Pith. sign in

REVIEW 4 major objections 4 minor 28 references

Estimating Signal-to-Noise Ratios for Multivariate High-dimensional Linear Models

T0 review · 4 major / 4 minor · reviewed 2026-08-07 · deepseek-v4-flash

Challenge this review Re-run · record.json Download PDF Read on arXiv ↗

Pith's one-line read This paper claims that the method-of-moments SNR estimator $\hat r^2 = \hat\rho^2/(\hat\rho^2+\hat\sigma^2)$ is asymptotically normal for multivariate fixed-effects and random-effects high-dimensional linear models, with explicit variance…

desk verdict The multivariate fixed-effects SNR CLT has a normalization error; the random-effects theory is the stronger half. read the letter →

arxiv 2506.10370 v1 pith:EARXCOOH submitted 2025-06-12 math.ST stat.MEstat.TH

Xiaohan Hu , Zhentao Li , Xiaodong Li This is my paper · ORCID
classification math.STstat.MEstat.TH MSC 62H1262J0562E2060B20
keywords signal-to-noiseratioestimationmethodofmomentsmultivariatelinearmodelheritabilityhigh-dimensionalasymptoticsrandomeffectsheteroskedasticnoiseWishart
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

This paper aims to extend method-of-moments estimation of the signal-to-noise ratio, $r^2 = \rho^2/(\rho^2+\sigma^2)$, from univariate to multivariate high-dimensional linear models, and to give the extension a full asymptotic inference theory. Its central claim is that the plug-in estimator $\hat r^2 = \hat\rho^2/(\hat\rho^2+\hat\sigma^2)$ is asymptotically normal at the $n^{1/2}$ rate, under fixed effects with Gaussian design, under random effects with general predictor covariance, and under heteroskedastic Gaussian noise, with an explicit variance in each case. If correct, the result converts multivariate SNR estimates, such as heritability of multiple correlated phenotypes in genomics, into fully quantified estimates with valid confidence intervals. The variance formulas make precise how the response covariance structure and, in the heteroskedastic case, the total degree of noise heterogeneity drive the estimator's uncertainty.

What carries the argument

The argument runs on the pair of method-of-moments estimators $(\hat\sigma^2, \hat\rho^2)$, each a linear combination of the two quadratic forms $\mathrm{tr}(Y^\top Y)$ and $\mathrm{tr}(Y^\top X X^\top Y)$; the fixed-effects coefficients are constants, while the random-effects coefficients are rational functions of the sample spectral moments $\hat g_k = \frac{1}{p}\mathrm{tr}(S_n^k)$ of the design, which converge to deterministic limits $g_k$ from random matrix theory. The proof has two load-bearing steps. First, Wishart cross-moment identities give the exact covariance of the two statistics: Lemma 6.6 extends the univariate moment results to the case $\alpha \neq \beta$ using the invariant moment formula for the Wishart distribution, and this is what accounts for the response correlation structure in the variance $V$. Second, a total-variation normal approximation converts that covariance into the central limit theorem: for the fixed-effects case the paper asserts the same $O(n^{-1/2})$ total-variation bound proved in the univariate theory, and for the random-effects case a Stein-type bound conditional on the design controls the approximation. For heteroskedasticity, leave-one-out analysis of the design shows the variance depends on the noise only through the average $\bar\Sigma_e$ and the scalar $\kappa_{\mathrm{tot}} = \sum_{k,l} \kappa_{kl}$.

What would settle it

Simulate the fixed-effects model with $q \geq 2$ responses, Gaussian design with $p/n \to \tau$, and fixed $B$ and $\Sigma_e$, then track the empirical distribution of $n^{1/2}(\hat r^2 - r^2)/\hat\sigma_r$ as $n$ grows: if the distribution fails to approach the standard normal, or the finite-sample covariance of $(\hat\sigma^2, \hat\rho^2)$ does not approach the $V$ of Theorem 3.1, the unproved total-variation step is the point of failure. A sharper check computes the Stein-type bound on the total-variation distance for the bivariate statistic directly and tests whether the $O(n^{-1/2})$ decay asserted at the end of Section 6.2 actually holds.

