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Regularized Multivariate Regression for Identifying Master Predictors with Application to Integrative Genomics Study of Breast Cancer

Jie Peng, Ji Zhu, Anna Bergamaschi, Wonshik Han, Dong-Young Noh, Jonathan R. Pollack, Pei Wang

arXiv:0812.3671v2stat.APstat.ME

TL;DR

The paper addresses challenges from high-dimensional predictors and variability in existing methods. It proposes remMap, a multivariate regression method using a MAP penalty, and identifies a 17q12-q21 trans-hub whose amplification influences 31 unlinked genes.

  • Problem

    High-dimensional predictors create new challenges, while existing methods can produce high-variability results.

  • Method

    remMap is a method for fitting multivariate regression models that uses a MAP penalty to address network-related challenges.

  • Results

    31 unlinked genes are influenced by amplification of a trans-hub region in cytoband 17q12-q21.

  • Takeaways & Limitations

    The identified findings warrant additional investigation.

  • Takeaways & Limitations

    The findings are preliminary, and combining CGH and expression arrays reveals only a small portion of the regulatory relationships.

Abstract

from arXiv · show

In this paper, we propose a new method remMap -- REgularized Multivariate regression for identifying MAster Predictors -- for fitting multivariate response regression models under the high-dimension-low-sample-size setting. remMap is motivated by investigating the regulatory relationships among different biological molecules based on multiple types of high dimensional genomic data. Particularly, we are interested in studying the influence of DNA copy number alterations on RNA transcript levels. For this purpose, we model the dependence of the RNA expression levels on DNA copy numbers through multivariate linear regressions and utilize proper regularizations to deal with the high dimensionality as well as to incorporate desired network structures. Criteria for selecting the tuning parameters are also discussed. The performance of the proposed method is illustrated through extensive simulation studies. Finally, remMap is applied to a breast cancer study, in which genome wide RNA transcript levels and DNA copy numbers were measured for 172 tumor samples. We identify a tran-hub region in cytoband 17q12-q21, whose amplification influences the RNA expression levels of more than 30 unlinked genes. These findings may lead to a better understanding of breast cancer pathology.

1 Introduction

Integrative DNA copy-number and RNA-expression analysis is needed to investigate regulatory relationships and distinguish primary cancer-driving alterations from secondary expression changes. The paper introduces remMap to address high-dimensional multivariate regression while identifying predictors that regulate many responses.

  • Motivation: DNA and RNA measurements provide complementary information for studying cancer-related regulatory relationships.RNA identifies over- and under-expression, whereas DNA captures gains and losses that can drive cancer.
  • Motivation: Trans-regulatory effects are difficult to establish because DNA alterations can influence RNA levels across many unlinked genomic regions.Potential mechanisms include altered transcriptional regulators and genome rearrangements affecting cis-regulatory elements.
  • Problem: The modeling problem combines high-dimensional predictors and responses, complicated response relationships, and interest in identifying master regulators.Separate regressions can produce high variability and overfitting.
  • Method: remMap combines overall coefficient sparsity with group sparsity to select predictors that affect multiple responses.Its group penalty constrains each predictor’s coefficient vector, facilitating detection of master predictors.
  • Results: Applied to breast cancer data, remMap identifies a trans-hub in cytoband 17q12-q21 whose amplification influences RNA levels of more than 30 unlinked genes.The analysis is presented alongside extensive simulation studies.
  • Limitations: The joint analysis reveals only a small portion of regulatory relationships, focusing on those explainable by DNA copy-number changes.The authors state that this subset should include many important relationships reflecting primary genetic alterations.

2 Method

remMap fits multivariate regression when both predictors and responses are high-dimensional, combining entrywise and row-wise regularization to identify master predictors. Its MAP penalty selects predictors across responses while limiting each predictor’s influence to a subset of responses.

