Tens of thousands of simultaneous hypothesis tests are routinely performed in genomic studies to identify differentially expressed genes. However, due to unmeasured confounders, many standard statistical approaches may be substantially biased. This paper investigates the large-scale hypothesis testing problem for multivariate generalized linear models in the presence of confounding effects. Under arbitrary confounding mechanisms, we propose a unified statistical estimation and inference framework that harnesses orthogonal structures and integrates linear projections into three key stages. It begins by disentangling marginal and uncorrelated confounding effects to recover the latent coefficients. Subsequently, latent factors and primary effects are jointly estimated through lasso-type optimization. Finally, we incorporate projected and weighted bias-correction steps for hypothesis testing. Theoretically, we establish the identification conditions of various effects and non-asymptotic error bounds. We show effective Type-I error control of asymptotic $z$-tests as sample and response sizes approach infinity. Numerical experiments demonstrate that the proposed method controls the false discovery rate by the Benjamini-Hochberg procedure and is more powerful than alternative methods. By comparing single-cell RNA-seq counts from two groups of samples, we demonstrate the suitability of adjusting confounding effects when significant covariates are absent from the model.

翻译：基因组研究中通常进行数万次同步假设检验，以识别差异表达基因。然而，由于未测量的混杂因素，许多标准统计方法可能产生显著偏差。本文研究了在存在混杂效应的情况下，多元广义线性模型的大规模假设检验问题。针对任意混杂机制，我们提出了一种统一的统计估计与推断框架，该框架利用正交结构并将线性投影整合到三个关键阶段。首先，通过分离边际和非相关混杂效应来恢复潜在系数。随后，通过套索型优化联合估计潜在因子与主效应。最后，我们引入投影加权偏差校正步骤进行假设检验。在理论上，我们建立了各类效应的识别条件与非渐近误差界。当样本量和响应变量维度趋于无穷时，我们证明了渐近z检验的有效第一类错误控制。数值实验表明，所提方法通过Benjamini-Hochberg过程控制了错误发现率，且效力优于替代方法。通过比较两组样本的单细胞RNA-seq计数数据，我们证明了当模型中缺乏显著协变量时调整混杂效应的适用性。