session 2: hazard functions, competing risks, cause...

2-1

Module20:SurvivalAnalysisforObserva9onalData

SummerIns9tuteinSta9s9csforClinicalResearchUniversityofWashington

July,2017

BarbaraMcKnight,Ph.D.

SESSION2:HAZARDFUNCTIONS,COMPETINGRISKS,CAUSE-SPECIFIC

HAZARDS,ANDANDCUMULATIVEINCIDENCE

2-2

OUTLINE

•  Nonparametrices9ma9onofhazardfunc9ons•  Compe9ngrisks:– Defini9on:whenthereismorethanonecauseofdeath/failure

– Cumula9vefunc9ons:•  Event-freesurvival•  Cumula9veIncidencees9mator

– Cause-specificandsub-distribu9onhazards– Fine-GrayandCoxregressionmodels–  Interpreta9onsubtle9es

SISCR2017Module20SurvivalObserva9onalB.McKnight

2-3

OUTLINE

•  Nonparametrices.ma.onofhazardfunc.ons•  Compe9ngrisks:– Defini9on:whenthereismorethanonecauseofdeath/failure




2-4

HAZARDFUNCTION

•  Instantaneousrateatwhichdeathoccursattinthosewhoarealiveatt

•  Examples:

–  Age-specificdeathrate–  Age-specificdiseaseincidencerate


2-5

CUMULATIVEHAZARDFUNCTION

⇧(t) =Z t

0⌃(s)ds

= area under the hazard function curve

between 0 and t.

= amount of “hazard" accumulated between 0 and t.

= � log(S(t))

Not usually of interest per se, but estimates useful for diagnostics.


2-6

EQUIVALENTCHARACTERIZATIONS

•  Anyoneofthesefourfunc9onsisenoughtodeterminethesurvivaldistribu9on.

•  Theyareeachfunc9onsofeachother:

• S(t) =R�t ƒ (s)ds = e�

R t0 �(s)ds

• ƒ (t) = � ddt S(t) = �(t)e�

R t0 �(s)ds

• �(t) = ƒ (t)S(t)

• ⇧(t) =R t0 �(s)ds = � log(S(t))


2-7


0 2 4 6 8

0.0

0.4

0.8

Survival Function

t

S(t)

0 2 4 6 8

0.00

0.10

0.20

0.30

Density Function

t

f(t)

0 2 4 6 8

0.0

0.5

1.0

1.5

Hazard Function

t

λ(t)

0 2 4 6 80

510

15

Cumulative Hazard Function

t

Λ(t)


2-8


0 2 4 6 8

0.0

0.4

0.8

Survival Function

t

S(t)

0 2 4 6 8

0.00

0.10

0.20

0.30

Density Function

t

f(t)

0 2 4 6 8

0.0

0.5

1.0

1.5

Hazard Function

t

λ(t)

0 2 4 6 8

0.0

1.0

2.0

Cumulative Hazard Function

t

Λ(t)


2-9

CUMULATIVEHAZARD

• Nelson - Aalen estimator:

⌥(t) =P

j:t(j)tD(j)N(j)

• Variance:

’V�r(⌥(t)) =P

j:t(j)tD(j)S(j)[N(j)]3

• Standard error:∆’V�r(⌥(t)) can be used to form pointwise CI’s.

• or could use ⌥(t) = � log(S(t))


2-10


0 1 2 3 4 5 6

time

Cum

ulat

ive H

azar

d

0.0

0.5

1.0

● ●● ●● ●


2-11


0 1 2 3 4 5 6

time

Cum

ulat

ive H

azar

d

0.0

0.5

1.0

0.17

0.25

0.33

0.5

● ●● ●● ●


2-12

SURVIVALANDIG

•  RandomsubsetofthedatafromA.Dispenzieri,J.Katzmann,R.Kyle,D.Larson,T.Therneau,C.Colby,R.Clark,G.Mead,S.Kumar,L.J.MeltonIII,andS.V.Rajkumar.Useofmonclonalserumimmunoglobulinfreelightchainstopredictoverallsurvivalinthegeneralpopula9on.MayoClinicProc,87:512–523,2012.

