Pith. sign in

REVIEW 4 major objections 3 minor 94 references

Uncertainty of high-dimensional genetic data prediction with polygenic risk scores

T0 review · 4 major / 3 minor · reviewed 2026-08-10 · deepseek-v4-flash

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

Pith's one-line read This paper proves CLTs for polygenic risk score predictions at individual and cohort levels and shows that ignoring prediction-induced uncertainty understates variance.

desk verdict Serious paper, but the central variance normalization in the key quadratic-form lemma is wrong, so the main CLTs are not established as written. read the letter →

arxiv 2412.20611 v1 pith:SWYOPZZP submitted 2024-12-29 stat.ME math.STstat.TH

Haoxuan Fu , Jiaoyang Huang , Zirui Fan , Bingxin Zhao This is my paper · ORCID
classification stat.MEmath.STstat.TH MSC 60F0562J0760B2062P10
keywords polygenicriskscorehigh-dimensionalpredictioncentrallimittheoremrandommatrixtheoryridgeregressionBerry-Esseenboundheritabilitylinkagedisequilibrium
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

The paper targets the uncertainty of polygenic risk scores in high-dimensional, non-sparse prediction, where thousands of weak genetic variants each contribute a little. It establishes asymptotic normality with explicit Berry-Esseen bounds for two quantities: a single individual's genetically predicted value and the cohort-level out-of-sample R-squared. The results cover marginal summary-statistic estimators and reference-panel ridge estimators, under a general linkage-disequilibrium covariance and without Gaussian distribution assumptions. The central practical claim is that standard practice, which treats the estimated PRS weights as fixed, underestimates true variance and produces overconfident confidence intervals and hypothesis tests. If correct, this gives formal justification for uncertainty-aware PRS inference and identifies sample size, number of variants, heritability, signal sparsity, and LD structure as the determinants of that uncertainty.

What carries the argument

The proofs rest on three named mechanisms. First, a leave-one-out decomposition rewrites each estimator as a sum of quadratic functionals ⟨a, X_0^T X_0 b⟩ and residual functionals ⟨a, ε⟩, whose randomness is isolated sequentially and controlled by Berry-Esseen-type bounds. Second, for the quadratic form β^TΣβ that enters cohort accuracy, a martingale central limit theorem with a Berry-Esseen bound O($m^{{-1/5}}$) handles the correlated off-diagonal terms without Gaussian assumptions. Third, for reference-panel ridge estimators the resolvent (W^T W + n_w λ I_p)^{-1} is handled by anisotropic local laws, whose limits are expressed through the Stieltjes transform m_w of the asymptotic spectral distribution and the tilting factor r_w = $m_w^{2}$/m'_w.

What would settle it

Simulate the marginal estimator with p = 461,488, n = 50,000, heritability 0.3, and m/p from 0.001 to 0.5 using real genotype data, comparing the empirical distribution of √n(z^Tβhat_M - z^TΣβ) against N(0,$σ_M^{2}$) for raw unnormalized rows, where ||z||_2 is about √p, and for rows normalized to ||z||_2 = 1; if only the normalized rows match the normal limit, the bounded-norm assumption is driving the result.

Watch

Extended reading notes

Core claim

The paper's central claim is that, conditional on the test subject's genotype z and the true effects β, the marginal estimator βhat_M = $n^{{-1}}$X^T y satisfies sup_t |P($σ_M^{{-1}}$√n(z^Tβhat_M - z^TΣβ) < t) - Φ(t)| ≤ C $n^{{-1/2}}$, with variance $σ_M^{2}$ = E($x_0^{4}$-3) Σ_i ($Σ^{{1/2}}$z)$_i^{2}$($Σ^{{1/2}}$β)$_i^{2}$ + $h_β^{{-2}}$||$Σ^{{1/2}}$z||^2||$Σ^{{1/2}}$β||^2 + 2(z^TΣβ)^2. Predicted values are therefore asymptotically Gaussian about their biased mean z^TΣβ, not about the true phenotype z^Tβ. The paper obtains analogous CLTs for the cohort accuracy A(βhat_M) and for reference-panel ridge estimators, with Berry-Esseen rates max($n_z^{{-1/2}}$, $n^{{-1/2}}$, $m^{{-1/5}}$) and $n^{{-1/5}}$ respectively, all under non-Gaussian genotypes. It also shows that a naive CLT treating βhat as fixed gives a variance smaller by a factor η ≥ 1, so true spread is larger whenever training-data randomness is ignored.

Load-bearing premise

The load-bearing assumption is that each testing individual's genotype vector has bounded Euclidean norm, a fixed constant independent of p, whereas real SNP rows have norm growing like √p, so the theorem as stated needs a normalization step that the variance formulas do not track.

Editorial extensions

If this is right

  • Individual-level PRS confidence intervals should use the paper's variance formula; ignoring the estimation noise in βhat understates the interval, especially in low-heritability traits or with small training samples.
  • Cohort-level R-squared A^2 is asymptotically normal, but its convergence rate is n_z^{1/2}η^{1/2} with η ≤ 1; when the testing cohort is large relative to training or the signal is very sparse, naive inference at the n_z^{1/2} rate is invalid.
  • For marginal estimators, signal sparsity m does not affect the first-order mean of A^2 but does inflate its variance and slow the CLT rate; with a general LD covariance, m enters the mean as well.
  • For reference-panel ridge estimators, the variance depends on eigenvectors of the LD matrix through the locations of causal variants, not only on eigenvalues, which first-order analyses miss.
  • For the traditional ridge estimator, the paper's supplement requires a Gaussian assumption and obtains the same qualitative conclusion—ignoring prediction variance understates uncertainty—but without an explicit rate.

