Source-linked AI summary
Tree-guided group lasso for multi-response regression with structured sparsity, with an application to eQTL mapping
Seyoung Kim, Eric P. Xing
TL;DR
The paper addresses how to incorporate relatedness among genes into eQTL association analysis rather than analyzing genes independently or aggregating them into module averages. It proposes tree lasso, which uses hierarchical clustering trees and overlapping structured penalties; simulations show it outperforms lasso and L1/L2-regularized multi-task regression for sparsity recovery.
Problem
eQTL analyses often treat genes independently despite co-expression and shared genetic bases, while module averaging can lose individual-gene and negative-correlation information.
Method
Tree lasso uses a hierarchical clustering tree to construct overlapping group penalties that encourage related responses to share relevant covariates.
Results
The method clearly outperforms lasso and L1/L2-regularized multi-task regression in recovering true nonzero regression coefficients across 50 simulated data sets.
Takeaways & Limitations
Tree lasso directly leverages multiscale gene relationships to identify SNPs with pleiotropic effects without replacing individual gene-expression measurements with module averages.
Takeaways & Limitations
Module-network approaches that average gene expressions can lose information about individual-gene activity and negative correlations within modules.
Abstract
from arXiv · showhide
We consider the problem of estimating a sparse multi-response regression function, with an application to expression quantitative trait locus (eQTL) mapping, where the goal is to discover genetic variations that influence gene-expression levels. In particular, we investigate a shrinkage technique capable of capturing a given hierarchical structure over the responses, such as a hierarchical clustering tree with leaf nodes for responses and internal nodes for clusters of related responses at multiple granularity, and we seek to leverage this structure to recover covariates relevant to each hierarchically-defined cluster of responses. We propose a tree-guided group lasso, or tree lasso, for estimating such structured sparsity under multi-response regression by employing a novel penalty function constructed from the tree. We describe a systematic weighting scheme for the overlapping groups in the tree-penalty such that each regression coefficient is penalized in a balanced manner despite the inhomogeneous multiplicity of group memberships of the regression coefficients due to overlaps among groups. For efficient optimization, we employ a smoothing proximal gradient method that was originally developed for a general class of structured-sparsity-inducing penalties. Using simulated and yeast data sets, we demonstrate that our method shows a superior performance in terms of both prediction errors and recovery of true sparsity patterns, compared to other methods for learning a multivariate-response regression.
1. Introduction.
The paper addresses the under-addressed problem of incorporating gene relatedness into eQTL analysis by introducing tree lasso, which uses hierarchical clustering structure to estimate structured sparsity. It combines information across related genes while balancing penalties across overlapping groups.
- eQTL mapping seeks genetic markers such as SNPs that influence gene-expression levels and may clarify genes’ roles in disease processes.
- Existing analyses often treat genes independently, despite co-expressed genes potentially sharing genetic bases, leaving gene relatedness underused during association testing.
- Unlike module-network approaches that average gene expressions, tree lasso retains detailed individual-gene activity, including negative correlations within modules.
- Tree lasso directly combines statistical strength across related genes using a hierarchical clustering tree whose leaves are genes and internal nodes represent multi-granularity clusters.
- The method uses overlapping hierarchically defined groups so related responses share relevant covariates in proportion to their relatedness, while systematically balancing coefficient penalization.
- Compared with graph-guided fused lasso, tree lasso has lower computational time and can analyze more than thousands of response variables because complexity depends on tree nodes rather than graph edges.
2. Background on multivariate regression approach for eQTL mapping.
The paper formulates eQTL mapping as sparse multivariate linear regression and reviews penalties for selecting relevant SNPs across gene-expression responses. Existing penalties either select responses independently, enforce one shared SNP set, or cannot represent hierarchical response groups.
- Multivariate regression model: eQTL data are modeled with an N × J SNP matrix X, an N × K gene-expression matrix Y, and coefficient matrix B linking covariates to responses.SNP entries in X are coded as 0, 1, or 2 according to minor-allele counts; B has dimensions J × K.
- Individual sparsity: The lasso is useful when J is large and few covariates are relevant, identifying sparse nonzero elements in B through an L1-penalized optimization.The tuning parameter λ controls sparsity in the solution.
- Limitations of separate selection: Standard lasso selects relevant covariates separately for each response and lacks a mechanism for jointly selecting common covariates across related responses.This motivates multi-task penalties that exploit response relatedness.
- Group sparsity: The L1/L2 group lasso selects covariates jointly across all responses, but eQTL data rarely support the assumption that every gene shares the same relevant SNP set.A selected covariate can receive nonzero coefficients across responses, while those coefficients may vary in magnitude.
