Optimal Spjøtvoll and sieve-grouped p-value weighting
Mathematical Methods AIMech:OptimalSpjotvollAndSieveGroupedPValueWeighting DRAFT
Aliases: Roeder-Wasserman Optimal and Grouped P-Value Weighting, Sieve-Principle Method-of-Moments Grouped P-Value Weighting, Maximin Binary and Oracle Spjøtvoll P-Value Weighting, Worst-Case Robust Weighted Bonferroni Testing
A frequentist family-wise error rate (FWER) p-value weighting and robustness framework developed by Kathryn Roeder and Larry Wasserman (2009, Statistical Science 24(4):398-413, arXiv:1010.4637v1, DOI:10.1214/09-STS289) for testing m hypotheses H_j: xi_j = 0 versus xi_j > 0 with standardized statistics T_j ~ N(xi_j, 1) and one-sided p-values P_j = Phi_bar(T_j), rejecting H_j when P_j <= alpha w_j / m (equivalently T_j > z_{alpha w_j / m}) for weights w_j >= 0 satisfying m^{-1} sum_{j=1}^m w_j = 1 (Lemma 2.1) or E_H[W_j] <= 1 for random auxiliary weights W_j = g(V_j, c) independent of P_j under H_j (Lemma 2.2). Maximizing average power m_1^{-1} sum_{j: xi_j > 0} Phi_bar(z_{alpha w_j / m} - xi_j) yields the one-parameter Spjøtvoll/Rubin-Dudoit-van der Laan oracle weight family w_j = rho_c(xi_j) = (m/alpha) Phi_bar(xi_j/2 + c/xi_j) for xi_j > 0 (and 0 for xi_j = 0), where c = c(theta) solves m^{-1} sum_{j=1}^m rho_c(xi_j) = 1 and gives test-specific critical cutoffs T_j > xi_j/2 + c/xi_j and oracle power Phi_bar(c/xi_j - xi_j/2) (Theorem 3.1, Lemma 3.2). Roeder and Wasserman prove two fundamental robustness properties explaining why poorly specified weights still outperform unweighted Bonferroni: (Property I) for worst-case binary weights B = min{w_j : w_j > 1} and b = min{w_j}, the robustness function (power gain minus power loss) R(xi) = [pi(xi, B) - pi(xi, 1)] - [pi(xi, 1) - pi(xi, b)] is positive iff Phi_bar(z_{B alpha/m} - xi) + Phi_bar(z_{b alpha/m} - xi) > 2 Phi_bar(z_{alpha/m} - xi) (Theorem 3.4) and remains positive on a non-empty safe zone [0, xi_*] even when b = 0 and B >= 2 (Lemma 3.5); and (Property II) although the optimal weight functional Q -> rho_Q(xi) is discontinuous in Kolmogorov-Smirnov distance d(Q, Q_tilde) = gamma when a fraction gamma of nulls are misspecified as positive mean u > 0 (Lemma 3.6), the worst-case power inf_{u >= 0} pi_u(xi) occurs at the least-favorable false-positive mean u_* = sqrt(2 c_*) solving gamma Phi_bar(sqrt(2 c_*)) + a Phi_bar(xi/2 + c_*/xi) = alpha/m and equals Phi_bar(c_*/xi - xi/2), which still beats unweighted Bonferroni for xi < xi_* and has asymptotic power deficit bounded by gamma/(1 - a) + O(a) (Theorems 3.7-3.8). To construct practical weights, the method provides both closed-form maximin binary external weights (Theorems 4.1-4.3) and a sieve-principle data-driven grouped estimator that partitions tests into K << m prior groups G_k of size r_k >= 20-30, estimates group mixture parameters (pi_k, xi_k) under T_{ik} ~ (1 - pi_k) N(0, 1) + pi_k N(xi_k, 1) via method of moments pi_hat_k = Y_k^2 / (Y_k^2 + S_k^2 - 1) and xi_hat_k = Y_k / pi_hat_k (Equation 12; or Equation 13 for two-sided chi_1^2 mixtures), and forms linearly smoothed group weights w_hat_k = (1 - gamma) w(xi_hat_k) + gamma K^{-1} sum_{k'} w(xi_hat_{k'}) with gamma in {0.01, 0.05} (Section 5).