Reading between the lines

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

  • A practical corollary the paper leaves implicit is that ranking subjects by PRS may be more robust to the bias term than absolute prediction, since a monotone bias shifts all values similarly, whereas tests on absolute predicted values should target z^TΣβ, not z^Tβ.
  • The bounded-norm assumption on z suggests that the theorems as stated apply to normalized genotype rows; an explicit normalization of raw SNP data would be needed to make the variance formulas directly usable for p around 461,000, and the effect of that scaling on heritability and σ_M^2 deserves a worked-out check.
  • The m^{-1/5} Berry-Esseen rate from the martingale CLT suggests that sparse genetic architectures are the hardest for coverage; testing whether sharper martingale conditions yield m^{-1/2} is a natural next step.
  • The same decomposition machinery should transfer to other sparse-free high-dimensional predictions such as transcriptome or metabolome risk scores, since the genetics-specific ingredients enter only through the covariance, heritability, and sparsity pattern.
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 / 3 minor

Summary. This paper develops Berry-Esseen-type central limit theorems for polygenic risk score (PRS) predictions in high-dimensional, sparsity-free settings. It studies the marginal estimator and a reference-panel ridge estimator, and derives asymptotic normality for both individual-level predicted values and cohort-level out-of-sample R-squared, under general covariance structure and non-Gaussian predictor distributions. The proof strategy decomposes the prediction into quadratic and residual functionals, applies leave-one-out and martingale CLT arguments, and uses anisotropic local laws for the ridge resolvent. The theoretical results are supported by numerical experiments on UK Biobank genotype data.

Significance. If correct, the results would be a substantial contribution: explicit convergence rates for PRS uncertainty, no Gaussian assumption on the genotype matrix, general LD covariance, and direct relevance to confidence intervals for individual PRS values and for prediction accuracy. The paper is clearly written, the numerical work is extensive, and the supplementary material contains detailed proofs. However, the central variance computation in the quadratic-form CLT is algebraically wrong, and this error propagates into the main theorems. The stated CLTs do not follow from the proof as written, and the numerical normality checks do not test the claimed variance formulas. The main theoretical contribution is therefore not currently established.

major comments (4)
  1. [Supplementary S.3, Lemma S1/S2] The variance in Lemmas S1 and S2 is the uncentered second moment of (a^T x)(b^T x), not the variance of the centered summand. For iid entries with mean 0 and variance 1, E[(a^T x)^2(b^T x)^2] = (E[x_0^4]-3) Σ a_i^2 b_i^2 + 2(a^T b)^2 + ||a||^2 ||b||^2, which is what the lemma uses as σ^2. But the random variables X_{n,i} in the proof are centered by subtracting a^T b, so their variance is E[(a^T x)^2(b^T x)^2] - (a^T b)^2 = (E[x_0^4]-3) Σ a_i^2 b_i^2 + (a^T b)^2 + ||a||^2 ||b||^2. Thus the paper's σ^2 overstates the true variance by (a^T b)^2. Consequently Σ E(X_{n,i}^2) = 1 - (a^T b)^2/σ^2, not 1, and Lyapunov's second condition fails whenever a^T b ≠ 0. The stated CLT and Berry-Esseen bound in Lemmas S1 and S2 are not valid as written.
  2. [Theorem 1 and Corollary 3.1] The variance miscalibration directly enters Theorem 1. With a = Σ^{1/2} z and b = Σ^{1/2} β, the correct centered variance for the quadratic part is (E[x_0^4]-3) Σ_i (Σ^{1/2}z)_i^2(Σ^{1/2}β)_i^2 + (z^T Σ β)^2 + ||Σ^{1/2}z||^2 ||Σ^{1/2}β||^2, plus the independent noise contribution. The theorem instead uses a coefficient of 2 on (z^T Σ β)^2. Hence the normalized statistic σ_M^{-1} √n (z^T βhat_M - z^T Σ β) has asymptotic variance strictly less than 1 whenever z^T Σ β ≠ 0, and its Kolmogorov distance to the standard normal cannot vanish. Corollary 3.1 inherits the same error with (z^T β)^2.
  3. [Theorems 3, 4, 5 and Corollaries 3.2, 4.1, 4.2] The same flawed quadratic-functional CLT is used in the numerator analysis of the cohort-level results and in the ridge-type individual-level results. For example, in Section S.6.3 the variance σ_1^2 for the testing-data quadratic form contains 2{β^T Σ X^T(Xβ+ϵ)}^2, and in Section S.8.3 the analogous quantity contains 2{β^T Σ R X^T y}^2. These terms are the (a^T b)^2 over-counts from Lemma S1. The stated η_M and η_W formulas in Theorems 3 and 5, and their Σ = I_p corollaries, therefore inherit the miscalibration. The Berry-Esseen bounds for the normalized cohort accuracy statistics do not follow from the supplied proofs.
  4. [Assumption 1(a) and Section 5] Assumption 1(a) requires ||z||_2 ≤ C for a fixed constant C. For the UK Biobank data described in Section 5, p = 461,488 and an unnormalized genotype row has Euclidean norm of order √p, so this assumption does not hold for the raw data as described. The concentration arguments in Sections S.4 and S.7 repeatedly use boundedness of z to control quadratic functionals such as z^T Σ z and quartic sums. If the authors intend a row normalization, it must be stated and tracked through the variance formulas; as written, the theorems do not cover the data regime claimed in the numerical experiments.
minor comments (3)
  1. [Section 5] The Shapiro-Wilk tests in Figures 1 and 2 only check the shape of the empirical distribution after standardization; they do not compare the empirical variance with the theoretical σ^2 or η formulas. A plot of empirical versus theoretical variance would directly test the main variance claims.
  2. [Section 1.1] The symbol I_m is described as a p×p diagonal matrix with only the first m diagonal entries equal to 1, which conflicts with the standard use of I_m for an m×m identity matrix. A different notation, such as D_m, would avoid confusion.
  3. [Throughout] There are several typographical issues, including 'donates' for 'denotes', inconsistent use of p in limiting statements, and undefined scaling in the caption of Figure 3 ('scaled sd(PRS_i)'). These should be corrected in any revision.