Watch

Extended reading notes

Core claim

This paper claims that the method-of-moments SNR estimator $\hat r^2 = \hat\rho^2/(\hat\rho^2+\hat\sigma^2)$, formed from unbiased moment estimates of the signal variance $\rho^2$ and noise variance $\sigma^2$, is asymptotically normal in the multivariate linear model $Y = XB + E$. Theorem 3.1 gives $n^{1/2}(\hat r^2 - r^2)/\sigma_r \Rightarrow N(0,1)$ under fixed effects, Theorem 3.2 under random effects, and Theorem 4.1 under heteroskedastic noise, with $\sigma_r^2$ defined explicitly in each case from the displayed $2\times2$ covariance matrix $V$ of $(\hat\sigma^2, \hat\rho^2)$. The $q=1$ case recovers the univariate estimator's known asymptotic behavior, and the fixed-effects theory imposes no sparsity on the coefficient matrix. In the heteroskedastic setting the noise covariance enters the variance only through its average $\bar\Sigma_e$ plus a single scalar $\kappa_{\mathrm{tot}}$ measuring total heterogeneity; the homogeneous case has $\kappa_{\mathrm{tot}}=0$ and reduces to Theorem 3.2. A further consistency result, Theorem 4.2, shows that plugging covariance estimates into the variance formulas is legitimate when $q = o(n)$.

Load-bearing premise

The proof of the fixed-effects central limit theorem (Theorem 3.1) assumes, without deriving it, that the two-dimensional statistic $(\hat\sigma^2, \hat\rho^2)$ satisfies the same $O(n^{-1/2})$ total-variation normal approximation that was proved for the univariate estimator, and this unproved step is what turns the computed covariance into the CLT.

Editorial extensions

If this is right

  • Plug-in standard errors from the explicit variance formulas give asymptotically valid confidence intervals for the multivariate SNR $r^2$, so heritability of multivariate phenotypes can be reported with uncertainty bounds.
  • The random-effects theory allows the number of responses $q$ to diverge, so the estimator scales to large phenotyping panels rather than a single trait.
  • Heteroskedastic noise, including heavy-tailed errors, Gaussian mixtures, and subgroup-structured noise, needs no new estimator: the same $\hat r^2$ stays asymptotically normal, with the variance formula gaining only the $\kappa_{\mathrm{tot}}$ heterogeneity term.
  • The fixed-effects results hold without any sparsity assumption on the coefficient matrix $B$, covering both dense and sparse signal regimes.

Reading between the lines

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

  • A Wald test comparing SNR estimates across groups or across trait sets could be built directly from the explicit variances given here, although the paper does not discuss such tests.
  • The clean coverage seen under SNP-style non-Gaussian designs in the simulations suggests the Gaussian-design condition in Theorem 3.1 may be removable; a sub-Gaussian proof would plausibly preserve the same variance formula.
  • For fully general heteroskedasticity the paper estimates $\kappa_{\mathrm{tot}}$ only in structured models (scalar heterogeneity, subgroups); whether $\kappa_{\mathrm{tot}}$ is identifiable for arbitrary $\Sigma_i$ marks the boundary between SNR estimation and noise-model estimation, a question the paper leaves open.
Share X Bluesky LinkedIn Reddit HN

Editorial analysis

A structured set of objections, weighed in public.

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

Referee Report

4 major / 4 minor

Summary. The paper extends method-of-moments SNR estimation to multivariate high-dimensional linear models under both fixed and random effects. It defines the SNR as r² = ρ²/(ρ²+σ²), proposes estimators (σ̂², ρ̂²) based on quadratic-form moment equations, and states asymptotic normality results with explicit covariance matrices in Theorem 3.1 (fixed effects), Theorem 3.2 (random effects), and Theorem 4.1 (heteroskedastic random effects). The paper also provides plug-in standard-error procedures, an operator-norm consistency result for covariance estimators under q = o(n) in Theorem 4.2, and extensive simulations for Gaussian and SNP-like designs, sparse and dense coefficients, and several noise heterogeneity models.

