Mixed-Effects Models for Conditional Quantiles with Longitudinal Data

Volume 5, Issue 1 2009 Article 28

The International Journal ofBiostatistics

Mixed-Effects Models for ConditionalQuantiles with Longitudinal Data

Yuan Liu, Medical University of South CarolinaMatteo Bottai, University of South Carolina

Recommended Citation:Liu, Yuan and Bottai, Matteo (2009) "Mixed-Effects Models for Conditional Quantiles withLongitudinal Data," The International Journal of Biostatistics: Vol. 5: Iss. 1, Article 28.DOI: 10.2202/1557-4679.1186

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Mixed-Effects Models for ConditionalQuantiles with Longitudinal Data

Yuan Liu and Matteo Bottai

Abstract

We propose a regression method for the estimation of conditional quantiles of a continuousresponse variable given a set of covariates when the data are dependent. Along with fixedregression coefficients, we introduce random coefficients which we assume to follow a form ofmultivariate Laplace distribution. In a simulation study, the proposed quantile mixed-effectsregression is shown to model the dependence among longitudinal data correctly and estimate thefixed effects efficiently. It performs similarly to the linear mixed model at the central locationwhen the regression errors are symmetrically distributed, but provides more efficient estimateswhen the errors are over-dispersed. At the same time, it allows the estimation at different locationsof conditional distribution, which conveys a comprehensive understanding of data. We illustratean application to clinical data where the outcome variable of interest is bounded within a closedinterval.

KEYWORDS: asymmetric Laplace distribution, longitudinal data, mixed-effects model, MonteCarlo Expectation Maximisation (MCEM) algorithm, multivariate Laplace distribution, quantileregression

Author Notes: We thank the two anonymous reviewers for their insightful comments and Dr.Sharon Yeatts for her help with the editing.



1 Introduction

Popularizedby KoenkerandBassett(1978),quantileregressionhasgraduallybe-comea well establishedtechniquein a wide rangeof applications.It representsanimportantcomplementto theclassicmeanregressionandcharacterizesthewholeconditionaldistributionof a responsevariablegivenasetof covariates.Robustnessagainstoutliers,efficiencyfor a wide rangeof errordistributions,andequivarianceto monotonetransformationare someof the other desirablefeaturesof quantileregression.Medianregression,a specialcaseof quantileregressionanda typeofrobustregression,hasbeenfamiliar to mostresearchers.Quantileregressionhasre-centlybeenappliedis avarietyof areas,includingeconomics(MachadoandMata,2005;Angristetal.,2006),medicine/publichealth(AustinandSchull,2003;Austinet al., 2005;Wei et al., 2006),ecology(Cadeet al., 2005),survival dataanalysis(Yin andCai, 2005)andmicroarraydataanalysis(Sohnet al., 2008;Li andZhu,2007;Huangetal., 2008).Formoredetailsaboutthequantileregressionapproach,onecanrefer to Buchinsky(1998),KoenkerandHallock (2001),Yu et al. (2003),andKoenker(2005).

In thecaseof classicquantileregression,therearevery few assumptionsabouttheform of errordistribution,andtheestimationof parametersis obtainedby min-imizing thesumof weightedabsoluteresidualsthroughlinearprogrammingmeth-ods. Recently,an asymmetricLaplacedistribution(ALD), which is characterizedby apeakat themodeandthick tails,hasbeenusedin Bayesianquantileregressionfor errordistribution(Yu andZhang,2005;Yu andMoyeed,2001).Theconnectionbetweenmaximizinga likelihood functioncomposedof independentlydistributedALD andminimisationof theobjectivefunctionin quantileregressionwasshownby Yu andMoyeed(2001)andusedto developthelikelihoodratio testfor quantileregression(KoenkerandMachado,1999),andto applyquantileregressionto lon-gitudinaldata(GeraciandBottai,2007).In themeanregression,a similar connec-tion existsbetweenMLE basedonnormaldistributionandleast-squaresestimation.Kotz etal. (2001)andKozubowskiandNadarajah(2008)reviewedseveraldifferentformsof theLaplacedistribution.

In this paper,we focuson exploringtheuseof quantileregressionfor analysisof longitudinaldata. Thesedataarecharacterizedby repeatedmeasurementsonthe samesubjectover time, as may be collectedin clinical trials, epidemiologicstudies,etc. By using the connectionbetweenALD andquantileregression,wedevelopa likelihood-basedapproachto estimateparametersof conditionalquantilefunctionswith the randomeffectsby adoptinganALD for the residualerrorsandamultivariatedistributionfor therandomeffectsthatis not restrictedto benormal.This quantilemixed-effectsmodel is analogousto the linear mixed model (Ver-bekeandMolenberghs,2000;Demidenko,2004). Thewithin-subjectdependence

1

Liu and Bottai: Quantile Mixed-Effect Model



amongdatais takeninto accountthroughincorporationof randomeffectsto avoidbiasin theparameterestimate.Theproposedapproachallowstheestimationatdif-ferentquantilesof conditionaldistribution,andhenceleadsto a robustestimationof parametersandconveysacomprehensiveunderstandingof data.

Thereis a relatively small amountof literatureaboutthe extensionof quan-tile regressionto longitudinaldataor dependentdata. Threetypesof modelscanbe identifiedamongthem: the marginalmodel,penalizedmodelandconditionalmodel. Jung(1996)proposedquasi-likelihoodbasedestimatingequationsfor me-dianregression,andbasedonJung’swork,Lipsitzetal. (1997)describedaweightedGEEmodelin quantileregressionfor longitudinaldatawith randomdrop-off. Karls-son(2008)examineda weightedversionof quantileregressionestimatoradjustedto the caseof nonlinearlongitudinaldata. Thesemethodsarebasicallymarginalmodelswhich capturethe overall trend amongall subjectsfor a given quantile.Koenker(2004)proposeda penalizedquantileregressionwith a largenumberofsubject-specificfixed effects,in which the inflation effectwascontrolledby a reg-ularization,or shrinkage,whosedegreeneedsto be chosensuitably. GeraciandBottai (2007)proposedaconditionalmodelfor quantileregressionwith randomin-tercept.In this paper,wegeneralizetheconditionalmodelproposedby GeraciandBottai (2007)andaquantileregressionwith multiple randomeffectsis developed.

