[Paper Review] REMI: Regression with marginal information and its application in genome-wide association studies

Jian Huang, Yuling Jiao+2 more|arXiv (Cornell University)|May 3, 2018

Genetic Associations and EpidemiologyBiochemistry, Genetics and Molecular Biology39 references4 citations

TL;DR

This paper proposes REMI, a novel $μatext{1}$-penalized regression method that estimates high-dimensional coefficients using only marginal summary statistics and a small reference panel, achieving estimation error comparable to Lasso with full individual-level data. It establishes theoretical guarantees showing REMI attains the minimax optimal error rate under high-dimensional settings with limited access to raw data.

ABSTRACT

In this study, we consider the problem of variable selection and estimation in high-dimensional linear regression models when the complete data are not accessible, but only certain marginal information or summary statistics are available. This problem is motivated from the Genome-wide association studies (GWAS) that have been widely used to identify risk variants underlying complex human traits/diseases. With a large number of completed GWAS, statistical methods using summary statistics become more and more important because of restricted accessibility to individual-level data sets. Theoretically guaranteed methods are highly demanding to advance the statistical inference with a large amount of available marginal information. Here we propose an $\ell_1$ penalized approach, REMI, to estimate high dimensional regression coefficients with marginal information and external reference samples. We establish an upper bound on the error of the REMI estimator, which has the same order as that of the minimax error bound of Lasso with complete individual-level data. In particular, when marginal information is obtained from a large number of samples together with a small number of reference samples, REMI yields good estimation and prediction results, and outperforms the Lasso because the sample size of accessible individual-level data can be limited. Through simulation studies and real data analysis of the NFBC1966 GWAS data set, we demonstrate that REMI can be widely applicable. The developed R package and the codes to reproduce all the results are available at https://github.com/gordonliu810822/REMI

Motivation & Objective

• To address the challenge of high-dimensional variable selection and estimation when individual-level data are unavailable, relying only on marginal summary statistics.
• To develop a theoretically grounded method for genome-wide association studies (GWAS) that leverages external reference panels to estimate regression coefficients.
• To achieve estimation accuracy comparable to Lasso using complete individual-level data, despite limited access to raw genotype and phenotype data.
• To provide a scalable and privacy-preserving alternative to traditional regression methods in large-scale genetic studies.

Proposed method

• REMI formulates a regularized optimization problem using only marginal statistics: $\mathbf{X}_j^T\mathbf{y}$ and $\mathbf{X}_j^T\mathbf{X}_j$ for each predictor $j$, along with a reference panel $\mathbf{X}_r$ to estimate the covariance structure.
• The method employs an $\ell_1$-penalized estimator that minimizes a loss function based on marginal correlations and estimated variances, with the reference panel used to approximate the design matrix covariance.
• It derives an upper bound on estimation error that matches the minimax rate of Lasso when full data are available, under standard high-dimensional assumptions.
• Theoretical analysis relies on restricted eigenvalue conditions on the reference panel's covariance matrix and probabilistic bounds on estimation errors in the marginal statistics.
• The method is robust to noise in marginal estimates and maintains consistency when the reference panel size $n_r$ is small relative to the number of predictors $p$, provided $n_r \ll p$.
• An R package is implemented and publicly available for reproducibility and application to real GWAS data.

Experimental results

Research questions

• RQ1 Can high-dimensional regression coefficients be accurately estimated using only marginal summary statistics and a small reference panel, without access to individual-level data?
• RQ2 Does the proposed method achieve estimation error comparable to Lasso with complete data, even when individual-level data are inaccessible?
• RQ3 What theoretical guarantees can be established for such a method under high-dimensional asymptotics with limited data access?
• RQ4 How does the method perform in practice on real-world GWAS data, particularly when the reference panel is small relative to the number of predictors?
• RQ5 Can the method maintain prediction accuracy and variable selection consistency in the presence of noisy or biased marginal estimates?

Key findings

• REMI achieves an estimation error bound that matches the minimax optimal rate of Lasso when complete individual-level data are available, under standard high-dimensional assumptions.
• The method's error bound scales as $\mathcal{O}\left(\sigma_\epsilon \sqrt{\frac{s \log p}{n}} + \sqrt{\frac{s \log p}{n_r}}\right)$, where $s$ is the sparsity, $n$ is the sample size of the main study, and $n_r$ is the reference panel size.
• Theoretical analysis shows that with high probability ($\geq 1 - 3/p^2 - 1/p^3$), the estimator satisfies the required conditions for error control.
• Simulation studies confirm that REMI outperforms standard Lasso when only marginal statistics are available, especially when the reference panel is small.
• Real data analysis on the NFBC1966 GWAS dataset demonstrates that REMI achieves competitive prediction and variable selection performance compared to methods using full data.
• The method maintains robustness and consistency even when marginal statistics are estimated from large but noisy summary data, thanks to the use of a reference panel for covariance estimation.