Circularity Check

0 steps flagged · score 0.0 of 10

No significant circularity: central CLTs are proven from stated assumptions; self-citations to [62,63] are non-load-bearing.

full rationale

The derivation chain is self-contained. Theorem 1 and Corollary 3.1 are proven in Section S.4 from Lemma S2, a quantitative quadratic-functional CLT, and Lemma S6, a residual-functional CLT, via an explicit leave-one-out decomposition; no parameter is fitted to the quantity whose distribution is being predicted. The cohort-level means in Corollary 3.2, Theorem 3, and Theorem 5 are derived in Theorems S4 and S6 rather than assumed; citations to [62,63] are used only to note consistency with prior first-order limits, and those prior limits do not carry the distributional content of the present paper. The ridge-type results similarly use external anisotropic local laws [35], the fixed-point equation of [6], and an external de-biasing framework [8] for the Gaussian case; the tilting factor r_w is derived from the Stieltjes transform rather than imported as an ansatz from the authors' prior work. The UK Biobank section evaluates the shapes of empirical distributions with Shapiro-Wilk tests and does not fit variance parameters to force agreement with the theorems. The variance normalization issue raised about Lemma S1 is an internal mathematical calibration concern: if the uncentered second moment is used where a centered variance is needed, the standardized statistic would not have unit variance. That is a correctness problem, not a circular reduction, and it does not make any theorem equivalent to its own inputs by construction. No self-definitional, fitted-input-called-prediction, uniqueness-importation, or renamed-known-result pattern is present.

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

The central claim rests on random matrix assumptions and a random-effects model for genetic effects. No constants are fitted to data in the theory; quantities such as Σ, h2, m, σ2, and λ are exogenous inputs rather than free parameters tuned to make the theorems hold.

assumptions (5)
  • domain assumption Assumption 1: rows of X0, Z0, W0 have i.i.d. entries with mean 0, variance 1, bounded eighth moment; Σ has eigenvalues bounded away from 0 and infinity; ||z||2 ≤ C.
    Invoked at the start of all proofs. The bounded row norm is unrealistic for raw SNP data unless rows are normalized.
  • domain assumption Assumption 2: β has i.i.d. entries with mean 0, variance σβ^2/p, only first m nonzero, fourth moment O(p^-1); errors i.i.d. mean 0.
    Underlies the quadratic-form CLT and the heritability definition. This random-effects causal model is standard in PRS literature but is not verified from data.
  • domain assumption Assumption 4: n ≈ nz ≈ nw ≈ p ≈ m for the reference-panel ridge estimator, with n/nw and p/nw converging.
    Needed for Stieltjes transform and anisotropic local law arguments in Section 4.
  • domain assumption Assumption S1: Gaussian entries for the traditional ridge estimator.
    Limits the traditional ridge results to Gaussian data; the headline Gaussian-free claim does not cover βhat_R.
  • standard math Berry-Esseen theorem, martingale CLT, anisotropic local laws, eigenvalue interlacing and Weyl inequalities.
    Used throughout the supplementary proofs and assumed correct as background mathematics.

how reviews work

0 comments
Cite this review

Pith. "Pith review of Uncertainty of high-dimensional genetic data prediction with polygenic risk scores." pith.science (2026). https://pith.science/paper/SWYOPZZP

@misc{pith2026241220611,
  author       = {Pith},
  title        = {Pith review of: Uncertainty of high-dimensional genetic data prediction with polygenic risk scores},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/SWYOPZZP}},
  note         = {Machine review of arXiv:2412.20611}
}
abstract

In many predictive tasks, there are a large number of true predictors with weak signals, leading to substantial uncertainties in prediction outcomes. The polygenic risk score (PRS) is an example of such a scenario, where many genetic variants are used as predictors for complex traits, each contributing only a small amount of information. Although PRS has been a standard tool in genetic predictions, its uncertainty remains largely unexplored. In this paper, we aim to establish the asymptotic normality of PRS in high-dimensional predictions without sparsity constraints. We investigate the popular marginal and ridge-type estimators in PRS applications, developing central limit theorems for both individual-level predicted values (e.g., genetically predicted human height) and cohort-level prediction accuracy measures (e.g., overall predictive $R$-squared in the testing dataset). Our results demonstrate that ignoring the prediction-induced uncertainty can lead to substantial underestimation of the true variance of PRS-based estimators, which in turn may cause overconfidence in the accuracy of confidence intervals and hypothesis testing. These findings provide key insights omitted by existing first-order asymptotic studies of high-dimensional sparsity-free predictions, which often focus solely on the point limits of predictive risks. We develop novel and flexible second-order random matrix theory results to assess the asymptotic normality of functionals with a general covariance matrix, without assuming Gaussian distributions for the data. We evaluate our theoretical results through extensive numerical analyses using real data from the UK Biobank. Our analysis underscores the importance of incorporating uncertainty assessments at both the individual and cohort levels when applying and interpreting PRS.

Figures

Figures reproduced from arXiv: 2412.20611 by the authors.

Figure 1
Figure 1. Asymptotic normality of individual-level genetically predicted values. Based on the real genetic data from the UK Biobank study, we illustrate the empirical distribution of genetically predicted values z ⊤βb for βbM, βbW(λ), and βbR(λ) (from left to right). We assess the asymptotic normality with the Shapiro-Wilk test [48]. Here we set p = 461, 488, heritability h 2 β = h 2 βz = 0.3, sparsity m/p ranging from 0.001 … view at source ↗
Figure 2
Figure 2. Asymptotic normality of cohort-level prediction accuracy. Based on the real genetic data from the UK Biobank study, we illustrate the empirical distribution of cohort-level prediction accuracy A(βb) for βbM, βbW(λ), and βbR(λ) (from left to right). We assess the asymptotic normality with the Shapiro-Wilk test [48]. Here we set p = 461,488, heritability h 2 β = h 2 βz = 0.3, sparsity m/p ranging from 0.001 to 0.5, n … view at source ↗
Figure 3
Figure 3. Simulation results estimating the standard error of individual-level genetically predicted values across the training data sample size (left) and heritability (right). Based on the real genetic data from the UK Biobank study, we estimate the standard error of z ⊤βb with the LDpred2-based method proposed in [20]. Here we set p = 461,488 and sparsity m/p = 0.1. In the left panel, the training sample size ranges from 5… view at source ↗

Discussion (0). Continue with ORCID to comment.