In addition,insteadof normaldistributionfor therandomeffects,heavy-taileddistributionsare usuallyconsideredto handleoutliersandunduly largeobserva-tionsin robustmixedmodels(Langeet al., 1989;Pinheiroet al., 2001;Songet al.,2007). Inspiredby therobustnessof quantileregressionandheavy-tailedpropertyfor ALD, wedescribeandevaluatetheuseof asymmetricmultivariateLaplacedis-tribution (Kotz et al., 2001)for therandomeffects,which is characterizedby peakat zerosand thick tails on the edges. Multivariate Laplacedistributionhasbeenmainly usedin speechandimagedata,andto the bestof our knowledgeit is thefirst timethatthisdistributionis beingconsideredfor therandomeffectsin amixedmodel.

This paperis organizedas follows: Section2 gives the connectionbetweenALD andquantileregression,thequantilemixed-effectsmodel,a proposedsemi-parametricMonte Carlo ExpectationMaximization(MCEM) algorithmfor pointestimation,and an introductionto a multivariateLaplacedistribution; Section3conductsa simulationstudy aimedto evaluatethe performanceof the proposedmodel;Section4 illustratesanapplicationto clinical trial data;Section5 makesaconclusion.

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The International Journal of Biostatistics, Vol. 5 [2009], Iss. 1, Art. 28

DOI: 10.2202/1557-4679.1186



2 Model and Method

2.1 IndependentData

Suppose(xTi , yi), i = 1, . . . , N , is anindependentrandomsamplefrom somepop-

ulation,wherexi is ap × 1 vectorof regressorsandyi is a scalarresponsevariablewith conditionalcumulativedistributionfunctionFy, whoseshapeis unspecified.FollowingKoenkerandBassett(1978),theτ -th quantileof datais modeledas

Qyi(τ |xi) = xT

i βτ , i = 1, . . . , N,

whereτ ∈ (0, 1), Qyi(·) ≡ F−1

yi(·), andβτ ∈ R

p is a columnvectorof unknownfixedparameterswith lengthp. Alternatively,theaboveexpressioncanberewrittenas

yi = xTi βτ + ǫi, with Qǫi

(τ |xi) = 0,

whereǫi is theerror termwhosedistributionis restrictedto havetheτ -th quantileto bezero.Throughanumericalmethod,e.g.linearprogramming,theestimatorβτ

is obtainedby solving

βτ = arg minβ∈Rp

N∑

i=1

ρτ (yi − xTi βτ ), (1)

whereρτ (v) = v(τ − I(v 6 0)) is thelossfunctionwith v beingarealnumberandI(·) is the indicatorfunction. For a specialcaseof τ = 0.5 (medianregression),equation(1) simplifiesto

β0.5 = arg minβ∈Rp

N∑

i=1

|yi − xTi β0.5|.

Theparameterβτ andits estimatorβτ dependon thequantileτ . For simplicity,wewill omit thesubscriptin theremainderof thepaper.

A three-parameterALD (Yu andZhang,2005)providesa naturallink betweenminimizationof the sum of weightedabsoluteresidualsin equation(1) and themaximumlikelihood theory.Otherformsof Laplacedistributionweresummarisedby Kotz etal. (2001)andKozubowskiandNadarajah(2008).

A randomvariableY follows an ALD if its correspondingprobabilitydensityis givenby

f(y|µ, σ, τ) =τ(1 − τ)

σexp

{−ρτ

(y − µ

σ

)}, (2)

whereρτ (v) is definedin equation(1), σ > 0 is the scaleparameter,and−∞ <µ < +∞ is thelocationparameterwhich is alsothemodeandtheτ -th quantileof

3




y. Let µi = xTi β, andfor anygivenvalueof τ , we assumethatǫi ∼ ALD(0, σ, τ),

which restrictstheτ -th quantileof residualsto bezero. Thenthelikelihood for Nindependentobservationsis written,

L(β, σ; y, τ) ∝ σ−N exp

{

−

N∑

i=1

ρτ

(yi − xT

i β

σ

)}

. (3)

Consideringσ a nuisanceparameter,themaximisationof the likelihood in (3)with respectto the parameterβ is equivalentto the minimisationof the objectivefunctionin (1).

2.2 Longitudinal Data

Considerlongitudinaldatawith repeatedmeasurementsin the form (yij, xTij), for

j = 1, . . . , ni, and i = 1, . . . , N , whereyij is the jth scalarmeasurementof acontinuousrandomvariableon the ith subject,xT

ij arerow p−vectorsof a knowndesignmatrix andβ is a p × 1 vectorof fixed regressioncoefficients.We followthesimilar notationasthe linear-mixedmodelanddefinethe linearmixed-effectsquantilefunctionof responseyij as

yij = xTijβ + zT

ijui + ǫij , with Qǫij(τ |xij , ui) = 0, (4)

wherezij is aq×1 subsetof xij with randomeffects;ui is aq×1 vectorof randomregressioncoefficients;theerrortermǫij , for j = 1, . . . , ni andi = 1, . . . , N , is as-sumedto beindependentlydistributedasALD. Therandomregressioncoefficentsui, for i = 1, . . . , N , which accountfor the correlationamongobservations,areassumedto be mutually independentandto follow somemultivariatedistributionfu(0, Σ). We discussthechoiceof thedistributionfor randomeffects,fu(0, Σ), insection2.5.Wealsoassumeindependencebetweenui andǫij andbetweentheran-dom regressioncoefficientsui andthe explanatoryvariablesxT

ij. The conditionaldensityfunctionof yij|ui canbewrittenas

f(yij|ui, xij ; β, σ) =τ(1 − τ)

σexp

{−ρτ

(yij − µij

σ

)},

whereµij = xTijβ + zT

ijui is thelinearpredictorof theτ th quantilefunction,andτis fixed andknown.

