Analyzing recurrent events when the history of previous episodes is unknown or not taken into account: proceed with caution

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GacSanit.2017;31(3):227–234

Original

article

Analyzing

recurrent

events

when

the

history

of

episodes

is

unknown

or

not

taken

into

account:

proceed

with

caution

Albert

Navarro

a,∗

_,

_Georgina

_Casanovas

a

_,

_Sergio

_Alvarado

b,c

_,

_David

_{Mori ˜na}

a,d

a_GRAAL-Unitat_de_{Bioestadística,}_Facultat_de_Medicina,_Universitat_Autònoma_de_Barcelona,_Bellaterra,_Barcelona,_Spain b_Programa_de_Salud_Ambiental,_Escuela_de_Salud_Pública,_Facultad_de_Medicina,_Universidad_de_Chile,_Chile

c_Facultad_de_Ciencias_de_la_Salud,_Universidad_de_Tarapacá,_Arica,_Chile

d_Unit_of_Infections_and_Cancer_(UNIC),_Cancer_Epidemiology_Research_Program_(CERP),_Catalan_Institute_of_Oncology_{(ICO)-IDIBELL,}_Barcelona,_Spain

a

r

t

i

c

l

e

i

n

f

o

Articlehistory: Received23June2016 Accepted8September2016 Availableonline15November2016 Keywords: Recurrence Cohortstudies Riskassessment Survivalanalysis Bias

a

b

s

t

r

a

c

t

Objective:Researchersinpublichealthareofteninterestedinexaminingtheeffectofseveralexposures

ontheincidenceofarecurrentevent.Theaimofthepresentstudyistoassesshowwellthe

common-baselinehazardmodelsperformtoestimatetheeffectofmultipleexposuresonthehazardofpresenting

anepisodeofarecurrentevent,inpresenceofeventdependenceandwhenthehistoryofprior-episodes

isunknownorisnottakenintoaccount.

Methods:Throughacomprehensivesimulationstudy,usingspecific-baselinehazardmodelsasthe

refer-ence,weevaluatetheperformanceofcommon-baselinehazardmodelsbymeansofseveralcriteria:bias,

meansquarederror,coverage,confidenceintervalsmeanlengthandcompliancewiththeassumptionof

proportionalhazards.

Results:Resultsindicatethatthebiasworsenaseventdependenceincreases,leadingtoaconsiderable

overestimationoftheexposureeffect;coveragelevelsandcompliancewiththeproportionalhazards

assumptionareloworextremelylow,worseningwithincreasingeventdependence,effectstobe

esti-mated,andsamplesizes.

Conclusions: Common-baselinehazardmodelscannotberecommendedwhenweanalyserecurrent

eventsinthepresenceofeventdependence.Itisimportanttohaveaccesstothehistoryofprior-episodes

persubject,itcanpermittoobtainbetterestimationsoftheeffectsoftheexposures

BY-NC-NDlicense(http://creativecommons.org/licenses/by-nc-nd/4.0/).

Análisis

de

eventos

recurrentes

cuando

la

historia

de

episodios

previos

es

desconocida

o

no

se

tiene

en

cuenta:

proceder

con

cautela

Palabrasclave: Recurrencia Estudiosdecohortes Medicióndelriesgo Análisisdesupervivencia Sesgo

r

e

s

u

m

e

n

Objetivo:Amenudolosinvestigadoresensaludpúblicaestáninteresadosenexaminarelefectodevarias

exposicionesenlaincidenciadeuneventorecurrente.Elobjetivodeesteestudioesevaluarel

fun-cionamientodelosmodelosderiesgobasalcomúnalestimarelefectodemúltiplesexposicionessobre

elriesgodepresentarunepisodiodeuneventorecurrente,cuandoexistedependenciadeleventoylos

antecedentesdelosepisodiosporsujetosondesconocidosobiennosetienenencuenta.

Métodos:Medianteunestudioexhaustivodesimulación,utilizandomodelosderiesgobasalespecífico

comoreferencia,seevalúaelrendimientodelosmodelosderiesgobasalcomúnatravésdediversos

cri-terios:sesgo,errorcuadráticomedio,cobertura,longituddelosintervalosdeconfianzaycompatibilidad

conelsupuestoderiesgosproporcionales.

Resultados:Elsesgoempeoraamedidaqueaumentaladependenciadelevento,llevandoauna

sobreesti-maciónconsiderabledelefectodelaexposición;losnivelesdecoberturayelcumplimientodelsupuesto

deriesgosproporcionalessonbajosomuybajos,loqueempeoraconelaumentodeladependenciadel

evento,elefectoaestimaryeltama ˜nomuestral.

Conclusiones:Elusodemodelosderiesgobasalcomúnnopuederecomendarsecuandoanalizamos

even-tosrecurrentesenpresenciadedependenciadelevento.Esimportanteteneraccesoalosantecedentes

deepisodiospreviosporsujeto,yaqueellopuedepermitirobtenermejoresestimacionesdelosefectos

delasexposiciones.

CCBY-NC-ND(http://creativecommons.org/licenses/by-nc-nd/4.0/).

∗ Correspondingauthor.

E-mailaddress:albert.navarro@uab.cat(A.Navarro).

Introduction

Theoutcomeofinterestinabiomedicalorepidemiologicalstudy is often an event that can occurmore than once in a subject. http://dx.doi.org/10.1016/j.gaceta.2016.09.004

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228 A.Navarroetal./GacSanit.2017;31(3):227–234 Therefore, identifyinga statisticalmethodsuitable for studying

recurrenteventsisofgreatinteresttothefield.