•  Arehighfree-chainIglevelsassociatedwithsurvival?– Popula9on-basedOlmsteadCountyexample– Menandwomen50+yearsofage


2-13

TOPDECILEFLC

0 1000 2000 3000 4000 5000

0.0

0.2

0.4

0.6

0.8

1.0

Days

Prob

abilit

y Su

rviva

l

Bottom 9Top Decile


2-14

FLCEXAMPLE


0 1000 2000 3000 4000 5000

0.0

0.2

0.4

0.6

0.8

1.0

1.2

Days

Cum

ulat

ive H

azar

d

Bottom 9Top Decile

2-15

FLCEXAMPLE


10 20 50 100 200 500 1000 2000 5000

−8−6

−4−2

0

Days

Log

Cum

ulat

ive H

azar

dBottom 9Top Decile

2-16

HAZARDFUNCTION

IDEA:

At each time t, let estimate of �(t) be a weighted average of jumpsin ⇥(t) at nearby times.

Steps:

1. Choose a "bandwidth" ±b outside of which observations arenot averaged.

2. Choose a "kernel" or weight function K(·).

x

K(x)

−1 0 1

00.5

1

x

K(x)

−1 0 1

00.5

1

x

K(x)

−1 0 1

00.5

1


2-17

HAZARDFUNCTION

3. Calculate the estimate:

• �(t) = 1b

PJj=1 K(

t�t(j)b )D(j)N(j)

• se(�(t)) = 1b

⇢PJj=1 K

2( t�t(j)b )D(j)N2(j)