Significance. If the results hold, the paper provides a useful multivariate extension of Dicker's univariate SNR estimator, with explicit variance formulas that could support heritability-type inference on multiple correlated phenotypes. The random-effects and heteroskedastic parts are substantially worked out: the proofs use a published quadratic-form Gaussian approximation lemma, and the variance formulas are parameter-free in the sense that no fitted constants enter the derivation. The simulation study is broad and generally reports good coverage. However, the fixed-effects central limit theorem currently contains a demonstrable normalization error and an unsupported total-variation step, so the central fixed-effects claim needs substantial repair before the results can be relied upon.

major comments (4)
  1. [Theorem 3.1; Eq. (3.1); Lemma 6.7] The normalization in Theorem 3.1 is incorrect. The displayed V is the finite-sample covariance matrix of (σ̂², ρ̂²): from Lemma 6.7 and (6.4), V11 = (2 + o(1))/(q²(n+1)² n){(n²+np)∥BᵀB∥²_F + ...} = Θ(1/n), and similarly for the other entries. Hence V^{-1/2} = Θ(n^{1/2}), so n^{1/2}V^{-1/2}(θ̂-θ) = Θ(n) · O_P(n^{-1/2}) = O_P(n^{1/2}), which diverges. The subsequent statement n^{1/2}(r̂²-r²)/σ_r with σ_r² built from (3.1) has the same problem: σ_r = Θ(n^{-1/2}) makes the normalized statistic O_P(n^{1/2}). The theorem becomes correct only if either the n^{1/2} multipliers are removed (with V and σ_r² understood as finite-sample variances) or V is redefined as n times the displayed matrix, so that V = Θ(1) is the asymptotic covariance of n^{1/2}(θ̂-θ). This is a load-bearing error, not a typo: any implementation following the printed formula would produce z-statistics inflated by a factor of roughly n^{1/2}.
  2. [Section 6.2, proof of Theorem 3.1] The CLT for (σ̂², ρ̂²) is not proved. After deriving the covariance in (6.4), the proof states 'By an argument similar to that used in the proof of Theorem 1 in Dicker [2014] ... d_TV(h(σ̂², ρ̂²), w) = O(n^{-1/2})' and then asserts asymptotic normality. Dicker's theorem is for a univariate estimator; here the joint law of two quadratic-form statistics with Wishart cross-moments is needed. Lemma 6.6 provides second moments only, and moments do not imply convergence in distribution. The paper needs either a direct CLT argument (e.g., Stein/Lindeberg or the Lemma 6.4 approximation used in Section 6.3) or a precise reduction to a published multivariate result. As it stands, Theorem 3.1 lacks support exactly at the step that turns the covariance computation into a CLT.
  3. [Theorem 3.2 and Section 6.3, asymptotic distribution paragraph] The proof of Theorems 3.2 and 4.1 requires the asymptotic covariance V to be nonsingular and bounded away from zero, but the invertibility condition is asserted rather than assumed. The text says 'Assumption 2.1 guarantees g2g4 - g3², g1g3 - g2² and g2 - g1² are lower bounded...', while Assumptions 2.1 and 2.4 do not exist in the paper; the assumptions are numbered 3.1–4.1. A precise non-degeneracy condition on the limiting spectral distribution H and τ, together with a proof (or a reference) that V_n is uniformly positive definite, must be added before V^{-1/2} is well-defined in the statement.
  4. [Remark 2 and Section 5.1] The paper does not establish consistency of the plug-in variance estimator in the fixed-effects setting. The matrix V in Theorem 3.1 depends on ∥BᵀB∥²_F, tr(ΣeBᵀB), and tr(Σe²). Remark 2 proposes to estimate these from ĉWb and bΣe in (2.2)–(2.3), but no theorem shows that these matrix estimators are consistent in the required norms for fixed q and p/n → τ. By contrast, the random-effects case has Theorem 4.2, but there is no fixed-effects analogue. Without such a result, the claimed confidence intervals in Tables 1–2 are not formally justified, though the simulations suggest the desired behavior.
minor comments (4)
  1. [Throughout] There are several typos: 'symmtric' in Theorem 3.2, 'entris' in Lemma 6.5, 'heterogerous' in the Introduction, and 'mutivariate' in Section 4.
  2. [Table 4, t7 row (500, 1000)] The empirical standard error is reported as 0.473 while the average estimated standard error is 4.45; the decimal is almost certainly misplaced (it should be 4.73). Please correct this entry.
  3. [Section 6.2] The notation d_TV(h(σ̂², ρ̂²)), w) is malformed. Also, h is introduced as a twice continuously differentiable function h : R → R, but it is applied to the bivariate vector (σ̂², ρ̂²); the notation should be clarified.
  4. [Section 5.2.2, display preceding (5.1)] The formula for E[(y_iᵀ y_i)² | X] is asserted without proof or reference. Since the estimator η̂ depends on this formula, a short derivation or citation would improve verifiability.

