q̄_1/ψ_1 − Q̄_0/ψ_1)², where the second denominator should be ψ_0. More importantly, the clever covariates H_{1,Q̄,P} and H_{0,Q̄,P} in Eq. (14) are written with P_0 expectations rather than expectations under the current distribution P_ε; the universal least favorable submodel property in Eq. (19) requires the current distribution, so an implementation that literally uses P_0 would break the identity in Eq. (19). These appear to be correctable typos, but they must be fixed and verified against the released code before the targeting update can be trusted. The appendix also defers the component EIFs to a book chapter references, so the manuscript alone does not allow an independent check of the derivation.
  3. [Appendix C and Section 4.1] The remainder analysis in Appendix C.1 concludes that asymptotic efficiency requires both Q̄ and g to converge at an n^{-1/4} L2 rate, achievable by HAL, and the paper explicitly notes the estimator is not doubly robust. However, the simulation implementation uses SuperLearner libraries (GLM, elastic net, BART, GAM) rather than HAL, and no rate condition is verified for these libraries in the simulation settings. Consequently, the coverage and Type-I error results in Section 4.2 are finite-sample empirical findings, not consequences of Theorem 2; the manuscript should state this distinction clearly and either add HAL-based simulations or temper the asymptotic efficiency claim.
minor comments (6)
  1. [Eq. (6) and Eq. (7)] The second term in Eq. (6) and the second term in Eq. (7) are missing the square on ψ_0 in the denominator; compare with Eq. (4) where the factor is 1/(ψ_0)². In Eq. (7), the notation g*_n(0|W) should be g^{(**)}_n(0|W) for consistency.
  2. [Section 3.2.4] The step size dε = 0.001 is introduced without any sensitivity analysis or guidance on choosing it; the manuscript should report whether results are sensitive to this tuning choice.
  3. [References and citations] The introduction attributes the iterative TMLE for the variance of the treatment-specific mean to Coyle et al. and cites reference [9], but [9] is the van der Laan and Rose book; the specific chapter or paper by Coyle and colleagues should be cited directly.
  4. [Appendix B] The derivation says the component EIFs were obtained using the method in 'A.3 in the targeted learning book [10]', but reference [10] is the van der Laan and Gruber paper; the intended book reference should be corrected.
  5. [Table 1 and Section 4.1] Table 1 reports the proportion of observations with propensity scores truncated at 0.025, but the text in Section 4.1 says truncation is within [0.025, 0.975]; the table caption should refer to the estimated propensity scores rather than g_0 and should clarify the truncation interval.
  6. [Section 5] The real-data simulation says 10,000 repeated samples of size 500 are drawn, but the analysis is then described as 'based on 140 data samples'; the relationship between these numbers should be explained.

Circularity Check

0 steps flagged · score 0.0 of 10

No significant circularity: the variance estimator is a plug-in TMLE for a defined estimand, and the simulation comparisons use external Monte Carlo truth.

full rationale

The paper defines the target parameter Σ²(P) as the variance of the efficient influence function of log(CRR) (Eq. 4), derives the EIF of this parameter (Theorem 1), and constructs a one-step TMLE as a plug-in estimator with a targeting step (Eqs. 7, 16-19). Using the EIF of the target parameter to build an efficient estimator of that parameter is the standard TMLE construction, not a circular reduction: the EIF is the derivative of the same functional, and estimating a variance from its EIF is not 'predicting' the input. The claimed improved coverage and Type I error are evaluated in simulations relative to a Monte-Carlo approximation of Σ²(P0) (Section 4), which is external to the estimator's construction. The paper does rely on prior work by the same authors for the universal least favorable submodel [8] and for component EIFs [9,10], but those citations supply general methodology or previously derived EIFs for other parameters (treatment-specific means), not the paper's conclusion; they are not load-bearing in the sense of assuming the target result. The stopping-criterion mismatch with the Theorem 2 condition and apparent EIF typos are correctness risks, not circularity. No step reduces by definition, by fitted input, or by self-citation chain to its own inputs.

Assumptions & free parameters 4 free parameters · 5 assumptions · 0 invented entities

The method introduces no new physical entities or parameters beyond tuning constants. The load-bearing technical inputs are the nonparametric model, the standard causal assumptions for interpretation, the regularity conditions for TMLE efficiency, and the n^{-1/4} rate condition that is not satisfied by the simulation library. The paper does not disclose this mismatch between theory and simulation.