From a statistical point of view, recurrent event analysis presentstwomajorchallenges.Thefirstisindividualheterogeneity, i.e.theunmeasuredeffectsproducedbybetween-subject variabil-ity,presumablyduetounobservedexposures.Forinstance,imagine thatastudymeasuringthenumberofrespiratorycrisesisnot ask-ingforsmokingstatus.Itislikelythatsmokerswillhaveadifferent patternfromnon-smokers,resultinginheterogeneityacrossthe subjectsthatcan’tbeattributedtoanyknownfactorassmoking statuswasnotrecorded.Thisissueisusuallytackledusingfrailty models,whichincorporaterandomeffecttermstoaccountforthis “extra”variability.Thesecondproblemiswithin-subject correla-tionsattributabletoasinglesubjectsufferingmultipleepisodesof theevent.Thesecorrelationsareespeciallyproblematicin situa-tionscomplicatedbyeventdependence,inotherwords,whenthe riskofhavinganewepisodedependsonthenumberofprevious episodes.Thisisthecaseofthenumberofsickleavessufferedby workers:Ahistoryofsickleavesincreasestheriskofasubsequent episode.Reisetal.1_quantified_the_extent_of_this_increase._If_we_fail_to

accountforeventdependence,ourresultingestimatorswillbe inef-ficientandpotentiallybiased.AsdiscussedinBox-Steffensmeier etal.,2_{common-baseline}_hazard_models_average_the_effects_across

alleventsnottakingstrataintoaccount,beingthisaveragesbiased in a predictabledirection. In cohort studies, event dependence canbecontrolledbyusingsurvivalmodelswithspecific-baseline hazardsforeachepisodethatthesubjectfaces.3

AmorimandCai4_provide_an_excellent_review_of_approaches_to

recurrenteventanalysis.Thearticledescribestheapplicable sta-tisticalmethodsforepidemiological studiesof recurrentevents, workingoffoftheassumptionthatresearchershaveaccesstoall oftheinformationrequiredbyeachmodel.Inpractice,however, muchofthisdataistypicallyunavailable.Specific-baselinehazard modelsassumethattheexactnumberofpreviousepisodessuffered byeachsubjectisknown,butinrealityitistypicallyimpracticalto obtainanexhaustivehistoryforeachpatient.Thisleavesuswithout amethodtodirectlyaddresseventdependence.Theusualpractice insuchcasesistofitmodelswithacommon-baselinehazard.

The aim of the present study is to assess how well these common-baselinehazardmodelsperformwhentheyareusedto estimatetheeffectof multipleexposures onthehazardof pre-sentinganepisodeofarecurrenteventwhentheprevioushistory isnottakenintoaccount.

Methods Simulations Example

We illustrate this work by reproducing a study from the literature5_to_analyze_long-term_sickness_absence_(SA)_frequency_in

acohortofDutchworkers.Wewillusethesamebaselinehazardas intheDutchstudy,0.0021perworker-week.Thebetween-episodes hazardratios(HR)donotcorrespondexactlytothoseofany spe-cificstudy,althoughReisetal.1_provide_values_for_a_wide_range_of

SA-relateddiagnoses.SAisacommonly-usedoutcomein occupa-tionalhealthstudiesbecauseitisconsideredamajoreconomicand publichealthissue,6–8_resulting_in_a_growing_interest_in

identify-ingthebestmethodtoquantitativelyandefficientlyanalyzethis phenomenon.5,9–13

Generationofpopulations

Sixdifferentpopulationsof250000workers,eachwith20years of history, weregenerated using thesurvsim14,15 _package _in _R

2.15.3(RFoundation forStatisticalComputing,Vienna,Austria).

Table1

Characteristicsofthesimulatedpopulations.

Baselinehazard HR i Worker-days Worker-weeks Population1 ˇ01=8.109 0.000301 0.002106 1 None ˇ02=7.927 0.000361 0.002526 1.20 ˇ03=7.745 0.000433 0.003030 1.44 Population2 ˇ01=8.109 0.000301 0.002106 1 _Gamma (1,0.1) ˇ02=7.927 0.000361 0.002526 1.20 ˇ03=7.745 0.000433 0.003030 1.44 Population3 ˇ01=8.109 0.000301 0.002106 1 None ˇ02=7.703 0.000451 0.003160 1.50 ˇ03=7.298 0.000677 0.004738 2.25 Population4 ˇ01=8.109 0.000301 0.002106 1 _Gamma (1,0.1) ˇ02=7.703 0.000451 0.003160 1.50 ˇ03=7.298 0.000677 0.004738 2.25 Population5 ˇ01=8.109 0.000301 0.002106 1 None ˇ02=7.193 0.000752 0.005263 2.50 ˇ03=6.276 0.001881 0.013166 6.25 Population6 ˇ01=8.109 0.000301 0.002106 1 Gamma (1,0.1) ˇ02=7.193 0.000752 0.005263 2.50 ˇ03=6.276 0.001881 0.013166 6.25

ˇ03referstoˇ0forthethirdandsubsequentepisodes.

HR:hazardratio.

Foreachsubjecti,thehazardofthenextepisodekwassimulated throughanexponentialdistribution:

hik(t) =i·e(−ˇ0k+ˇ1X1+ˇ2X2+ˇ3X3) (1)

whereh0k(t) =e−ˇ0k,i.e.thebaselinehazardforsubjectsexposed

toepisodek.ThemaximumnumberofSAepisodesthataworker maysufferwasnotfixed,althoughthebaselinehazardwas con-sideredconstantwhenk≥3.X1,X2,andX3arethethreecovariates

thatrepresenttheexposure,withXi∼Bernoulli(0.5).ˇ1,ˇ2,andˇ3

aretheparametersofthethreecovariatesthatrepresenttheeffect, setindependentlyoftheepisodektowhichtheworkerisexposed, as:ˇ1=0.25,ˇ2=0.5,andˇ3=0.75inordertorepresenteffects

ofdifferentmagnitudes.iisarandomeffect.

