Simultaneous Inference for Generalized Linear Models with Unmeasured Confounders

Core Concepts

This paper proposes a unified statistical estimation and inference framework that harnesses orthogonal structures and integrates linear projections to effectively mitigate confounding effects and elucidate primary effects in generalized linear models with unmeasured confounders.

Abstract

The paper addresses the challenge of simultaneous hypothesis testing in the presence of unmeasured confounders, which can substantially bias standard statistical approaches. The proposed method, gcate (generalized confounder adjustment for testing and estimation), consists of three main steps:

Estimation of uncorrelated latent components: The authors use joint maximum likelihood estimation to obtain initial estimates of the marginal effects and uncorrelated latent components by projecting the latent factors onto the orthogonal space of the observed covariates.

Estimation of latent coefficients: The authors use the condensed singular value decomposition of the normalized latent components to obtain the final estimates of the latent coefficients.

Estimation of latent factors and direct effects: The authors employ ℓ1-regularization to simultaneously estimate the sparse direct effects and the latent factors, while removing the influence of the unmeasured confounders.

The authors establish conditions for identifying the latent coefficients and direct effects, and provide non-asymptotic estimation error bounds for these estimated quantities in high-dimensional scenarios. They also derive the asymptotic normality of their proposed bias-corrected estimator and show the proper control of statistical errors, enabling the construction of valid confidence intervals and hypothesis tests.
The proposed method is demonstrated to be effective in controlling the false discovery rate and more powerful than alternative methods through numerical experiments. The authors also showcase the suitability of the method in adjusting for confounding effects when significant covariates are absent from the model, using a real-world analysis of single-cell RNA-seq data on systemic lupus erythematosus disease.

Stats

The paper does not provide specific numerical data, but rather focuses on the theoretical and methodological development of the proposed framework.

Quotes

"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."
"To the best of our knowledge, the proposed method is the first estimation and inference framework capable of (1) accommodating general relationships between observed covariates and unmeasured confounders, allowing for arbitrary confounding mechanisms; (2) utilizing generalized linear models, allowing for nonlinear modeling; and (3) incorporating information from multiple outcomes."

Key Insights Distilled From

Simultaneous inference for generalized linear models with unmeasured confounders

by Jin-Hong Du,... at arxiv.org 04-23-2024

https://arxiv.org/pdf/2309.07261.pdf

Deeper Inquiries

How can the proposed method be extended to handle time-varying confounders or dynamic causal relationships between the observed covariates, latent factors, and the response variables

To extend the proposed method to handle time-varying confounders or dynamic causal relationships, we can incorporate a time series component into the model. This would involve modifying the generalized linear model to include lagged variables or time-dependent covariates. By introducing time-related features, we can capture the temporal dynamics of the confounders and their impact on the response variables over different time points. Additionally, we can utilize dynamic modeling techniques such as autoregressive integrated moving average (ARIMA) models or state-space models to account for the time-varying nature of the confounding effects. This adaptation would enable the framework to analyze data with sequential observations and evolving relationships between the variables.

What are the potential limitations of the orthogonal projection-based approach, and how can it be further improved to handle more complex confounding structures

One potential limitation of the orthogonal projection-based approach is its reliance on the assumption of orthogonality between the confounders and the observed covariates. In real-world scenarios, confounding structures may be more complex and may not adhere strictly to orthogonality. To address this limitation, the method can be further improved by incorporating more flexible modeling techniques that can capture non-linear relationships and interactions between the variables. This could involve using non-linear projection methods, such as kernel-based approaches or neural networks, to better capture the intricate relationships in the data. Additionally, ensemble methods or Bayesian modeling techniques could be employed to handle uncertainty and variability in the confounding structures. By enhancing the flexibility and adaptability of the modeling approach, we can better accommodate the complexities of real-world data and improve the accuracy of the inference.

Can the proposed framework be adapted to other high-dimensional statistical problems beyond differential expression analysis, such as causal inference or transfer learning

The proposed framework can be adapted to other high-dimensional statistical problems beyond differential expression analysis by modifying the model specifications and estimation procedures to suit the specific characteristics of the new problem domain. For causal inference tasks, the framework can be extended to include causal modeling techniques such as instrumental variable analysis or propensity score matching to account for confounding variables and estimate causal effects. Transfer learning applications can benefit from the framework by incorporating domain adaptation methods to transfer knowledge from one dataset to another while adjusting for confounding factors that may affect the transferability of information. By customizing the framework to the requirements of causal inference or transfer learning tasks, it can be effectively applied to a wide range of high-dimensional statistical problems beyond the scope of genomic studies.