多基因風險評分迴歸中的誤差統計方法：動差估計與正規化

Abstract

多基因風險評分（PGS）是量化個體特定表型遺傳傾向的重要指標。然而，由於其效應值需從全基因組關聯分析（GWAS）的匯總統計數據中估計，PGS 不可避免地包含測量誤差。在迴歸分析中，若直接使用這些帶有誤差的 PGS，將導致系統性的統計偏誤，例如衰減偏誤。

近年來，研究逐漸採用多重 PGS（multi-PGS）框架以提升預測準確度，但多數方法並未處理其內在的測量誤差問題。特別是在高維度設定下，忽略測量誤差不僅會導致估計偏誤，亦會扭曲變數選擇結果，使得標準正則化方法（如 Lasso）錯誤地選取雜訊而非真正的基因訊號。

為彌補此一方法學缺口，本研究提出一套完整的統計框架，用以修正 PGS 模型中的測量誤差。我們的核心策略為利用連鎖不平衡分數迴歸（LDSC）直接估計潛在真實 PGS 的共變異數矩陣，並以此作為動差法（MoM）進行偏誤修正的基礎。然而，在高維度情境下，該共變異數矩陣的估計往往存在數值不穩定的問題，且由 LDSC 推得的矩陣亦不保證為半正定，進而導致變數選擇問題轉為非凸優化。

為克服上述挑戰，我們進一步引入修正正則化方法，包括 Corrected Lasso（RCL）與 CoCoLasso。透過將非正定的共變異數矩陣投影至最近的半正定空間，此方法可確保優化問題維持嚴格凸性，同時達成穩健的誤差修正與變數選擇。 透過模擬研究，我們發現傳統 MoM 在高維度設定下表現非常不穩；相較之下，CoCoLasso 架構則能有效恢復模型參數，並維持變數選擇的一致性。在實證分析中，我們將此方法應用於包含 50 個 PGS 預測變數的模型，重新分析重度憂鬱症（MDD）患者對選擇性血清素再回收抑制劑（SSRI）的治療反應。研究結果辨識出心率 (heart rate) 為影響藥物反應的關鍵基因因子 。此結果突顯了本研究處理測量誤差對於提升臨床基因預測模型之穩定性與生物解釋性的絕對必要性 。

Polygenic scores (PGS) are critical indicators for quantifying an individual's genetic predisposition to a specific phenotype. However, since their effect sizes must be estimated from Genome-Wide Association Study (GWAS) summary statistics, PGS inevitably contain measurement errors. In regression analysis, direct use of these error-in PGS leads to systematic statistical biases, such as attenuation bias.

In recent years, research has increasingly adopted multi-PGS frameworks to enhance predictive accuracy, yet most existing methods do not address the inherent measurement error issue. Particularly in high-dimensional settings, ignoring measurement error not only introduces complex estimation biases but also distorts variable selection results, causing standard regularization methods like the Lasso to mistakenly select noise instead of true genetic signals.

To bridge this methodological gap, this study proposes a comprehensive statistical framework for correcting measurement errors in PGS-based models. Our core strategy involves leveraging Linkage Disequilibrium Score Regression (LDSC) to directly estimate the covariance matrix of the latent true PGS, which serves as the foundation for bias correction via the Method of Moments (MoM). However, in high-dimensional scenarios, the estimation of this covariance matrix often encounters numerical instability, and matrices derived from LDSC are not guaranteed to be positive semi-definite, causing the variable selection problem to become a non-convex optimization.

To overcome these challenges, we further introduce corrected regularization methods, including the regularized corrected lasso (RCL) and CoCoLasso. By projecting the non-positive semi-definite covariance matrix onto the nearest positive semi-definite space, this approach ensures that the optimization problem maintains strict convexity while achieving robust error correction and variable selection.

Through simulation studies, we found that traditional MoM performs unstably in high-dimensional settings; in contrast, the CoCoLasso framework effectively recovers model parameters and maintains variable selection consistency. In the empirical analysis, we applied this method to a model with 50 PGS predictors, re-analyzing the treatment response to Selective Serotonin Reuptake Inhibitors (SSRI) in patients with Major Depressive Disorder (MDD). The results identified heart rate as a critical genetic factor influencing drug response. These findings underscore the absolute necessity of addressing measurement errors to enhance the stability and biological interpretability of clinical genetic prediction models.