A Direct Variance Estimation (DiVE) for Meta-Analysis of Median Differences

Kenichi Hayashi; Masataka Taguri; Tadahisa Okuda

arxiv: 2605.23208 · v1 · pith:3A37UIR3new · submitted 2026-05-22 · 📊 stat.ME · stat.AP

A Direct Variance Estimation (DiVE) for Meta-Analysis of Median Differences

Tadahisa Okuda , Masataka Taguri , Kenichi Hayashi This is my paper

Pith reviewed 2026-05-25 04:08 UTC · model grok-4.3

classification 📊 stat.ME stat.AP

keywords meta-analysismedian differencevariance estimationdirect estimationsimulation studytwo-stage methodspooled estimate

0 comments

The pith

DiVE directly estimates the variance of a pooled median difference from study medians and sample sizes alone.

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

Meta-analyses of two-group studies reporting median differences often drop studies that lack quartiles or ranges because standard methods need those extra dispersion numbers to compute variances. DiVE bypasses this requirement by calculating the variance of the overall pooled difference straight from the reported median differences and the sample sizes of each study. Simulations covering many distributional shapes show the new estimator matches or beats the usual two-stage procedures and gains an edge when only a handful of studies are available. Re-analyses of existing meta-analyses confirm that DiVE lets previously excluded studies re-enter the synthesis.

Core claim

DiVE directly estimates the variance of the pooled median difference using only study-level median differences and their sample sizes. This avoids the need for dispersion measures such as quartiles or ranges that two-stage methods require. A comprehensive simulation study across a wide range of distributional scenarios shows that DiVE performs comparably to or better than conventional two-stage methods, with clear advantages when the number of studies is small. A re-analysis of published meta-analyses demonstrates that DiVE enables the inclusion of studies lacking dispersion statistics, leading to a more comprehensive and potentially less biased synthesis of evidence.

What carries the argument

The Direct Variance Estimation (DiVE) formula that computes the variance of the pooled median difference directly from the vector of study medians and study sample sizes.

If this is right

Studies lacking quartiles or ranges can now be retained rather than dropped from the meta-analysis.
Pooled estimates become feasible from a larger and potentially less selected set of evidence.
Performance gains appear most clearly when the total number of studies is small.
The same median-difference data can be re-analyzed under both DiVE and two-stage methods for comparison.

Where Pith is reading between the lines

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

DiVE could be adapted to other location measures that are commonly reported without accompanying scale statistics.
Journals might begin to require only medians and sample sizes for certain trial summaries if DiVE becomes standard.
Past meta-analyses that excluded many studies for missing dispersion data could be revisited with the new estimator.
A practical next step would be to apply DiVE to a registry of published median-difference trials and compare the resulting pooled intervals with those from restricted two-stage analyses.

Load-bearing premise

The direct variance formula remains valid and unbiased across the range of real-world distributions and study sizes where median differences are reported, without requiring the dispersion statistics that two-stage methods use.

What would settle it

A simulation in which DiVE produces empirical coverage rates for the pooled median difference that fall well below the nominal level in heavy-tailed distributions or with very small per-study sample sizes.

Figures

Figures reproduced from arXiv: 2605.23208 by Kenichi Hayashi, Masataka Taguri, Tadahisa Okuda.

**Figure 2.** Figure 2: Distribution of relative errors under the challenging skewed outcome scenario (log [PITH_FULL_IMAGE:figures/full_fig_p009_2.png] view at source ↗

read the original abstract

Meta-analyses of two-group studies that report median differences typically rely on methods that require, in addition to the median difference and sample size, summary measures of dispersion such as quartiles or ranges. Studies that do not report such statistics are often excluded from the meta-analysis. Existing two-stage approaches first estimate the asymptotic variance of the median difference within each study under parametric assumptions, and then combine these study-specific estimates to obtain the pooled median difference and its variance. We propose Direct Variance Estimation (DiVE), a method that directly estimates the variance of the pooled difference using only study-level median differences and their sample sizes. A comprehensive simulation study across a wide range of distributional scenarios shows that DiVE performs comparably to or better than conventional two-stage methods, with clear advantages when the number of studies is small. A re-analysis of published meta-analyses demonstrates that DiVE enables the inclusion of studies lacking dispersion statistics, leading to a more comprehensive and potentially less biased synthesis of evidence.

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.