Circularity Check

0 steps flagged · score 0.0 of 10

No significant circularity: the SNR estimators and their asymptotic variances are derived from model moments, not fitted to the target.

full rationale

The paper's central derivation is self-contained in the relevant sense: the SNR r² is defined directly from the model parameters ρ² and σ², and the estimators (2.4)–(2.5) are method-of-moments plug-ins obtained from the moment identities in Section 2. The asymptotic covariance matrices in Theorems 3.1, 3.2, and 4.1 are explicit functions of model unknowns, computed via Wishart moment expansions (Lemmas 6.6 and 6.7) and leave-one-out analyses (Lemma 6.9), rather than constants fitted to data. Plugging consistent estimators into these variance formulas for confidence intervals is standard delta-method inference, not circular prediction. The only imported ingredient, the total-variation approximation 'By an argument similar to that used in the proof of Theorem 1 in Dicker [2014]' (end of Section 6.2), cites an external univariate result; even if that analogy is a proof gap, it does not reduce the multivariate conclusion to its own inputs. The self-citation to Hu and Li (2022) appears only as contextual related work and is not load-bearing. A possible normalization/scaling error in Theorem 3.1 is a correctness concern, not circularity. Accordingly, no circular step can be exhibited.

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

The central results are derived from standard random matrix and Wishart moment tools plus explicit Gaussian assumptions. No free parameters are fitted in the theoretical derivation; the SNR, ρ², σ² and κ_tot are defined directly from model quantities. The paper introduces no new physical or statistical entities.

assumptions (5)
  • standard math Wishart moment formulas of Letac and Massam (2004) and Graczyk et al. (2005), as used in Lemma 6.6, provide exact E[α⊤Wαβ⊤Wβ]-type moments.
    Invoked in Appendix B to prove Lemma 6.6, which is the basis for the fixed-effects covariance calculations.
  • standard math Quadratic-form CLT bound in Lemma 6.4 (Dicker and Erdogdu 2017) holds for sub-Gaussian vectors and supplies the random-effects asymptotic normality.
    Used in Section 6.3 to approximate the conditional law of the quadratic forms by a normal distribution.
  • domain assumption Gaussianity of B and E (Assumptions 3.5, 3.6, 4.1) with bounded operator norms, so Gaussian fourth-moment identities with kurtosis 3 hold.
    Lemma 6.2 and the variance expressions use E[(x⊤Ax)^2] = 2tr(AΣAΣ) + tr²(AΣ); non-Gaussian kurtosis would change the variance formulas.
  • domain assumption Design X has i.i.d. standard Gaussian entries in the fixed-effects setting (Assumption 3.1) and sub-Gaussian rows with bounded covariance and limiting spectral distribution in the random-effects setting (Assumption 3.4).
    Gaussian design enables exact Wishart moments for Theorem 3.1; the random-effects theory requires the spectral convergence in (3.2).
  • domain assumption Heteroskedastic noise model e_i ~ N(0, Σ_i) with uniformly bounded ∥Σ_i∥, as in Assumption 4.1.
    Introduced to extend the random-effects result; κ_tot measures the total noise heterogeneity.

how reviews work

0 comments
Cite this review

Pith. "Pith review of Estimating Signal-to-Noise Ratios for Multivariate High-dimensional Linear Models." pith.science (2026). https://pith.science/paper/EARXCOOH

@misc{pith2026250610370,
  author       = {Pith},
  title        = {Pith review of: Estimating Signal-to-Noise Ratios for Multivariate High-dimensional Linear Models},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/EARXCOOH}},
  note         = {Machine review of arXiv:2506.10370}
}
read the original abstract

Signal-to-noise ratios (SNR) play a crucial role in various statistical models, with important applications in tasks such as estimating heritability in genomics. The method-of-moments estimator is a widely used approach for estimating SNR, primarily explored in single-response settings. In this study, we extend the method-of-moments SNR estimation framework to encompass both fixed effects and random effects linear models with multivariate responses. In particular, we establish and compare the asymptotic distributions of the proposed estimators. Furthermore, we extend our approach to accommodate cases with residual heteroskedasticity and derive asymptotic inference procedures based on standard error estimation. The effectiveness of our methods is demonstrated through extensive numerical experiments.