Let f(yi|ui, xi; β, σ) =∏ni

j=1 f(yij|ui, xij ; β, σ) be thedensityfor the ith sub-ject conditionalon the randomeffect ui, whereyi = (yi1, . . . , yini

)T and xi =

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(xi1, . . . , xini)T . The completedatadensityof (yi, ui), for i = 1, . . . , N , is then

givenby

f(yi, ui|xi; η) = f(yi|ui, xi; β, σ)f(ui|xi; Σ)

= f(yi|ui, xi; β, σ)f(ui|Σ),

whereweassumethatui is independentof xi, f(ui|Σ) is thedensityof ui, andη =(β, σ, Σ) is theparameterof interest.If we let y = (y1, . . . , yN), x = (x1, . . . , xN)andu = (u1, . . . , uN), thejoint densityof (y, u) basedonN subjectsis givenby

f(y, u|x; η) =N∏

i=1

f(yi|ui, xi; β, σ)f(ui|Σ). (5)

For simplicity, in the next sectionwe will usef(y, u|η) andf(y|η) to denotef(y, u|x; η) andf(y|x; η) .

2.3 Estimation

Weobtainthemaximumlikelihoodestimatesfor theparameterη by maximisingthemarginaldensityf(y|η), whichis calculatedby integratingouttherandomeffectsuin equation(5). Thatis,L(η; y) =

∫f(y|u; η)f(u; Σ)du. Only underlimited cases,

however,is a closedform solutionavailable;generally,this integralis intractable.We proposea MonteCarloExpectationMaximisation(MCEM) algorithm,whichhasbeenwidely appliedto generalizedlinear mixed models(Booth andHobert,1999).Within thisalgorithm,randomeffectsareconsideredasunobserved,missingvaluesandasimulationmethodis usedto evaluatetheintractableintegralin theE-step.

2.3.1 E-step

Wewrite theE-stepfor theith subjectat iteration(t + 1)

Oi(η|η(t)) = E{l(η; yi, ui)|yi; η

(t)}

=

∫{log f(yi|β, ui, σ) + log f(ui|Σ)}f(ui|yi, η

(t))dui, (6)

whereη(t) = (β(t), σ(t), Σ(t)), l(η; yi, ui) is the log likelihood for the ith subject,andtheexpectationis takenwith respectto thedistributionof theunobserveddataui giventheobserveddatayi, whosedensityis

f(ui|yi, η(t)) ∝ f(yi|ui; β

(t), σ(t))f(ui|Σ(t)). (7)

5




By drawinga randomsamplevi = (vi1, . . . , vimi) of sizemi from the condi-

tionaldistributionof f(ui|yi, η(t)) in (7), theexpectationin (6) canbeapproximated

by

Oi(η|η(t)) ≈

1

mi

mi∑

k=1

{log f(yi|vik; β, σ) + log f(vik|Σ)}. (8)

Basedon theassumptionthat theobservationsareindependentacrosssubject,themarginaldensityof log f(y|η) is approximately

O(η|η(t)) ≈

N∑

i=1

Oi(η|η(t))

≈N∑

i=1

1

mi

mi∑

k=1

{log f(yi|vik; β, σ) + log f(vik|Σ)}. (9)

TheMonteCarloM-stepis usuallyrelativelysimpleaspointedout by McCul-loch (1994).Thereasonis thatO(η|η(t)) in equation(9) is thesumof a likelihoodinvolving only β andσ in thefirst termanda secondterminvolving only Σ. In thefirst term, if we let yijk = yij − zT

ijvik, thenwe mayconsideryijk to be indepen-dent,andthemaximisationof (9) with respectto β andσ leadsto a linearquantileregressionwith responsevariable yijk. The maximisationwith respectto Σ cansometimesbe written in closedform dependingon thedistributionof the randomeffects.

2.3.2 M-step

To obtainthemaximumlikelihoodestimateof theparameterη for theτ th quantilefunction,weproposethefollowing iterativeprocedure:

Step1: Initialize the parametersη(t) = (β(t), σ(t), Σ(t)), set t = 0 andsubstituteη(t) in (7);

Step2: Foreachsubjecti = 1, . . . , N , independentlydrawasamplev(t)i = (v(t)

i1 , . . . ,

v(t)imi

) for k = 1, . . . , mi, from equation(7) with samplesizemi by a MCMCsample,e.g.via theGibbssamplerwith theadaptiverejectionsamplingalgo-rithm (Gilks etal., 1995).

Step3: Solve

β(t+1) = arg minβ∈B

N∑

i=1

1

mi

mi∑

k=1

ni∑

j=1

˜ρτ (y(t)ikj − xT

ijβ),

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wherey(t)ikj = yij − zT

ijv(t)ik . Set

σ(t+1) =1

∑Ni=1 nimi

N∑

i=1

mi∑

k=1

ni∑

j=1

˜ρτ (y(t)ikj − xT

ijβ(t+1))

andΣ(t+1) = Φ(v(t)

i , . . . , v(t)N ),

whereΦ is the maximumlikelihood estimateof Σ, which dependson thedistributionof randomeffects;

Step4: Sett = t + 1 andrepeatsteps1− 3 until theparameterη achievesconver-gence.