� 12


2-18


0 1 2 3 4 5 6

time

Cum

ulat

ive H

azar

d

0.0

0.5

1.0

● ●● ●● ●


2-19

KERNELHAZARDESTIMATE

0 1 2 3 4 5 6

time

Cum

ulat

ive H

azar

d In

crem

ents

0.0

0.5

1.0

● ●● ●● ●


2-20


0 1 2 3 4 5 6

time

Cum

ulat

ive H

azar

d In

crem

ents

0.0

0.5

1.0

● ●● ●● ●


2-21


0 1 2 3 4 5 6

time

Cum

ulat

ive H

azar

d In

crem

ents

0.0

0.5

1.0

● ●● ●● ●


2-22


0 1 2 3 4 5 6

time

Cum

ulat

ive H

azar

d In

crem

ents

0.0

0.5

1.0

● ●● ●● ●


2-23


0 1 2 3 4 5 6

time

Cum

ulat

ive H

azar

d In

crem

ents

0.0

0.5

1.0

● ●● ●● ●


2-24


0 1 2 3 4 5 6

time

Cum

ulat

ive H

azar

d In

crem

ents

0.0

0.5

1.0

● ●● ●● ●


2-25


0 1 2 3 4 5 6

time

Cum

ulat

ive H

azar

d In

crem

ents

0.0

0.5

1.0

● ●● ●● ●


2-26


0 1 2 3 4 5 6

time

Cum

ulat

ive H

azar

d In

crem

ents

0.0

0.5

1.0

● ●● ●● ●


2-27


0 1 2 3 4 5 6

time

Cum

ulat

ive H

azar

d In

crem

ents

0.0

0.5

1.0

● ●● ●● ●


2-28


0 1 2 3 4 5 6

time

Cum

ulat

ive H

azar

d In

crem

ents

0.0

0.5

1.0

● ●● ●● ●


2-29


0 1 2 3 4 5 6

time

Cum

ulat

ive H

azar

d In

crem

ents

0.0

0.5

1.0

● ●● ●● ●


2-30


0 1 2 3 4 5 6

time

Cum

ulat

ive H

azar

d In

crem

ents

0.0

0.5

1.0

● ●● ●● ●


2-31


0 1 2 3 4 5 6

time

Cum

ulat

ive H

azar

d In

crem

ents

0.0

0.5

1.0

● ●● ●● ●


2-32


0 1 2 3 4 5 6

time

Cum

ulat

ive H

azar

d In

crem

ents

0.0

0.5

1.0

● ●● ●● ●


2-33

FLCEXAMPLE


0 1000 2000 3000 4000 5000

0e+0

02e−0

44e−0

46e−0

4

Days since Enrollment

Smoo

thed

Haz

ard

Top DecileOther 9

2-34

OUTLINE

•  Nonparametrices9ma9onofhazardfunc9ons•  Compe.ngrisks:– Defini.on:whenthereismorethanonecauseofdeath/failure

– Cumula.vefunc.ons:•  Event-freesurvival•  Cumula.veIncidencees.mator



2-35

COMPETINGRISKS

•  Whenthereismorethanonecauseoffailure:– Recurrenceordeathbeforerecurrence– MI,stroke,PEordeathfromothercauses

•  Thedifferenttypesoffailurearecalled“compe9ngrisks”.– They“compete”tobethefirsttomakesubjectsexperienceanevent


2-36

MONOCLONALGAMMOPATHY

•  241MayoClinicPa9ents(MonoclonalGammopathyofUndeterminedSignificance)

•  20-40yearsoffollow-upanerDx•  64developedplasmacellmalignancy(PCM),163diedwithoutit.

•  PCManddeathwithoutPCMarecompe9ngrisksRKyle,Benignmonoclonalgammopathy–aner20to35yearsoffollow-up,MayoClinicProc1993;68:26-36


2-37


•  Insitua9onslikethis,ithasbeencommonprac9cetoapplytheKMmethodtoes9mate“survival”func9ons:– ProbabilityofavoidingPCMover9me– Probabiltyofavoidingdeathw/oPCM

•  ForPCMcurve,treatdeathsw/oPCMascensored•  Fordeathw/oPCM,treatPCMsascensored•  Conceptualdifficul9esindescribingdistribu9onofthe9meofaneventthatmaynotoccur.


2-38


0 10 20 30 40

0.0

0.2

0.4

0.6

0.8

1.0

Probability no PCM

years


2-39



0 10 20 30 40

0.0

0.2

0.4

0.6

0.8

1.0

Probability no Death without PCM

2-40



0 10 20 30 40

0.0

0.2

0.4

0.6

0.8

1.0

years

PCMDeath without PCM

Probability of avoiding PCM or death w/o PCM

Whatiswrongwiththispicture?

2-41

CUMULATIVEINCIDENCE


0 10 20 30 40

0.0

0.2

0.4

0.6

0.8

1.0

years

Death without PCMPCM

2-42

CUMULATIVEINCIDENCE


0 2000 4000 6000 8000 10000 12000 14000

0.0

0.2

0.4

0.6

0.8

1.0

Days Since Diagnosis

Cum

ulat

ive In

cide

nce

DeathPCM

Cumulative IncidenceKM

2-43

ESTIMATINGCUMULATIVEINCIDENCE


• We can write

1� S(k)(t) =X

j:t(j)t

D(k)(j)N(j)

S(k)(t(j�1))

• At the second failure time of type k,

1� S(k)(t(2)) = 1� N(1)�D(k)(1)N(1)

· N(2)�D(k)(2)

N(2)=

D(k)(1)N(1)+

D(k)(2)N(2)· N(1)�D

(k)(1)

N(1)

• If any failures of another type have occurred between t(1) and

t(2), theN(1)�D(k)(1)

N(1)term is too big.

• This bias will accumulate and get larger, as we move to largerand larger t(j).