  • Model setting: remMap targets multivariate regression with both predictor and response dimensions larger than the sample size.In the breast cancer example, 172 samples are contrasted with hundreds of genes and chromosomal regions after pre-screening.
  • Regularization: The model applies an ℓ1 penalty to coefficient entries to control overall sparsity.This regularization addresses nonuniqueness of ordinary least squares when P > N.
  • Regularization: An ℓ2 penalty groups each predictor’s coefficients across responses, shrinking entire rows to zero and selecting predictors jointly.A predictor is excluded when its corresponding row norm is shrunk to zero.
  • Master predictors: The row-wise penalty favors predictors that affect relatively many response variables, enabling identification of master predictors.The paper motivates this structure by the existence and interest of master regulators in genetic regulatory networks.
  • Prior structure: The pre-specified indicator matrix C incorporates prior knowledge by exempting known predictor–response coefficients from penalization.Without prior information, C may be set to a constant matrix with cpq ≡ 1.
  • Estimator: The combined ℓ1 and ℓ2 terms form the MAP penalty, and its coefficient estimator is called remMap.The ℓ1 term induces coefficient sparsity, while the ℓ2 term induces row sparsity in C · B.
  • Evaluation: Simulations report that remMap outperforms the joint ℓ1-only method when large hubs exist, while performing similarly otherwise.The authors attribute the advantage to simultaneous variable selection from the ℓ2 penalty and report little cost when hubs are absent.
  • Evaluation: Joint modeling borrows information across regressions and improves model efficiency, whereas separate lasso regressions are described as prone to variability and over-fitting.The comparison concerns remMap and joint versus sep, which controls sparsity separately for each response.

3 Simulation

The simulations evaluate remMap against joint and separate regression methods under varied network, signal, dimensionality, and correlation settings. remMap methods generally provide the strongest trans-edge and master-predictor selection performance, while cv.vote reduces false positives with limited increases in false negatives.

  • Simulation design: Nine methods combine remMap, joint, or separate regressions with cross-validation, cv.vote, or BIC tuning strategies.Cross-validation uses 10 folds, and cv.vote uses Va = 5.
  • Simulation results: remMap.cv and remMap.cv.vote perform best on total false detections, followed by remMap.bic.The three separate-regression methods perform worst in this comparison.
  • Simulation results: cv.vote decreases false-positive detections while only slightly increasing false-negative counts.This pattern is reported across the simulation comparisons and is also described as controlling false-positive rates without sacrificing much power.
  • Simulation results: Larger signal sizes improve performance, whereas increasing dimensionality produces more false-negative detections.These effects are reported across methods.
  • Simulation design: 151 trans-edges and 60 trans-predictors define the uniform-network setting used to compare detection performance.This setting uses P = Q = 600, N = 200, s = 0.25, ρx = 0.4, and ρǫ = 0.
  • Simulation results: In the five-large-hub setting, remMap.cv.vote performs best for both edge detection and master-predictor prediction.For the no-big-hubs setting, remMap and joint methods perform similarly and both outperform separate methods.

4 Real application

The breast cancer application uses copy-number intervals, RNA expression data, and subtype indicators to model regulatory relationships. After cross-validation and false-positive filtering, the analysis identifies chromosome 17 trans-regulatory regions influencing 31 unlinked RNAs.