Eventdependence

Event dependencewasaddressedby usingvariousvalues of h_0k(t),specifyingdifferentˇ_0k.Table1presentsthespecifications forthegeneratedpopulationsintermsofthebaselinehazardsby SAepisodeandrandomeffectsused.Table1alsopresentstheHR resultingfromthecomparisonofthebaselinehazardwiththatof thefirstepisode,whichgivesustheeventdependenceforthe phe-nomenon.Notethatforpopulations1and2,theHR=1.20and1.44, respectively,forthesecondSAepisode,aswellasforthethirdand subsequentSAepisodeswithrespecttothefirst.Thismeansthat betweenthesecondandthirdSAepisodes,thebaselinehazardwas alsoincreasedbyafactorof1.20.TheHR=1.50betweenepisodes twoandthreeforpopulations3and4,and2.50forpopulations 5and6.Wechosetosimulatephenomenawithincreasingevent dependence,giventhatReisetal.1_demonstrated_that_the_hazard

alwaysincreasesinthepresenceofpreviousSA. Individualheterogeneity

Individual heterogeneity was addressed by introducing i,

therandom effect. This effectwas held constant over the var-ious episodesfor a given subject but varied between subjects.

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A.Navarroetal./GacSanit.2017;31(3):227–234 229 Specifically, we established: a) absence of any random effect

(i=1),whichleadstoaperfectlyspecifiedpopulationoncethe

subjectcovariatesareset,andb)individualheterogeneity,where i∼Gammawithmean=1andvariance=0.1.

Table1showsthesimulatedpopulations. Cohortdesign

Althoughthepopulationswith20yearsofhistorywere gener-ated,aprocedurewassubsequentlyappliedtolimittheeffective follow-upperiodsto1,3,and5years,withsomesubjectshaving sufferedapriorepisodebeforethefollow-upperiodbegan.

Thiswasachievedasfollows.

Weselectedthesubjectswhowereeitherpresentat15years offollow-uporincorporatedafterthatdate.Follow-uptimewas thenre-scaled,settingt=0at15yearsforsubjectsalreadypresent inthepopulationandt=0atthebeginningofthefollow-upperiod forthoseincorporatedlater.Thepurposeofthisprocedurewasto obtainacohortinwhichsomesubjectshadaworkhistoryprior tothe15-yearpointthatincludedpreviousepisodes,whichwere treatedasunknown.Thefigureof15yearswaschosenasa rep-resentativelengthofworkhistoryfortypicalcorporateemployee. Usingthissubpopulation,wethengeneratedthethreesub-bases correspondingtodifferentstudyend-points:at16years(1yearof effectivefollow-up,fromthe15thto16thyear),at18years(three yearsoffollow-up),andat20years(fiveyearsoffollow-up). Sampleselectionandmodelfitting

Foreachofsub-base,B=500randomsamplesweredrawnwith sizesn1=500, n2=1000,andn3=3000,and for eachselected

subject,theepisodeswithintheeffectivefollow-upperiodwere recorded.Finally,themodelswerefittedtoeachofthesesamples. Models

Allofthemodelsconsideredarenon-parametricandare exten-sionsoftheCoxproportionalhazardsmodel.16,17_For_all_workers,

weusetherealpreviousepisodeswhenfittingspecific-baseline models,andwecompletelyignoretheminthecommon-baseline models.

Modelsfornon-individualheterogeneitycontext

1)Specific-baselinehazardapproach:Prentice-Williams-Peterson (PWP)

Forstudiesofrecurrentphenomenainvolvingevent depend-encebutnotindividualheterogeneity,PWPisthesurvivalmodel ofreference.18_PWP_addresses_event_dependence_by_stratifying

accordingtonumberofpreviousepisodes,therebyassigninga specific-baselinehazardtoeachpotentialepisode.Whenthe i-thsubjectisatriskofthek-thepisode,thehazardfunctionis definedas:

hik(t) =h0k(t) eXiˇ (2)

whereXiˇrepresentthevectorsofcovariatesandtheregression

coefficients.

2)Common-baselinehazardapproach:Andersen-Gill(AG) AG19 _is_based _on_counting_processes_and _assumes_that_the

baselinehazardiscommonacrossallepisodes,independentof thenumberofpreviousepisodes.Ithasthefollowinghazard function:

hi(t) =h0(t) eXiˇ (3)

whereh0(t) =e−ˇ0 andisthereforethesameforallepisodes,

k.AGtreatsdifferentepisodeswithinagivensubjectasthough theywereindependent,subsequentlyobtainingarobust “sand-wich”estimatorofthevariance.20

Modelsforindividualheterogeneitycontext

1)Specific-baseline hazardapproach: Conditional FrailtyModel (CFM)

Whenindividualheterogeneitycomesintoplay,thereference modelbecomesCFM.21_This_model_addresses_individual

hetero-geneitybyassumingalatentmultiplicativeeffectonthehazard function:

hik(t) =Uih0k(t) eXiˇ (4)

Uiisanindividualrandomeffectwhichisassumedtohaveunit

meanandfinitevariance,whichisestimatedfromthedata.22

SinceUiisamultiplicativeeffect,wecanthinkthisfrailtyasa

representationofthecumulativeeffectofoneormoreomitted covariates.22,23_The_most_{commonly-adopted}_frailty_terms24–26

areE [Ui] =1andV [Ui] =.

2)Common-baselinehazardapproach:SharedFrailtyModel(SFM) Amongotherapplications,SFM27–29_may_be_used_in_the

con-textofrecurrentevents,wherewithin-subjectepisodessharea frailtytermthatisindependentofthoseforotherindividuals. Itshazardfunctionis:

hi(t) =Uih0(t) eXiˇ (5)

wherethebaselinehazardisindependentoftheepisodekto whichthesubjectisexposed.UiisparameterizedasinCFM.