In step2, MonteCarlo randomsamplesfor randomeffectsaregeneratedby aGibbssamplerandusedto approximatetheE-step,andtheminimisationproblemin step3 is equivalentto aweightedquantileregressionwith anoffset.Then,ateachiteration,η(t+1) is themaximumlikelihood estimateof η, givenη(t). ThechoiceoftheMonteCarlosamplesizemi andconvergencecriteriaareimportantissuesin theimplementationof theMCEM algorithm.It is well knownthat thesimulationsizemi shouldbeincreasedwith thenumberof iterations.Severalapproacheshavebeenproposedto increasethesimulationsizeover iterations.BoothandHobert(1999)suggesteda rule for increasingthenumberof simulationswhenthechangein theparametervalueis swampedby Monte Carlo error, aswell asa rule for stoppingwhenthechangein theparameterestimatesrelativeto theirstandarderrorsis smallenough.Instead,Eickhoff et al. (2004)proposeda likelihood-distance-basedalgo-rithm, in which the changein the estimatedlog likelihood over iterationsis eval-uated,andthealgorithmis stoppedwhensucha likelihood distanceis lessthanacertainsmallnumberδ with probabilitylargerthan1−ǫ (e.g.,ǫ = 0.05or 0.01).WeadoptEickhoff’s algorithmto assessconvergenceof the MCEM algorithm,sincethe likelihood at eachiterationcanbe approximatedby randomsamplesdirectly.Thefinal estimatedlog likelihoodcanalsobeusedto calcualtetheAIC to evaluatethemodelfit.

2.4 ConfidenceIntervals for Parameters

Constructionof confidenceintervals for the parametersis usually basedon theasymptoticnormalityof themaximumlikelihoodestimator(MLE). Theasymptotictheory for quantileregressionis well studied,but the developmentof convenientinferenceprocedureshasbeenchallenging,astheasymptoticcovariancematrix ofquantileestimatesinvolves the unknownerror densityfunction which cannotbe

7




estimatedreliably. In our case,the error term hasbeenset to be ALD, andfor agivenτ , themodeof ALD is locatedat theτ -th quantileof residuals.Maximizingthelikelihoodleadsto unbiasedpointestimate,butALD maynot representthetruedistributionof error density. At the sametime the densityfunction f(y|η) mightnotbedifferentiablewith respectto η. Alternatively,someothermethodsareavail-ableto provideinferencefor quantileregressionwith longitudinaldata,suchastherank scoretest proposedby Wang and Fygenson(2009) recently,and the blockbootstrapmethodwhich hasbeenappliedin the works of Buchinsky(1995)andLipsitz et al. (1997). In this study,we considerthe lattermethodto constructtestsandthe confidenceintervalsfor βτ . The bootstrapmethodhasbeenwidely usedin applicationsof quantileregression.To retainthedependentstructurein a longi-tudinaldata,independentsubjectsareassumedandthexy-pairsfrom eachsubject{(yi, xi), for i = 1, . . . , N} aretreatedasbasicresamplingunitsandsampledfromoriginal datawith replacementB times.Thepracticalquestionaboutchoosingthenumberof replicationsB wasaddressedby AndrewsandBuchinsky(2000,2001).

2.5 Multivariate LaplaceDistribution for Random Effects

In this sectionwe focuson the distributionfor the randomeffectsin our model.Randomeffectsin the linearmixedmodelaretypically assumedto follow a mul-tivariatenormaldistribution,which sometimesis believedto be too restrictivetorepresentthedataandvulnerableto outliers.MultivariateStudent’st distributionisa classicalternative,which is usefulwhenthedataareoverdispersedwith respectto the normaldistribution. Othermultivariatedistributionswith heaviertails thannormalmay alsoprovidealternatives.Inspiredby the robustnesspropertyof theLaplacedistribution,weconsideramultivariateLaplacedistribution.

As discussedby Kotz et al. (2001),let X bea q-dimensional,zero-meanGaus-sianvariablewith covariancematrix equalto Σ, andZ be a standardexponentialvariablewhich is independentof X; thenthe representationY = mZ + Z

1/2X

is distributedasmultivariateLaplacedistributionwith parameterm andΣ andde-notedasY ∼ ALq(m, Σ), whosedensitycanbeexpressedas

f(y; m, Σ) =2eyT Σ−1m

(2π)q/2 |Σ|1/2

(yT Σ−1y

2 + mT Σ−1m

)ν/2

×K ν

{√(2 + mT Σ−1m)(yT Σ−1y)

}, (10)

wherem is a q-vector,Σ is a q × q covariancematrix,ν = (2 − q)/2 andK ν(u) isthemodifiedBesselfunctionof thethird kind which is givenby

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K ν(u) =1

2

(u

2

)ν∫

∞

1

t−λ−1 exp

(−t −

u2

4t

)dt, u > 0.

Thedensityin equation(10) representsaseriesof skewedmultivariatedistribu-tion with E(Y) = m andCOV (Y) = Σ + mmT , in which parameterm controlsboth locationandskewness.For m = 0, it simplifiesto a symmetricmultivariateLaplacedistributionwhich is elliptically contoured.Also, whenq = 1, equation(10) is anotherform of asymmetricLaplacedistributiondiscussedby Kozubowskiand Nadarajah(2008). Kotz et al. (2001, page233) showedthe heaviertail ofsymmetricbivariateLaplacedistribution relative to a Gaussianbivariatedensityundersamesettingof m andΣ. Becauseof theavailability of thecorrelationma-trix, LindseyandLindsey(2006)appliedit to modeldependenceamongrepeatedmeasurementsdata.Eltoft et al. (2006)presenteda similar symmetricmultivariateLaplacedistributionwith applicationof speechandimagedata.