Identity Notes & Concept Scope
No normalized identity match for Optimal Spjøtvoll and sieve-grouped p-value weighting across all 442 pinned AIO records and 592 maintained IDs, names, aliases and scoped synonyms. Only frontier reference matches in Round 194-197 discovery artifacts were found. Distinct from AIMech:ConstrainedConvexPValueWeightingInMonotoneLikelihoodRatioFamilies (Dobriban 2018), which solves an order- and box-constrained convex ROC optimization across general monotone likelihood ratio families via an O(J) tridiagonal-plus-rank-1 interior-point solver; distinct from AIMech:BayesOptimalPValueWeightingUnderGaussianPrior (Dobriban, Fortney, Kim, & Owen 2015), which integrates over heterogeneous Gaussian prior uncertainty mu_i ~ N(eta_i, sigma_i^2); distinct from AIMech:OptimalMultiWeightedStepUpProcedureForFDRControl (Roquain & van de Wiel 2009), which optimizes multi-weighted step-up critical values for false discovery rate (FDR) control; and distinct from AIMech:AdaptiveDataDrivenOptimalPValueWeighting (Durand 2019), which adapts multi-weighted step-up FDR procedures via group-ordered isotonic optimization, whereas Roeder and Wasserman (2009) establish the worst-case robustness function R(xi) (Theorems 3.3-3.4, Lemma 3.5), the Kolmogorov-Smirnov discontinuity of optimal weights vs. least-favorable worst-case power at u_* = sqrt(2 c_*) (Lemma 3.6, Theorems 3.7-3.8), closed-form maximin binary external weights (Theorems 4.1-4.3), and sieve-principle method-of-moments grouped p-value weighting (Section 5, Equations 12-13).
Claims & Grounded Evidence
4 Claims
Mechanism
#one-parameter-spjotvoll-optimal-weights-and-oracle-power
For standardized one-sided Gaussian test statistics T_j ~ N(xi_j, 1) and weighted Bonferroni rejection rule P_j <= alpha w_j / m at family-wise error rate alpha, maximizing the average power m_1^{-1} sum_{j: xi_j > 0} Phi_bar(z_{alpha w_j / m} - xi_j) subject to w_j >= 0 and m^{-1} sum_{j=1}^m w_j = 1 yields the one-parameter optimal weight family w_j = rho_c(xi_j) = (m/alpha) Phi_bar(xi_j/2 + c/xi_j) for xi_j > 0 and w_j = 0 for xi_j = 0 (Theorem 3.1), where c = c(theta) is the unique root of m^{-1} sum_{j=1}^m rho_c(xi_j) = 1, corresponding to test-specific critical value T_j > xi_j/2 + c/xi_j and oracle power Phi_bar(c/xi_j - xi_j/2) (Lemma 3.2).
Mechanism
#sparse-weight-robustness-function-and-zero-weight-safe-zone
For any weight vector w with m^{-1} sum_{j=1}^m w_j = 1, minimum up-weight B = min{w_j : w_j > 1}, and minimum down-weight b = min{w_j}, the worst-case robustness function R(xi) = [pi(xi, B) - pi(xi, 1)] - [pi(xi, 1) - pi(xi, b)] (power gain from up-weighting minus power loss from down-weighting at alternative mean xi) satisfies R(xi) > 0 if and only if Phi_bar(z_{B alpha/m} - xi) + Phi_bar(z_{b alpha/m} - xi) > 2 Phi_bar(z_{alpha/m} - xi), with Lebesgue measure mu({xi : R(xi) < 0}) -> 0 and inf_xi R(xi) -> 0 as b -> 1 (Theorem 3.4); moreover, even in the extreme case b = 0 (where down-weighted hypotheses have zero power) with B >= 2, R_{B,0}(xi) > 0 on a positive safe zone [0, xi_*] containing the Mill's-ratio interval [0, z_{alpha/m} - 1/(z_{alpha/m} - z_{B alpha/m})] (Lemma 3.5).