Discussion (0). Continue with ORCID to comment.

Reference graph

Works this paper leans on

28 extracted references · 26 canonical work pages

  1. [1]

    Variance estimation in high-dimensional linear models

    Lee H Dicker. Variance estimation in high-dimensional linear models. Biometrika, 101 0 (2): 0 269--284, 2014

    work page 2014

  2. [2]

    Maximum likelihood for variance estimation in high-dimensional linear models

    Lee H Dicker and Murat A Erdogdu. Maximum likelihood for variance estimation in high-dimensional linear models. In Artificial Intelligence and Statistics, pages 159--167. PMLR, 2016

    work page 2016

  3. [3]

    Flexible results for quadratic forms with applications to variance components estimation

    Lee H Dicker and Murat A Erdogdu. Flexible results for quadratic forms with applications to variance components estimation. Annals of Statistics, 45 0 (1): 0 386--414, 2017

    work page 2017

  4. [4]

    High-dimensional asymptotics of prediction: Ridge regression and classification

    Edgar Dobriban and Stefan Wager. High-dimensional asymptotics of prediction: Ridge regression and classification. Annals of Statistics, 46 0 (1): 0 247--279, 02 2018

    work page 2018

  5. [5]

    Variance estimation using refitted cross-validation in ultrahigh dimensional regression

    Jianqing Fan, Shaojun Guo, and Ning Hao. Variance estimation using refitted cross-validation in ultrahigh dimensional regression. Journal of the Royal Statistical Society Series B: Statistical Methodology, 74 0 (1): 0 37--65, 2012

    work page 2012

  6. [6]

    Multidimensional heritability analysis of neuroanatomical shape

    Tian Ge, Martin Reuter, Anderson M Winkler, Avram J Holmes, Phil H Lee, Lee S Tirrell, Joshua L Roffman, Randy L Buckner, Jordan W Smoller, and Mert R Sabuncu. Multidimensional heritability analysis of neuroanatomical shape. Nature communications, 7 0 (1): 0 13291, 2016

    work page 2016

  7. [7]

    Graczyk, G

    P. Graczyk, G. Letac, and H. Massam. The hyperoctahedral group, symmetric group representations and the moments of the real wishart distribution. Journal of Theoretical Probability, 18 0 (1): 0 1--42, 2005. doi:10.1007/s10959-004-0579-9. URL https://doi.org/10.1007/s10959-004-0579-9

    work page doi:10.1007/s10959-004-0579-9 2005

  8. [8]

    Optimal estimation of genetic relatedness in high-dimensional linear models

    Zijian Guo, Wanjie Wang, T Tony Cai, and Hongzhe Li. Optimal estimation of genetic relatedness in high-dimensional linear models. Journal of the American Statistical Association, 114 0 (525): 0 358--369, 2019

    work page 2019

Show all 28 references
  1. [9]

    Misspecification analysis of high-dimensional random effects models for estimation of signal-to-noise ratios

    Xiaohan Hu and Xiaodong Li. Misspecification analysis of high-dimensional random effects models for estimation of signal-to-noise ratios. arXiv preprint arXiv:2202.06400, 2022

    2022 arXiv

  2. [10]

    Eigenprism: inference for high dimensional signal-to-noise ratios

    Lucas Janson, Rina Foygel Barber, and Emmanuel Candes. Eigenprism: inference for high dimensional signal-to-noise ratios. Journal of the Royal Statistical Society. Series B, Statistical methodology, 79 0 (4): 0 1037, 2017

    2017

  3. [11]

    Reml estimation: Asymptotic behavior and related topics

    Jiming Jiang. Reml estimation: Asymptotic behavior and related topics. Annals of Statistics, pages 255--286, 1996

    1996

  4. [12]

    On high-dimensional misspecified mixed model analysis in genome-wide association study