Modelassessmentcriteria

Thecriteriausedtoevaluatemodelperformancewere:1) per-centage bias: ı=

ˆ ˇ−ˇ ˇ

×100, where ˇ is the truevalue for

estimateofinterest, ˆˇ = B

j=1 ˆ ˇ_j

B,whereBisthenumberof

simula-tionsperformed;2)percentagemeansquarederror(MSE):MSE= ( ˆˇj−ˇ)

2

+V

ˇˆj

ˇ ×100, forj=1,...,B,where ˆˇjistheestimate ofinterestwithineachofthej=1,...,BsimulationsandV

ˇˆj

is thevarianceoftheestimateofinterestwithineach simulation; 3)coverage:percentageoftimesthatthe95%confidenceinterval

ˆ ˇj±z1−˛/2SE

ˆ ˇj

includesˇ,forj=1,...,B,whereSE

ˇˆj

isthe standarderroroftheestimateofinterestwithineachsimulation; 4)confidenceintervalsaveragelength;5)proportionalhazards: Percentageoftimesthattheassumptionofproportionalhazards cannotberejected,forj=1,...,B,accordingtothetestproposedby GrambschandTherneau.30

All models were fitted using the coxph function from the survival31_package_in_R.

Results

Theresultspresentedherereferonlytothe5-yearfollow-up cohorts.Resultsforthecohortswith1and3yearsoffollow-upare availableassupplementarydataonline,butarenotdetailedhere, asthefindingswerequitesimilar.

Regardingthesituationswithno-individualheterogeneity,we canseethattheaveragebiasinthecommon-baselinehazard mod-elsis11−16%forpopulationwithloweventdependence,risingto 42−51%forthosewithhigheventdependence(Table2).Ingeneral, thebiasdoesnotchangemarkedlyintermsoftheeffectassociated withˇ,samplesize,orheterogeneityofthepopulation.Higher sam-plesizemeanslowerMSEand,forcommon-baselinemodels,MSE increaseswiththeexposureeffect(Table3).Intermsofcoverage, Table4showsthatAGonlyachievesperformancesapproaching95%

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230 A. Navarro et al. / Gac Sanit. 2017; 31(3) :227–234 Table2

Percentageofbias(95%confidenceinterval).

n=500 n=1000 n=3000

Population Model Biasˇ1 Biasˇ2 Biasˇ3 Biasˇ1 Biasˇ2 Biasˇ3 Biasˇ1 Biasˇ2 Biasˇ3

1 PWP -3.5(-6.6,-0.4) -5.6(-7.2,-4) -3.6(-4.6,-2.5) -2.2(-4.3,0) -6.2(-7.2,-5.1) -4.4(-5.2,-3.6) -3.1(-4.4,-1.8) -6.2(-6.8,-5.6) -4.2(-4.6,-3.7) 1 AG 14.5(11,18) 12.3(10.5,14.1) 14.9(13.7,16.1) 16.4(13.9,18.9) 11.8(10.6,13) 14.3(13.4,15.1) 15.8(14.4,17.3) 11.7(11.1,12.4) 14.3(13.9,14.8) 2 CFM -7.3(-10.4,-4.1) -9.2(-10.9,-7.6) -6.7(-7.9,-5.6) -8.8(-11,-6.7) -9.1(-10.2,-8) -8.1(-8.9,-7.4) -8.7(-10,-7.4) -8.3(-9,-7.7) -7.8(-8.2,-7.3) 2 SFM 13.5(9.8,17.1) 10.7(8.7,12.6) 14.2(12.9,15.5) 11.3(8.8,13.8) 11.1(9.9,12.4) 12.7(11.8,13.5) 12.2(10.7,13.7) 12.1(11.3,12.8) 13(12.6,13.5) 3 PWP -9.9(-12.7,-7.1) -4.2(-5.6,-2.7) -5.3(-6.4,-4.2) -8.7(-10.7,-6.7) -5.5(-6.5,-4.5) -5.3(-6,-4.6) -9.1(-10.2,-7.9) -5.2(-5.8,-4.6) -4.2(-4.6,-3.8) 3 AG 15.9(12.3,19.5) 25.2(23.4,27.1) 24.8(23.5,26.1) 17.7(15.3,20.2) 23.9(22.6,25.2) 24.5(23.6,25.4) 17.3(15.8,18.9) 23.8(23,24.5) 26.1(25.7,26.6) 4 CFM -2.3(-5.3,0.7) -5.9(-7.5,-4.3) -8.2(-9.2,-7.2) -3.3(-5.3,-1.3) -6.1(-7.1,-5) -7.7(-8.4,-6.9) -3.5(-4.6,-2.3) -6.1(-6.8,-5.5) -7.8(-8.2,-7.3) 4 SFM 28.5(24.7,32.2) 24.5(22.6,26.5) 22.2(21,23.4) 27.3(24.6,29.9) 23.7(22.4,25.1) 22.7(21.7,23.7) 27.1(25.5,28.6) 23.2(22.4,24.1) 22.3(21.8,22.8) 5 PWP -2.4(-5,0.2) -4.1(-5.4,-2.8) -3(-3.9,-2.1) -2.2(-4.1,-0.4) -4.4(-5.3,-3.4) -3.4(-4,-2.7) -1.3(-2.3,-0.2) -3.4(-4,-2.8) -3.3(-3.7,-2.9) 5 AG 50.8(46,55.6) 43.9(41.6,46.3) 48(46.4,49.5) 51.7(48.4,55) 42.7(41.1,44.4) 46.9(45.7,48.1) 51.5(49.6,53.4) 44.7(43.7,45.6) 46.7(46.1,47.3) 6 CFM -7.6(-10.6,-4.6) -2.6(-4.1,-1.1) -3.3(-4.3,-2.3) -6.4(-8.5,-4.3) -2.6(-3.6,-1.6) -3(-3.7,-2.3) -6.8(-8,-5.6) -2.1(-2.7,-1.5) -3.8(-4.2,-3.4) 6 SFM 45.8(40.4,51.1) 53.7(51,56.4) 53.5(51.7,55.4) 50.1(46.5,53.8) 53.8(51.9,55.7) 53.7(52.5,54.9) 47.4(45.4,49.5) 54.3(53.3,55.4) 53(52.3,53.7) Table3