- Structured response groups: Sparse group lasso permits individual coefficients within selected groups to become zero, yet still cannot incorporate response groups organized at multiple granularities.This leaves a gap for hierarchical structures among related genes.
3. Tree lasso for exploiting hierarchical clustering tree in eQTL mapping.
The tree lasso uses a hierarchical clustering tree to impose structured sparsity in multivariate-response regression, encouraging related responses to share relevant covariates. Its overlapping-group penalty balances joint and separate selection across tree levels while supporting extensions beyond simple trees.
- Tree-guided penalty: The tree lasso assigns each tree node an overlapping group containing the responses in that node’s descendant leaves.Leaf nodes correspond to individual responses, while internal nodes represent response clusters.
- Tree-guided penalty: Weights derived from node heights balance separate covariate selection within child groups against joint selection across related responses.s_v = h_v and g_v = 1−h_v; larger g_v favors joint selection for highly related responses.
- Balanced weighting: The penalty interpolates between separate and shared covariate selection, reducing to the lasso when g_v = 0 and to shared multi-task selection when s_v = 0.The L1 and L2 components respectively support separate and joint selection for response groups.
- Balanced weighting: The weighting scheme ensures that each regression coefficient receives equal aggregate penalization across nested overlapping groups.The sum of weights for all groups containing a response is one, avoiding the imbalance associated with arbitrary overlapping-group weights.
- Extensions: The regularization extends to trees with different branching factors, forests, and pruned trees whose leaves contain variable groups.This broadens the penalty’s applicability to related structured-response settings.
- Optimization: Smoothing proximal gradient optimization is used because overlapping, nonseparable L2 terms prevent closed-form coordinate-descent updates.The method smooths the structured penalty and applies an accelerated gradient procedure to optimize the resulting objective.
4. Experiments.
Simulations and yeast eQTL analyses compare tree lasso with lasso and L1/L2-regularized multi-task regression. Across both settings, tree lasso better recovers structured sparsity and achieves lower prediction errors while revealing pleiotropic associations.
- 4.1. Simulation study: The simulation compares lasso, L1/L2-regularized multi-task regression, and tree lasso using hierarchical response structure.Figure 3 displays true and estimated coefficient matrices, with responses as rows and covariates as columns.
- 4.1. Simulation study: Tree lasso suppresses false positives and better recovers relevant covariates for correlated responses than the competing methods.Lasso produces scattered false positives, whereas L1/L2 regularization selects covariates across all responses and forms vertical stripes.
- 4.1. Simulation study: Across 50 simulated data sets, tree lasso clearly outperforms lasso and L1/L2-regularized multi-task regression in ROC-based recovery of true nonzero coefficients.The advantage is especially significant when the signal-to-noise ratio is low.
- 4.1. Simulation study: Tree lasso has lower prediction errors than all other compared methods on simulated test data.Prediction errors are averaged over 50 simulated data sets using an additional 50 samples as test data.
- 4.2. Analysis of yeast data: In the yeast data, tree lasso reveals clearer groupings of gene–SNP associations than lasso or L1/L2-regularized multi-task regression.The analysis uses chromosome 3 with 21 SNPs and 3,684 genes, and tree-lasso regularization parameters ρ = 0.9 and 0.7.
- 4.2. Analysis of yeast data: Tree lasso performs significantly better in yeast test-set prediction errors and generally finds more significant GO enrichment than L1/L2 regularization.The GO results provide evidence of pleiotropic effects, but lack of ground truth prevents direct demonstration that tree lasso produces more significant findings.
- 4.2. Analysis of yeast data: Tree-lasso GO results mostly recover previously reported enrichments while identifying many additional enrichments.Table 1 compares tree-lasso enrichments with categories reported using conventional single-SNP/single-gene tests.
- 4. Experiments: Overall, simulated and yeast experiments show increased power for detecting weak signals and reduced false positives with tree lasso.The method jointly identifies covariates relevant to multiple related responses by leveraging a hierarchical clustering tree.
SUPPLEMENTARY MATERIAL
The supplementary material documents the tree-lasso weighting scheme and adds experimental comparisons with other sparse regression methods.
- SUPPLEMENTARY MATERIAL: The tree-lasso weighting scheme is designed to penalize all regression coefficients in a balanced manner.The supplementary material states that this property is proved.
- SUPPLEMENTARY MATERIAL: Table 1 organizes enriched GO categories by tree-lasso estimates and compares them with previously reported categories.Columns 1–4 use tree-lasso coefficients; the final column reports Zhu et al. (2008) categories across biological-process, molecular-function, and cellular-component groups.
- SUPPLEMENTARY MATERIAL: Additional simulated-data experiments compare tree lasso with other sparse regression methods.These results supplement the main comparisons.