[Paper Review] Simultaneous inference for generalized linear models with unmeasured confounders

Jin‐Hong Du, Larry Wasserman+1 more|PubMed|Sep 13, 2023

Gene expression and cancer classification4 citations

TL;DR

This paper proposes a unified framework for simultaneous inference in multivariate generalized linear models with unmeasured confounders, leveraging orthogonal structures and projected bias correction to recover latent coefficients and estimate primary effects via lasso-type optimization. The method ensures asymptotically valid Type-I error control and effective false discovery rate control, outperforming existing approaches in power and robustness under high-dimensional, non-normal response settings.

ABSTRACT

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.

Motivation & Objective

• To address the challenge of unmeasured confounders in large-scale hypothesis testing for multivariate generalized linear models in genomics.
• To develop a unified statistical framework that identifies and adjusts for confounding effects without requiring knowledge of the causal relationship between observed covariates and latent factors.
• To ensure valid statistical inference—particularly Type-I error control and false discovery rate (FDR) control—under arbitrary confounding mechanisms and non-normal response distributions.
• To extend existing methods beyond linear models and single-outcome settings to handle high-dimensional, multivariate, and nonlinear data common in modern omic studies.

Proposed method

• The method disentangles marginal and uncorrelated confounding effects through orthogonal decomposition, enabling consistent estimation of latent coefficients.
• It jointly estimates latent factors and primary effects using a lasso-type optimization framework that regularizes both the coefficient matrix and the factor loadings.
• Projected and weighted bias-correction steps are integrated to correct for estimation bias in the primary effects, enabling valid asymptotic z-tests.
• The approach leverages linear projections and non-asymptotic analysis to establish identification conditions and error bounds under arbitrary confounding mechanisms.
• Sample splitting is used in the inference stage to ensure independence between estimation and testing, though the method remains robust even without it.
• The framework is applicable to generalized linear models with multivariate responses, accommodating non-normal distributions such as those in single-cell RNA-seq data.

Experimental results

Research questions

• RQ1 Can we achieve valid simultaneous inference in multivariate generalized linear models when unmeasured confounders are present, even without knowing their causal relationship with observed covariates?
• RQ2 How can we consistently estimate primary effects and latent confounders in high-dimensional settings with non-normal, sparse, or overdispersed responses?
• RQ3 Does the proposed method maintain asymptotic Type-I error control and effective false discovery rate (FDR) control under arbitrary confounding mechanisms?
• RQ4 To what extent does the method outperform existing approaches in terms of statistical power and robustness in real-world genomics data?

Key findings

• The proposed method controls the false discovery rate (FDR) effectively under the Benjamini-Hochberg procedure, with median FDP (false discovery proportion) values of 0.191–0.219 across simulations.
• The method achieves high statistical power, with median power reaching 0.987 when no sample splitting is used, and 0.963 with 80% of data reserved for inference.
• Type-I error is well-controlled, with median rates of 0.050–0.051 across all simulation settings, indicating asymptotic validity of the z-tests.
• Non-asymptotic error bounds are established for the estimated coefficient matrix, with $ Vertm{B} - m{B}^* Vert_{ ext{F}} \lesssim 1/√{n \wedge p}$, showing consistent estimation under high-dimensional regimes.
• The method remains robust to sample splitting ratios, with minimal performance degradation even when only 20% of data is reserved for inference.
• In single-cell RNA-seq data from lupus patients, the method successfully adjusted for confounding effects even when significant covariates were unobserved, demonstrating practical utility in real-world genomics applications.