Percentagemeansquarederror(95%confidenceinterval).

n=500 n=1000 n=3000

Population Model MSEˇ1 MSEˇ2 MSEˇ3 MSEˇ1 MSEˇ2 MSEˇ3 MSEˇ1 MSEˇ2 MSEˇ3

1 PWP 6.1(2.7,18.7) 3.4(1.5,9.7) 2.4(1.1,7.2) 3(1.4,9.3) 1.7(0.8,5.1) 1.3(0.6,3.9) 1(0.5,3.3) 0.7(0.3,2.2) 0.5(0.2,1.5) 1 AG 8.3(3.6,24.2) 4.9(1.8,15.9) 4.4(1.3,14.8) 4.6(1.8,14.2) 2.6(0.9,7.8) 3(0.7,9) 1.9(0.6,6.6) 1.3(0.3,3.8) 2(0.4,4.9) 2 CFM 6.3(2.7,20.1) 3.7(1.5,12) 2.8(1.1,10.1) 3.1(1.4,9.9) 1.9(0.7,6.7) 1.6(0.6,5) 1.2(0.5,4) 0.9(0.2,3) 0.8(0.2,2.8) 2 SFM 7.8(2.8,29.2) 4.6(1.4,18.1) 4.2(1,14.8) 3.8(1.4,14.5) 2.3(0.7,8.5) 2.5(0.5,7.8) 1.6(0.5,5.3) 1.3(0.2,4) 1.7(0.2,4.5) 3 PWP 5.4(2.4,14.6) 2.9(1.2,9.7) 2.4(0.9,7.5) 2.8(1.2,8.1) 1.5(0.6,4.7) 1.2(0.5,3.7) 1(0.4,3.2) 0.6(0.2,1.8) 0.5(0.2,1.4) 3 AG 9.2(4,27) 7.6(2,26) 7.7(1.5,23.2) 4.9(2,15.6) 5(1.1,15.7) 6(0.9,15.5) 2.2(0.7,7.1) 3.6(0.6,8.7) 5.6(2.1,10.6) 4 CFM 5.4(2.3,18) 3.1(1.2,10) 2.4(0.9,7.4) 2.5(1.2,8.2) 1.5(0.6,4.7) 1.5(0.5,4.8) 0.9(0.4,3) 0.7(0.2,2.3) 0.8(0.2,2.3) 4 SFM 9.2(2.5,33.3) 6.8(1.3,26.4) 6.1(0.9,20.3) 5.4(1.3,22.3) 4.7(0.7,16.9) 5.3(0.6,18) 3(0.4,10.4) 3.3(0.3,8.9) 4.1(1.1,8.6) 5 PWP 4.4(1.8,13.4) 2.3(0.9,6.7) 1.7(0.7,4.8) 2.2(1,6.7) 1.2(0.5,4) 0.9(0.4,2.8) 0.7(0.3,2) 0.4(0.2,1.4) 0.4(0.1,1.1) 5 AG 20.9(6.7,72.3) 16.7(3.5,52.1) 21.8(3.5,49.4) 13.7(3.4,43.6) 12.6(2,33.9) 19(4.5,41.8) 8.9(1.4,22.5) 11.2(3,22.5) 17.1(8.6,26.7) 6 CFM 5(1.9,18.6) 2.5(1,8.2) 1.8(0.7,6.6) 2.5(0.9,7.6) 1.2(0.5,4.5) 1(0.4,3.3) 0.9(0.3,3.4) 0.4(0.2,1.6) 0.4(0.1,1.3) 6 SFM 17(2.5,72) 20.5(1.4,64.5) 25.9(1.8,70.1) 11.9(1.3,47.5) 17.4(1.3,45.3) 23.5(6.5,49.3) 7.4(0.6,22.9) 15.7(5,29.3) 21.7(10.9,36)

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A. Navarro et al. / Gac Sanit. 2017; 31(3) :227–234 231 Table4

Coverage:percentageoftimesthatthetrueparametervalueisincludedinthe95%confidenceinterval.

n=500 n=1000 n=3000

Population Model Coverageˇ1 Coverageˇ2 Coverageˇ3 Coverageˇ1 Coverageˇ2 Coverageˇ3 Coverageˇ1 Coverageˇ2 Coverageˇ3

