REVIEW 3 major objections 2 minor 3 cited by
Two-Sample Testing with Missing Data via Energy Distance: Weighting and Imputation Approaches
T0 review · 3 major / 2 minor · reviewed 2026-08-05 · deepseek-v4-flash
Pith's one-line read This paper claims that a weighted energy-distance test provides valid two-sample testing under missing data, with two resampling procedures and a special bootstrap for imputation-based tests.
desk verdict A plausible methods paper on energy-distance testing under missingness; the key question is whether weight estimation is accounted for in the null distribution. read the letter →
The pith
A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.
The reading
What carries the argument
The central object is the weighted energy distance statistic. The energy distance between two samples is computed from pairwise distances between observations, and the paper modifies it by assigning weights to each pair so that all available data contribute, with pairs weighted according to how likely they were to be observed. The analytic result is the asymptotic null distribution of this weighted statistic; it carries the argument because it justifies the two resampling procedures. A second mechanism is the imputation-specific bootstrap, which treats imputed values as random rather than fixed, so p-values reflect the extra uncertainty introduced by imputation.
What would settle it
A simulation in which missingness is set to depend on unobserved variables (missing not at random), so the assumed observation probabilities are misspecified; if the weighted test's type I error exceeds its nominal level by a non-negligible margin, the claim that the method works under a variety of missingness mechanisms would be falsified. An analytical derivation showing that the weighted statistic's expectation is nonzero under misspecified weights would also settle the question.
Extended reading notes
Core claim
The paper's central claim is that a weighted modification of the energy distance statistic—where pairwise contributions are weighted to compensate for missing observations—has a well-defined asymptotic null distribution under a range of missingness mechanisms. Based on this distribution, the authors construct two resampling methods that produce approximate p-values for the weighted test. For imputation-based tests, they develop a new bootstrap that accounts for the randomness introduced when missing values are filled in. An extensive simulation study compares complete-case, weighted, and imputation-based approaches, evaluating type I error control and statistical power across sample sizes, d
Load-bearing premise
The weighted test's null distribution is valid only if the weights reflect the true probabilities that observations are missing; if the missingness model is wrong, the test can reject too often.
Editorial extensions
If this is right
- A weighted energy distance test can be used in place of complete-case analysis, preserving information from partially observed records and potentially increasing power.
- The two resampling procedures give practitioners a way to compute p-values without relying on an unverified large-sample approximation.
- The imputation-specific bootstrap provides a calibration tool for tests on imputed data, addressing a gap in the standard imputation workflow.
- Simulation-based recommendations identify which test—complete-case, weighted, or imputation-based—should be preferred under different combinations of sample size, dimension, distribution, missingness mechanism, and missingness rate.
Reading between the lines
- The weighting scheme is likely transferable to other pairwise-distance two-sample statistics, such as kernel maximum mean discrepancy, because the same pairwise-sum structure appears there.
- If the weights are estimated from data rather than assumed known, the asymptotic null distribution may require an additional adjustment; the paper does not state whether its theory covers estimated weights, so extending it in that direction is a natural next step.
- The imputation-specific bootstrap could serve as a template for uncertainty quantification after imputation beyond testing, such as confidence intervals for a treatment effect.
- Because the weighted statistic is computed on all available pairs, its computational cost is similar to the complete-case version, which may make it scalable to large datasets with blockwise missingness.
Editorial analysis
A structured set of objections, weighed in public.
Referee Report
Summary. The paper addresses two-sample testing with energy distance in the presence of missing data. It considers the standard complete-case approach and proposes a modified test statistic that uses all available data through weighting. The authors state that they derive the asymptotic null distribution of the weighted statistic and propose two resampling procedures for p-values. They also propose a new bootstrap method for test statistics based on multiply imputed data. The claims are evaluated through simulations across sample sizes, dimensions, distributions, missingness mechanisms, and missingness rates, with general recommendations for practice.
Significance. If the claims are correct, the paper contributes a practical extension of the energy-distance two-sample test to missing-data settings, a common problem in applied statistics. A weighted complete-data test with a valid asymptotic null distribution and accompanying resampling procedures would be useful. The proposed bootstrap for imputation-based tests also addresses a gap, since standard bootstrap methods may not properly reflect imputation uncertainty. However, the validity of these contributions cannot be assessed from the abstract alone; no equations, proofs, or simulation results are available in the material under review.
major comments (3)
- [Abstract (first paragraph)] The central validity condition for the weighted test is not stated. The authors mention 'a variety of missingness mechanisms' and 'appropriate weights,' but do not specify whether the missingness is assumed MAR/MCAR, whether the weights are known or must be estimated, and if estimated, whether the asymptotic null distribution accounts for estimation error. The load-bearing assumption is that the weights represent (or consistently estimate) the true selection probabilities; without this, the weighted statistic is biased under the null. The manuscript should state these assumptions explicitly and, if weights are estimated, indicate how the asymptotic distribution is adjusted.
