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In Section 2 we provide a briefdiscussion of previousworkon sparse regressionestimation.In Section 3 we introducethetreelasso and describean efficienotptimizationmethodbased on SPG

Tree-guided group lasso for multi-response regression with structured sparsity, with an application to eQTL mapping

ANNALS OF APPLIED STATISTICS, no. 3 (2012): 1095-1117

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Lassostructured sparsityhigh-dimensional regressiongenetic association mappingeQTL analysis

Abstract

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 ...More

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Introduction

Recent advances in high-throughputtechnologyforprofilinggene expressionsand assayinggeneticvariationsat a genome-wide scale have providedresearchersan unprecedentedopportunityto comprehensivelystudythe geneticcauses of complex diseases such as asthma,diabetes,and cancer.Expression quantitativetraitlocus mappingconsidersgene expressionmeasurements, knownas gene-expressiontraits, as intermediatpehenotypes,and aims to identifythegeneticmarkerssuch as single nucleotidepolymorphisms(SNPs) thatinfluencetheexpressionlevels of genes, whichgives rise to thevariabilityin

ReceiveDdecembe2r009;reviseFdebruar2y012. 2SSuuppppoorrtteeiidnndppaarrttbbyyNONIHR1NR00010G14M0901807765984N,.
The lasso estimationin (2.2) is equivalentto selectingrelevantcovariatesfor each oftheK responsesseparatelya,nddoes notprovideanymechanismtoenforce a joint selectionof commonrelevantcovariatesformultiplerelatedresponses.In theliteraturoefmulti-tasklearning,an L ' ¡Li penalty, knownas a grouplasso penalty[Yuan and Lin (2006)], has been adoptedin multivariate-responsreegression to takeadvantageof therelatednessof theresponsevariablesand recoverthe unionsuppor-t thepatternof nonzeroregressioncoefficientsharedacross all of theresponses[Obozinski,Wainwrightand Jordan(2008)]
This methodis widely known as the Li/L2-regularized multi-taskregressionin the machine learning communitya,nd itsestimateforregressioncoefficientiss givenas (2.3)

Highlights

Recent advances in high-throughputtechnologyforprofilinggene expressionsand assayinggeneticvariationsat a genome-wide scale have providedresearchersan unprecedentedopportunityto comprehensivelystudythe geneticcauses of complex diseases such as asthma,diabetes,and cancer.Expression quantitativetraitlocus mappingconsidersgene expressionmeasurements, knownas gene-expressiontraits, as intermediatpehenotypes,and aims to identifythegeneticmarkerssuch as single nucleotidepolymorphisms(SNPs) thatinfluencetheexpressionlevels of genes, whichgives rise to thevariabilityin

ReceiveDdecembe2r009;reviseFdebruar2y012. 2SSuuppppoorrtteeiidnndppaarrttbbyyNONIHR1NR00010G14M0901807765984N
In Section 2 we provide a briefdiscussion of previousworkon sparse regressionestimation.In Section 3 we introducethetreelasso and describean efficienotptimizationmethodbased on SPG
Let us assume thatdata are collected for J SNPs and K gene-expressiontraits overN individuals.Let X denotetheN x J matrixof SNP genotypesforcovariates,and Y x K matrixof gene-expressionmeasurementsforresponses.In eQTL mapping,each elementoftheX takesvalues from{0, 1,2} accordingto the numberof minoralleles at thegivenlocus in each individual.,we assume a linearmodel forthefunctionalmappingfromcovariatesto responsevariables: (2.1)
||i is thematrixL' norm,and A.is a tuning parameterthatcontrolstheamountof sparsityin thesolution.SettingA.to a small value leads to a smallernumberofnonzeroregressioncoefficients
In thisarticlewe proposeda novel regularizedregressionapproach,called thetreelasso, thatidentifiescovariatesrelevantto multiplerelated responsesjointlyby leveragingthecorrelationstructurein responsesrepresented as a hierarchicalclusteringtree.We discussed how thisapproachcan be used in eQTL analysis to learn SNPs withpleiotropoiceffectsthatinfluencethe activities of multipleco-expressedgenes
For optimization,we adoptedthe smoothing proximalgradientapproach thatwas originallydeveloped fora generalclass of structured-sparsity-inducpinengalties,as thetree-lassopenaltycan be viewed as a special case

Methods

The authors demonstratethe performanceof the methodon simulateddata setsand theyeastdata setofgenotypesand gene expressions,and compare the resultswiththose fromthe lasso and the Li /¿^-regularizedmulti-task regressionthatdo not assume any structureover responses.In all of the experiments,the authors determinetheregularizationparameterXby fittingmodels on a training setfora rangeof values forX,computingthepredictionerrorof each model on a validationset,and thenselectingthevalue of a regularizationparameterthatgives thelowestpredictionerror.The authors evaluatethesemethodsbased on two criteria,sensitivity/specificiitnydetectingtruerelevantcovariatesand predictionerrorson test data sets.The authors notethatthe 1-and sensitivityare equivalentto typeI errorrateand 1-,respectivelyT. esterrorsare obtainedas mean squared differencesbetweenthepredictedand observedresponsemeasurements based on testdata setsthatare independentoftrainingand validationdata sets.

4. 1.
The authors demonstratethe performanceof the methodon simulateddata setsand theyeastdata setofgenotypesand gene expressions,and compare the resultswiththose fromthe lasso and the Li /¿^-regularizedmulti-task regressionthatdo not assume any structureover responses.In all of the experiments,the authors determinetheregularizationparameterXby fittingmodels on a training setfora rangeof values forX,computingthepredictionerrorof each model on a validationset,and thenselectingthevalue of a regularizationparameterthatgives thelowestpredictionerror.The authors evaluatethesemethodsbased on two criteria,sensitivity/specificiitnydetectingtruerelevantcovariatesand predictionerrorson test data sets.The authors notethatthe 1-and sensitivityare equivalentto typeI errorrateand 1-,respectivelyT.
To illustratethe behavior of differenmt ethods,the authors fitthe lasso, the L1/L2regularizedmulti-taskregression,and the methodto a single data set simulated withthenonzeroelementsof B setto 0.4, and show theresultsin Figure3(c)-(e), respectivelyS.
S a result,once a covariate is selectedas relevantfora response,itgetsselectedforall oftheotherresponses, and the authors observeverticalstripesof nonzerovalues in Figure 3(d).
When thehierarchical clusteringstructurein Figure 3(a) is available as priorknowledge,it is visuallyclear fromFigure3(e) thatourmethodis able to suppressfalsepositives, and to recoverthe truerelevantcovariatesforcorrelatedresponses significantly betterthanothermethods

Conclusion

In thisarticlewe proposeda novel regularizedregressionapproach,called thetreelasso, thatidentifiescovariatesrelevantto multiplerelated responsesjointlyby leveragingthecorrelationstructurein responsesrepresented as a hierarchicalclusteringtree.The authors discussed how thisapproachcan be used in eQTL analysis to learn SNPs withpleiotropoiceffectsthatinfluencethe activities of multipleco-expressedgenes.
The authors' resultson boththesimulatedand yeastdata sets showed a clear advantageof thetreelasso in increasingthepowerof detectingweak signals and reducingfalsepositives

Tables

Table1: EnricheGdOcategorifeosrgeneswhosexpressiolenvelsareinfluencbeydthesameSNPinthe yeasteQTLdataset.Theresultisncolumn1s-4arebasedonthetree-lasseostimatoefregression

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Reference

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