In our quantilemixed-effectsmodel,for a givenquantileτ , therandomeffectsareassumedto bedistributedaroundβτ with zero-mean,soasymmetricmultivari-ateLaplacedistributionis adoptedfor therandomeffectsin model(5) as

f(ui; Σ) =2

(2π)q/2 |Σ|1/2

(uT

i Σ−1ui

2

)ν/2

K ν

(√2(uT

i Σ−1ui)

). (11)

Theimplementationof this distributioninto theMCEM algorithmdescribedinsection2.3 is straightforward,andthemodifiedBesselfunctionin thelikelihood isavailablein thestatisticalsoftwareR. After generatingrandomsamplefor randomeffects(v(t)

i , . . . , v(t)N ) in step2 of the MCEM algorithm, we are able to get the

maximumlikelihood estimateΣ(t+1) by an iterativeEM-type approach(Eltoft etal., 2006). Alternatively,by methodof moments,we cancalculatethat Σ(t+1) =1N

∑Ni=1 v(t)

i v(t)Ti (Kollo andSrivastava,2004).

3 Simulation Study

We evaluatetheperformanceof theproposedquantilemixed-effectsmodelat sev-eral quantilesof dataunderdifferent data-generatingscenarios.The probabilitydistributionsfor theerrortermaregeneratedfrom theNormal,theStudent’sT3 andtheχ2

2 distribution,andthelattertwo representover-disperseddataandskeweddatarespectively.Theprobabilitydistributionsfor therandomeffectsareselectedfromeithertheNormalor theT3 distribution,andthecorrelationamongtherandomef-fectsis setto bezero.Hencethereareatotalof six combinationsof distributionfortheerrortermandtherandomeffects.

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We useN = 20 subjectsandni = n = 20, for i = 1, . . . , N , measurementswithin eachsubject.Thedataaregeneratedby themodel

yij = 1 + 2xij + u0i + u1ixij + ǫij , j = 1, . . . , 20, i = 1, . . . , 20, (12)

wherexij is theindependentvariablegeneratedfrom auniformdistributionbetween0 and20. Therandomeffectsfor intercept,u0i, andslope,u1i, andtheerror term,ǫij , areindependentlygeneratedandtheselecteddistributionisstandardizedto havemeanzeroandvarianceone.

In eachscenario,1000simulateddatasetsaregenerated,andfor eachdataset,weestimatethefixedregressioncoefficientsβ for threedifferentquantilefunctions,τ = (0.25, 0.50, 0.75), by usingtheproposedmodelin whichaLaplacedistributionfor errortermandamultivariateLaplacedistributionfor randomeffectsareusedtoconstructthelikelihood. Forcomparison,wealsoprovidetheparametersestimatedby thelinearmixedmodel(LMM).

Basedon1000MonteCarloreplications,thebiasandvarianceof theestimatorsof thefixedregressioncoefficients,β0 andβ1, for eachquantileτ arecalculatedas

Bias = βl − βl

Variance =1

1000

1000∑

k=1

(β(k)l − βl)

2,

whereβl =∑1000

k=1 β(k)l /1000 andβl, l = 0, 1 arethequantile-dependenttrueval-

uesof theinterceptandslope.Accordingto thedata-generatingprocedure,thetruevalueof β0 is obtainedby addingtheτ -th quantilevalueof standardizederrordis-tribution to the true interceptvalue,which is 1 in this simulationstudy. The truevalueβ1 = 2 is constantacrossquantiles.

Theleft panelof Figure1 showsthescatterplot of thedatafrom a singlesim-ulateddatasetgeneratedfrom standardizednormaldistributionsfor both theerrorterm andthe randomeffects. The varianceof y increasesalongwith x. First weerroneouslytreattheseasindependentdataandestimatetheslopeof y givenx byclassicallinearquantileregressionsfor 9 quantiles,τ = 0.1, 0.2, . . . , 0.9. Thees-timatedslopeβ1 increasesalongwith τ (blackdotsin right panel).Thenwe applytheproposedquantilemixed-effectsmodel.Theestimatesβ1 areconsistentlynearthetruevalue(hollow trianglesin right panel).This indicatesthat,for longitudinaldata,thedependencewithin subjectsshouldbeaccountedfor to avoidseriousbiasin theestimationof quantileregressionmodels.

Table1 summarizesthe resultsfrom six scenarios.Thefixed regressionslopeβ1 estimatedby the proposedquantilemixed-effectsmodeldoesnot vary across

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0

10

20

30

40

50

60

0 5 10 15 20x

y

0.0 0.2 0.4 0.6 0.8 1.0

1.2

1.4

1.6

1.8

2.0

2.2

2.4

τ

β1

Figure1: A plot of simulateddatageneratedby normaldistributionsfor both theerror term andthe randomeffects(left panel). On the right panel,the black dotsindicatetheestimatedslopeof y givenx by assumingindependentobservationsat9 quantiles(τ = 0.1, 0.2, . . . , 0.9), andthehollow trianglesindicatetheestimatedslopeby theproposedquantilemixed-effectsmodel.

quantilesandis consistentlyaroundits true valuewith small biasandvarianceinall the six scenarios.For the scenarioswith symmetricerror distributions(ǫ ∼Normal,ǫ ∼ T3), in which themeanandmedianareequal,wecomparethemedianmixed-effectsmodelQ0.50 with LMM. Thetwo methodsshowedcomparablebiasandvariance.

