# Simultaneous inference for generalized linear models with unmeasured confounders

arXiv: https://arxiv.org/abs/2309.07261 · HTML: https://mlsanity.com/paper/2309.07261/

## Key facts

- Cited by 3 papers in the map (data through 14 Oct 2026).
- Builds on 10 of its 29 references that are in the map.
- First posted on arXiv 13 Sep 2023.

## Builds on (most cited first, 10 of 10)

- [Confidence Intervals and Hypothesis Testing for High-Dimensional Regression](https://mlsanity.com/paper/1306.3171/) (2013)
- [A Unified Computational and Statistical Framework for Nonconvex Low-Rank Matrix Estimation](https://mlsanity.com/paper/1610.05275/) (2016)
- [Doubly Debiased Lasso: High-Dimensional Inference under Hidden Confounding](https://mlsanity.com/paper/2004.03758/) (2020)
- [Bi-cross-validation for factor analysis](https://mlsanity.com/paper/1503.03515/) (2015)
- [Accounting for unobserved covariates with varying degrees of estimability in high dimensional biological data](https://mlsanity.com/paper/1801.00865/) (2018)
- [Inference in High-dimensional Multivariate Response Regression with Hidden Variables](https://mlsanity.com/paper/2201.08003/) (2022)
- [Empirical Bayes Shrinkage and False Discovery Rate Estimation, Allowing For Unwanted Variation](https://mlsanity.com/paper/1709.10066/) (2017)
- [Estimating and accounting for unobserved covariates in high dimensional correlated data](https://mlsanity.com/paper/1808.05895/) (2018)
- [Treatment Effect Estimation with Unobserved and Heterogeneous Confounding Variables](https://mlsanity.com/paper/2207.14439/) (2022)
- [High-Dimensional Inference for Generalized Linear Models with Hidden Confounding](https://mlsanity.com/paper/2209.03482/) (2022)

## Built on by (most cited first, 3 of 3)

- [Causal Inference for Genomic Data with Multiple Heterogeneous Outcomes](https://mlsanity.com/paper/2404.09119/) (2024)
- [Network-based Neighborhood regression](https://mlsanity.com/paper/2407.04104/) (2024)
- [Assumption-Lean Post-Integrated Inference with Surrogate Control Outcomes](https://mlsanity.com/paper/2410.04996/) (2024)

## Abstract (arXiv)

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.