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ModelingModeling clusteredclustered survivalsurvival
datadata
TheThe differentdifferent approachesapproaches
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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
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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
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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
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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
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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
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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
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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
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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
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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))
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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
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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 ≥= :
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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
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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
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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
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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
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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
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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
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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
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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 ∂∂
∂
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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 +
++
=
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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
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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
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Simulation Simulation resultsresults
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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
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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
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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
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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??
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Simulation Simulation resultsresults
jackknifejackknife
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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=
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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=
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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
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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