ModelingModeling clusteredclustered survivalsurvival

datadata

TheThe differentdifferent approachesapproaches

Alternative Alternative approachesapproaches for for

modelingmodeling clusteredclustered survivalsurvival datadata

nn TheThe fixedfixed effectseffects modelmodel

nn TheThe stratifiedstratified modelmodel

nn TheThe frailtyfrailty modelmodel

nn TheThe marginal marginal modelmodel

ExampleExample ofof bivariatebivariate survivalsurvival datadata

nn TimeTime to reconstitution to reconstitution ofof bloodblood--milkmilk barrierbarrier

afterafter mastitismastitis

n Two quarters are infected with E. coli

n One quarter treated locally, other quarter not

n Blood milk-barrier destroyed

n Milk Na+ increases

n Time to normal Na+ level

Time to reconstitution dataTime to reconstitution data

6.50*0.98…2.626.50*0.41Placebo

4.930.66…4.786.50*1.9Treatment

01…011Heifer

10099…321Cow number

TimeTime to reconstitution figurativeto reconstitution figurative

Cow number

Tim

e to

re

so

lutio

n

0 20 40 60 80 100

01

23

45

6

TheThe parametricparametric fixedfixed effectseffects modelmodel

nn IntroduceIntroduce fixedfixed cowcow effecteffect

nn WeWe parameteriseparameterise baselinebaseline hazardhazard

nn E.g.E.g. WeibullWeibull: :

( )iijij cxthth += βexp)()( 0

Baseline hazard Treatment effect

Fixed cow effect,c1=0

( )ρλξλρ ρ ,ith w)( 1

0 == −tth

TheThe proportionalproportional hazardshazards modelmodel

nn FromFrom thethe modelmodel

itit followsfollows thatthat thethe hazardhazard ratio ratio ofof twotwo

individualsindividuals isis givengiven byby

andand thisthis ratio ratio isis thusthus constant constant overover timetime

( )iijij cxthth += βexp)()( 0

( )( )kxl

iij

kl

ij

cxth

cxth

th

th

++

=ββ

exp)(

exp)(

)(

)(

0

0

FixedFixed effectseffects modelmodel likelihoodlikelihood

nn SurvivalSurvival likelihoodlikelihood: : hazardhazard andand survivalsurvival functionsfunctions

requiredrequired

nn Maximise log Maximise log likelihoodlikelihood to to findfind estimatesestimates for for ββ, , ccii andand ξξ

( ) ( )( )iijiij

t

ij cxtHcxthtS +−=

+−= ∫ ββ exp)(expexp)(exp)( 0

00

( ) )(log)(log )()(1

2

11

2

1

∑∑∏∏= == =

+==s

i j

ijijijfixij

s

i j

ijfix tSthltSthL ij δδ

( )iijij cxthth += βexp)()( 0

ParameterParameter estimatesestimates

fixedfixed effectseffects modelmodel

nn ParameterParameter estimateestimate for for trttrt effecteffect β β withwith ρρ=1=1

nn AdditionallyAdditionally anotheranother 99 (!) 99 (!) parametersparameters for for thethe

differentdifferent cowscows

0.1900.1900.1850.185FixedFixed effectseffects

SE(SE(ββ))ββModelModel

DisadvantagesDisadvantages fixedfixed effectseffects modelmodel

nn EstimatesEstimates a large set a large set ofof nuisance nuisance parametersparameters

nn No No estimateestimate for for thethe cowcow to to cowcow variabilityvariability

nn OnlyOnly handleshandles covariatescovariates thatthat change change withinwithin clusterclusternn E.g.E.g. heiferheifer effecteffect cancan notnot bebe studiedstudied in in fixedfixedeffectseffects modelmodel

nn LessLess efficient efficient thanthan frailtyfrailty modelmodel ((seesee laterlater))

