Technical Summary: Univariate-Guided Sparse Regression for Biobank-Scale High-Dimensional Omics Data

Problem Statement

The paper addresses the challenge of performing sparse regression on ultra-high-dimensional genomic data, specifically within the context of Polygenic Risk Score (PRS) construction using biobank-scale datasets like the UK Biobank. In these settings, the number of predictors (SNPs, $p$) vastly exceeds the number of observations ($n$), often with $p > 1,000,000$ and $n \approx 500,000$.

Standard regression approaches face three core difficulties in this domain:

1. Computational Infeasibility: Fitting multivariate models on millions of SNPs for hundreds of thousands of individuals is computationally prohibitive, particularly when using cross-validation for hyperparameter tuning.
2. Correlated Predictors (Linkage Disequilibrium): SNPs exist in blocks of high correlation. Standard multivariate methods often struggle to decide whether to select a sparse set of representatives or distribute ("smear") effects across many correlated variables, leading to dense, hard-to-interpret models.
3. Data Privacy and Integration: In biomedical research, raw individual-level data is often restricted due to privacy regulations. Researchers frequently rely on summary statistics from external cohorts (e.g., FinnGen) rather than full datasets, making it difficult to integrate this external information into a unified sparse multivariate model.

Historically, PRS construction has relied on a two-step pipeline: a univariate Genome-Wide Association Study (GWAS) followed by a post-hoc shrinkage or selection method (e.g., clumping and thresholding, LDpred, PRS-CS). The authors argue this approach is suboptimal because the first step treats SNPs in isolation, lifting entire neighborhoods of correlated SNPs, while the second step attempts to correct for this double-counting, often resulting in dense predictors with flattened effect sizes.

Methodology: Omni-adapted uniLasso

The authors propose a scalable adaptation of Univariate-Guided Sparse Regression (uniLasso), originally introduced by Chatterjee et al. [2025], tailored for biobank-scale data. The method is a two-stage penalized regression procedure that leverages univariate coefficients to guide the multivariate selection.

The Algorithm

1. Stage 1: Univariate Pre-computation:
   For each SNP $j$, the algorithm computes univariate intercepts ($\hat{\beta}_{0j}$) and slopes ($\hat{\beta}_j$). Crucially, it calculates Leave-One-Out (LOO) fitted values ($\hat{y}^{-i}_{ij}$) to create a feature matrix $F$. Using LOO values rather than raw estimates forces the subsequent selection step to rely on out-of-sample predictive signals rather than in-sample noise.

   • Optimization: To handle the scale, the authors vectorized these computations and parallelized them across CPU cores using the Adelie Python package.

2. Stage 2: Guided Sparse Regression:
   A non-negative Lasso is fitted using the LOO predictions in $F$ as features to predict the response $y$.

   • Sign Constraint: The coefficients $\theta_j$ in this stage are constrained to be non-negative. The final multivariate coefficients are recovered as $\hat{\gamma}_j = \hat{\beta}_j \hat{\theta}_j$. This ensures that the sign of the final coefficient matches the sign of the univariate coefficient (or is zero), preventing sign reversals that complicate interpretation.
   • Handling Binary Responses: For binary outcomes, the method uses penalized logistic regression. Since closed-form LOO solutions do not exist for logistic loss, the authors employ Iteratively Reweighted Least Squares (IRLS) with an approximation for LOO predictions proposed by Rad and Maleki [2020].

Scalability and Stability Modifications

To make uniLasso feasible for the UK Biobank, the authors introduced several critical engineering and statistical modifications:

• Data Splitting: Instead of $k$-fold cross-validation (which is too expensive for $p \gg n$), they used a 70/10/20 train/validation/test split to tune the regularization parameter $\lambda$.
• MAF Filtering: SNPs with Minor Allele Frequency (MAF) < 0.0005 were excluded to remove rare variants that contribute little predictive power and cause instability.
• Variance Stabilization: For SNPs with low variance (but MAF $\ge$ 0.0005), the denominator in the univariate slope calculation ($S_{xx}$) was regularized by adding the fifth percentile of the empirical distribution of $S_{xx}$ values. This prevents coefficient explosion in near-constant columns.
• External Score Integration (uniLasso ES): To incorporate external data (e.g., from FinnGen) without raw data access, the authors adapted the penalty term. They use external univariate coefficients ($\tilde{\beta}_j$) to define adaptive weights ($|\tilde{\beta}_j|$) and sign constraints in the $\ell_1$ penalty of the second-stage regression, rather than computing LOO estimates from the training data.