Desk Editor's Note private letter to a colleague

DiVE claims a direct variance estimate from medians and ns alone, but that core step looks difficult to justify without assumptions or a proxy that mixes heterogeneity into the variance.

read the letter

The main thing to know is that this paper proposes DiVE as a way to estimate the variance of a pooled median difference using only the study medians and sample sizes, without any dispersion statistics. That would let more studies stay in a meta-analysis, which is a practical win if the method holds up. They position it as bypassing the usual two-stage process that first estimates per-study variances under parametric assumptions. The simulations across various distributions report performance that matches or exceeds the standard approaches, with an edge when the number of studies is small. The re-analysis section shows it pulling in extra studies that lack quartiles or ranges. That part is straightforward and addresses a real exclusion issue in applied work. The soft spot sits at the variance formula. The asymptotic variance of a median difference requires the density at the median, which cannot be recovered from the median value and n without either a shape assumption or using spread across studies as a stand-in for within-study dispersion. The second option would fold heterogeneity into the variance estimate, and that proxy becomes unstable exactly when k is small—the case where they highlight the biggest gains. If the simulations held dispersion constant across studies, they would miss this. This is for applied statisticians and meta-analysts working with skewed outcomes in clinical or similar data. A reader who routinely drops studies for missing dispersion stats might find the idea worth examining, but they would need to check the actual derivation before using it. It deserves peer review. The problem is common enough and the simulations provide some concrete evidence, so referees can test whether the direct formula stands up on its own terms.

Referee Report

3 major / 2 minor

Summary. The manuscript introduces Direct Variance Estimation (DiVE) for meta-analysis of two-group median differences. Unlike conventional two-stage methods that first estimate study-specific asymptotic variances (requiring dispersion statistics such as quartiles) and then pool, DiVE claims to estimate the variance of the pooled median difference directly from the vector of study medians and sample sizes alone. Simulations across distributional scenarios are reported to show DiVE performs comparably or better than two-stage estimators, with advantages at small k; a re-analysis of published meta-analyses is used to illustrate that DiVE permits inclusion of studies lacking dispersion data.

Significance. If the direct estimator is unbiased and its performance generalizes beyond the simulated designs, the method would allow meta-analysts to retain studies that currently must be dropped for lack of quartiles or ranges, thereby reducing selection bias and increasing the number of includable studies, especially in small-k settings common in median-difference meta-analyses.

major comments (3)

[§2, Eq. (4)] §2, Eq. (4): the claimed direct variance formula is presented without an explicit derivation showing how the term involving the unknown density f(m) at the median is recovered from medians and n_i alone; the asymptotic variance 1/(4n f(m)^2) per arm cannot be obtained from the median value and sample size without either a shape assumption or an implicit proxy that uses cross-study median variation, which would conflate sampling variance with heterogeneity.
[§4] Simulation design (described in §4): all scenarios appear to hold the underlying dispersion (hence f(m)) constant across studies within each replicate; this does not test the heterogeneous-scale case that would be required to establish robustness when real-world studies report medians on different measurement scales or with different spreads.
[Table 3] Table 3, k=5 rows: the reported coverage probabilities and MSE advantages for DiVE are shown only under the constant-dispersion design; without results under scale heterogeneity, the claim of “clear advantages when the number of studies is small” remains conditional on an assumption that may not hold in the target application.

minor comments (2)

[§2] Notation for the pooled estimator and its variance should be introduced once in §2 and used consistently thereafter; occasional reuse of “V” for both study-specific and pooled quantities is confusing.
[§5] The re-analysis section would benefit from a table listing, for each meta-analysis, how many additional studies are recovered by DiVE and the resulting change in the pooled point estimate and CI width.

Simulated Author's Rebuttal

3 responses · 0 unresolved

We thank the referee for the constructive comments on our manuscript. We address each major comment below and indicate where revisions will be made to improve clarity and robustness.

read point-by-point responses

Referee: [§2, Eq. (4)] §2, Eq. (4): the claimed direct variance formula is presented without an explicit derivation showing how the term involving the unknown density f(m) at the median is recovered from medians and n_i alone; the asymptotic variance 1/(4n f(m)^2) per arm cannot be obtained from the median value and sample size without either a shape assumption or an implicit proxy that uses cross-study median variation, which would conflate sampling variance with heterogeneity.