Reference graph

Works this paper leans on

94 extracted references · 78 canonical work pages

  1. [1]

    A global reference for human genetic variation

    1000-Genomes-Consortium. A global reference for human genetic variation. Nature, 526(7571):68– 74, 2015

    2015

  2. [2]

    R.B. Ash. Probability and Measure Theory. Academic Press, London/San Diego, 2nd edition, 2000

    2000

  3. [3]

    Bai and Jack W

    Z. Bai and Jack W. Silverstein. Clt for linear spectral statistics of large-dimensional sample covariance matrices. The Annals of Probability, 32(1):553–605, 2004

    2004

  4. [4]

    Z. D. Bai and Jack W. Silverstein. No eigenvalues outside the support of the limiting spectral distri- bution of large-dimensional sample covariance matrices. The Annals of Probability, 26(1):316 – 345, 1998

    1998

  5. [5]

    Z. D. Bai and Jack W. Silverstein. CLT for linear spectral statistics of large-dimensional sample covariance matrices. The Annals of Probability, 32(1A):553 – 605, 2004

    2004

  6. [6]

    Spectral analysis of large dimensional random matrices , vol- ume 20

    Zhidong Bai and Jack W Silverstein. Spectral analysis of large dimensional random matrices , vol- ume 20. Springer, 2010

    2010

  7. [7]

    Large sample covariance matrices without independence structures in columns

    Zhidong Bai and Wang Zhou. Large sample covariance matrices without independence structures in columns. Statistica Sinica, 18(2):425–442, 2008. 19

    2008

  8. [8]

    Bellec and Cun-Hui Zhang

    Pierre C. Bellec and Cun-Hui Zhang. Debiasing convex regularized estimators and interval estimation in linear models. The Annals of Statistics, 51(2):391 – 436, 2023

    2023

Show all 94 references
  1. [9]

    Bentkus, F

    V . Bentkus, F. G ¨otze, and W. R. van Zwet. An Edgeworth expansion for symmetric statistics. The Annals of Statistics, 25(2):851 – 896, 1997

    1997

  2. [10]

    Approximately independent linkage disequilibrium blocks in human populations

    Tomaz Berisa and Joseph K Pickrell. Approximately independent linkage disequilibrium blocks in human populations. Bioinformatics, 32(2):283–285, 2016

    2016

  3. [11]

    Bolthausen

    E. Bolthausen. Exact convergence rates in some martingale central limit theorems. The Annals of Probability, 10(3):672 – 688, 1982

    1982

  4. [12]

    An expanded view of complex traits: from polygenic to omnigenic

    Evan A Boyle, Yang I Li, and Jonathan K Pritchard. An expanded view of complex traits: from polygenic to omnigenic. Cell, 169(7):1177–1186, 2017

    2017

  5. [13]

    B. M. Brown. Martingale central limit theorems. The Annals of Mathematical Statistics , 42(1):59 – 66, 1971

    1971

  6. [14]

    Ld score regression distinguishes confounding from polygenicity in genome-wide association studies

    Brendan K Bulik-Sullivan, Po-Ru Loh, Hilary K Finucane, Stephan Ripke, Jian Yang, Nick Patterson, Mark J Daly, Alkes L Price, Benjamin M Neale, Schizophrenia Working Group of the Psychiatric Genomics Consortium, et al. Ld score regression distinguishes confounding from polygen...

    2015

  7. [15]

    The uk biobank resource with deep phenotyping and genomic data

    C Bycroft, C Freeman, D Petkova, G Band, LT Elliott, K Sharp, A Motyer, D Vukcevic, O Delaneau, J O’Connell, et al. The uk biobank resource with deep phenotyping and genomic data. Nature, 562(7726):203–209, 2018

    2018

  8. [16]

    Challenges of the inconsistency regime: Novel debiasing methods for missing data models

    Michael Celentano and Martin J Wainwright. Challenges of the inconsistency regime: Novel debiasing methods for missing data models. arXiv preprint arXiv:2309.01362, 2023

    2023 arXiv

  9. [17]

    Tutorial: a guide to performing polygenic risk score analyses

    Shing Wan Choi, Timothy Shin-Heng Mak, and Paul F O’Reilly. Tutorial: a guide to performing polygenic risk score analyses. Nature Protocols, 15(9):2759–2772, 2020

    2020

  10. [18]

    Accuracy of predicting the genetic risk of disease using a genome-wide approach

    Hans D Daetwyler, Beatriz Villanueva, and John A Woolliams. Accuracy of predicting the genetic risk of disease using a genome-wide approach. PLoS One, 3(10):e3395, 2008

    2008

  11. [19]

    A central limit theorem for generalized quadratic forms

    Peter de Jong. A central limit theorem for generalized quadratic forms. Probability Theory and Related Fields, 75:261–277, 1987

    1987

  12. [20]

    Large uncertainty in individual polygenic risk score estimation impacts prs-based risk stratification

    Yi Ding, Kangcheng Hou, Kathryn S Burch, Sandra Lapinska, Florian Priv ´e, Bjarni Vilhj ´almsson, Sriram Sankararaman, and Bogdan Pasaniuc. Large uncertainty in individual polygenic risk score estimation impacts prs-based risk stratification. Nature Genetics, 54(1):30–39, 2022

    2022

  13. [21]

    One-shot distributed ridge regression in high dimensions

    Edgar Dobriban and Yue Sheng. One-shot distributed ridge regression in high dimensions. arXiv preprint arXiv:1903.09321, 2019

    1903 arXiv

  14. [22]

    High-dimensional asymptotics of prediction: Ridge regression and classification

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

    2018

  15. [23]

    Power and predictive accuracy of polygenic risk scores

    Frank Dudbridge. Power and predictive accuracy of polygenic risk scores. PLoS Genetics , 9(3):e1003348, 2013

    2013

  16. [24]

    The spectral norm of random inner-product kernel matrices

    Zhou Fan and Andrea Montanari. The spectral norm of random inner-product kernel matrices. Proba- bility Theory and Related Fields, 173:27–85, 2019. 20

    2019

  17. [25]

    A gene- based association method for mapping traits using reference transcriptome data

    Eric R Gamazon, Heather E Wheeler, Kaanan P Shah, Sahar V Moza ffari, Keston Aquino-Michaels, Robert J Carroll, Anne E Eyler, Joshua C Denny, GTEx Consortium, Dan L Nicolae, et al. A gene- based association method for mapping traits using reference transcriptome data. Nature G...

