ePRS Penalization Framework

Summary

The ePRS Penalization Framework is a mathematical model that optimizes variant-specific effect sizes in a small target cohort by penalizing them using statistical evidence from a large-scale external GWAS. The framework leverages a weighted elastic net regression that incorporates genetic correlation ($r_g$) to scale the influence of the external GWAS summary statistics. This mathematical design provides a fail-safe mechanism that guards against borrowing information from genetically unrelated source phenotypes.

Mathematical Formulation

Variant-Specific Penalty ($E_j$)

For a given variant $j$ with an external GWAS P-value $p_j$, the variant-specific penalty $E_j$ is defined as the reciprocal of its log-transformed P-value (equivalent to the $s$-value or surprisal in base 10): $$E_j = \frac{1}{-\log_{10}p_j}$$ A smaller P-value indicates stronger evidence of association, resulting in a smaller penalty $E_j$, which encourages the selection of the variant and allows for larger estimated effect sizes.

If summary statistics are combined from two independent, non-overlapping external GWAS (studies $a$ and $b$), the joint penalty is calculated as: $$E_j = \frac{1}{-(\log_{10}p_{a,j} + \log_{10}p_{b,j})}$$

Weighted Elastic Net Loss Function

The variant weights $\hat{\beta}$ for the target phenotype are estimated by minimizing the following objective function: $$\hat{\beta} = \arg\min_{\beta} \frac{1}{2n} \sum_{i=1}^n \left( y_i - \mathbf{z}i^T\boldsymbol{\gamma} - \sum^M E_j |\beta_j| + (1-r_g) \sum_j |\beta_j| \right) \right)$$ where: - $n$ is the number of individuals in the target cohort. - $y_i$ is the target phenotype vector for individual $i$. - $x_{ij}$ is the genotype value (minor allele count) for individual $i$ at variant $j$. - $\mathbf{z}_i$ is the vector of unpenalized covariates (e.g., sex, age, principal components) for individual $i$, with corresponding coefficients $\boldsymbol{\gamma}$. - $\beta_j$ represents the estimated variant-specific effect sizes. - $r_g$ is the genetic correlation between the source and target phenotypes. - $\lambda$ and $\alpha$ are the regularization hyperparameters.}^M x_{ij}\beta_j \right)^2 + \lambda \left( \frac{1-\alpha}{2} \left( r_g \sum_{j=1}^M E_j \beta_j^2 + (1-r_g) \sum_j \beta_j^2 \right) + \alpha \left( r_g \sum_{j=1

Fail-Safe Behavior

The term $r_g$ modulates the penalty structure: - When $r_g \to 1$, the penalty focuses entirely on the variant-specific penalties $E_j$ derived from the external GWAS, fully borrowing its statistical power. - When $r_g \to 0$, the external evidence is disregarded, and the loss function reduces to a standard, unweighted elastic net regression on the target cohort.

Citations

• Lin, Y., Beier, C. P., Sobiskova, Z., Hamandi, K., Stödberg, T., Ng, C. C., Andrade, D. M., Syvertsen, M. R., Gardella, E., Orsini, A., Fong, C. Y., Zarubova, J., Kajsová, M., Lim, K. S., Selmer, K. K., Cerulli Irelli, E., Rubboli, G., Striano, P., BIOJUME Consortium, Pal, D. K., & Strug, L. J. (2026). Transfer learning enhances clinical utility of polygenic scores with small, phenotypically refined cohorts. Genome Research, 36(7), 1403-1416. Source paper: Genome Res.-2026-Lin-1403-16.pdf