Authors: We agree that an explicit derivation of Equation (4) is required. In the revised manuscript we will insert a full derivation that shows how the density term is recovered under the fixed-effect model using only the vector of medians and the n_i values. The approach relies on the fact that, when all studies share the same underlying distribution, the observed dispersion among the study medians supplies an estimate of the common within-study variance component; we will explicitly state that this construction assumes absence of heterogeneity and will add a paragraph discussing the consequences if that assumption is violated. revision: yes
Referee: [§4] Simulation design (described in §4): all scenarios appear to hold the underlying dispersion (hence f(m)) constant across studies within each replicate; this does not test the heterogeneous-scale case that would be required to establish robustness when real-world studies report medians on different measurement scales or with different spreads.

Authors: The referee is correct that the reported simulations maintain constant dispersion within each replicate. This design isolates performance under the model assumptions used to derive DiVE. To address the concern, the revised manuscript will include an additional set of simulations in which scale (and therefore f(m)) varies across studies within each replicate, allowing direct assessment of robustness under heterogeneous dispersion. revision: yes
Referee: [Table 3] Table 3, k=5 rows: the reported coverage probabilities and MSE advantages for DiVE are shown only under the constant-dispersion design; without results under scale heterogeneity, the claim of “clear advantages when the number of studies is small” remains conditional on an assumption that may not hold in the target application.

Authors: We acknowledge that the advantages reported for small k in Table 3 are obtained under constant dispersion. In the revision we will either qualify the claim to reflect this scope or, preferably, augment Table 3 (or add a supplementary table) with the corresponding results from the new heterogeneous-scale simulations, thereby making the small-k advantage claim less conditional. revision: partial

Circularity Check

0 steps flagged

No circularity detected; derivation is self-contained

full rationale

The paper derives DiVE as a standalone formula for the variance of the pooled median difference that uses only the reported medians and sample sizes per study, without fitting parameters to the target pooled quantity or invoking self-citations for uniqueness. No equations reduce the claimed variance estimator to a re-expression of the input medians by construction, nor does any simulation step serve as the derivation itself. The method is presented as mathematically independent of two-stage dispersion-based estimators, and external validation via simulation does not create a fitted-input loop. This is the normal case of a self-contained statistical derivation.

Axiom & Free-Parameter Ledger

0 free parameters · 1 axioms · 0 invented entities

The method rests on the assumption that a direct formula for pooled variance exists and is estimable solely from medians and sample sizes; no free parameters or invented entities are mentioned in the abstract.

axioms (1)

domain assumption The variance of the pooled median difference estimator can be obtained directly from study-level medians and sample sizes without intermediate per-study variance estimates.
This is the defining premise of the DiVE approach as stated in the abstract.

pith-pipeline@v0.9.0 · 5700 in / 1182 out tokens · 21156 ms · 2026-05-25T04:08:57.894462+00:00 · methodology

discussion (0)

Reference graph

Works this paper leans on

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LetR denote the number of replicates per design setting. Within a replicate and studyi, letσ 2 i be the theoretical variance of the study-level median difference, computed from the population densities at the group-specific medians and the real- ized sample sizes (standard large-sample formula; see McGrath et al. (2020)). Given a target heterogeneityI 2,τ...

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Bahadur, R. R. (1966). A note on quantiles in large samples.Ann. Math. Stat.37 (3), 577–580. Bland, M. (2014). Estimating mean and standard deviation from the sample size, three quartiles, minimum, and maximum.Int. J. Stat. Med. Res.4 (1), 57–64. Borenstein, M., Hedges, L. V., Higgins, J. P. T., and Rothstein, H. R. (2009).Introduction to meta-analysis. H...

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Calatayud, D., Arora, S., Aggarwal, R., Kruglikova, I., Schulze, S., Funch-Jensen, P., and Grantcharov, T. (2010). Warm-up in a virtual reality environment improves performance in the operating room.Ann. Surg.251 (6), 1181–1185. Chowdhry, A. K., Dworkin, R. H., and McDermott, M. P. (2016). Meta-analysis with missing study-level sample variance data.Stat. ...

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and Laird, N

DerSimonian, R. and Laird, N. (1986). Meta-analysis in clinical trials.Control. Clin. Trials7 (3), 177–188. Desender, L., Van Herzeele, I., Lachat, M., Duchateau, J., Bicknell, C., Teijink, J., Heyligers, J., Vermassen, F., and PAVLOV Study Group (2017). A multicentre trial of patient specific rehearsal prior to EVAR: impact on procedural planning and tea...

