{"id":"551288f9-70e3-44ac-b208-3ff849f5b12d","arxiv_id":"2412.20611","paper_version":1,"verdict":"REJECT","confidence":"MODERATE","novelty_score":6.0,"correctness_risk":"high","formal_verification":"none","parameter_count":0,"one_line_summary":"Marginal and reference-panel ridge PRS estimators are claimed to be asymptotically normal with explicit variance formulas, but a key quadratic-form lemma appears to use the wrong variance normalization.","lead":"This paper derives central limit theorems for polygenic risk score predictions, giving variance formulas for individual predicted values and cohort-level accuracy. It argues that standard practice ignoring training-data uncertainty produces overconfident confidence intervals and significance tests.","discovery_kind":"extension","skeptic_critique":{"model":"deepseek-v4-flash","headline":"Lemma S1/S2 variance normalization is wrong: σ² equals E[(aᵀx)²(bᵀx)²], not the centered variance; it overstates by (aᵀb)², and Theorem 1's σ_M² inherits this, invalidating the stated CLT.","rationale":"The paper aims to derive second-order CLTs for polygenic risk score predictions, with the marginal estimator analysis resting on the quadratic-form CLT in Lemma S1/S2. My direct computation shows that the claimed σ² is the uncentered second moment of (aᵀx)(bᵀx), not the variance; the missing subtraction of (aᵀb)² is exactly the term that breaks unit limiting variance whenever zᵀΣβ ≠ 0. This is an internal inconsistency, independent of any debate about data-generating assumptions. The same error propagates to Theorem 1 and to the cohort-level results, so the central claims as stated are false. I credit the paper for extensive supplementary proofs and real-data simulations, but the simulations only examine normality shapes (Shapiro–Wilk) and do not check the variance normalization, so they cannot detect the denominator error. The bounded-row assumption is a separate external-validity issue; the variance miscalibration is more fundamental because it invalidates the theorems under the paper's own assumptions. The paper is potentially salvageable by replacing σ² with the centered variance, which would alter the normalizer and likely preserve the shape of the CLT, but the submitted claims are incorrect as written. Hence I concur with the reader's reject verdict, though I identify a different weakest point than the reader's stated row-norm concern.","tokens_in":98469,"tokens_out":10617,"duration_ms":97212,"concrete_test":"Compute Var((aᵀx)(bᵀx)) analytically in the simplest case: p=1, a=b=1, x~N(0,1). Then ⟨a, X₀ᵀX₀b⟩ = Σᵢ₌₁ⁿ xᵢ² and aᵀb=1. The paper's Lemma S1 gives σ² = (E[x⁴]−3)+2+1 = 3 (since E[x⁴]=3), while the true centered variance is E[(x²−1)²]=2. Thus the statistic (Σ xᵢ² − n)/(√(3n)) converges to N(0,2/3) by the chi-square CLT, contradicting the claimed N(0,1) and the n^{−1/2} Berry-Esseen bound. A quick simulation with n=10⁴ would show the empirical CDF is the dilation Φ(t/√(2/3)), not Φ(t).","verdict_should_be":"UNCHANGED","load_bearing_attack":"The most load-bearing defect is in the quadratic-form CLT, Lemma S1 (and its quantitative version S2). For iid entries with mean 0 and variance 1, the variance of (aᵀx)(bᵀx) is Var = ‖a‖²‖b‖² + (aᵀb)² + (E[x₀⁴]−3)Σᵢ aᵢ²bᵢ², because E[(aᵀx)²(bᵀx)²] = ‖a‖²‖b‖² + 2(aᵀb)² + (E[x₀⁴]−3)Σᵢ aᵢ²bᵢ² and the mean is aᵀb. Lemma S1 instead uses σ² = E[x₀⁴−3]Σ aᵢ²bᵢ² + 2(aᵀb)² + ‖a‖²‖b‖², i.e., the raw second moment rather than the centered variance. The error is exactly (aᵀb)². The proof claims Lyapunov's second condition follows trivially, but it fails unless aᵀb=0. Theorem 1 inherits the error: with a=Σ^{1/2}z, b=Σ^{1/2}β, aᵀb = zᵀΣβ, so σ_M² contains +2(zᵀΣβ)² where the correct centered variance requires +(zᵀΣβ)². Hence the normalized statistic σ_M⁻¹√n(zᵀβ̂_M − zᵀΣβ) has limiting variance (correct σ²)/(correct σ² + (zᵀΣβ)²) < 1 whenever (zᵀΣβ)²>0, so sup_t |F_M(t)−Φ(t)| cannot vanish. The same flawed σ² feeds Corollary 3.2, Theorem 3, and the ridge-type analyses via the quadratic functional; all main CLT statements inherit the miscalibration. This is not a matter of outside-consensus disagreement; it is an internal identity error.","agreement_with_reader":"disagree"},"referee_report":{"model":"deepseek-v4-flash","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.","tokens_in":98999,"tokens_out":14866,"duration_ms":141939,"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":[{"comment":"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.","section":"Supplementary