    2015

  18. [26]

    Polygenic prediction via bayesian regression and continuous shrinkage priors

    Tian Ge, Chia-Yen Chen, Yang Ni, Yen-Chen Anne Feng, and Jordan W Smoller. Polygenic prediction via bayesian regression and continuous shrinkage priors. Nature Communications, 10(1):1–10, 2019

    2019

  19. [27]

    G ¨otze and A

    F. G ¨otze and A. Tikhomirov. Asymptotic distribution of quadratic forms and applications. Journal of Theoretical Probability, 15:423–475, 2002

    2002

  20. [28]

    On the rate of convergence in the central limit theorem for martingales with discrete and continuous time

    Erich Haeusler. On the rate of convergence in the central limit theorem for martingales with discrete and continuous time. The Annals of Probability, 16(1):275 – 299, 1988

    1988

  21. [29]

    Tibshirani

    Trevor Hastie, Andrea Montanari, Saharon Rosset, and Ryan J. Tibshirani. Surprises in high- dimensional ridgeless least squares interpolation. The Annals of Statistics, 50(2):949–986, 2022

    2022

  22. [30]

    A community driven gwas summary statistics standard

    James Hayhurst, Annalisa Buniello, Laura Harris, Abayomi Mosaku, Christopher Chang, Christo- pher R Gignoux, Konstantinos Hatzikotoulas, Mohd Anisul Karim, Samuel A Lambert, Matt Lyon, et al. A community driven gwas summary statistics standard. bioRxiv, pages 2022–07, 2022

    2022

  23. [31]

    Handbook of Linear Algebra

    Leslie (Ed.) Hogben. Handbook of Linear Algebra . Chapman and Hall /CRC, 2nd edition, 2013. https://doi.org/10.1201/b16113

    2013 doi

  24. [32]

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

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

    2016

  25. [33]

    A resource-efficient tool for mixed model association analysis of large-scale data

    L Jiang, Z Zheng, T Qi, KE Kemper, NR Wray, PM Visscher, and J Yang. A resource-efficient tool for mixed model association analysis of large-scale data. Nature Genetics, 51(12):1749–1755, 2019

    2019

  26. [34]

    Polygenic prediction of weight and obesity trajectories from birth to adulthood

    Amit V Khera, Mark Cha ffin, Kaitlin H Wade, Sohail Zahid, Joseph Brancale, Rui Xia, Marina Dis- tefano, Ozlem Senol-Cosar, Mary E Haas, Alexander Bick, et al. Polygenic prediction of weight and obesity trajectories from birth to adulthood. Cell, 177(3):587–596, 2019

    2019

  27. [35]

    Anisotropic local laws for random matrices

    Antti Knowles and Jun Yin. Anisotropic local laws for random matrices. Probability Theory and Related Fields, 169:257–352, 2017

    2017

  28. [36]

    Eigenvectors of some large sample covariance matrix ensembles

    Olivier Ledoit and Sandrine P ´ech´e. Eigenvectors of some large sample covariance matrix ensembles. Probability Theory and Related Fields, 151(1):233–264, 2011

    2011

  29. [37]

    Predicting unobserved phenotypes for complex traits from whole-genome snp data

    Sang Hong Lee, Julius HJ Van Der Werf, Ben J Hayes, Michael E Goddard, and Peter M Visscher. Predicting unobserved phenotypes for complex traits from whole-genome snp data. PLoS Genetics, 4(10):e1000231, 2008

    2008

  30. [38]

    Asymptotic normality and confidence intervals for pre- diction risk of the min-norm least squares estimator

    Zeng Li, Chuanlong Xie, and Qinwen Wang. Asymptotic normality and confidence intervals for pre- diction risk of the min-norm least squares estimator. InInternational Conference on Machine Learning, pages 6533–6542. PMLR, 2021

    2021

  31. [39]

    Genetic prediction of complex traits with polygenic scores: a statistical review

    Ying Ma and Xiang Zhou. Genetic prediction of complex traits with polygenic scores: a statistical review. Trends in Genetics, 37(11):995–1011, 2021. 21

    2021

  32. [40]

    Polygenic scores via penalized regression on summary statistics

    Timothy Shin Heng Mak, Robert Milan Porsch, Shing Wan Choi, Xueya Zhou, and Pak Chung Sham. Polygenic scores via penalized regression on summary statistics. Genetic Epidemiology, 41(6):469– 480, 2017

    2017

  33. [41]

    Significance tests for r2 of out- of-sample prediction using polygenic scores

    Md Moksedul Momin, Soohyun Lee, Naomi R Wray, and S Hong Lee. Significance tests for r2 of out- of-sample prediction using polygenic scores. The American Journal of Human Genetics, 110(2):349– 358, 2023

    2023

  34. [42]

    On the rate of convergence in the martingale central limit theorem

    Jean-Christophe Mourrat. On the rate of convergence in the martingale central limit theorem. Bernoulli, 19(2):633–645, 2013

    2013

  35. [43]

    Dissecting the genetics of complex traits using summary associ- ation statistics

    Bogdan Pasaniuc and Alkes L Price. Dissecting the genetics of complex traits using summary associ- ation statistics. Nature Reviews Genetics, 18(2):117–127, 2017

    2017

  36. [44]

    Polygenic risk scores for schizophrenia and bipolar disorder predict creativity

    Robert A Power, Stacy Steinberg, Gyda Bjornsdottir, Cornelius A Rietveld, Abdel Abdellaoui, Michel M Nivard, Magnus Johannesson, Tessel E Galesloot, Jouke J Hottenga, Gonneke Willem- sen, et al. Polygenic risk scores for schizophrenia and bipolar disorder predict creativity. N...

