Analyse et interpretation des risquesconcurrents :

consensus et problemes ouverts

Aurelien LatoucheConservatoire national des arts et metiers

IHP, 16 octobre 2014, Paris

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Analyse de Survie :

Processus (Xt)t≥0, Xt ∈ {0, 1}, P (X0 = 0) = 1

0 1

Delai de survie T = inf{t |Xt 6= 0}Delai de censure C independant de T :

On observe : T ∧ C, 1(T ≤ C)

Le risque instantane est l’objet central

λ(t)dt = P (T ∈ dt |T ≥ t) = P (Xt+dt = 1 |Xt− = 0)

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Du risque cumule a la fonction de Survie

Λ(t) =

∫ t

0λ(u)du

Comme

λ(t) =f(t)

S(t)=d(log(F (t))

dt

λ(t) = −d(log(S(t))

dt

⇒S(t) = exp−Λ(t)

En Survie univariee ⇒ Equivalence risque et Probabilite +Censure Independante

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Competing risks : Formalisme

Pour alleger la presentation

0Initial

1

2

Processus Multi-etat Xt ∈ {0, 1, 2}, P (X0 = 0) = 1

Delai de survie T = inf{t |Xt 6= 0},Type d’evenement XT ∈ {1, 2}Les risques cause-specifique λi(t) determinenttotalement la dynamique du processus

λi(t)dt = P (T ∈ dt,XT = i |T ≥ t), i = 1, 2.

Delai de survie et Type d’evenement

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Competing risks : Relations fondamentales

Survie et Risques Cause-specifique

P (T > t) = exp(−∫ t

0λ1(u) + λ2(u)︸ ︷︷ ︸

λ(u)

du)

En particulier exp(−∫ t0 λi(u)du) n’est

plus une probabilite

Incidence cumulee pour l’evt de type i=1,2

P (T ≤ t,XT = i) = Fi(t) =

∫ t

0P (T > u)λi(u) du

L’incidence cumulee pour l’evt 1 depend donc des 2risques cause-specifiques

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Relation entre Survie et Incidences cumulees

P (T ≤ t) = P (T ≤ t,XT = 1)︸ ︷︷ ︸F1(t)

+P (T ≤ t,XT = 2)︸ ︷︷ ︸F2(t)

Donc

P (T ≤ t) ≥ Fi(t)

S(t) = 1− (F1(t) + F2(t))

1− KM(t) va sur-estimer l’incidence cumulee

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Illustration : Sur-estimation Evt 0 1 2# 24 64 112

0.0 0.5 1.0 1.5 2.0 2.5

0.0

0.2

0.4

0.6

time

1−KMCIF

Donnees simulees n=2007 / 21

Illustration : Sur-estimation Evt 0 1 2# 70 107 23

0.0 0.5 1.0 1.5 2.0 2.5 3.0

0.0

0.2

0.4

0.6

time

1−KMCIF

Donnees simulees n=2008 / 21

Subdistribution hazard : Un exemple fondateur

Risque de sous-repartition

Idee : retablir la relation hasard / probabilite en presence derisques concurrents

Survie univariee : equivalence entre risque et probabilite

Risque concurrents : exp(−∫ t0 λi(u)du) n’est plus une

probabilite

Gray (Annals of Stat, 1988) introduit le risque de sousrepartition α1(t) tel que

P (T ≤ t,XT = 1) =

∫ t

0P (T > u−)λ1(u) du

!= 1− exp(−

∫ t

0α1(u)du)

Puis un modele a risque proportionnel pour α1(t)9 / 21

Effet d’une covariables sur les risques cause-specifique etl’incidence cumulee

Exemple :

2 evenements : Recidive locale / Recidive Distante

1 variable binaire :

Traitement chirurgie et radiotherapie (=1)Chimiotherapie (=2)

Risque Cause-specifique Evt 1 : λ1•λ11 = 3λ12 = 2

Risque Cause-specifique Evt 2 : λ2•λ21 = 3λ22 = 1

Donc λ11(t) > λ12(t) ∀t

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Effet d’une covariable sur les risques cause-specifique etl’incidence cumulee (2)

F11(t) =

∫ t

0P1(T > u)λ11(u) du

=

∫ t

0exp(−6u)3 du

Incidence cumulee de recidive locale

Chir + radio : F11(t) = (1− exp(−6t))/2

Chimio : F12 = 2(1− exp(−3t))/3

Q: F11 > F12 ∀t ?