Key Contributions

1. Scalable uniLasso Implementation: The paper presents the first adaptation of uniLasso capable of handling biobank-scale data ($p \approx 10^6, n \approx 5 \times 10^5$) using individual-level data, overcoming the memory and time barriers of the original formulation.
2. External Data Integration: A novel framework for integrating external summary statistics into a sparse multivariate model, bypassing the need for raw data sharing while improving predictive accuracy.
3. Binary Response Extension: An extension of the algorithm to binary phenotypes using logistic regression and IRLS-based approximations for LOO fits.
4. Empirical Validation on UK Biobank: A comprehensive evaluation across four phenotypes (Height, BMI, CHD, Asthma) comparing the method against standard Lasso, PRS-CS, and other baselines.

Results

The authors evaluated their methods on the UK Biobank (restricted to White British individuals) across continuous (Height, BMI) and binary (CHD, Asthma) phenotypes.

• Sparsity vs. Performance:
  • uniLasso achieved predictive performance comparable to standard Lasso (e.g., $R^2$ for Height: 0.707 vs. 0.713) but selected substantially fewer variants.
  • Specifically, uniLasso selected 38% fewer SNPs for Height and 43% fewer for BMI compared to Lasso, with negligible drops in $R^2$ (0.006 and 0.016, respectively).
  • For binary traits like CHD, uniLasso was over 50% sparser than Lasso with only a 0.001 increase in test error.
• Interpretability:
  • A key finding is the elimination of sign changes. In standard Lasso, coefficients often flip signs relative to their univariate effects due to correlation (e.g., 12% of Height coefficients changed signs). uniLasso, by construction, ensures all non-zero coefficients retain the sign of their univariate effect, significantly enhancing interpretability.
• External Scores (uniLasso ES):
  • Integrating external scores from FinnGen (uniLasso ES) yielded the best predictive performance across all four phenotypes.
  • For Height, uniLasso ES achieved an $R^2$ of 0.716 (vs. 0.713 for Lasso).
  • Despite using external data, the models remained significantly sparser than Lasso and PRS-CS.
• Comparison to PRS-CS:
  • PRS-CS, a dense Bayesian method, underperformed all Lasso-based methods on continuous traits and matched them only on CHD. PRS-CS produced dense models (using all SNPs), whereas uniLasso ES maintained high sparsity.

Significance and Claims

The paper claims that uniLasso offers a superior trade-off between predictive accuracy, sparsity, and interpretability for biobank-scale genomic data.

• Challenging the "Infinitesimal" View: The results provide an empirical counterpoint to strictly "pangenomic" or infinitesimal models (like PRS-CS or LDpred-inf) which assume every variant has a non-zero effect. The authors argue that for practical prediction, a "sparse-polygenic" view—where a few tens of thousands of variants capture most of the signal—is sufficient and more robust.
• Mechanistic Clarity: By concentrating weights on (or near) truly causal variants rather than spreading them across correlated LD blocks, uniLasso reduces the "noise" contribution from non-causal tag SNPs, potentially improving the robustness of individual risk estimates.
• Practical Utility: The method demonstrates that it is possible to fit multivariate models directly on individual-level biobank data, moving beyond the limitations of the traditional two-step GWAS-plus-PRS pipeline. The integration of external summary statistics further enhances this utility in privacy-constrained environments.

The authors conclude that while dense models have value when only summary statistics are available, the sparse multivariate approach of uniLasso is highly effective for settings where interpretability and computational efficiency are paramount.