1 PWP 95.4(93.4,97.2) 93.6(91.4,95.6) 95(93,96.8) 94.8(92.8,96.6) 93.6(91.4,95.6) 91.6(89,94) 94.8(92.8,96.6) 86.4(83.4,89.4) 86.4(83.4,89.4) 1 AG 94.8(92.8,96.6) 89.8(87,92.4) 81.6(78.2,85) 90(87.2,92.6) 87.8(84.8,90.6) 70(66,74) 82.6(79.2,85.8) 70(66,74) 28.2(24.2,32.2) 2 CFM 94.4(92.4,96.4) 90.2(87.6,92.8) 91(88.4,93.4) 92.2(89.8,94.4) 90.2(87.6,92.8) 85.2(82,88.2) 89(86.2,91.6) 78.6(75,82.2) 67.4(63.2,71.4) 2 SFM 93.2(91,95.4) 90(87.2,92.6) 81.8(78.4,85.2) 93.6(91.4,95.6) 89.6(86.8,92.2) 75(71.2,78.8) 87(84,89.8) 69.2(65.2,73.2) 32.4(28.4,36.6) 3 PWP 94.4(92.4,96.4) 94(91.8,96) 89.8(87,92.4) 93(90.6,95.2) 91.4(88.8,93.8) 89(86.2,91.6) 89.4(86.6,92) 86.8(83.8,89.6) 85.4(82.2,88.4) 3 AG 93.8(91.6,95.8) 78.2(74.6,81.8) 58(53.6,62.4) 91(88.4,93.4) 65(60.8,69.2) 30.6(26.6,34.6) 80.8(77.2,84.2) 20(16.6,23.6) 0.4(0,1) 4 CFM 93(90.6,95.2) 91.2(88.6,93.6) 89(86.2,91.6) 96(94.2,97.6) 91.4(88.8,93.8) 81.2(77.8,84.6) 94.8(92.8,96.6) 86(82.8,89) 63.6(59.4,67.8) 4 SFM 91.8(89.4,94.2) 81.2(77.8,84.6) 69.2(65.2,73.2) 86.4(83.4,89.4) 67(62.8,71) 42.8(38.4,47.2) 67.4(63.2,71.4) 26.6(22.8,30.6) 2.8(1.4,4.4) 5 PWP 92.8(90.4,95) 94.8(92.8,96.6) 95.2(93.2,97) 93.4(91.2,95.4) 93.4(91.2,95.4) 92.6(90.2,94.8) 95.8(94,97.4) 90(87.2,92.6) 86.2(83.2,89.2) 5 AG 85.4(82.2,88.4) 62(57.8,66.2) 21.6(18,25.2) 72(68,75.8) 37.6(33.4,41.8) 4.6(2.8,6.6) 33.6(29.4,37.8) 1.6(0.6,2.8) 0(0,0) 6 CFM 92(89.6,94.2) 93.8(91.6,95.8) 92.8(90.4,95) 93.4(91.2,95.4) 94(91.8,96) 92.2(89.8,94.4) 92.4(90,94.6) 92.6(90.2,94.8) 86.2(83.2,89.2) 6 SFM 89.8(87,92.4) 55.2(50.8,59.6) 25.2(21.4,29) 81.4(78,84.8) 30(26,34) 2.2(1,3.6) 57(52.6,61.4) 0.4(0,1) 0(0,0) Table5

Percentageoftimesthattheassumptionofproportionalhazardsisnotrejected(95%confidenceinterval).

n=500 n=1000 n=3000 Population Model PHˇ1 PHˇ2 PHˇ3 PHˇ1 PHˇ2 PHˇ3 PHˇ1 PHˇ2 PHˇ3 1 PWP 94.2(92,96.2) 95.2(93.2,97) 95.8(94,97.4) 95.4(93.4,97.2) 93(90.6,95.2) 95.2(93.2,97) 94.2(92,96.2) 91.8(89.4,94.2) 90.2(87.6,92.8) 1 AG 90.6(88,93) 89.2(86.4,91.8) 90.4(87.8,93) 91.4(88.8,93.8) 88.8(86,91.4) 88.8(86,91.4) 89.4(86.6,92) 83.2(79.8,86.4) 82(78.6,85.4) 2 CFM 91.4(88.8,93.8) 95.6(93.8,97.4) 90(87.2,92.6) 92.6(90.2,94.8) 93.2(91,95.4) 91.8(89.4,94.2) 89.4(86.6,92) 91.4(88.8,93.8) 85(81.8,88) 2 SFM 87.2(84.2,90) 90.2(87.6,92.8) 89(86.2,91.6) 87.6(84.6,90.4) 87(84,89.8) 89(86.2,91.6) 90.4(87.8,93) 86(82.8,89) 81.2(77.8,84.6) 3 PWP 94.4(92.4,96.4) 94.2(92,96.2) 92.6(90.2,94.8) 92.8(90.4,95) 93.2(91,95.4) 94.4(92.4,96.4) 93.4(91.2,95.4) 92.2(89.8,94.4) 88(85,90.8) 3 AG 84.6(81.4,87.6) 82.4(79,85.6) 82.6(79.2,85.8) 80.6(77,84) 80.8(77.2,84.2) 74.6(70.8,78.4) 80.4(76.8,83.8) 73(69,76.8) 60.4(56,64.6) 4 CFM 92.4(90,94.6) 93(90.6,95.2) 91(88.4,93.4) 92.4(90,94.6) 92.2(89.8,94.4) 89(86.2,91.6) 89.6(86.8,92.2) 90.8(88.2,93.2) 87.6(84.6,90.4) 4 SFM 84.2(81,87.4) 81.4(78,84.8) 81.2(77.8,84.6) 82.8(79.4,86) 81.6(78.2,85) 78.6(75,82.2) 79(75.4,82.6) 76.8(73,80.4) 64.2(60,68.4) 5 PWP 94.4(92.4,96.4) 93.8(91.6,95.8) 92.2(89.8,94.4) 95.2(93.2,97) 93(90.6,95.2) 93.2(91,95.4) 96.2(94.4,97.8) 91.8(89.4,94.2) 88.8(86,91.4) 5 AG 51.2(46.8,55.6) 50.8(46.4,55.2) 46.2(41.8,50.6) 47.4(43,51.8) 44.6(40.2,49) 34(29.8,38.2) 28.4(24.4,32.4) 29.4(25.4,33.4) 9.2(6.8,11.8) 6 CFM 91.2(88.6,93.6) 93(90.6,95.2) 93.4(91.2,95.4) 92.2(89.8,94.4) 89(86.2,91.6) 92.4(90,94.6) 91.2(88.6,93.6) 90.4(87.8,93) 90.6(88,93) 6 SFM 55.4(51,59.8) 49.8(45.4,54.2) 34(29.8,38.2) 52.6(48.2,57) 42.6(38.2,47) 17.4(14.2,20.8) 31.4(27.4,35.6) 19.2(15.8,22.8) 1.2(0.4,2.2)

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232 A.Navarroetal./GacSanit.2017;31(3):227–234 Table6