2-44

ESTIMATINGCUMULATIVEINCIDENCE


• Letting D(k)(j) = the number of failures of types other than k att(j), an unbiased estimate of F(k)(t) is given by

X

j:t(j)t

D(k)(j)N(j)

j�1Y

�=1

N(�) �D(k)(�) �D(k)(�)

N(�)=X

j:t(j)t

D(k)(j)N(j)

j�1Y

�=1

N(�) �D(k)(�)N(�)

·N(�) �D(k)(�)

N(�)

"no ties between failures of different types

• Compare to biased upward

1� S(k)(t) =X

j:t(j)t

D(k)(j)N(j)

S(k)(t(j�1)) =X

j:t(j)t=D(k)(j)N(j)

j�1Y

�=1

N(�) �D(k)(�)N(�)

.

2-45

PREFERRED:TOGETHER

0 2000 4000 6000 8000 10000 12000 14000

0.0

0.2

0.4

0.6

0.8

1.0


Cum

ulat

ive P

roba

bilit

y Dead

Plasma Cell Malignancy

Alive and Malignancy−Free


2-46

CHOICEOFOUTCOMEEVENT

•  Maynotwantcause-specificeventasprimaryoutcome.

•  Interpreta9oncloudy,par9cularlyforsurvivalcurves,sincethosewhodieearlymaycountas“survivors”ofacompe9ngevent.


2-47

CUMULATIVEFUNCTIONS


• When there are competing risks, functions that describe thedistribution of the event-specific time Tk do not make sense:

– If the subject fails of another cause before t, Tk is not de-fined.

• Some other cumulative functions do make sense, depending oncontext:

– The probability that a subject is alive and event-of-interest-free at t.

∗ This means re-defining the event of interest to be theoriginal event of interest or death.

– The probability that an event of type k has (or has not)occurred by time t.

∗ It has not occurred if the subject dies before t.

2-48

CUMULATIVEFUNCTIONS


Event-free Survival:

Estimating the probability a subject is alive and event-of-interest-free at time t is easy:

1. Redefine the event of interest to be either the original event ofinterest or death

�� =⇢1 event of interest or death from any cause0 censored

T� = time to event of interest, death or censoring

2. Compute the KM estimate of S(t) in the usual way with (T�, ��)data.

2-49

CUMULATIVEINCIDENCE


0 2000 4000 6000 8000 10000 12000 14000

0.0

0.2

0.4

0.6

0.8

1.0


Cum

ulat

ive In

cide

nce

DeathPCMPCM or DeathCumulative IncidenceKM

2-50

MALIGNANCY-FREESURVIVAL

0 2000 4000 6000 8000 10000 12000 14000

0.0

0.2

0.4

0.6

0.8

1.0


Mal

igna

ncy−

Free

Sur

viva

l


2-51

EXAMPLE


MillerK,WangM,GralowJ,DicklerM,CobleighM,PerezEA,ShenkierT,CellaD,DavidsonNE.Paclitaxelplusbevacizumabversuspaclitaxelaloneformetasta9cbreastcancer.NewEnglandJournalofMedicine.2007;357(26):2666–2676.

2-52

SOMESUBTLETIES

•  Cumula9veincidence:theprobabilitythataneventoftypekhasoccurredby9met:– Makessensewithoutrequiringthata9metothekthtypeofeventbedefinedforallsubjects

–  Dependsonthepor9onofthepopula9ons9llatriskateach9me,soitsvaluewilldependnotonlyontheriskoftheeventofinterest,butalsoontheriskofalltheothercausesoffailure.

–  Isapopula9on-specificquan9tythatdependsonwhatotherrisksareopera9nginthepopula9onandhowtheyarerelatedtotheriskoftheeventofinterest.


2-53

OUTLINE



– Cause-specificandsub-distribu.onhazards– Fine-GrayandCoxregressionmodels–  Interpreta9onsubtle9es


2-54

REGRESSIONMODELS

•  Therearetwotypesofregressionmodelsforcompe9ngrisksoutcomedatathatcanhaveusefulinterpreta9ons– Fine-Graymodels,thatmodeltheassocia9onbetweenindependentvariablesandthehazardfunc9onassociatedwiththecumula9veincidencesub-distribu9onfunc9on

– Coxregressionmodel,thatmodelstheassocia9onbetweenindependentvariablesandthecause-specifichazardfunc9on


2-55

FINE-GRAYHAZARD


�FG(k)(t) = Pr[T 2 [t, t + �t), c = k|T � t or both T < t and c 6= k]/�t

Interpretations:

• the risk of failure of type k among those event free at t andthose who have experienced all events other than a type kevent. (Note if type k is not death, this would include subjectswho had already died.)