  • Data and preprocessing: The study analyzes 172 tumor samples with approximately 17K mapped genes in CGH arrays and approximately 18K mapped genes in expression arrays.The analysis focuses on 654 breast cancer-related genes and summarizes copy numbers into 384 CNAIs.
  • Data and preprocessing: CNAIs summarize genome regions whose genes tend to be amplified or deleted together within samples.For each CNAI, the analysis calculates the mean estimated copy number of its genes.
  • Model specification: The estimated RNA concentration network, Exp.Net.664, contains 664 gene pairs with significant partial correlations after accounting for other genes.This network is used to represent RNA relationships in the regression model.
  • Model specification: The remMap model includes 384 CNAIs, 654 gene expressions, and five subtype indicators as nominal predictors.Expression predictors not connected to each target in Exp.Net.664 are constrained to have zero coefficients.
  • Model selection: The selected tuning parameters are (λ1, λ2) = (355.1, 266.7), yielding 56 trans-regulations before cv.vote filtering.cv.vote with Va = 5 removes 13 inconsistent trans-edges.
  • Findings: The remaining 43 trans-edges involve three contiguous CNAIs on chromosome 17 and 31 distinct unlinked RNAs.The detected associations are reported as unlikely to result from stratification among the five tumor subtypes.
  • Findings: Amplification of CNAI-17q12 up-regulates eight linked clones and influences expression levels of 31 unlinked genes or clones.The authors suggest that transcription factors or co-activators in this region may mediate these effects; four such factors, including MED1 and MED24, are identified in the region.

5 Discussion

The discussion presents remMap as a joint, regularized multivariate regression framework for high-dimensional genomic data, emphasizing master-predictor detection, cross-validation selection, and breast-cancer findings. It also identifies extensions and a validation need for the inferred trans-hub.

  • Method: remMap fits multivariate regression models and uses a MAP penalty to identify master predictors in sparse genomic regulatory networks.The penalty combines overall sparsity with row-sparsity of the coefficient matrix.
  • Method: The combined ℓ1 and ℓ2 regularization improves model efficiency, edge detection, and master-predictor identification by borrowing information across response regressions.Joint modeling incorporates relationships among response variables into the model.
  • Model selection: cv.vote selects variables consistently identified across cross-validation folds, reducing false positives while only slightly increasing false negatives.The procedure treats each fold's training data as a bootstrap sample and selects variables appearing in a majority of folds.
  • Application: Applied to 172 breast cancer tumors, remMap suggests that amplification of cytoband 17q12-q21 influences RNA levels of 31 unlinked genes.The region is described as a breast-cancer hot region that harbors ERBB2.
  • Limitations: The authors caution that additional investigation is warranted, including sequence analysis for common motifs upstream of the 31 transcripts.The proposed motif search remains future work.
  • Extensions: The remMap idea is proposed for regulatory relationships among other biological molecules, grouped-variable regression, and vector autoregressive models; an R package is publicly available.Examples include SNP-to-RNA and RNA-to-protein relationships.

Appendix A: Proof of Theorem 1

Appendix A derives the remMap estimator's coordinate-wise solution by analyzing the penalized objective under fixed coefficients and separating positive and nonpositive cases. It then characterizes the global minimum using the resulting expressions.

  • Coordinate-wise minimization: The proof fixes all coefficient rows except βq and analyzes the resulting one-row minimization problem.The derivation treats the remaining βq′ terms as fixed.
  • Coordinate-wise minimization: For xyq > 0, the minimizing coefficient is nonnegative and follows a positive-part thresholding expression; for xyq ≤ 0, the analogous solution is nonpositive.The sign of xyq determines the sign of the minimizer.
  • Global minimization: The proof combines the coordinate-wise minima across β1,…,βQ to characterize the minimum of L(β) over ||β||2 > 0.The appendix denotes this minimum by β̂min.
  • Global minimization: The objective either attains its minimum at the nonzero candidate β̂min or at zero, depending on the relevant condition involving the penalty and the candidate norm.The proof concludes by comparing the nonzero and zero cases.

Appendix B: BIC criterion for tuning

Appendix B defines a BIC criterion for selecting remMap's two tuning parameters and derives degrees-of-freedom estimates under orthogonal design. The derivation links the criterion to residual error and shrinkage from the ℓ2 penalty.

