PRS-CS

PRS-CS employs a global-local scale mixture of normals prior on SNP effect sizes. Unlike discrete mixture models, this continuous shrinkage framework allows each variant to have its own adaptive penalty parameter. The horseshoe prior and Strawderman-Berger prior are key options, providing robust shrinkage of small effects while preserving large signals. This eliminates the need for arbitrary p-value thresholding or clumping steps inherent in C+T methods.

The method explicitly models linkage disequilibrium using an external reference panel (e.g., 1000 Genomes Project). By incorporating the LD structure as a block-diagonal matrix, PRS-CS jointly estimates effect sizes across correlated variants. This avoids double-counting of signals and accounts for the correlation between SNPs that confounds naive marginal effect summation. The LD matrix is partitioned into independent blocks for computational tractability.

PRS-CS uses a Markov Chain Monte Carlo (MCMC) Gibbs sampling algorithm to infer the posterior distribution of SNP effect sizes. Key outputs include:

• Posterior mean effect sizes for each variant
• Posterior inclusion probabilities indicating variant importance
• Full posterior distributions for uncertainty quantification The method operates directly on GWAS summary statistics without requiring individual-level genotype access.

A critical hyperparameter φ controls the overall sparsity of the genetic architecture. This global shrinkage factor is estimated from the data using a fully Bayesian approach with a half-Cauchy prior. The parameter adaptively determines the proportion of variants with non-zero effects, allowing PRS-CS to automatically adjust to polygenic architectures ranging from sparse (few large effects) to highly polygenic (many small effects).

PRS-CS-auto extends the base model by placing a gamma prior on the global shrinkage parameter, enabling fully automatic estimation without cross-validation. This variant learns the optimal shrinkage directly from the GWAS summary statistics. The auto version is particularly valuable when an external validation dataset is unavailable, making it suitable for biobank-scale applications where holdout data may be limited.

Despite its Bayesian complexity, PRS-CS achieves practical runtimes through:

• Block-diagonal LD matrix approximation reducing matrix operations
• Gibbs sampling with conjugate updates for most parameters
• Pre-computed LD blocks from reference panels Typical runtime for a GWAS with 1M SNPs is under 2 hours on a single compute node, making it scalable to biobank-scale analyses.

The core statistical innovation of PRS-CS. Unlike methods that discretely classify variants as causal or null, continuous shrinkage applies a global-local scale mixture of normals prior, allowing each SNP's effect size to be shrunk toward zero by a unique, adaptive amount. This better captures the polygenic architecture where many variants have tiny but non-zero effects, avoiding the hard thresholding of C+T or the discrete mixture of LDpred2.

The input fuel for PRS-CS. These aggregated results from genome-wide association studies contain, for each SNP:

• Effect allele and alternate allele
• Beta coefficient or odds ratio
• Standard error and p-value
• Sample size and allele frequency PRS-CS requires only these summary-level data, not individual genotypes, enabling analysis when raw data cannot be shared due to privacy constraints.

A critical external input for PRS-CS. The algorithm uses a population-matched LD reference (e.g., 1000 Genomes Project, UK Biobank) to model the correlation structure between SNPs. This allows PRS-CS to:

• Distinguish true causal signals from tagging variants
• Properly distribute effect sizes across correlated SNPs
• Avoid double-counting genetic effects The choice of reference panel significantly impacts cross-ancestry portability.

The statistical framework underpinning PRS-CS. Rather than specifying the shrinkage prior's hyperparameters arbitrarily, the method estimates them directly from the GWAS data using a variational Bayes algorithm. This adaptive approach:

• Learns the genetic architecture from the observed summary statistics
• Automatically tunes the amount of shrinkage per variant
• Avoids the need for cross-validation to select tuning parameters
• Enables computationally efficient posterior inference

A major challenge for PRS-CS and all polygenic methods. Scores trained on European GWAS often show substantial attenuation in non-European populations due to:

• Differences in LD structure between populations
• Divergent allele frequencies at causal variants
• Gene-environment interactions PRS-CSx extends the framework to multi-ancestry settings by jointly modeling summary statistics from diverse populations, improving transferability.