    2015

  37. [45]

    Plink: a tool set for whole- genome association and population-based linkage analyses.The American Journal of Human Genetics, 81(3):559–575, 2007

    Shaun Purcell, Benjamin Neale, Kathe Todd-Brown, Lori Thomas, Manuel AR Ferreira, David Bender, Julian Maller, Pamela Sklar, Paul IW De Bakker, Mark J Daly, et al. Plink: a tool set for whole- genome association and population-based linkage analyses.The American Journal of Hum...

    2007

  38. [46]

    A note on exact convergence rates in some martingale central limit theorems

    Joachim Renz. A note on exact convergence rates in some martingale central limit theorems. The Annals of Probability, 24(3):1616–1637, 1996

    1996

  39. [47]

    J. Schur. ¨Uber potenzreihen, die im innern des einheitskreises beschr ¨ankt sind. Journal f¨ ur die reine und angewandte Mathematik, 147:205–232, 1917

    1917

  40. [48]

    An analysis of variance test for normality (complete samples)

    Samuel Sanford Shapiro and Martin B Wilk. An analysis of variance test for normality (complete samples). Biometrika, 52(3/4):591–611, 1965

    1965

  41. [49]

    Genetics of 35 blood and urine biomarkers in the uk biobank

    Nasa Sinnott-Armstrong, Yosuke Tanigawa, David Amar, Nina Mars, Christian Benner, Matthew Aguirre, Guhan Ram Venkataraman, Michael Wainberg, Hanna M Ollila, Tuomo Kiiskinen, et al. Genetics of 35 blood and urine biomarkers in the uk biobank. Nature Genetics, 53(2):185–194, 2021

    2021

  42. [50]

    What are polygenic scores and why are they important? JAMA, 321(18):1820–1821, 2019

    Leo P Sugrue and Rahul S Desikan. What are polygenic scores and why are they important? JAMA, 321(18):1820–1821, 2019

    2019

  43. [51]

    Genetic architecture: the shape of the genetic contribution to human traits and disease.Nature Reviews Genetics, 19(2):110–125, 2018

    Nicholas J Timpson, Celia MT Greenwood, Nicole Soranzo, Daniel J Lawson, and J Brent Richards. Genetic architecture: the shape of the genetic contribution to human traits and disease.Nature Reviews Genetics, 19(2):110–125, 2018

    2018

  44. [52]

    The personal and clinical utility of polygenic risk scores

    Ali Torkamani, Nathan E Wineinger, and Eric J Topol. The personal and clinical utility of polygenic risk scores. Nature Reviews Genetics, 19(9):581–590, 2018

    2018

  45. [53]

    Genome-wide association studies

    Emil U ffelmann, Qin Qin Huang, Nchangwi Syntia Munung, Jantina de Vries, Yukinori Okada, Ali- cia R Martin, Hilary C Martin, Tuuli Lappalainen, and Danielle Posthuma. Genome-wide association studies. Nature Reviews Methods Primers, 59(1):1–21, 2021. 22

    2021

  46. [54]

    Modeling linkage disequilibrium increases accuracy of polygenic risk scores

    Bjarni J Vilhj ´almsson, Jian Yang, Hilary K Finucane, Alexander Gusev, Sara Lindstr ¨om, Stephan Ripke, Giulio Genovese, Po-Ru Loh, Gaurav Bhatia, Ron Do, et al. Modeling linkage disequilibrium increases accuracy of polygenic risk scores. The American Journal of Human Genetic...

    2015

  47. [55]

    Impact of individual level uncertainty of lung cancer polygenic risk score (prs) on risk stratification

    Xinan Wang, Ziwei Zhang, Yi Ding, Tony Chen, Lorelei Mucci, Demetrios Albanes, Maria Teresa Landi, Neil E Caporaso, Stephen Lam, Adonina Tardon, et al. Impact of individual level uncertainty of lung cancer polygenic risk score (prs) on risk stratification. Genome Medicine, 16(...

    2024

  48. [56]

    Theoretical and empirical quantification of the accuracy of polygenic scores in ancestry divergent populations

    Ying Wang, Jing Guo, Guiyan Ni, Jian Yang, Peter M Visscher, and Loic Yengo. Theoretical and empirical quantification of the accuracy of polygenic scores in ancestry divergent populations. Nature Communications, 11(1):1–9, 2020

    2020

  49. [57]

    Gcta: a tool for genome-wide complex trait analysis

    Jian Yang, S Hong Lee, Michael E Goddard, and Peter M Visscher. Gcta: a tool for genome-wide complex trait analysis. The American Journal of Human Genetics, 88(1):76–82, 2011

    2011

  50. [58]

    Concepts, estimation and interpretation of snp-based heritability

    Jian Yang, Jian Zeng, Michael E Goddard, Naomi R Wray, and Peter M Visscher. Concepts, estimation and interpretation of snp-based heritability. Nature Genetics, 49(9):1304–1310, 2017

    2017

  51. [59]

    Large Sample Covariance Matrices and High- Dimensional Data Analysis

    Jianfeng Yao, Shurong Zheng, and Zhidong Bai. Large Sample Covariance Matrices and High- Dimensional Data Analysis. Cambridge Series in Statistical and Probabilistic Mathematics. Cambridge University Press, 2015

    2015

  52. [60]

    A saturated map of common genetic variants associated with human height

    Lo ¨ıc Yengo, Sailaja Vedantam, Eirini Marouli, Julia Sidorenko, Eric Bartell, Saori Sakaue, Marielisa Graff, Anders U Eliasen, Yunxuan Jiang, Sridharan Raghavan, et al. A saturated map of common genetic variants associated with human height. Nature, 610:704–712, 2022

    2022

  53. [61]

    Estimating trans-ancestry genetic correlation with unbalanced data resources

    Bingxin Zhao, Xiaochen Yang, and Hongtu Zhu. Estimating trans-ancestry genetic correlation with unbalanced data resources. Journal of the American Statistical Association, 119(546):839–850, 2024