TheThe stratifiedstratified modelmodel

nn DifferentDifferent baselinebaseline hazardhazard for for eacheach cowcow

nn BaselineBaseline hazardhazard isis leftleft unspecifiedunspecified

nn WeWe use partial use partial likelihoodlikelihood ((CoxCox, 1972), 1972)

( )βijiij xthth exp)()( 0=

Baseline hazard Treatment effect

StratifiedStratified modelmodel likelihoodlikelihood

nn Partial Partial likelihoodlikelihood determineddetermined for for eacheach cowcow

separatelyseparately, , thenthen multipliedmultiplied ((independenceindependence))

nn Maximise partial log Maximise partial log likelihoodlikelihood to to findfind estimatesestimates

for for ββ alonealone

( )( )( )

∏∏ ∑= = ∈

s

i j yRl il

ij

ij

iji

x

x

1

2

1 exp

expδ

ββ

( ) { }ijiliji yylyR ≥= :

ParameterParameter estimatesestimates

stratifiedstratified modelmodel

nn ParameterParameter estimateestimate for for trttrt effecteffect β β withwith ρρ=1=1

0.2090.2090.1310.131StratifiedStratified

0.1900.1900.1850.185FixedFixed effectseffects

SE(SE(ββ))ββModelModel

DisadvantagesDisadvantages stratifiedstratified modelmodel

nn ==disadvantagesdisadvantages fixedfixed effectseffects modelmodel

nn EvenEven more inefficientmore inefficient

nn A A cowcow onlyonly contributescontributes to to thethe partial partial likelihoodlikelihood

if an if an eventevent isis observedobserved for for oneone quarter quarter whilewhile

thethe otherother quarter quarter isis stillstill atat riskrisk

( ) ( )( ) ( ) ( )( )( ) ( )∏∏

= = +<+<s

i j ii

iiiiii

xx

yyxyyx ii

1

2

1 21

122211

expexp

exp exp 21

ββββ δδ

11

TheThe frailtyfrailty modelmodel

nn DifferentDifferent frailtyfrailty termterm for for eacheach cowcow

nn BaselineBaseline hazardhazard isis assumedassumed to to bebe parametricparametric

nn WeWe makemake distributionaldistributional assumptionsassumptions for for uuiinn E.g.E.g. oneone parameterparameter gamma gamma frailtyfrailty densitydensity

( )βijiij xuthth exp )()( 0=

Baseline hazard

Treatment effect

Random cow effect

( ) ( )( )θθ

θθ

θ

1

exp1

11

Γ−=

−ii

iU

uuuf

FrailtyFrailty modelmodel likelihoodlikelihood

nn ConditionalConditional (on (on frailtyfrailty) ) survivalsurvival likelihoodlikelihood

nn Marginal Marginal survivalsurvival likelihoodlikelihood: : integrateintegrate out out frailtyfrailty

( )( )iijiij cxutHtS +−= βexp )(exp)( 0

( ) )(log)(log )()(1

2

11

2

1

∑∑∏∏= == =

+==s

i j

ijijijcondij

s

i j

ijcond tSthltSthL ij δδ

( )βijiij xuthth exp )()( 0=

( )iiUij

s

i j

ijm uduftSthL ij∫∏∏∞

= =

=0 1

2

1

arg )( )()(δ

ParameterParameter estimatesestimates

frailtyfrailty modelmodel

nn ParameterParameter estimateestimate for for trttrt effecteffect β β withwith ρρ=1=1

0.1680.1680.1710.171FrailtyFrailty

0.2090.2090.1310.131StratifiedStratified

0.1900.1900.1850.185FixedFixed effectseffects

SE(SE(ββ))ββModelModel

AdvantagesAdvantages frailtyfrailty modelmodel

nn ProvidesProvides an an estimateestimate ofof thethe cowcow to to cowcow variabilityvariability, , θθ or or thethe variance variance ofof thethe randomrandom effecteffect..nn In In ourour exampleexample, , θθ =0.286=0.286

nn ItIt willwill alsoalso givegive estimatesestimates for for covariatescovariates thatthat are are onlyonlychangingchanging fromfrom cluster to cluster, cluster to cluster, nn E.g.E.g. thethe heiferheifer variable changes variable changes fromfrom cowcow to to cowcow

nn ItIt uses uses thethe availableavailable information in information in thethe mostmost efficient efficient waywaynn ItIt uses uses allall information, information, eveneven if if withinwithin a cluster a cluster oneone observation observation isis missingmissing

nn Most efficient Most efficient eveneven for for balancedbalanced bivariatebivariate survivalsurvival datadata