    Jiming Jiang, Cong Li, Debashis Paul, Can Yang, and Hongyu Zhao. On high-dimensional misspecified mixed model analysis in genome-wide association study. Annals of Statistics, 44 0 (5): 0 2127--2160, 2016

    2016

  5. [13]

    Kakade, and Michael I

    Chi Jin, Praneeth Netrapalli, Rong Ge, Sham M. Kakade, and Michael I. Jordan. A short note on concentration inequalities for random vectors with subgaussian norm, 2019

    2019

  6. [14]

    All Invariant Moments of the Wishart Distribution

    Gérard Letac and Hélène Massam. All Invariant Moments of the Wishart Distribution . Scandinavian Journal of Statistics, 31 0 (2): 0 295--318, June 2004. doi:10.1111/j.1467-9469.2004. URL https://ideas.repec.org/a/bla/scjsta/v31y2004i2p295-318.html

    2004

  7. [15]

    The mahalanobis kernel for heritability estimation in genome-wide association studies: fixed-effects and random-effects methods

    Ruijun Ma and Lee H Dicker. The mahalanobis kernel for heritability estimation in genome-wide association studies: fixed-effects and random-effects methods. 2019

    2019

  8. [16]

    Distribution of eigenvalues for some sets of random matrices

    V A Mar c enko and L A Pastur. Distribution of eigenvalues for some sets of random matrices. Mathematics of the USSR -Sbornik , 1 0 (4): 0 457--483, apr 1967

    1967

  9. [17]

    Quasi-regression for heritability, 2012

    Art Owen. Quasi-regression for heritability, 2012

    2012

  10. [18]

    Hanson-wright inequality and sub-gaussian concentration

    Mark Rudelson and Roman Vershynin. Hanson-wright inequality and sub-gaussian concentration. Electronic Communications in Probability, 18: 0 1 -- 9, 2013

    2013

  11. [19]

    Silverstein and S.I

    J.W. Silverstein and S.I. Choi. Analysis of the limiting spectral distribution of large dimensional random matrices. Journal of Multivariate Analysis, 54 0 (2): 0 295--309, 1995

    1995

  12. [20]

    Hede: Heritability estimation in high dimensions by ensembling debiased estimators

    Yanke Song, Xihong Lin, and Pragya Sur. Hede: Heritability estimation in high dimensions by ensembling debiased estimators. arXiv preprint arXiv:2406.11184, 2024

    2024 arXiv

  13. [21]

    Scaled sparse linear regression

    Tingni Sun and Cun-Hui Zhang. Scaled sparse linear regression. Biometrika, 99 0 (4): 0 879--898, 2012

    2012

  14. [22]

    Joel A. Tropp. An introduction to matrix concentration inequalities, 2015

    2015

  15. [23]

    Vershynin

    R. Vershynin. High-Dimensional Probability: An Introduction with Applications in Data Science. Cambridge Series in Statistical and Probabilistic Mathematics. Cambridge University Press, 2018

    2018

  16. [24]

    Introduction to the non-asymptotic analysis of random matrices, 2011

    Roman Vershynin. Introduction to the non-asymptotic analysis of random matrices, 2011

    2011

  17. [25]

    Adaptive estimation of high-dimensional signal-to-noise ratios

    Nicolas Verzelen and Elisabeth Gassiat. Adaptive estimation of high-dimensional signal-to-noise ratios. 2018

    2018

  18. [26]

    Wainwright

    Martin J. Wainwright. High-Dimensional Statistics: A Non-Asymptotic Viewpoint. Cambridge Series in Statistical and Probabilistic Mathematics. Cambridge University Press, 2019

    2019

  19. [27]

    Gcta: a tool for genome-wide complex trait analysis

    J Yang, S H Lee, ME Goddard, and PM Visscher. Gcta: a tool for genome-wide complex trait analysis. American journal of human genetics, 88 0 (1): 0 76--82, 2011

    2011

  20. [28]

    Integrating multiple correlated phenotypes for genetic association analysis by maximizing heritability

    Jin J Zhou, Michael H Cho, Christoph Lange, Sharon Lutz, Edwin K Silverman, and Nan M Laird. Integrating multiple correlated phenotypes for genetic association analysis by maximizing heritability. Human heredity, 79 0 (2): 0 93--104, 2015

    2015

Pith tools

Reviewed August 7, 2026 · model on record in the stance chip above.