The point estimatesof the fixed regressioninterceptβ0 vary the quantilesandcapturethelocationof thedifferentquantilesof theconditionaldistribution.Whenthe error term is symmetricallydistributed(ǫ ∼ Normal, ǫ ∼ T3), the quantilefunction Q0.50 is moreefficient (smallerbiasandvariance)thanQ0.75 andQ0.25.Whenthe error term is χ2

2 distributed,β0 is moreefficient in the lower quartiles.Efficiencyis greateratquantilesthathaveahigherdensityof data.Thebiasof β1 inQ0.50 is similar to thatin LMM for thescenarioswith symmetricerrordistributions.Theestimatorβ0 is slightly moreefficientin LMM whentheerrortermis normallydistributed,while it is more efficient in Q0.50 when the error term follows a T3

distribution. This agreeswith the expectationthat meanregressionoutperformsmedianregressionwhentheerror termis normallydistributed,andis lessefficientwhenthedistributionhasheavytails.

We alsofitted the samesimulateddatawith quantilemixed-effectsmodelsin

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Table 1: Estimates,bias, andvarianceof β0 and β1 for the threequantilefunc-tionsQ0.25, Q0.50, andQ0.75, andthe linearmixedmodel(LMM) in six simulatedscenarios.Theresultsareaveragedover1000MonteCarloreplications.

β0 β1

LMM Q0.25 Q0.50 Q0.75 LMM Q0.25 Q0.50 Q0.75

ǫ ∼ normal u ∼ normalEst 0.996 0.306 0.994 1.687 1.994 1.994 1.994 1.993Bias -0.004 -0.019 -0.006 0.013 -0.006 -0.006 -0.006 -0.007Var 0.059 0.064 0.062 0.064 0.055 0.055 0.055 0.055

ǫ ∼ normal u ∼ T3

Est 1.009 0.317 1.008 1.698 1.991 1.991 1.991 1.991Bias 0.009 -0.008 0.008 0.023 -0.009 -0.009 -0.009 -0.009Var 0.060 0.073 0.070 0.069 0.047 0.047 0.047 0.047

ǫ ∼ T3 u ∼ normalEst 0.998 0.505 1.002 1.498 1.998 1.998 1.998 1.998Bias -0.002 -0.054 0.002 0.057 -0.002 -0.002 -0.002 -0.002Var 0.058 0.057 0.053 0.056 0.050 0.050 0.050 0.050

ǫ ∼ T3 u ∼ T3

Est 1.004 0.513 1.003 1.494 1.999 1.998 1.999 1.999Bias 0.004 -0.045 0.003 0.053 -0.001 -0.002 -0.001 -0.001Var 0.061 0.063 0.059 0.064 0.046 0.046 0.046 0.046

ǫ ∼ χ22 u ∼ normal

Est 1.005 0.329 0.763 1.478 2.009 2.010 2.009 2.009Bias 0.005 0.041 0.070 0.092 0.009 0.010 0.009 0.009Var 0.059 0.053 0.057 0.070 0.050 0.050 0.050 0.051

ǫ ∼ χ22 u ∼ T3

Est 0.987 0.313 0.741 1.448 2.007 2.008 2.008 2.009Bias -0.013 0.025 0.047 0.062 0.007 0.008 0.008 0.009Var 0.075 0.068 0.077 0.095 0.047 0.047 0.047 0.047

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which thedensitychosento modeltherandomeffectswasmultivariatenormalin-steadof multivariateLaplace.Theresults(notshown)wereverysimilar to thoseinTable1. In all scenariosconsideredin oursimulation,thechoiceof thedistributionfor randomeffectsappearsnot to becrucial.

4 RealData Analysis: Labour Pain Data

Thelabourpaindatawerereportedby Davis(1991)andanalysedby Jung(1996),GeraciandBottai (2007),andHe et al. (2003). The datasetconsistsof repeatedmeasurementsof self-reportedpainin labouronN = 83 women,of which43wererandomlyassignedto a pain medicationgroup and 40 to a placebogroup. Theresponsewasmeasuredevery30minutesona100-mmline, where0 meantnopainand100 extremepain. A nearlymonotonepatternof missingdatawasfound forthe responsevariable,and the maximumnumberof measurementsfor a womanwassix. Figure2 showsthebox-plotfor thesedata.Theskewnessin bothplaceboandpainmedicationgroupsis obvious.Themeanresponsecanbemodeledby thelinearmixedmodelfor thisdata,but it maynotbethebestlocationto representthedata. Instead,the quantilemixed-effectsmodelwe proposedwill providea moreinsightful alternativeby fitting the modelat different quantilesof data. Like thelinearmixedmodel,ourmodelcanhandletheimbalancein thisdataandmakeuseof all availabledata.Theamountof measuredpainis boundedbetween0 and100,andthepainscorein theplacebogroupincreasesvery quickly in thefirst 2 hoursandstabilisesafterwards.Thispatternsuggeststhatanon-linearmodelmayprovideamorerealisticfit.

Let yij betheamountof painfor patienti attimej, Ri bethetreatmentindicatortakingon value0 for placeboand1 for treatment,andlet Tij be themeasurementtime dividedby 30 minutesandcenteredat its mean.For i = 1, . . . , 83, andj =1, . . . ni, weconsiderfive models.

Model 1: Quantilelinearregressionwith no randomeffects

yij = xTijβ + ǫij , with Qǫij

(τ |xij) = 0,

wherexTij = (1, Tij, Ri, RiTij) andβ = (β0, β1, β2, β3)

T .

Model 2: Quantilelinearregressionwith randomintercept

yij = xTijβ + u0i + ǫij , with Qǫij

(τ |xij, u0i) = 0,

wherexTij = (1, Tij, Ri, RiTij), β = (β0, β1, β2, β3)

T .

13




minutes since randomization

labo

r pa

in

50% 50%25% 25%

75% 75%

30 60 90 120 150 180

0

20

40

60

80

100

Figure2: Box plot of labourpainscorefor theplacebo(hollow box) andthepainmedication(shadedbox) groupswith the fitted 25%, 50% and75% quantilefunc-tionsby Model 5 for theplacebo(solid line) andthepainmedication(dashedline)groups.