S.3, Lemma S1/S2"},{"comment":"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.","section":"Theorem 1 and Corollary 3.1"},{"comment":"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.","section":"Theorems 3, 4, 5 and Corollaries 3.2, 4.1, 4.2"},{"comment":"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.","section":"Assumption 1(a) and Section 5"}],"minor_comments":[{"comment":"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.","section":"Section 5"},{"comment":"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.","section":"Section 1.1"},{"comment":"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.","section":"Throughout"}],"recommendation":"reject","confidential_remarks":"To the editor: I see no indication of bad faith; the issue is an internal algebraic error in the variance normalization. The authors' reliance on their own earlier first-order results [62,63] is appropriate given that the paper explicitly extends those results. The numerical study is not designed to detect the error because it never checks the theoretical variance values. If the variance computations and all downstream CLT normalizations are re-derived, a future revision could be viable, but the central claims as stated are not established."},"author_rebuttal":null,"desk_editor":{"model":"deepseek-v4-flash","letter":"Bottom line: the paper has a real and original program, but a load-bearing technical error invalidates the main CLT statements as stated. The stress-test concern is correct. In Lemma S1/S2, for iid x with mean zero and variance one,\n\nVar((a^T x)(b^T x)) = (E[x0^4]-3) sum_i a_i^2 b_i^2 + ||a||^2 ||b||^2 + (a^T b)^2.\n\nThe paper defines sigma^2 as the raw second moment,\n\nsigma^2 = (E[x0^4]-3) sum_i a_i^2 b_i^2 + 2(a^T b)^2 + ||a||^2 ||b||^2,\n\nwhich is exactly the uncentered second moment, (a^T b)^2 too large. Dividing by sigma sqrt(n) leaves normalized sum variance 1 - (a^T b)^2 / sigma^2, not 1. The proof's claim that the second Lyapunov condition holds \"trivially\" is false unless a^T b = 0.\n\nThis is not cosmetic. Theorem 1's sigma_M^2 inherits the mistake: with a = Sigma^{1/2} z and b = Sigma^{1/2} beta, the term 2(z^T Sigma beta)^2 should be (z^T Sigma beta)^2, so the normalized statistic has limiting variance strictly below 1 whenever z^T Sigma beta != 0, and the stated sup-norm Berry-Esseen bound cannot hold. The same flawed sigma^2 feeds Corollary 3.1, Corollary 3.2, Theorem 3, and the ridge-type analyses through the quadratic functional. Theorem 2, for the beta-quadratic form, is centered correctly; the error is localized but it sits at the base of the main results.\n\nWhat the paper does well: it identifies a genuinely underdeveloped problem, develops a serious proof framework combining leave-one-out, martingale CLT, and anisotropic local laws, and it makes a real attempt to map sparsity, LD, and heritability into second-order behavior. The UK Biobank numerical work is a legitimate check, though the empirical standardization in the Shapiro-Wilk histograms would not expose the variance miscalibration.\n\nA secondary soft spot is Assumption 1(a): bounded row norm ||z|| <= C does not hold for raw SNP rows with p ~ 461,000 unless an unstated row normalization is applied, and that normalization needs to be tracked through the variance formulas. The self-citations to [62, 63] are for first-order limits and are not a problem.\n\nMy recommendation: this deserves a serious referee because the topic matters and the proof program is substantial, but the referee should not accept it as is. The variance normalization needs to be corrected and every downstream formula rechecked. I would not cite it in its current form.","headline":"Serious paper, but the central variance normalization in the key quadratic-form lemma is wrong, so the main CLTs are not established as written.","tokens_in":99513,"tokens_out":9362,"would_cite":false,"duration_ms":87396,"reading_group":"maybe","serious_thinker":"yes","would_accept_peer_review":true},"rs_alignment":null,"lean_confirmation":null,"pith_extraction":{"msc":["60F05","62J07","60B20","62P10"],"pacs":[],"model":"deepseek-v4-flash","headline":"This paper proves CLTs for polygenic risk score predictions at individual and cohort levels and shows that ignoring prediction-induced uncertainty understates variance.","keywords":["polygenic risk score","high-dimensional prediction","central limit theorem","random matrix theory","ridge