free parameters (4)
  • step_size_depsilon = 0.001
    Chosen step size for numerical integration of the universal least favorable submodel path; no theoretical justification is given and it affects the number of iterations and the final estimate.
  • propensity_truncation_sim = 0.025, 0.975
    Truncation bounds for estimated propensity scores in simulations, used to stabilize inverse weights in the variance formulas.
  • outcome_truncation_sim = 0.001, 0.999
    Truncation bounds for outcome regression estimates in simulations, used to keep predicted probabilities away from 0 and 1.
  • real_data_truncation = 0.036 (5/(sqrt(n) log(n)))
    Default truncation level in the tmle package used for the WASH Benefits data analysis.
assumptions (5)
  • domain assumption The statistical model M is nonparametric, so the tangent space is the full Hilbert space of mean-zero functions.
    Invoked in Section 3.2.3 to justify the orthogonal decomposition of the EIF (Eq. 9).
  • domain assumption Standard causal identification assumptions: no unmeasured confounding, positivity, and consistency.
    Stated in Section 2.1 to interpret the statistical parameter as a causal risk ratio. The authors note the variance estimator itself does not require these for construction.
  • standard math TMLE regularity conditions: Donsker class, second-order remainder o_p(n^{-1/2}), and L2 consistency.
    Assumed in Theorem 2 to establish asymptotic efficiency of the variance TMLE.
  • ad hoc to paper Initial estimators of Q and g converge to their true values at L2 rate n^{-1/4} or faster (achieved by HAL).
    Required in Appendix C.1.1 for the second-order remainder to vanish. The simulations use SuperLearner with GLM, BART, and XGBoost, which do not guarantee this rate.
  • standard math The universal least favorable submodel theory of van der Laan and Gruber (2016) is valid.
    Used in Section 3.2.4 as the basis for the one-step updating procedure; taken as a black box.

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Cite this review

Pith. "Pith review of Targeted Learning Estimation of Sampling Variance for Improved Inference." pith.science (2026). https://pith.science/paper/BZJ7X3OQ

@misc{pith2026250510624,
  author       = {Pith},
  title        = {Pith review of: Targeted Learning Estimation of Sampling Variance for Improved Inference},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/BZJ7X3OQ}},
  note         = {Machine review of arXiv:2505.10624}
}
read the original abstract

For robust statistical inference it is crucial to obtain a good estimator of the variance of the proposed estimator of the statistical estimand. A commonly used estimator of the variance for an asymptotically linear estimator is the sample variance of the estimated influence function. This estimator has been shown to be anti-conservative in limited samples or in the presence of near-positivity violations, leading to elevated Type-I error rates and poor coverage. In this paper, capitalizing on earlier attempts at targeted variance estimators, we propose a one-step targeted variance estimator for the causal risk ratio (CRR) in scenarios involving treatment, outcome, and baseline covariates. While our primary focus is on the variance of log(CRR), our methodology can be extended to other causal effect parameters. Specifically, we focus on the variance of the IF for the log relative risk (log(CRR)) estimator, which requires deriving the efficient influence function for the variance of the IF as the basis for constructing the estimator. Several methods are available to develop efficient estimators of asymptotically linear parameters. In this paper, we concentrate on the so-called one-step targeted maximum likelihood estimator, which is a substitution estimator that utilizes a one-dimensional universal least favorable parametric submodel when updating the distribution. We conduct simulations with different effect sizes, sample sizes and levels of positivity to compare the estimator with existing methods in terms of coverage and Type-I error. Simulation results demonstrate that, especially with small samples and near-positivity violations, the proposed variance estimator offers improved performance, achieving coverage closer to the nominal level of 0.95 and a lower Type-I error rate.

Figures

Figures reproduced from arXiv: 2505.10624 by the authors.

Figure 1
Figure 1. RMSE and absolute bias of each variance estimator compared to Σ [PITH_FULL_IMAGE:figures/full_fig_p011_1.png] view at source ↗
Figure 2
Figure 2. Coverage rates of the 95% CIs for each variance estimator for the TMLE of log(CRR) [PITH_FULL_IMAGE:figures/full_fig_p012_2.png] view at source ↗
Figure 3
Figure 3. Type I error rates for each variance estimator for the TMLE estimator of log(CRR) [PITH_FULL_IMAGE:figures/full_fig_p012_3.png] view at source ↗
Figures from the paper (10 more)
Figure 4
Figure 4. Figure 4: Distribution of estimates for each variance estimator from the WASH Benefits study. [PITH_FULL_IMAGE:figures/full_fig_p013_4.png]
Figure 5
Figure 5. Figure 5: Distribution of variance estimates of different variance estimators compared to the [PITH_FULL_IMAGE:figures/full_fig_p025_5.png]
Figure 6
Figure 6. Figure 6: Distribution of variance estimates of different variance estimators compared to the [PITH_FULL_IMAGE:figures/full_fig_p026_6.png]
Figure 7
Figure 7. Figure 7: RMSE and absolute bias of each variance estimator compared to Σ [PITH_FULL_IMAGE:figures/full_fig_p027_7.png]
Figure 8
Figure 8. Figure 8: Coverage rates for each variance estimator for the TMLE estimator of log(CRR) under [PITH_FULL_IMAGE:figures/full_fig_p027_8.png]
Figure 9
Figure 9. Figure 9: Type I error rates for each variance estimator for the TMLE estimator of log(CRR) [PITH_FULL_IMAGE:figures/full_fig_p028_9.png]
Figure 10
Figure 10. Figure 10: RMSE and absolute bias of each variance estimator compared to Σ [PITH_FULL_IMAGE:figures/full_fig_p029_10.png]
Figure 11
Figure 11. Figure 11: Distribution of variance estimates of different variance estimators compared to the [PITH_FULL_IMAGE:figures/full_fig_p029_11.png]
Figure 12
Figure 12. Figure 12: Distribution of variance estimates of different variance estimators compared to the [PITH_FULL_IMAGE:figures/full_fig_p030_12.png]
Figure 13
Figure 13. Figure 13: Coverage rates for each variance estimator for the TMLE estimator of log(CRR) [PITH_FULL_IMAGE:figures/full_fig_p030_13.png]

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