Mechanism
#ks-discontinuity-versus-least-favorable-worst-case-power
Under the two-point alternative distribution Q = (1 - a) delta_0 + a delta_xi (where the true optimal weight on xi is rho(xi) = 1/a), perturbing a fraction gamma of null hypotheses to a small false-positive mean u > 0 yields Q_tilde = (1 - a - gamma) delta_0 + gamma delta_u + a delta_xi at Kolmogorov-Smirnov distance d(Q, Q_tilde) = gamma; choosing xi = A + sqrt(A^2 - 2c) and u = B - sqrt(B^2 - 2c) with w_0 / w_1 = K reduces the weight ratio rho_tilde(xi)/rho(xi) to 1/(K + 1) when gamma = a (e.g. a factor of 1001 reduction for K = 1000), proving that the optimal weight functional Q -> rho_Q(xi) is discontinuous in Kolmogorov-Smirnov distance (Lemma 3.6); nevertheless, the worst-case power inf_{u >= 0} pi_u(xi) at the true signal xi is attained at the unique least-favorable false-positive mean u_* = sqrt(2 c_*) solving gamma Phi_bar(sqrt(2 c_*)) + a Phi_bar(xi/2 + c_*/xi) = alpha/m and equals inf_{u >= 0} pi_u(xi) = Phi_bar(c_*/xi - xi/2) = Phi_bar((z_{alpha/(m gamma)}^2 - xi^2)/(2 xi)) + O(a) (Theorem 3.8), which still beats unweighted Bonferroni for xi < xi_* = z_{alpha/m} + sqrt(z_{alpha/m}^2 - z_q^2) with q = alpha(1 - a)/(m gamma) (when u <= xi, Theorem 3.7) and has worst-case power deficit bounded by gamma/(1 - a) + O(a) for xi >= xi_*.
Mechanism
#maximin-binary-weights-and-sieve-grouped-method-of-moments-weighting
At the marginal effect size xi_0 = z_{alpha/m} (where unweighted power is 1/2), the binary-weight robustness function R(B, varepsilon) with w_1 = B/(varepsilon B + 1 - varepsilon) and w_0 = 1/(varepsilon B + 1 - varepsilon) is unimodal in B > 1 with a unique maximizer B^*(varepsilon) and turnaround point B_0(varepsilon) where R(B_0, varepsilon) = 0 (Theorem 4.1), while maximizing the minimum power min_j pi(xi_m, w_j) subject to at least a fraction varepsilon of hypotheses achieving power >= 1 - beta yields the closed-form binary weights c = Phi_bar(z_{alpha/m} + z_{1-beta}), B = c m(1 - varepsilon)/(alpha - varepsilon c m), w_1 = B/(varepsilon B + 1 - varepsilon), and w_0 = 1/(varepsilon B + 1 - varepsilon) (Theorem 4.2); when only a prior grouping G_1, ..., G_K of hypotheses into K << m clusters of size r_k >= 20–30 is available, the sieve-principle grouped procedure models T_{ik} ~ (1 - pi_k) N(0, 1) + pi_k N(xi_k, 1), estimates group parameters via method of moments pi_hat_k = Y_k^2 / (Y_k^2 + S_k^2 - 1) and xi_hat_k = Y_k / pi_hat_k when pi_hat_k > 1/r_k (Equation 12; or Equation 13 for chi_1^2 mixtures), and applies linear smoothing w_hat_k = (1 - gamma) w(xi_hat_k) + gamma K^{-1} sum_{k'} w(xi_hat_{k'}) with gamma in {0.01, 0.05} so every group receives strictly positive weight while preserving FWER control via the low-dimensional sieve partition K << m (Section 5).
⚠️ Curated Limitations & Negative Results
- By Lemma 3.6, the unsmoothed Spjøtvoll optimal weight functional Q -> rho_Q(xi) is discontinuous in the Kolmogorov-Smirnov distance and assigns zero weight w(xi_hat_k) = 0 whenever an estimated effect size is zero or near zero (|xi_hat_k| < delta), which can prevent any hypothesis in a down-weighted group from being rejected unless weights are regularized via linear smoothing w_hat_k = (1 - gamma) w(xi_hat_k) + gamma K^{-1} sum w(xi_hat_{k'}) with gamma > 0 (Section 3 Property II and Section 5).
- Sieve-principle grouped method-of-moments weighting (Section 5, Equations 12–13) uses the same test statistics T_{ik} both to estimate the group-level mixture parameters (pi_hat_k, xi_hat_k) and to perform the weighted tests, so its approximate FWER validity relies on the sieve condition K << m with sufficiently large group sizes (r_k >= 20–30); if groups are very small or chosen post-hoc to isolate individual extreme statistics, family-wise error rate can be inflated (Section 5 and Section 6.2).