• the hazard function associated with the sub-distribution givenby the cumulative incidence function for a type k failure

2-56

FINE-GRAYMODEL


Fine-Gray Hazard:

�FG(k)(t) = Pr[T 2 [t, t + �t), c = k|T � t or both T < t and c 6= k]/�t

Model:

�FG(k)(t|x) = �FG(k)(t|0)e�x

2-57

INTERPRETATION


Before, we noted that the interpretation of the Fine-Gray hazard isthe risk of failure of type k among those event free at t and thosewho have experienced all events other than a type k event.

Suppose a type k failure is not death, and high values of � are asso-ciated with higher risk of death. Then if � has no causal associationwith type k failure, it will still be negatively associated with the Fine-Gray hazard of type k failure, since subjects with high values of �will be less likely to live long enough to experience a type k failure.

• Cumulative incidence of type k events will be lower for highvalues of �.

• Appropriate model when concern is about population burden,cost of type k events or estimating probabilities of the differentpossible outcomes for a patient.

• Not appropriate for causal modeling.

2-58

COXCAUSE-SPECIFICHAZARD


�(k)(t) = Pr[T 2 [t, t + �t), c = k|T � t]/�tInterpretation:

• Risk of type k event among subjects who have not experiencedany of the types of events.

• Not influenced by the number of event-free subjects BUT can beinfluenced by who the event-free subjects are (Tsiatis, 1975).

• No real interpretation in terms of cumulative functions. Theanalogue would be the KM-survival curve, which is not appro-priate.

• Appropriate for causal modeling with interpretation restrictedto the population from which data are drawn (Prentice et al.,1978)

2-59

PROPERTIES


T = time to first "failure" of any type

�(k)(t) = lim�t!0

Pr[T 2 [t, t + �t), c = k|T � t]/�t

• The different events defined by c must be mutually exclusive

• The different events defined by c must be exhaustive

• The hazard function for the distribution of T is given by :

�(t) =KX

k=1�(k)(t)

2-60

ESTIMATION


Cause-specific Hazard Functions:

• For estimation, we can treat failures from other causes as cen-sored, and estimate the cause-specific hazard �(k)(t) and cu-mulative cause-specific hazard �(k)(t) in the usual way.

Q: Why does this work?

A:

Q: How are failures from other causes conceptually different fromthe censoring we have talked about earlier in this course?

A:

2-61

ESTIMATION


Cumulative Incidence:

• We can treat failures from other causes as censored, and esti-mate the cause-specific hazard �(k)(t) and cumulative cause-specific hazard �(k)(t) in the usual way.

• If we do the same thing for the cumulative incidence, the Kaplan-Meier 1� S(k)(t) is a biased estimate of F(k)(t).

Q: Why?

A:

2-62

MGUSDATA


0 2000 4000 6000 8000 10000 12000

0.00

000

0.00

005

0.00

010

0.00

015

0.00

020


Haz

ard

Rat

e

Death FirstPCM

2-63

OUTLINE



– Cause-specificandsub-distribu9onhazards– Fine-GrayandCoxregressionmodels–  Interpreta.onsubtle.es


2-64

IDENTIFIABILITYANDINTERPRETATION

•  Tsia9s(1975)showedthatwecannotiden9fyfrom(T,c=k)datawhethersubjectswhofailfromonecausewouldhavebeenmoreorlesssuscep9blelatertofailurefromanothercause,hadtheysurvived.–  Cannottellwhetherthosewhodiefromheartdiseasewouldhavebeenmoreorlesslikelytodevelopcancerlater.