regression","Berry-Esseen bound","heritability","linkage disequilibrium"],"falsifier":"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.","tokens_in":98260,"feed_emoji":"🧬","tokens_out":7423,"duration_ms":68543,"temperature":0.7,"pith_summary":"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.","feed_headline":"PRS prediction errors go Gaussian, but wider than naive intervals","feed_subtitle":"New CLTs for PRS values and cohort R-squared show when ignoring training noise breaks intervals.","key_machinery":"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.","core_discovery":"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.","pith_inferences":["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."],"forward_implications":["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."],"supporting_citations":[{"why":"Supplies the first-order limit of the marginal estimator's prediction accuracy that the CLT extends to a distributional result.","marker":"[63]"},{"why":"Supplies the first-order limits of reference-panel ridge-type PRS estimators that the new second-order results build on.","marker":"[62]"},{"why":"Provides the high-dimensional ridge prediction asymptotics and signal-to-noise parameterization used as the first-order baseline.","marker":"[22]"},{"why":"Supplies the anisotropic local laws used to control resolvents and reference-panel randomness.","marker":"[35]"},{"why":"Provides the martingale central limit theorem used for the quadratic form β^TΣβ.","marker":"[13]"},{"why":"Supplies the Berry-Esseen bound for martingales that yields the m^{-1/5} convergence rate.","marker":"[28]"},{"why":"Provides the de-biased estimator techniques adapted in the supplement for the traditional ridge estimator under Gaussian assumptions.","marker":"[8]"}],"fun_headline_variants":["PRS confidence intervals too narrow without new variance CLTs","High-dim PRS: Gaussian limits, but variance underestimated","New CLTs for PRS widen variance, prevent overconfidence","PRS prediction uncertainty larger than naive theory claims","Second-order CLTs show PRS intervals need more spread"],"cache_read_input_tokens":3200,"weakest_assumption_plain":"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.","fun_headline_variants_meta":{"raw":{"variants":["PRS confidence intervals too narrow without new variance CLTs","High-dim PRS: Gaussian limits, but variance underestimated","New CLTs for PRS widen variance, prevent overconfidence","PRS prediction uncertainty larger than naive theory claims","Second-order CLTs show PRS intervals need more spread"]},"model":"deepseek-v4-flash","effort":"low","cost_usd":0.000304,"raw_usage":{"total_tokens":1827,"prompt_tokens":1106,"completion_tokens":721,"prompt_tokens_details":{"cached_tokens":384},"prompt_cache_hit_tokens":384,"prompt_cache_miss_tokens":722,"completion_tokens_details":{"reasoning_tokens":651}},"tokens_in":722,"tokens_out":721,"duration_ms":7687,"temperature":1.0,"reasoning_tokens":651,"cache_read_input_tokens":384,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-08-10T23:17:05.790371+00:00","model_set":{"reader":"deepseek-v4-flash"},"falsifier":"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.","supporting_citations":[{"cited_title":"On polygenic risk scores for complex traits prediction","cited_arxiv_id":null,"evidence_quote":"Supplies the first-order limit of the marginal estimator's prediction accuracy that the CLT extends to a distributional result."},{"cited_title":"On blockwise and reference panel-based estimators for genetic data prediction in high dimensions","cited_arxiv_id":null,"evidence_quote":"Supplies the first-order limits of reference-panel ridge-type PRS estimators that the new second-order results build on."},{"cited_title":"High-dimensional asymptotics of prediction: Ridge regression and classification","cited_arxiv_id":null,"evidence_quote":"Provides the high-dimensional ridge prediction asymptotics and signal-to-noise parameterization used as the first-order baseline."},{"cited_title":"Anisotropic local laws for random matrices","cited_arxiv_id":null,"evidence_quote":"Supplies the anisotropic local laws used to control resolvents and reference-panel randomness."},{"cited_title":"On the rate of convergence in the central limit theorem for martingales with discrete and continuous time","cited_arxiv_id":null,"evidence_quote":"Supplies the Berry-Esseen bound for martingales that yields the m^{-1/5} convergence rate."}],"review_version":1}