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Effet d’une covariable sur les risques cause-specifique etl’incidence cumulee (3)

0.0 0.5 1.0 1.5 2.0

0.0

0.2

0.4

0.6

0.8

1.0

t

01

23

4

F11

F12

λ11

λ12

Effet indirecte du traitement sur l’evenement concurrent (+Test de Gray vs Test dulog-rank)

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Competing Risks Regression

Cox pour risque cause-specifique (Kalbfleisch et Prentice,1978)

λi(t;Z) = λi0(t) exp (βZ)

Cox pour subdistribution (Fine et Gray, 1999)

αi(t;Z) = αi0(t) exp (γZ)

Andersen et Klein (2005)

g(F1(t, Zi)) = β0(t) + β1Zi1 + . . .+ βpZip

g= clog log → Fine-Gray (1999)g =Id → additive model (Klein, 2006)g=log → Absolute risk model (Gerds–Scheike–Andersen,2012)

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Relation entre ces risques

λi(t) =dFi(t)

dt/S(t)

α1(t) =dF1(t)

1− F1(t)

⇒ λ1(t) =

(1 +

F2(t)

P (T > t)

)α1(t)

Un modele a risque proportionnel pour λ1(t) impliqueun modele a risque non-proportionnel pour α1(t)

0http://www.ncbi.nlm.nih.gov/pubmed/1675553314 / 21

Consensus

Necessaire !

Quasiment dans tous les domaines de l’epidemiologie(clinique) BMJ, IJE, AJE, JCE

Dernier en date :http://www.bmj.com/content/349/bmj.g5060.long

En general concordant !

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Summary box

In the presence of competing events, classic survival curves(Kaplan-Meier) overestimate cumulative risks and aretherefore inadequate

Risk accumulates differently when there are competingevents and should be displayed as cumulative incidencefunctions for each event. Subdistribution hazard ratiosmeasure the effect of exposures in terms of cumulativeincidence functions

Reporting one hazard ratio for the event of interest isincomplete, and it does not compare cumulative risks. Foran aetiological exploration it is necessary to report hazardratios also for the competing events

0http://www.bmj.com/content/349/bmj.g5060.long16 / 21

Recommendation

Employer une terminologie pour les risques : CSHR pourCox et SHR pour Fine–Gray

Presenter pour chaque risque le CSHR

Presenter le SHR pour l’evenement d’interet et pour lesevenements concurrents

Tester l’hypothese de proportionnalite aussi pour le modelede Fine et Gray

Fournir des graphiques des Incidences cumulees pour lesvariables categorielles

Permet de distinguer les effets directs d’une variable sur lerisque cause specifiqueDes effets indirects d’une variable sur le risque causespecifique de l’evt concurrent

0http://hal.archives-ouvertes.fr/hal-0079499517 / 21

Goodness of Fit

Back to basic : Because there is no intrinsic reason to expectproportional subdistribution hazards, a general diagnosticmethod to examine the proportional hazards assumption ishelpfull

H-H plot pour le subdistribution hazard

Residus de Schoenfeld

0Fine et Gray, JASA199918 / 21

H-H plot (SH) ; evt : Rechute; covariable : Status 1

0.00 0.05 0.10 0.15 0.20 0.25

0.0

0.2

0.4

0.6

Λ(t, Z = 0)

Λ(t,

Z =

1)

1https://hal.archives-ouvertes.fr/file/index/docid/794995/

filename/WebApp.pdf19 / 21

H-H plot (SH) ; evt : TRM; covariable : Status 2

0.00 0.05 0.10 0.15 0.20 0.25 0.30

0.00

0.05

0.10

0.15

0.20

0.25

Λ(t, Z = 0)

Λ(t,

Z =

1)

2https://hal.archives-ouvertes.fr/file/index/docid/794995/

filename/WebApp.pdf20 / 21

Question ouverte : Selection de variables

Recemment (Kuk et Varadhan, 2013)

crrstep : AIC, BIC pour le modele de Fine et Gray

�

Simulation seulement a partir de subdistributionhasard.