•  Pren9ceetal(1978)arguedthatthecause-specifichazardfunc9on(Coxmodel)wasthebestbasisforcausalinferenceinthepopula9onasitiscons9tuted,butcannotextendinterpreta9ontoanotherpopula9onwherecompe9ngrisksarenotopera9ng.–  Cannotsayhowxmightberelatedtocancerriskinapopula9onwheretherearenodeathsfromMI


2-65

INTERPRETATIONSUBTLETY


Cannot estimate what the hazard �(k) (or survival function S(k)(t))would be if competing causes of failure were removed.

Reason: �(k)(t) = risk of type k failure at t among thosestill at risk at t.

6= risk of type k failure at t among thosestill at risk at t if other causes were removed.

For these to be equal, population at risk at T would need to havethe same risk of event k whether or not other causes of failure wereremoved. This is a strong and unverifiable assumption (Tsiatis,1975).

2-66

INTERPRETATIONSUBTLETY

•  Cause-specifichazardisnonetheless,lessdependentontheriskofothertypesoffailureinthefollowingsense–  Ifsomeinterven9onincreasestheriskofanothertypeoffailure,butdoesnotchangehowsuscep9blesurvivorsofthatfailuretypearetothetypeofinterest,theinterven9onwillnotinfluencethecause-specifichazardfunc9on

–  Ifsomeinterven9onincreasestheriskofanothertypeoffailure,therewillbefewersubjectsatriskinthepopula9onatlater9mes,andthecumula9veincidenceoftheeventofinterestwillbelower.


2-67

COXMODEL

Cause-specific Hazard:

�(k)(t) = Pr[T 2 [t, t + �t), c = k|T � t]/�t

Cox Model:

�(k)(t|x) = �(k)(t|0)e�x


2-68

CHOOSINGACOMPETINGRISKSMODEL

•  Coxmodelwithcause-specifichazardgoodwheninterestedinwhatcausestheeventofinterestinthepopula9onasitiscons9tuted.

•  Fine-Graygoodwheninterestedinpredic9ngpa9entprognosisorpopula9ondisease/costburden.

•  Botharepropor9onalhazardsmodels,butfordifferenthazardfunc9ons


2-69

EXAMPLE

•  CoxandFine-Graymodelsfortheassocia9onofsexwithPCMandDeathbeforePCMintheMonoclonalGammopathydata.

•  Willshow– Cause-specifichazardfunc9onsbysexandcause– Cumula9veincidencefunc9onsbysexandcause– Es9matedHazardra9os(maletofemale)bycauseunderbothmodels


2-70

CAUSE-SPECIFICHAZARDESTIMATES

0 5 10 15 20 25 30

0.00

0.02

0.04

0.06

0.08

0.10

Years

Haz

ard

Rat

e

malesfemales

Plasma Cell Malignancy


2-71

CAUSE-SPECIFICHAZARDESTIMATES


0 5 10 15 20 25 30

0.00

0.02

0.04

0.06

0.08

0.10

Years

Haz

ard

Rat

emalesfemales

Deaths from Other Causes

2-72

COXMODELS

OutcomeType M/FHazardRa9o 95%CI P-valuePlasmaCellMalignancy 0.95 (0.58,1.56) 0.8441DeathfromOtherCauses 1.55 (1.13,2.14) 0.0064


2-73

CUMULATIVEINCIDENCE


0 10 20 30 40

0.0

0.2

0.4

0.6

Female Death no PCMMale Death no PCMFemale PCMMale PCM

2-74

CUMULATIVEINCIDENCE


0 10 20 30 40

0.0

0.2

0.4

0.6

0.8

1.0

Years

Prob

abilit

y

MalesFemales

Alive PCM−free

PCM

Death no PCM

2-75

FINE-GRAYMODELS

OutcomeType M/FHazardRa9o 95%CI P-valuePlasmaCellMalignancy 0.71 (0.44,1.16) 0.17

DeathfromOtherCauses 1.45 (1.06,1.97) 0.02


PCMhazardra9ofartherfromoneherebecausemenaremorelikelytodiefromothercausesandnotsurvivetodevelopPCM.