  • BIC criterion: BIC selects (λ1, λ2) by combining N × log(RSSq) with a log N penalty weighted by the regression's degrees of freedom.The criterion is defined separately for each response regression and then combined overall.
  • Derivation: The derivation defines RSSq from residual sums of squares and obtains the estimator through derivatives of the fitted coefficients with respect to the response.The calculation uses the orthogonality relation X^T X = D^-1.
  • Degrees of freedom: The degrees-of-freedom estimator is derived for remMap when the columns of X are orthogonal, under a normal-residual assumption.The appendix uses Stein's identity to obtain an unbiased estimator.
  • Degrees of freedom: Under orthogonal design and λ2 = 0, the degrees of freedom equal the total number of nonzero coefficients because the model becomes separate lasso regressions.The regressions share the same λ1 penalty parameter.
  • Degrees of freedom: When λ2 is nonzero, remMap's degrees of freedom are smaller than the number of nonzero coefficients because the ℓ2 penalty adds shrinkage.This reduction is reflected in the derived degrees-of-freedom formula.

C.1 Preprocessing for array CGH data

The array-CGH preprocessing standardizes, smooths, normalizes, and clusters measurements into genomic copy-number intervals, then filters breast-cancer gene sets to define the expression analysis data.

  • Array preprocessing: Each array's log2 ratios is standardized to median 0, smoothed, and normalized according to its estimated noise level.Noise is estimated from regions with inferred copy number equal to 2.
  • CNAI construction: FOC builds a hierarchical tree in genome order, joins only adjacent clusters, and uses measurement similarity rather than spatial distance.The method generalizes CLAC while preserving the fixed order of genomic leaves.
  • CNAI construction: 384 non-overlapping CNAIs are defined from all 172 CGH arrays, with sample-correlation heatmaps provided for the intervals.The intervals are illustrated in Figure S-1 and their correlations in Figure S-2.

C.3 Interactions among RNA expressions

The analysis constructs a sparse RNA interaction network with a power-law-oriented degree distribution, then incorporates subtype information to reduce confounding in remMap.

  • RNA interaction network: SPACE estimates RNA interactions by identifying non-zero partial correlations under an overall sparsity assumption.The resulting network is used to account for RNA interactions when investigating CNAI–RNA regulations.
  • RNA interaction network: The SPACE tuning parameter is chosen so the estimated network has a power parameter around 2.This choice reflects the reported power-law degree distribution of genetic-regulatory networks.
  • RNA interaction network: 664 edges connect hub genes and other genes in the resulting RNA concentration network.Seven nodes have at least 10 edges; listed hubs include PLK1, PTTG1, AURKA, ESR1, and GATA3.
  • RNA interaction network: The hub genes include proteins and regulators linked in the text to genome stability, proliferation, metastasis, estrogen signaling, and breast cancer.The cited examples are PLK1, PTTG1, AURKA, ESR1, and GATA3.
  • Subtype adjustment: Subtype differences can create strong population-wide CNAI–expression correlations even when within-subtype correlations are weak.The analysis derives five expression-based breast cancer subtypes and includes subtype indicators as additional remMap predictors.
  • Subtype adjustment: The five groups correspond to Luminal A, Luminal B, ERBB2-overexpressing, Basal, and Normal Breast-like subtypes.Subtype labels are derived from normalized expression patterns using k-means clustering across 172 samples.

C.5 Comments on the results of the remMap analysis.

The remMap analysis identifies a trans-hub region on chromosome 17 whose amplification influences 31 unlinked genes. Several affected genes have reported cancer-related functions or potential therapeutic relevance.

  • Biological relevance: The 31 affected genes or clones include genes reported to have functions directly related to cancer.The paper notes that some may serve as potential drug targets.
  • Biological relevance: AGTR1 is described as a breast-cancer-associated receptor that is possibly druggable.Its genetic polymorphisms have been reported to associate with breast cancer risk.
  • Biological relevance: CDH3 encodes a cell-cell adhesion glycoprotein considered a candidate tumor-suppressor gene.The text links disturbed intracellular adhesion to invasion and metastasis of tumor cells.
  • Biological relevance: PEG3 is described as a mediator between p53 and Bax in DNA damage-induced neuronal death and may function as a tumor suppressor gene.
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