    2024

  54. [62]

    On blockwise and reference panel-based estimators for genetic data prediction in high dimensions

    Bingxin Zhao, Shurong Zheng, and Hongtu Zhu. On blockwise and reference panel-based estimators for genetic data prediction in high dimensions. The Annals of Statistics, 52(3):948–965, 2024

    2024

  55. [63]

    On polygenic risk scores for complex traits prediction

    Bingxin Zhao and Fei Zou. On polygenic risk scores for complex traits prediction. Biometrics, 78(2):499–511, 2022

    2022

  56. [64]

    Global biobank meta-analysis initia- tive: Powering genetic discovery across human disease

    Wei Zhou, Masahiro Kanai, Kuan-Han H Wu, Humaira Rasheed, Kristin Tsuo, Jibril B Hirbo, Ying Wang, Arjun Bhattacharya, Huiling Zhao, Shinichi Namba, et al. Global biobank meta-analysis initia- tive: Powering genetic discovery across human disease. Cell Genomics, 2(10):100192, ...

    2022

  57. [65]

    Pn i=1 E(X2 n,i)→ 1 when n→∞ ; and 29

  58. [66]

    The first two conditions follow trivially from our definition of Xn,i

    Pn i=1 E(X4 n,i)→ 0 when n→∞ . The first two conditions follow trivially from our definition of Xn,i. Now we examine the fourth moment of Xn,i. Notice that nX i=1 E(X4 n,i) = E n (a⊤x0b⊤x0− a⊤b)4o σ4n ≤ M n E n (a⊤x0)4(b⊤x0)4o + (a⊤b)4o σ4n for some large constant M. Moreover,...

  59. [67]

    By takingλ =∥b∥4 2/∥a∥4 2, we have ME n (a⊤x0)4(b⊤x0)4o σ4n ≤ 1 n MN r2 → 0, as n→∞

    = 1 n MN r2 λ ∥a∥4 2 ∥b∥4 2 + 1 λ ∥b∥4 2 ∥a∥4 2 . By takingλ =∥b∥4 2/∥a∥4 2, we have ME n (a⊤x0)4(b⊤x0)4o σ4n ≤ 1 n MN r2 → 0, as n→∞ . Since σ2≥ Op{(a⊤b)2}≥ s(a⊤b)2 for some small s, we have (a⊤b)4 σ4n ≤ (a⊤b)4 s2(a⊤b)4n = 1 s2n→ 0 as n→∞ . Combining two results a...

  60. [68]

    Pn i=1 E X2 n,i → 1 when n→∞ ; and

  61. [69]

    Pn i=1 E X4 n,i ≤ Op(n−1). Define Lyapunov coefficients as Lt = nX i=1 E Xn,i t , and by using the Berry-Esseen Theorem with Lyapunov coefficient L4, we have the following result sup t∈R |Fn(t)− Φ(t)|≤ cL3≤ c p L4 =⇒ sup t∈R |Fn(t)− Φ(t)|≤ c r τ n for some constantτ decided by...

  62. [70]

    Denote Xn,i = aix⊤ i b q E ∥a∥2 2 ∥b∥2 , we proceed by checking Lyapunov’s conditions

  63. [71]

    Pn i=1 E X2 n,i = nPn i=1 E a2 i ∥b∥2 2 o / n E ∥a∥2 2 ∥b∥2 2 o = 1; and

  64. [72]

    31 □ Lemma S4 (Quantitative CLT for linear functional)

    Pn i=1 E X4 n,i = hPn i=1 E a4 i n E x4 0− 3 Pp j=1 b4 j + 3∥b∥4 2 oi / E ∥a∥2 2 2 ∥b∥4 2 = Op Pn i=1 E a4 i ∥b∥4 2 / E ∥a∥2 2 2 ∥b∥4 2 → 0 as n→∞. 31 □ Lemma S4 (Quantitative CLT for linear functional). Under Assumption 1, assume X0 is a n× p matrix and some non-deterministic...

  65. [73]

    Pn i=1 E X2 n,i = 1; and

  66. [74]

    Pn i=1 E X4 n,i = Op Pn i=1 E a4 i /E ∥a∥2 2 2 . Define Lyapunov coefficientss as Lt = nX i=1 E Xn,i t , using Berry-Esseen Theorem with the Lyapunov coefficient L4, we have the following result sup t∈R |Fn(t)− Φ(t)|≤ cL3≤ c p L4 =⇒ sup t∈R |Fn(t)− Φ(t)|≤ c vuuutτPn i=1 E a4 i...

  67. [75]

    Pn i=1 E X2 n,i = nPn i=1 E ϵ2 i a2 i o / σ2 ϵ∥a∥2 2 = 1; and

  68. [76]

    □ Lemma S6 (Quantiative CLT for residue functional)

    Pn i=1 E X4 n,i = n E ϵ4 i Pn i=1 a4 i o / σ4 ϵ∥a∥4 → 0 as n→∞ . □ Lemma S6 (Quantiative CLT for residue functional). Under Assumption 1, assume a to be a deterministic vector, and assume thatϵ is a n×1 random vector where each entry are i.i.d with bounded second and fourth mo...

  69. [77]

    Pn i=1 E X2 n,i = nPn i=1 E ϵ2 i a2 i o / n σ2 ϵ a⊤a o = 1; and

  70. [78]

    Pn i=1 E X4 n,i = n E ϵ4 i Pn i=1 a4 i o / n σ4 ϵ∥a∥4 2 o . Define Lyapunov coefficients as Lt = nX i=1 E Xn,i t , using Berry-Esseen Theorem with the Lyapunov coefficient L4, we have the following result sup t∈R |Fn(t)− Φ(t)|≤ cL3≤ c p L4 =⇒ sup t∈R |Fn(t)− Φ(t)|≤ c vutτE ϵ4 ...