UndadjustedUndadjusted modelmodel

nn FinallyFinally considerconsider unadjustedunadjusted modelmodel

nn Are Are observedobserved resultsresults for for ourour exampleexample coincidencecoincidence

or do or do theythey reflectreflect a a particularparticular pattern?pattern?

( )βijij xthth exp)()( 0=

0.1620.1620.1760.176UnadjustedUnadjusted

0.1680.1680.1710.171FrailtyFrailty

0.2090.2090.1310.131StratifiedStratified

0.1900.1900.1850.185FixedFixed effectseffects

SE(SE(ββ))ββModelModel

AsymptoticAsymptotic variancevariance

nn TheThe asymptoticasymptotic variance variance ofof thethe estimateestimate ofof ββ isisgivengiven as a diagonal as a diagonal elementelement ofof thethe inverse inverse ofof

observedobserved or or expectedexpected information information matrixmatrix

nn TheThe expectedexpected (Fisher) information (Fisher) information matrixmatrix isis

withwith HH((ζζζζζζζζ) ) thethe HessianHessian matrixmatrix ((ζζζζζζζζ isis parameterparameter vectorvector))

withwith (q,r)(q,r)thth elementelement

( )( )HE)( −=I

)( 2

lrq ςς ∂∂

∂

AsymptoticAsymptotic efficiencyefficiency (1)(1)

nn UnadjustedUnadjusted modelmodel

nn FixedFixed effectseffects modelmodel

nn FrailtyFrailty modelmodel

( )∑=

+=s

i

iiu xx1

2

21)(I β

( ) ( )( )∑=

−+−=s

i

iiiifix xxxx1

2

.2

2

.13

2)(I β

)(I31

3)(I

31

1)(I β

θβ

θθβ fixufrail +

++

=

AsymptoticAsymptotic efficiencyefficiency (2)(2)C

ontr

ibution u

nadju

ste

d m

odel

0 1 2 3 4

0.0

0.2

0.4

0.6

0.8

1.0

θ0.33

0.5

Small Small samplesample sizesize efficiencyefficiency by by

simulationsimulation

nn GenerateGenerate 2000 data sets 2000 data sets withwith 100 pairs 100 pairs ofof

twotwo subjectssubjects withwith λλ=0.23, =0.23, ββ=0.18, =0.18, θθ=0.3=0.3

nn ThreeThree differentdifferent settingssettings

nn 100 % balance100 % balance

nn 80 % balance80 % balance

nn 80 % 80 % uncensoreduncensored

nn Look Look atat medianmedian andand coveragecoverage

Simulation Simulation resultsresults

TheThe marginal marginal modelmodel

nn Assume Assume frailtyfrailty modelmodel isis truetrue underlyingunderlying modelmodel

nn FittingFitting modelmodel withoutwithout takingtaking clusteringclustering intointo

accountaccount, , likelihoodlikelihood contributions are contributions are basedbased onon

nn ThereforeTherefore, , thisthis isis calledcalled thethe marginal marginal modelmodel

