FDR-Controlled Variable Selection for Generalized Linear Models and Cox Regression with Virtual Dummies

• With far more candidate predictors than observations but few active variables, false positives can limit reproducibility and prompt costly follow-up on inactive variables.Source: §1 · p. 1
• Benjamini-Hochberg and Benjamini-Yekutieli p-values are invalid in multivariate models with more predictors than observations, while model-X knockoffs become prohibitively costly beyond tens of thousands of variables.Source: §1 · p. 1
• T-Rex scales to millions of variables by using dummy competition in early-terminated forward-selection runs, and virtual dummies remove the need to store large dummy matrices.The virtual-dummy construction had not yet been specified beyond linear models, potentially costing power for nonlinear models.Source: §1 · p. 1
• The paper extends virtual dummies to score-based forward selection for Bernoulli, Poisson, multinomial, and Cox responses, preserving equivalence to explicit dummies and reporting simulation and genomic application results.Simulations find higher power with correct specification than with a misspecified linear model.Source: §1 · p. 1

2. PRELIMINARIES

• The goal is to recover the sparse active set with as many true positives as possible while keeping FDR at or below target α.Predictors are centered and normalized, and the response depends on them through a linear predictor.Source: §2 · p. 1

2.1. The T-Rex Selector

• T-Rex runs forward selection with L random dummies and stops each experiment when T dummies enter, then thresholds variables’ relative occurrence across experiments at voting level v.The threshold and stopping parameters are calibrated to maximize selected variables while the estimated false discovery proportion stays below α.Source: §2.1 · pp. 1–2
• Although its computational complexity is linear in n and p, T-Rex’s memory bottleneck is storing an n × L dummy matrix per experiment, with L ≥ p.Source: §2.1 · p. 2

2.2. Virtual Dummies

• Virtual dummies avoid storing the dummy matrix by retaining adaptively growing projections onto the revealed subspace, realizing a full vector only when selected.This construction makes the virtual and augmented selection paths identical in distribution, carrying over the FDR guarantee.Source: §2.2 · p. 2

3.1. Score-Based Forward Selection

• For canonical-link GLMs, forward selection replaces OMP’s residual with a score contribution and ranks candidates by inner products after ML refits; AFS can instead take fractional steps.Source: §3.1 · p. 2
• The method supports Bernoulli and Poisson GLMs, multinomial responses ranked by a reference-invariant norm across class scores, and Cox regression using martingale-residual scores.Source: §3.1 · p. 2

3.2. Virtual Dummies for Score-Based Selection

• Dummies enter ranking through projections onto prior scores; a selected dummy is fully realized, then ML refits update and orthonormalize scores to extend the basis.Source: §3.2 · p. 2
• For multinomial responses, all C−1 class scores enter the ranking and must lie in the virtual-dummy basis.Source: §3.2 · pp. 2–3
• The final selected set maximizes the number of variables whose voting scores exceed the chosen threshold, subject to the estimated FDP staying below the target α.Source: §3.2 · p. 3
• Under mutual independence of null predictors and their independence from active predictors, the virtual-dummy selector is equal in distribution to explicit augmentation and inherits its FDR guarantee.Source: §3.2 · p. 3

4.1. Equivalence and Misspecification

• AD-GLM and VD-GLM coincide in FDP and TPP across the simulation grid and control FDR below target; misspecification costs most power for multinomial and least for Bernoulli responses.VD-Gaussian has TPP 0.3 for multinomial responses while VD-Multinomial recovers nearly all active predictors; under Cox, the gap closes only at the highest SNR.Source: §4.1 · p. 3

4.2. Memory and Runtime

• For Poisson, Bernoulli, and Cox, VD-GLM adds around 205 MB and runs 2.2–2.5× faster; multinomial uses 411 MB and runs 2.1× faster than explicit augmentation.The multinomial projection block remains two orders of magnitude below AD-Multinomial; the comparison uses n = 10^4, p = 10^5, and L = 5p.Source: §4.2 · p. 3

4.3. Application 1: Disease Subtype Classification

• In HAPNEST disease classification, VD-Multinomial has the highest power among methods keeping FDP below target; VD-Gaussian achieves less than half its power, while one-vs-rest and knockoffs select nothing.The knockoff filter takes 23.4 hours per replicate versus 6.5 minutes for VD-Multinomial; marginal procedures largely exceed target FDP because they test marginal rather than joint-model hypotheses.Source: Table 2 · p. 3, §4.3 · p. 4

4.4. Application 2: Survival in Clear-Cell Renal Cell Carcinoma

• For TCGA-KIRC survival, VD-AFS-Cox selects eight genes versus one for the misspecified linear model on log t; selection counts reflect differing error criteria across methods.BH and BY select several thousand genes, while the knockoff filter selects none; Cox Lasso selects 28 but provides no FDR control.Source: §4.4 · p. 4, Table 3 · p. 4

5. CONCLUSION

• The extension covers Bernoulli, Poisson, multinomial, and Cox responses; its selection path equals that of explicit augmentation in distribution, carrying over the T-Rex FDR-control result.The basis grows along the centered score of the refitted model.Source: §5 · p. 4
• On genomic data, multinomial selection recovers more than twice as many causal SNPs as any competitor within target FDR; Cox selects eight genes versus one for the linear model on log t.Misspecification reduces TPR most for multinomial responses and least for Bernoulli responses.Source: §5 · p. 4
• Future work could extend the approach to robust forward selectors such as LAD or Huber, and to path algorithms with variable deletion or more general pursuit rules.Source: §5 · p. 4