- All optimality and worst-case robustness derivations (Theorems 3.1–4.3) are formulated for weighted Bonferroni FWER control under Gaussian shift models T_j ~ N(xi_j, 1) (or chi_1^2 mixtures in Equation 13) rather than step-up false discovery rate (FDR) procedures or general non-Gaussian monotone likelihood ratio families.
- All 818 lines of Roeder & Wasserman (2009, Statistical Science 24(4):398-413, arXiv:1010.4637v1, DOI:10.1214/09-STS289) were read. Constructed numerical checks in research/discovery/optimal-spjotvoll-and-sieve-grouped-p-value-weighting-boundaries.py verify (1) Theorem 3.1 & Lemma 3.2 one-parameter Spjøtvoll oracle weights and power identity (c^* = 3.6167495309, budget error = 2.22e-16, max KKT stationarity error = 6.592e-17, max Lemma 3.2 power identity error = 3.608e-16, average oracle power = 0.53897493 > 0.39544288 unweighted Bonferroni), (2) Theorem 3.4 & Lemma 3.5 sparse-weight robustness at b = 0 and B = 4.0 (Mill's-ratio sufficient threshold = 1.18817712 < exact zero-crossing xi_* = 2.41591146 < z_{alpha/m} = 4.41717341, with R_{B,0} = +0.00051182 > 0), (3) Section 3 Property II & Lemma 3.6 Kolmogorov-Smirnov discontinuity (at gamma = 0.1, K = 1000, c = 0.1: u = 0.030529, xi = 9.763185, weight reduced from rho(xi) = 10.0 to rho_tilde(xi) = 0.00999001, ratio = 0.000999 = 1/1001) versus Theorems 3.7–3.8 worst-case power at xi = 4.2 < xi_* = 5.91887833 (least-favorable u_* = 3.44915472 gives worst-case power 0.75292715 > 0.62149445 unweighted Bonferroni), and (4) Theorems 4.1–4.2 optimal binary weights (unique mode B^* = 9.459504, turnaround B_0 = 111.65303, and maximin weights w_1 = 9.183082, w_0 = 0.569311 achieving exact target power 0.7) and Section 5 Equation (12) sieve-principle grouped method-of-moments estimation (max parameter recovery errors 5.551e-17 and 4.441e-16, smoothed weights [2.875555, 0.969733, 0.104712, 0.05]).
Artificial Intelligence Ontology (AIO) Grounding
Mapped AIO Classes
| AIO Target Class | Mapping Relation | Mapping Rationale |
|---|---|---|
AIO:DataDredgingBias
Data Dredging Bias
|
related_match | Optimal Spjøtvoll and sieve-grouped p-value weighting allocates differential per-hypothesis significance thresholds alpha w_j / m using external priors or low-dimensional group mixtures while maintaining family-wise error rate control against multiplicity-induced false positives (AIO:DataDredgingBias). |
Evidence Sources & Primary Literature
3 Sources| Source ID | Publication / Document | Type / Access | Reference & Checksum |
|---|---|---|---|
roeder-wasserman2009-ar5iv
|
Kathryn Roeder, Larry Wasserman
|
primary paper
full_text
|
ID: https://doi.org/10.1214/09-STS289
sha256:a7ac40415cd7…
|
roeder-wasserman2009-crossref
|
Kathryn Roeder, Larry Wasserman
|
primary paper
abstract_only
|
ID: https://doi.org/10.1214/09-STS289
sha256:26d259113110…
|
roeder-wasserman2009-abs
|
Kathryn Roeder, Larry Wasserman
|
primary paper
abstract_only
|
ID: https://doi.org/10.48550/arXiv.1010.4637
sha256:b3d1954b604c…
|
Ontology & Grounding Topology
Open in interactive network graph →Curation Provenance
Curator: Antigravity
Method: ai_assisted
Model: Gemini 3
Created: 2026-10-17T15:20:00Z
Updated: 2026-10-17T15:20:00Z
Source File:
data/curated/OptimalSpjotvollAndSieveGroupedPValueWeighting.yaml
· Schema:
curated_concept.yaml
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