Confidenceintervalsmeanlength.

n=500 n=1000 n=3000

Population Model Lengthˇ1 Lengthˇ2 Lengthˇ3 Lengthˇ1 Lengthˇ2 Lengthˇ3 Lengthˇ1 Lengthˇ2 Lengthˇ3

1 PWP 0.339 0.349 0.367 0.241 0.247 0.259 0.139 0.142 0.149 1 AG 0.385 0.392 0.403 0.273 0.277 0.285 0.157 0.160 0.164 2 CFM 0.346 0.355 0.372 0.244 0.250 0.261 0.140 0.144 0.150 2 SFM 0.400 0.403 0.409 0.282 0.284 0.288 0.161 0.162 0.165 3 PWP 0.313 0.323 0.337 0.221 0.229 0.239 0.127 0.131 0.138 3 AG 0.404 0.410 0.418 0.285 0.290 0.295 0.164 0.167 0.171 4 CFM 0.328 0.335 0.348 0.233 0.238 0.247 0.135 0.138 0.143 4 SFM 0.438 0.437 0.436 0.309 0.308 0.308 0.178 0.178 0.177 5 PWP 0.284 0.289 0.305 0.201 0.205 0.214 0.116 0.118 0.124 5 AG 0.519 0.518 0.521 0.367 0.367 0.368 0.212 0.212 0.213 6 CFM 0.317 0.322 0.331 0.224 0.228 0.234 0.130 0.132 0.135 6 SFM 0.617 0.603 0.586 0.432 0.424 0.411 0.248 0.243 0.237

forpopulationswithsmallormoderateeventdependence (popu-lations1and3)andforˇ1=0.25.Fortheotherscenarios,coverage

fallsnotably,worseningwithincreasingeventdependence,effect toestimate,and samplesize. Forexample, inpopulation5,the 95%CIincludedthetrueparametervalueforˇ3inamere0-4.6%of

sampleswhenn=1000orn=3000.AsshowninTable5,AG demon-stratedoveralllowcompliancewiththeassumptionofproportional hazards,worsening withincreasingeventdependence,effectto estimate,andsamplesize.Compliancereachedlevelsapproaching 90%onlyinpopulation1,fallingdramaticallyforpopulation5.

Resultsforheterogeneouspopulationspresentanalmost iden-ticalpattern.Slightdifferencesareobservedregardingthe95%CI: SFMCI95%wasgenerallybroader(Table6),translatingintoaslight riseincoveragelevel(Table4).

Thespecific-baselinehazardapproachesshowedmuchbetter results than the common-baseline approaches, both in homo-geneous and heterogeneous contexts. For populations free of heterogeneity,thepercentageofbiasremainedbelow10%andwas generallynegative,i.e.slightlyunderestimatingtheeffectand cov-eragelevelswerearound85−95%.Overall,morethan90%ofthe simulatedsamplescompliedwiththeassumptionofproportional hazards.Inpresenceofindividualheterogeneity,whenthereislow eventdependence,thebiasslightlyfallswiththeincreaseofthe effecttoestimateandsamplesize.

Discussion

Statisticalanalysisofrecurrentoutcomeswithevent depend-enceisnottrivial,asitrequiresmethodsthatcanaccountforthis dependencetoobtainefficientandunbiasedestimates.Although includingthenumber ofpreviousepisodesasatime-dependent covariatewould addresstheproblem,10_{episode-specific} _hazard

functionsaremorecoherentwiththenatureofrecurrentevents. Inanycase,todeployeitheralternative,itisnecessarytoknow howmanypreviousepisodeseachsubjecthashad,whichisoften impossible. As a result, some epidemiologists often recur to a common-baselinehazardfunctionthatisindependentofprevious episodes.Thepresent paperassesses how wellthese common-baselinehazardmodelsperform,incomparisontosomeofthemost commonspecific-baselinehazardmodels,whenappliedto situ-ationscomplicatedbyeventdependenceandwhentheprevious episodesarenottakenintoaccount.

It is worth noting that the results obtained here may be indicative of thebehavior of phenomena with“positive” event dependence(riskofpresentinganewepisodeincreasesinfunction ofthenumberofpreviousepisodes),notwheneventdependence is“negative”(whichinouropinionismuchlesscommoninthe studyofpublichealthphenomena).Similarly,themagnitudeofthe

bias,coveragelevels,etc.,dependsonotherspecificaspectsofeach study,astheintensityoftheeventdependence,samplesize,etc.

Itisimportanttohighlightthattherewerealmostnodifferences betweenthepattern ofbehavior ofcommon-baseline approach versusspecific-baselineapproach,inheterogeneousor homoge-nouspopulationsintermsofbias,coverage,orcompliancewith theproportionalhazardsassumption.

Theperformance of thecommon-baseline approaches wors-enedaseventdependence increased,producing lowercoverage and increasingly overestimating the effect. Subjects in the previously-exposedgrouphadmoreeventoccurrencesand there-foremore recurrentepisodes,and theysufferedtheseepisodes earlierthansubjectsinthenon-exposedgroup.Thus,theexposed subjectsarrivedatahigherbaselinehazardsoonerandingreater numbers.Thismeansthatifspecific-baselinehazardsarenotused, theincreasedbaselinehazardwouldbelargelyattributabletothe exposedgroup.

Astheeffecttobeestimatedincreases,performanceofmodels withcommon-baselinehazardworsens.Theexplanationissimilar totheoneabove:thelargertheeffect,thegreaterthedifferencein riskbetweensubjectsinexposedandnon-exposedgroups;hence, thenumbersandrecurrenceratesamongexposedsubjectsbecome progressivelygreatercomparedtothoseoftheunexposedsubjects. Thus,asinthecaseofeventdependence,thebaselinehazardeffect isdisproportionallyattributabletoexposure.

