LASSO Regression

The defining characteristic of LASSO is the L1 penalty term added to the ordinary least squares loss function. This penalty is proportional to the absolute value of the coefficients:

• The constraint region forms a diamond shape in parameter space
• The loss contours hit the corners of the diamond, forcing coefficients to exactly zero
• This produces a sparse model where only a subset of genetic variants are retained
• The tuning parameter lambda (λ) controls the strength of regularization
• As λ increases, more coefficients are driven to zero, simplifying the model

The geometric property of the L1 norm is what distinguishes LASSO from ridge regression (L2 penalty), which shrinks coefficients toward zero but never eliminates them entirely.

LASSO intentionally introduces bias into coefficient estimates to dramatically reduce variance, improving out-of-sample prediction accuracy:

• Unbiased estimates from standard GWAS suffer from high variance when p >> n (more variants than samples)
• LASSO shrinks large coefficients and eliminates small ones, trading a small amount of bias for a large reduction in variance
• The optimal λ is typically chosen via k-fold cross-validation to minimize prediction error
• In PRS applications, this tradeoff prevents overfitting to the discovery GWAS sample
• The resulting model generalizes better to independent target cohorts

This regularization is essential in genomic prediction where hundreds of thousands of SNPs are tested but only a small fraction have true non-zero effects.

LASSO is computationally feasible for genome-wide data due to the coordinate descent optimization algorithm:

• Updates one coefficient at a time while holding all others fixed
• Uses soft-thresholding to apply the L1 penalty efficiently
• Each update has a closed-form solution, making iterations extremely fast
• Cycles through all predictors repeatedly until convergence
• Scales to millions of genetic variants with reasonable compute time
• Implemented in popular packages like glmnet in R and scikit-learn in Python

The algorithm exploits the fact that most coefficients remain zero throughout optimization, allowing sparse matrix operations that dramatically reduce memory and computational requirements for PRS construction.

Despite its utility, LASSO has specific limitations when applied to polygenic risk score modeling:

• Correlated predictors: LASSO tends to arbitrarily select one variant from a group of highly correlated SNPs in linkage disequilibrium, discarding others that may be equally predictive
• Saturation: LASSO can select at most n predictors (sample size) before saturating, which may be insufficient for highly polygenic traits influenced by thousands of causal variants
• Effect size bias: Selected coefficients are shrunk toward zero, underestimating true effect sizes
• Elastic net (combining L1 and L2 penalties) often outperforms pure LASSO for complex traits by retaining correlated predictors
• Methods like LDpred2 and PRS-CS were specifically developed to address these limitations while maintaining sparsity benefits

Building a LASSO-based polygenic risk score involves a structured pipeline:

• Quality control: Filter SNPs by minor allele frequency, imputation quality, and Hardy-Weinberg equilibrium
• LD pruning: Optionally reduce redundancy before LASSO to improve stability, though LASSO can handle some correlation
• Standardization: Center and scale genotypes to have mean zero and unit variance so the penalty applies uniformly
• Cross-validation: Use 5-fold or 10-fold CV to select λ, typically choosing lambda.min (minimum CV error) or lambda.1se (most regularized model within one standard error)
• Refitting: Some practitioners refit an unpenalized model using only LASSO-selected variants to reduce shrinkage bias
• Scoring: Apply the estimated coefficients to target cohort genotypes to calculate individual PRS values

LASSO occupies a specific niche in the PRS methodology landscape:

• C+T (Clumping + Thresholding): Simpler and faster but uses arbitrary p-value cutoffs rather than data-driven selection
• Ridge regression: Uses L2 penalty, retains all variants with small coefficients, often better for highly polygenic traits but produces dense models
• Elastic net: Combines L1 and L2 penalties, balancing sparsity with the ability to retain correlated predictors
• LDpred2: Bayesian approach modeling the full genetic architecture with a point-normal prior, generally superior for complex traits but computationally heavier
• PRS-CS: Continuous shrinkage prior that adapts to the LD structure, often outperforming LASSO in cross-ancestry prediction
• LASSO remains valuable for biomarker discovery applications where identifying a small, interpretable set of variants is more important than maximizing predictive R²