Covariable peut avoir un effet tres different sur le SH et lesCSHs (cf exemple Gray)

En l’absence d’un etude de simulation a partir des risquescause-specifiques : utilisation discutable et a comparer ad’autres approches (penMSM, coxBoost)

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References I

P. K. Andersen, R. B. Geskus, T. de Witte, and H. Putter.

Competing risks in epidemiology: possibilities and pitfalls.Int J Epidemiol, 41(3):861–870, 2012.

J. Beyersmann, A. Allignol, and M. Schumacher.

Competing risks and multistate models with R.Springer, New York, 2011.

R. J. Gray.

A class k-sample tests for comparing the cumulative incidence of a competing risk.Annals of Stat, 116:1141–1154, 1988.

John P Klein.

Modelling competing risks in cancer studies.Statistics in Medicine, 25(6):1015–1034, 2006.

M. T. Koller, H. Raatz, E. W. Steyerberg, and M. Wolbers.

Competing risks and the clinical community: irrelevance or ignorance?Stat Med, 31(11-12):1089–1097, 2012.

D. Kuk and R. Varadhan.

Model selection in competing risks regression.Stat Med, 32(18):3077–3088, 2013.

A. Latouche, A. Allignol, J. Beyersmann, M. Labopin, and J. P. Fine.

A competing risks analysis should report results on all cause-specific hazards andcumulative incidence functions.J Clin Epidemiol, 66(6):648–653, 2013.

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References II

A. Latouche, J. Beyersmann, and J. P. Fine.

Comments on ’Analysing and interpreting competing risk data’.Stat Med, 26(19):3676–3679, 2007.

B. Lau, S. R. Cole, and S. J. Gange.

Competing risk regression models for epidemiologic data.Am. J. Epidemiol., 170:244–256, 2009.

M. Noordzij, K. Leffondre, K. J. van Stralen, C. Zoccali, F. W. Dekker, and K. J. Jager.

When do we need competing risks methods for survival analysis in nephrology?Nephrol. Dial. Transplant., 28(11):2670–2677, 2013.

M. Wolkewitz, B. S. Cooper, M. J. Bonten, A. G. Barnett, and M. Schumacher.

Interpreting and comparing risks in the presence of competing events.BMJ, 349:g5060, 2014.

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Une recherche Pubmed 3

3 Link

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R : Competing risks

The package cmprsk estimates the cumulative incidence functions, but they can be comparedin more than two samples. The package also implements the Fine and Gray model forregressing the subdistribution hazard of a competing risk. crrSC extends the cmprsk packageto stratified and clustered data. The kmi package performs a Kaplan-Meier multipleimputation to recover missing potential censoring information from competing risks events,permitting to use standard right-censored methods to analyse cumulative incidence functions.The crrstep package implements stepwise covariate selection for the Fine and Gray model.Package pseudo computes pseudo observations for modelling competing risks based on thecumulative incidence functions. timereg does flexible regression modelling for competing risksdata based on the on the inverse-probability-censoring-weights and direct binomial regressionapproach. riskRegression implements risk regression for competing risks data, along with otherextensions of existing packages useful for survival analysis and competing risks data. TheCprob package estimates the conditional probability of a competing event, aka., the conditionalcumulative incidence. It also implements a proportional-odds model using either the temporalprocess regression or the pseudo-value approaches. Packages survival (via survfit) and prodlimcan also be used to estimate the cumulative incidence function. The compeir package estimatesevent-specific incidence rates, rate ratios, event-specific incidence proportions and cumulativeincidence functions. randomSurvivalForest provides summary measures for competing risk suchas ensemble CIF. The NPMLEcmprsk package implements the semi-parametric mixture modelfor competing risks data. The wtcrsk package implements a proportional subdistributionhazards model with adjustment for covariate-dependent censoring. The MIICD packageimplements Pan’s (2000) multiple imputation approach to the Fine and Gray model for intervalcensored data.

3http://cran.r-project.org/web/views/Survival.html4 / 4