Model 3: Quantilelinearregressionwith randominterceptandrandomslope

yij = xTijβ + zT

ijui + ǫij , with Qǫij(τ |xij , ui) = 0,

wherexTij = (1, Tij , Ri, RiTij), β = (β0, β1, β2, β3)

T , zTij = (1, Tij) and

ui = (u0i, u1i)T .

Model 4: Quantilecubicregressionwith randominterceptandrandomslope

yij = xTijβ + zT

ijui + ǫij , with Qǫij(τ |xij , ui) = 0,

wherexTij = (1, Tij, Ri, RiTij , T

2ij, T

3ij , RiT

2ij , RiT

3ij), β = (β0, β1, β2, β3,

β4, β5, β6, β7)T , zT

ij = (1, Tij) andui = (u0i, u1i)T .

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Model 5: Quantilelogit regressionwith randomeffects

yij = 100 ·exT

ijβ+zTijui

1 + exTij

β+zTij

ui

+ ǫij , with Qǫij(τ |xij , ui) = 0,

wherexTij = (1, Tij , Ri, RiTij), β = (β0, β1, β2, β3)

T , zTij = (1, Tij) and

ui = (u0i, u1i)T .

Model1 is thelinearquantileregressiondefinedby KoenkerandBassett(1978),in which independenceof observationsis assumed.Model 2 is thequantileregres-sionappliedby GeraciandBottai (2007),in which thedependenceamongdataisaccountedfor by a subject-specificrandomintercept.Model 3, a linearmodel,andmodel4, a cubic polynomialmodel,both includea randominterceptanda ran-dom slopefor time. Sincethe outcomevariableis boundedbetween0 and 100and the trend is nonlinear,we utilize the logistic function in model 5. Insteadof estimatingthe complicatednonlinearmodel5 directly, we makeuseof a dis-tinctive featureof quantileregression,namelythat its inferenceis “equivariant”to monotonetransformation.If we let h be a monotonefunction on R, then forany randomvariableY , we haveQτ{h(Y )|x} = h{Qτ (Y |x)}, or equivalently,Qτ (Y |x) = h−1{Qτ (h(Y )|x)}. We first apply a logit transformationto the painscoreyij anddefiney∗

ij = logit(yij/100). For thelogit to bedefined,wereplacethevalues0 and1 with 0.025and0.975,respectively.Thenweregressy∗

ij onthelinearmodel,asin model3.

In all five models,we assumethat the residualsfollow the ALD(0, σǫ, τ) asdefinedin section2.2andtherandomeffectsfollow themultivariateLaplacedistri-bution

[u0i

u1i

]∼ ALq

([00

],

[σ2

u0σu0u1

σu0u1σ2

u1

]).

We estimatethe fixed regressioncoefficientsβ and the variancecomponents(σǫ, σ

2u0

, σu0u1, σ2

u1) by the MCEM algorithm proposedand provide the standard

deviationby theblock bootstraptechniquewith a sampleof sizeB = 500. For agivenτ , AIC is calculatedas−2 log L + 2p for modelcomparison,wherelog L istheestimatedlog likelihoodfrom theMCEM stepin section2.3andp is thenumberof parametersin themodel.

Table 2 summarizesthe estimatedfixed regressioncoefficientsfor the threequantilefunctionsτ = (0.25, 0.50, 0.75) for all five models. Firstly, we comparethemodelsaccordingto theAIC. Amongthelinearmodels(models1 - 3), model3,whicht includesmultiplerandomeffectsprovidesthebestfit. Thenon-significanthigherpowertermsin thecubicmodel4 do not improvethefit overmodel3. The

15




nonlinearmodel5 hasfewer parametersthanmodel4 andprovidesthe bestfit ofall threequantilefunctions.

The interpretationof the parametersin model5 is similar to that in model3,exceptthat the outcomeof model5 is the transformedpain score. The intercept,β0, representsthe τ th quantileof the pain scoreyij in models3 or of the logittransformedpain scorey∗

ij in model5 for womenin the placebogroup at meanfollow-up time. It’s estimateincreasesalongwith thequantiles.Thecoefficientβ2

representsthedifferencein logit transformedpainscorebetweentheplacebogroupand the treatmentgroup at the meantime, and suchdifferenceis largestin thethird quartile. Theestimatesfor thecoefficientβ1 showthat the logit transformedpain scoresignificantly increasesover time in the placebogroupand the rateofchangeis greaterin higherquartile. The rateof changein pain is quite smallerin the medicationgroup. The interactionterm T × R is significantat the 0.05level in the threequartilefunctions,which suggeststhat the treatmentis effective.Also, themagnitudeof thecoefficientassociatedwith T × R or β3 is largerin thehigherquantiles(τ = 0.5 andτ = 0.75) of the labourpainscore,which indicatesthat the treatmentis moreeffectivewhen the labour pain is worse. In model1,the effectsof T andT × R in the third quartilearesubstantiallysmallerthaninthe other two quartiles,but in models2, 3, and5, their magnitudeis comparableacrossall quartilefunctions. model1 assumesindependenceof observations,andits inconsistentinferencesuggeststhatoverlookingthedependenceamongdataina quantileregressionmayleadto biasedestimation.Thefitted curvesfrom model5 areplottedin Figure2.