- [Abstract (second paragraph)] The two resampling procedures for the weighted statistic are not described, so it is unclear whether they re-estimate the weights in each bootstrap replicate. If the bootstrap resamples observed data without re-estimating the weights, the resulting p-values may be anti-conservative even under correctly specified weights, because the variability of weight estimation is ignored. The proposed method must be specified in enough detail to determine whether this issue is addressed.
- [Abstract (general)] The manuscript as provided is an abstract only; there are no equations, theorem statements, proofs, or simulation tables. The claims that the asymptotic null distribution is derived and that simulations support the recommendations cannot be checked. For a rigorous review, the full text is required. In particular, the expression for the asymptotic null distribution and representative type I error/power tables are necessary to verify the central claims.
minor comments (2)
- [Abstract (first paragraph)] The phrase 'appropriate weights' is vague. A brief statement about the form of the weights (e.g., inverse probability weights based on a missingness model) would improve clarity.
- [Abstract (second paragraph)] The abstract mentions 'a variety of missingness mechanisms' but does not list them. Specifying whether MNAR is included would help the reader assess the generality of the recommendations.
Circularity Check
No circularity identified from abstract-only review; derivation claims are about asymptotic null distributions and resampling, not fits or self-citations.
full rationale
The available text is the abstract only. It claims a weighted modification of the energy-distance statistic, derivation of its asymptotic null distribution, and two resampling procedures, plus a bootstrap for imputation-completed samples. Nothing in the abstract indicates that the proposed statistic or its null distribution is defined in terms of the very quantity it is meant to predict or test. There is no fitted parameter renamed as a prediction, no load-bearing self-citation, and no uniqueness theorem imported from the authors' prior work. The only substantive caveat—that validity depends on correct specification of the missingness mechanism—is a correctness/assumption risk, not a circularity argument. Under the hard rules, concerns about untested assumptions or the absence of full-text derivations do not constitute exhibited circular steps, so the honest finding is no significant circularity with score 0.
Assumptions & free parameters
assumptions (3)
- standard math Energy distance is a valid metric for distributions
- domain assumption Missingness mechanisms are correctly specified or consistently estimated
- standard math Asymptotic normality of U-statistics under the null
Cite this review
Pith. "Pith review of Two-Sample Testing with Missing Data via Energy Distance: Weighting and Imputation Approaches." pith.science (2026). https://pith.science/paper/EBTVZZOE
@misc{pith2026250811421,
author = {Pith},
title = {Pith review of: Two-Sample Testing with Missing Data via Energy Distance: Weighting and Imputation Approaches},
year = {2026},
howpublished = {\url{https://pith.science/paper/EBTVZZOE}},
note = {Machine review of arXiv:2508.11421}
}
read the original abstract
In this paper, we address the problem of two-sample testing in the presence of missing data under a variety of missingness mechanisms. Our focus is on the well-known energy distance-based two-sample test. In addition to the standard complete-case approach, we propose a modification of the test statistic that incorporates all available data, utilizing appropriate weights. The asymptotic null distribution of the test statistic is derived and two resampling procedures for approximating the corresponding p-values are proposed. We also propose a new bootstrap method specifically designed for a test statistic based on samples completed via common imputation methods. Through an extensive simulation study, we compare all methods in terms of type I error control and statistical power across a set of sample sizes, dimensions, distributions, missingness mechanisms, and missingness rates. Based on these results, we provide general recommendations for each considered scenario.
Forward citations
Cited by 3 Pith papers
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A unified U-statistic framework tests equality of smooth parameters across k populations via Wald and ANOVA statistics, with fixed-d asymptotics, weighted bootstrap, and normal limits when d grows slower than n.
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Testing independence in the presence of missing data: high-dimensional case
Two new modifications to a Kendall tau-based test are proposed and analyzed for independence testing in high-dimensional data with missing observations, backed by theory and simulations.
Reviewed August 5, 2026 · model on record in the stance chip above.
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