  71. [79]

    ξt is adapted to the filtration Ft

  72. [80]

    Tm = Pm t=1ξt = peβ⊤Σ′eβ−σ2 βTr(Σ′); and

  73. [81]

    o ff-diagonal

    var (Tm) = p2 n E(eβ4)− 3σ4 β/p2oPm i=1 Σ′ i,i 2 + 2σ4 βTr(Σ′2) = Θp(m). Denote the conditional variance S m = var(Tm)−1 Pm i=1 E ξ2 t|Ft−1 , we aim to check E|S m− 1|2 → 0 as m→∞ , which leads to S m p → 1 in martingale CLT. We also aim to quantify the convergence behavior in...

  74. [82]

    Pm i=1 E ξ2 m,i = 1; and

  75. [83]

    Recall Lyapunov coefficients are defined as Lt = nX i=1 E Xn,i t

    Pm i=1 E ξ4 m,i ≤ Op(m−1). Recall Lyapunov coefficients are defined as Lt = nX i=1 E Xn,i t . Using the Berry-Esseen Theorem [2] with the Lyapunov coefficient L4, we have the following result sup t∈R P  mX i=1 ξi < t − Φ(t) ≤ CL3≤ C p L4 =⇒ sup t∈R P  mX i...

  76. [84]

    Moreover, we have EZ0 y⊤XZ⊤ZX⊤y = Op(nzn2κ1) and EX0 ∥z⊤ 0iΣ1/2X⊤y∥4 2 = E z4 0− 3 pX j=1 ΣX⊤y 4 j + 3∥Σ1/2X⊤y∥4 2 = Op(n4κ2 1)

    Using Lemma S7, for ∀δ∈ (0, 1/2) and∀ϵ1 > 0, we have P y⊤XZ⊤ZX⊤y− nzy⊤XΣX⊤y < n1/2+δ z n2κ1ϵ1 ≥ 1− C ϵ2 1n2δz . Moreover, we have EZ0 y⊤XZ⊤ZX⊤y = Op(nzn2κ1) and EX0 ∥z⊤ 0iΣ1/2X⊤y∥4 2 = E z4 0− 3 pX j=1 ΣX⊤y 4 j + 3∥Σ1/2X⊤y∥4 2 = Op(n4κ2 1). Similarly, note that E z⊤ 0 ΣX⊤y 8 =...

  77. [85]

    A B C D #−1 =

    It helps us eliminate all higher moment terms in our later analysis. Proposition S3. (Originally proposed in [4]) Let x = (x1,··· , xn)⊤ be i.i.d. standardized entries and A be an n× n matrix. For any p≥ 2, we have E x⊤Ax− Tr(A) p ≤ Kp n E|x1|4Tr(AA⊤) op/2 + E|x1|2pTr(AA⊤)p/2 ...

  78. [86]

    eκ2 = Tr(Σ1/2RΣRΣ1/2)

  79. [87]

    eκ5 = Tr n (ΣRΣIm)2o ; and

  80. [88]

    Notice that for β∈ Ξ5, we can further simplify our Berry-Esseen upper bounds in eq

    eκ6 = Tr(ΣRΣIm). Notice that for β∈ Ξ5, we can further simplify our Berry-Esseen upper bounds in eq. (S.10) as follows Op  vuuuuuut nz n2 neω1 +eκ2∥Σ1/2β∥2 2 2 + n2eκ2 2 + Op(p1/2+δ) n2z n2 neω1 +eκ2∥Σ1/2β∥2 2 2 + n2eκ2 2 + Op(p3/2+δ)  = Op ...

  81. [89]

    −Σ(1 + mnΣ)−1 0 0 mn # , and Σ =

    Tr n (ΣRΣIm)2o . The matrix Im does not cause trouble for the first three quantities as we only sum up tom terms. However, the last quantity requires more non-trivial analysis, as we cannot avoid the matrix Im in its complicated structure. This makes the analysis much di ffere...

  82. [90]

    de-biased

    First, we decompose∥y− Xbβ∥2 2 as follows ∥y− Xbβ∥2 2 = y⊤y− 2y⊤X(X⊤X + nλIp)−1X⊤y + y⊤X(X⊤X + nλIp)−1X⊤X(X⊤X + nλIp)−1X⊤y. The concentration of the first term can be obtained by applying Proposition S3, where we have y⊤y n β⊤Σβ +σ2ϵ p → 1. The second term can be further decom...

  83. [91]

    y⊤X(X⊤X + nλIp)−1Σ(X⊤X + nIp)−1X⊤y and

  84. [92]

    S.10.4.1 Concentration for the first quantity In this section, we provide concentration for y⊤X(X⊤X + nλIp)−1Σ(X⊤X + nλIp)−1X⊤y in three steps

    β⊤Σ(X⊤X + nλIp)−1X⊤y. S.10.4.1 Concentration for the first quantity In this section, we provide concentration for y⊤X(X⊤X + nλIp)−1Σ(X⊤X + nλIp)−1X⊤y in three steps. Step 1: Decomposing y⊤X(X⊤X + nλIp)−1Σ(X⊤X + nλIp)−1X⊤y, we have y⊤XRΣRX⊤y = β⊤X⊤XRΣRX⊤Xβ + ϵ⊤XRΣRX⊤ϵ + 2ϵ⊤XRΣR...

  85. [93]

    ( ϵ⊤ z + β⊤Z)(Z⊤β + ϵz) and

  86. [94]

    For the first quantity (ϵ⊤ z + β⊤Z)(Z⊤β + ϵz), recall in eq

    y⊤X(X⊤X + nλIp)−1Z⊤Z(X⊤X + nλIp)−1X⊤y. For the first quantity (ϵ⊤ z + β⊤Z)(Z⊤β + ϵz), recall in eq. (S.13) we have shown that P ∥Zβ + ϵz∥2 2− nz(σ2 βγ1 +σ2 ϵz) ≤ Op(n1/2+δ z ) ≥ 1− Op(n−2δ z ). (S.7) For the second quantity y⊤X(X⊤X + nλIp)−1Z⊤Z(X⊤X + nλIp)−1X⊤y, we have shown ...

Pith tools

Reviewed August 10, 2026 · model on record in the stance chip above.