( )mijmmij xthth βexp)()( ,0, =

)()()( and )()()(0

,

0

, ∫∫∞∞

== iiUijmijiiUijmij duufththduuftStS

Marginal Marginal modelmodel parameterparameter

estimatesestimates

nn TheThe estimateestimate isis a consistent a consistent estimatorestimator for for ββnn SeeSee Wei, Lin Wei, Lin andand WeissfeldWeissfeld (1989) (1989)

nn ItsIts asymptoticasymptotic variance variance mightmight notnot bebe correct correct

because because nono adjustmentadjustment donedone for for correlationcorrelation

nn WeWe mightmight use use eithereither

nn JackknifeJackknife estimatorsestimators

nn Sandwich Sandwich estimatorsestimators

mβ̂

JackknifeJackknife estimatorestimator

nn GenerallyGenerally givengiven byby

nn WeWe use use groupedgrouped jackknifejackknife techniquetechnique

nn LeftLeft--outout observations observations independentindependent ofof remainingremaining

( )( )∑=

−− −−

− N

i

T

iiN

pN

1

ˆˆˆˆ

( )( )∑=

−− −−

− s

i

T

iis

ps

1

ˆˆˆˆ

JackknifeJackknife versus sandwichversus sandwich

nn LipsitzLipsitz (1994) (1994) demonstratesdemonstrates

correspondencecorrespondence betweenbetween jaccknifejaccknife andand

sandwich sandwich estimatorestimator

nn In In thethe timetime to to bloodblood milkmilk reconstitutionreconstitution

nn UnadjustedUnadjusted modelmodel: SE = 0.176: SE = 0.176

nn GroupedGrouped jackknifejackknife estimatorestimator: SE = 0.153: SE = 0.153

nn GroupedGrouped jackknifejackknife estimatorestimator leadsleads to to

smallersmaller variance!! variance!! IsIs thisthis alwaysalways soso??

Simulation Simulation resultsresults

jackknifejackknife

AcceleratedAccelerated failurefailure timetime modelsmodels

nn AFT AFT modelmodel (for (for binarybinary covariatecovariate))

nn φφ isis acceleratoraccelerator factorfactor:: φφ>1 >1 acceleratesaccelerates processprocess in in

treatmenttreatment groupgroup

)()( tStS CT φ=

Su

rviv

al

0 1 2 3 4 5 6

0.0

0.2

0.4

0.6

0.8

1.0

1.0

0.8

0.6

0.4

0.2

0.0

ControlTreated

Re

co

ve

ry

e.g.

( ) 09.22log1

50, == ρλCt

%50)( =tSC

)18.4()09.22(

)()09.2(

CC

CT

SS

tSS

=×== φ

TC MM φ=

ProportionalProportional hazardshazards (PH) versus (PH) versus

acceleratedaccelerated failurefailure timetime (AFT)(AFT)

nn PH PH modelmodel (for (for binarybinary covariatecovariate))

nn AFT AFT modelmodel (for (for binarybinary covariatecovariate))

)()( tStS CT φ=

( )βijij xthth exp)()( 0= )()( 0 ththC =

( )βexp )()( 0 ththT = )exp()(

)(

ratio Hazard

β

ψ

=

=

th

th

C

T

)()( 0 ththC =

)()( 0 ththT φφ=( )( ) ( )ββ ijijij xtxhth expexp)( 0=

Survival

02

46

0.0 0.4 0.8

φ=2

φ=2

1.0

52.1

1.6

3.2

Hazard

02

46

0.0 1.0 2.0 3.0

Survival

02

46

0.0 0.4 0.8

Hazard

02

46

0.0 1.0 2.0 3.0

HR

=2

HR

=2

0.5

6

1.2

0

0.8

5

1.7

0

LogLog--linearlinear modelmodel representationrepresentation

nn In In mostmost packages (SAS, R) packages (SAS, R) survivalsurvival modelsmodels

((andand theirtheir estimatesestimates) are ) are parametrizedparametrized as log as log

linearlinear modelsmodels

nn If If thethe errorerror termterm eeijij hashas extremeextreme value value

distribution, distribution, thenthen thisthis modelmodel corresponds tocorresponds to

nn PH PH WeibullWeibull modelmodel withwith

nn AFT AFT WeibullWeibull modelmodel withwith

ijijij exT log σαµ ++=

σαβσρσµλ −==−= − )exp( 1

αβσρσµλ −==−= − )exp( 1