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J., Gottschalk, C., Grafeneder, J., Schmitz, S., Kraker, S., Ganslmeier, M., Muth, A., Seitel, A., Maier-Hein, L., Benedetti, A., Larmann, J., Weigand, M

Katzenschlager, S., Zimmer, A. J., Gottschalk, C., Grafeneder, J., Schmitz, S., Kraker, S., Ganslmeier, M., Muth, A., Seitel, A., Maier-Hein, L., Benedetti, A., Larmann, J., Weigand, M. A., McGrath, S., and Denkinger, C. M. (2021). Can we predict the severe course of COVID-19 - a systematic review and meta-analysis of indicators of clinical outcome?PLoS O...

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Early supported dis- charge services for people with acute stroke.Cochrane Database Syst

Langhorne, P., Baylan, S., and Early Supported Discharge Trialists (2017). Early supported dis- charge services for people with acute stroke.Cochrane Database Syst. Rev.7 (7), CD000443. Lehmann, E. L. and Casella, G. (1998).Theory of point estimation. 2nd ed. New York, NY: Springer. Luo, D., Wan, X., Liu, J., and Tong, T. (2018). Optimally estimating the ...

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J., McKenzie, J

Page, M. J., McKenzie, J. E., Bossuyt, P. M., Boutron, I., Hoffmann, T. C., Mulrow, C. D., Shamseer, L., Tetzlaff, J. M., Akl, E. A., Brennan, S. E., Chou, R., Glanville, J., Grimshaw, J. M., Hr´ objartsson, A., Lalu, M. M., Li, T., Loder, E. W., Mayo-Wilson, E., McDonald, S., McGuinness, L. A., Stewart, L. A., Thomas, J., Tricco, A. C., Welch, V. A., Whi...

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J., Lindsley, K

Saldanha, I. J., Lindsley, K. B., Money, S., Kimmel, H. J., Smith, B. T., and Dickersin, K. (2020). Outcome choice and definition in systematic reviews leads to few eligible studies included in meta-analyses: a case study.BMC Med. Res. Methodol.20 (1),

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M., Grantcharov, T

Shore, E. M., Grantcharov, T. P., Husslein, H., Shirreff, L., Dedy, N. J., McDermott, C. D., and Lefebvre, G. G. (2016). Validating a standardized laparoscopy curriculum for gynecology residents: a randomized controlled trial.Am. J. Obstet. Gynecol.215 (2), 204.e1–204.e11. Sutton, A. J. and Higgins, J. P. T. (2008). Recent developments in meta-analysis.St...

work page 2016

[9] [9]

J., Butcher, I., Assi, V., Lewis, S

Weir, C. J., Butcher, I., Assi, V., Lewis, S. C., Murray, G. D., Langhorne, P., and Brady, M. C. (2018). Dealing with missing standard deviation and mean values in meta-analysis of continuous outcomes: a systematic review.BMC Med. Res. Methodol.18 (1),

work page 2018

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T., Conteh, L., Cibulskis, R., and Ghani, A

White, M. T., Conteh, L., Cibulskis, R., and Ghani, A. C. (2011). Costs and cost-effectiveness of malaria control interventions: a systematic review.Malar. J.10 (1),

work page 2011

[11] [11]

W., Klassen, T

Wiebe, N., Vandermeer, B., Platt, R. W., Klassen, T. P., Moher, D., and Barrowman, N. J. (2006). A systematic review identifies a lack of standardization in methods for handling missing variance data.J. Clin. Epidemiol.59 (4), 342–353. 17 Appendix A Direct estimator forVar(ˆµ s): derivation and proof A.1 Notation and assumptions Recall ˆµs = PN i=1 ˜ws i ...

work page 2006

[12] [12]

LetR denote the number of replicates per design setting. Within a replicate and studyi, letσ 2 i be the theoretical variance of the study-level median difference, computed from the population densities at the group-specific medians and the real- ized sample sizes (standard large-sample formula; see McGrath et al. (2020)). Given a target heterogeneityI 2,τ...

work page 2020

[13] [13]

It is set to a pre-specified non-zero constant in the scenarios, following the mean-shift approach of McGrath et al. (2020). Group 2 receives no shift, so the true median difference equalsc. Skew-normal parameters follow the direct parameterisation (location, scale, shape). Log-normal parameters are on the natural-log scale. 23 I² = 0% I² = 25% I² = 50% I...

work page 2020