Forthesemodels,coverageisaffectedbysamplesize, worsen-ingassamplesizeincreases.Clearlythisisaspuriousrelationship; whatreallyhappensisthatlargersamplesizesprovidegreater pre-cision,butsincetheestimatesobtainedarebiased,greaterprecision meanspoorercoverage.32

Asexpected,PWPwasclearlysuperiortoAGinsituations com-plicatedbyeventdependence.Evenso,coverageandcompliance withtheproportionalhazardsassumptionremainedunacceptably lowinthefaceofsignificanteventdependenceandlargeeffectsto beestimated.Note,however,thatourresultsshowthatPWP over-alltendstoslightlyunderestimatethevalueof␤.Thisisprobably becausetheupperstrata,representingsubjectswithgreater num-bersofrecurrences,concentratemembersoftheexposedgroup. Furtherstudiestoinvestigatethebeststrategytouseintheupper stratawouldbehelpful.Inordertokeepallepisodesinthe analy-sis,wepooledallepisodesbeyondthesecondrecurrence.Itwould beinterestingtoseewhether“truncating”thenumberofepisodes or,alternatively,notgroupingthemtogetheratall,wouldimprove performance.Thefirstoptionhasthedisadvantageofeliminating someepisodes,whereasthesecondproducesstratawithveryfew subjectsandconsequentlyunstableestimates.27_All_the_above

com-mentsarealsovalidforCFM.Ontheotherhand,Torá-Rocamora etal.13_show_that_fitting_the_CFM_when_dealing_with_very_large

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A.Navarroetal./GacSanit.2017;31(3):227–234 233 alternativecouldbetheconditionalfrailtyPoissonmodelwhich

producessimilarresultsbutdecreasesthetimesubstantially.We shouldalsomentionthattheapproachespresentedinthispaper arenottheonlyonesthatcouldbeusedfortheanalysisof recur-rentevents.Alternativesincludemultilevelmixedeffectssurvival parametricmodels33_,_flexible_parametric34_or_multistate_models.35

Insummary,informationaboutpreviousepisodesis fundamen-talforsoundanalysisofrecurrentevents,buttherequireddata is not always available. Allthe common-baseline hazard mod-elsthatweevaluatedperformedalmostequallypoorly,making itimpossibletorecommendoneoveranother.Theoneexception inwhichacommon-baselinehazardmodelmaybeareasonable optionforevent-dependentanalysisisasituationinwhichthelevel ofeventdependenceisvery lowandtheeffecttobeestimated issmall.Althoughthisestimatewouldstillbesomewhatbiased, coverageandcompliancewiththeproportionalhazards assump-tionmightbewithintherealmofacceptability.Inothersituations, thesemodelsareclearlyinappropriate,producinglowcoverage, loworextremelylowcompliancewiththeproportionalhazards assumption,andblatantoverestimationoftheeffectofexposure. Inpractice,themagnitudeofthisproblemmayevenbegreater.Reis etal.1_showed_that_event_dependence_for_SA_is_often_higher_than

thefiguresusedinoursimulations,meaningthatthe common-baselinehazardsmodelswould performevenmore poorly.The authorsshowed,forexample,thattheHRforthesecondandthird episodesofsickleaveduetomentalandbehavioraldisorderswere 9.52and20.26,respectively,withrespecttothefirstepisode.

Fromthis paperwe mayderivetwo mainconclusions: first, availabilityofthehistoryofpreviousepisodespersubjectisvery importantandtherefore,anefforttothispurposeshouldbemade inthefieldwork;second,ifwedon’thavethis information,itis importanttofindvalidalternativestotackleanalysesofthistype. Oneoptionthatweconsiderworthinvestigatingisimputingthe numberofpreviousepisodes,which wouldallowfortheuseof modelswithspecific-hazardfunctions.

Whatisknownaboutthetopic?

One of the main challenges in recurrent event analy-sis is accounting for within-subject correlations. Failure to properlyaddressthesecorrelationscancreateserious prob-lems,especiallyin thepresence of event dependence; that is,when theriskof havinganew episodedepends onthe number of previous episodes suffered by the subject. The specific-baselinehazardmodelcanbeusedtoaddressevent dependenceandobtainefficientestimators.However,usinga specific-baselinehazardmodelrequiresknowingthenumber ofpreviousepisodesexperiencedbyeachsubject;inpractice, thesedatais oftenunavailable. Under this situation, many researcherschoosetousecommon-baselinehazardmodels toanalysethiskindofdata.

Whatdoesthisstudyaddtotheliterature?

Thisstudyprovidesaquantificationofthemagnitudeof theconsequencesofusingcommon-baselinehazardmodels whenthereiseventdependence,inseveralscenariosbasedon arealisticexample.Inthiscontext,acommon-baselinehazard modelisnotappropriate,asthemodelproducesinefficient andbiased estimations.Thetrueparameter valuedoesnot fallwithintheconfidenceintervalatanacceptablefrequency, andcompliancewiththeassumptionofproportionalhazards isunacceptablylow.

Editorincharge

María-VictoriaZunzunegui. Transparencydeclaration

Thecorrespondingauthoronbehalfoftheotherauthors guar-antee the accuracy, transparency and honesty of the data and informationcontainedinthestudy,thatnorelevantinformation hasbeenomittedandthatalldiscrepanciesbetweenauthorshave beenadequatelyresolvedanddescribed.

Authorshipcontributions

Allauthors contributedto theconception and design of the work,thedesignofthesimulations,theanalysisand interpreta-tionofthedata,thewritingofthepaperanditscriticalreviewwith importantintellectualcontributions,andtotheapprovalofthefinal versionforitspublications.

Funding None.

Conflictsofinterests None.

AppendixA. Supplementarydata

Supplementarydataassociatedwiththisarticlecanbefound,in theonlineversion,atdoi:10.1016/j.gaceta.2016.09.004.

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