5 Conclusion

In this paper,a likelihood-basedquantileregressionis proposedfor longitudinaldataby adoptingtheasymmetricLaplacedistributionfor theerror term, in whichmultiple randomeffectsareincorporatedinto themodelto accountfor thedepen-denceamongdata.Thisapproachis analogousto thetraditionallinearmixedmodelfor themeanbut allowstheestimationat differentquantilesof theconditionaldis-tributionandhenceprovidesamorerobustestimatorandoffersamorecomprehen-siveunderstandingof the data. This methodis moregeneralthanthat previouslydescribedby GeraciandBottai (2007)andpermitsgreaterflexibility in theanaly-sisof longitudinaldata. In additionto longitudinaldata,theproposedmethodcanbeappliedto othertypesof dependentdata,suchascluster,hierarchical,andspa-tial data. In the simulationstudy, the proposedquantilemixed-effectsregressioncorrectlyhandlesthedependenceamonglongitudinaldataandestimatesthefixedregressioncoefficientsefficiently. Its observedbiasandsamplingvariancearecom-

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Table2: Parameterestimatesfor thelabourpaindatafor thefive differentregressionmodels.

Paramter Q0.25(s.d.) Q0.5(s.d.) Q0.75(s.d.)

Model1β0 22.35(2.88)* 46.7(2.9)* 80.44(3.03)*β1 11.01(1.58)* 16.64(1.31)* 8.37(1.63)*β2 -21.7(2.95)* -38.04(3.34)* -49.47(3.91)*β3 -10.81(1.63)* -15.46(1.64)* -4.29(2.28)*AIC 3330.0 3425.4 3545.1

Model2β0 36.59(3.97)* 48.95(4.33)* 64.43(4.6)*β1 9.81(0.92)* 11.23(1.12)* 12.18(1.19)*β2 -26.85(5.54)* -30.76(5.93)* -36.58(6.47)*β3 -9.12(1.03)* -9.95(1.28)* -9.78(1.42)*AIC 2838.9 2894.5 2932.6

Model3β0 41.29(4.1)* 50.25(4.46)* 59.54(4.44)*β1 11.09(1.28)* 11.66(1.37)* 12.06(1.4)*β2 -29.27(5.56)* -31.44(6.23)* -32.75(6.19)*β3 -9.06(1.73)* -9.67(1.87)* -9.55(1.94)*AIC 2574.9 2578.1 2623.3

Model4β0 40.41(4.16)* 50.08(4.58)* 58.69(4.71)*β1 13.76(1.78)* 13.56(1.86)* 13.51(1.92)*β2 -29.18(5.62)* -32.88(6.51)* -34.47(6.57)*β3 -12.51(2.32)* -12.45(2.55)* -12.25(2.67)*β4 0.75(0.48) 0.67(0.48) 0.68(0.47)β5 -0.55(0.27) -0.39(0.27) -0.3(0.27)β6 -0.58(0.58) -0.18(0.6) 0.15(0.6)β7 0.69(0.32) 0.58(0.34) 0.51(0.35)AIC 2576.5 2575.6 2622.4

Model 5β0 -0.46(0.28) 0.09(0.27) 0.64(0.28)*β1 0.72(0.08)* 0.76(0.09)* 0.77(0.09)*β2 -1.92(0.39)* -2.01(0.37)* -2.06(0.38)*β3 -0.58(0.11)* -0.63(0.12)* -0.62(0.12)*AIC 2567.8 2541.5 2579.8

*significantat 5% level.

17




parablewith thoseof thelinearmixedmodelat thecentrallocationwhentheerrorsaresymmetricallydistributed.By fitting a setof quantilefunctions,the proposedmethodeffectivelydescribesanyunderlyingconditionaldistributionandprovidesmoreefficientestimatesthanthelinearmixedmodelfor themeanwhentheerrorsareover-dispersed.

Theuseof theALD error permitsembeddingour modelwithin the likelihoodframework.Althoughfully parametric,theproposedmodelallowsfor semi-parame-tric estimationof theconditionalquantilesin thatit is valid for anyunderlyingdis-tribution of the regressionresidualerror, providedthat the model (4) is correctlyspecified.Theuseof thebootstrapfurtherallows for inferencethat is free of dis-tributionalassumptions,asit is illustratedin thesimulationstudy. Inferenceaboutconditionalquantiles,whetherwith independentor dependentdata,facilitatesun-derstandingof theentireconditionaldistributionof theoutcomegiventheexplana-tory variables.

The proposedmodelassumesthat the randomcoefficientsareindependentoftheexplanatoryvariables.Thisassumption,whichalsounderliesthepopularmixedeffectsregressionon themean,maybevalid in ourapplicationto labourpaindata,wheretheindependentvariableis aproductof experimentalrandomization.In gen-eral,however,regressionmodels,whetheradditiveor multiplicative,thatallow forarbitrarydependenceof the randomcoefficientsandexplanatoryvariables,mightbepreferablewhile beingalsomoredemandingon thesizeof datasets.

TheproposedMCEM algorithmandbootstrapprovideconvenientsolutionsforestimationand inferencebut canbe computationallydemanding.Computationaltime increasessubstantiallywith the numberof subjects. We recommendcau-tion whenselectingthe likelihood distanceδ in Eickhoff’s algorithm. Too smalla valuemayexcessivelyincreasethesimulationsizeandmakeconvergencediffi-cult to achieve.

TheasymmetricmultivariateLaplacedistributionALq(m, Σ) (Kotzetal.,2001),a thick tailedmultivariatedistribution,is consideredfor therandomeffectsfor thepurposeof robustness.Eventhoughwe do not observesubstantialgain over thetraditionalmultivariatenormaldistributionin thesimulationstudy,it is worthwhileto keepconsideringthis distributionandmakeuseof its asymmetricscenariostocovertheskewedrandomeffectsdistributions.Alternatively,Visk (2009)presentedathree-parameterasymmetricmultivariateLaplacedistributionALq(a, µ, Σ) whichhasseparateshift (a) andshape(µ) parametersandmaybemoreappealingto ac-commodateskewedrandomeffectsdistributions.

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Mixed-Effects Models for Conditional Quantiles with Longitudinal Data

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