2-76

COMMONMISINTERPRETATION

•  TheFine-Graymodelisnotthemodelthat“accountsforcompe9ngrisks”

•  BoththeCoxmodelwithcause-specifichazardfunc9onsandtheFine-Graymodelaccountforcompe9ngrisks,butaswehaveseentheirtargetsofinferencearedifferent.– Hazardamongthoses9llatriskoftheevent(Cox)– Hazardamongthosewhohavenotyetexperiencedtheeventofinterest,butcouldhaveexperiencedothers,includingdeath,already(F-G)


2-77

EXAMPLE

•  Ashburneretal(2017)studiedacohortof13,559subjectsdiagnosedwithatrialfibrilla9on(AF)atKaiserNorthernCalifornia–  1092thromboembolismevents(1017ischemicstrokes)–  4414experienceddeathwithoutthromboembolismevent–  Thromboemolism-freeDeathratewas5.5/100PYamongwarfarintakersand8.1/100PYamongnon-takers

–  Non-takerswereolderhadhigherstroke-riskscores•  TheycomparedCoxandF-Gregressionwith9me-dependent

currentwarfarinuseastheexposure


2-78

EXAMPLE

•  TheyconcludedthattheFine-Graymodelthat“accountedfor”compe9ngrisksgaveabeter“real-world”assessmentofthebenefitofwarfarin.

•  Whatareyourthoughts?

AshburnerJM,GoAS,ChangY,FangMC,FredmanL,ApplebaumKM,SingerDE.JAmGeriatrSoc.2017Jan1;65(1):35–41.


Event

Model

AdjustedHazardRa9o

95%CI

Thromboembolism Cox 0.57 (0.50,0.65)

Fine-Gray 0.87 (0.77,0.99)

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SOMECOMPETINGRISKSREFERENCES•  Aus9nPC,LeeDS,FineJP.Introduc9ontotheAnalysisofSurvivalDatainthe

PresenceofCompe9ngRisks.Circula9on.2016Feb9;133(6):601–609.•  FineJP,GrayRJ.APropor9onalHazardsModelfortheSubdistribu9onofa

Compe9ngRisk.JournaloftheAmericanSta9s9calAssocia9on.1999Jun1;94(446):496–509.

•  LauB,ColeSR,GangeSJ.Compe9ngriskregressionmodelsforepidemiologicdata.AmericanjournalofEpidemiology.2009;170(2):244–256.

•  NoordzijM,LeffondréK,StralenV,JK,ZoccaliC,DekkerFW,JagerKJ.Whendoweneedcompe9ngrisksmethodsforsurvivalanalysisinnephrology?NephrolDialTransplant.2013Nov1;28(11):2670–2677.

•  Pren9ceRL,KalbfleischJD,PetersonJrAV,FlournoyN,FarewellVT,BreslowNE.Theanalysisoffailure9mesinthepresenceofcompe9ngrisks.Biometrics.1978;541–554.

•  Tsia9sA.Anoniden9fiabilityaspectoftheproblemofcompe9ngrisks.ProceedingsoftheNa9onalAcademyofSciences.1975;72(1):20–22.


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TOWATCHOUTFOR

•  Interpreta9oninthepresenceofcompe9ngriskscanbesubtleandrequirescare.

–  S(t)definedintermsoftheprobabilitydistribu9onofTkdoesnotmakesense

–  Cannotinterpretfunc9onsofthecause-specifichazardasapplyinginapopula9onwithoutcompe9ngriskspresent.

–  1–KMes9matorcangiveupwardbiasedes9mateofcumula9veincidence.

–  Cannotinterpretcumula9veincidenceasapplyinginapopula9onwithoutcompe9ngriskspresent

–  Fine-GraymodelisnotTHEwaytoaccountforcompe9ngrisks.


session 2: hazard functions, competing risks, cause...

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