Download The analysis of case cohort design in the

The analysis of case cohort
design in the presence of
competing risks
Melania Pintilie, Yan Bai, Lingsong Yun and
David C. Hodgson
Ontario Cancer Institute, University of Toronto, ICES, Princess Margaret Hospital
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Hodgkin’s disease
• Hodgkin’s disease is a type of cancer
which affects the young adults (median
age=37).
• 5 year survival 84% in USA
• Treatment: Radiation and/or
chemotherapy
• Treatment may be cardiotoxic.
• Aim: Study the incidence of cardiac events
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HD cohort
• In Ontario over 16 year period (1988-2003)
~3000 HD
• ~300 cardiac events
• ~ 600 die without a cardiac event
• Time to event analysis the number of events
determines the power of the study
• To study the cardiac events a large amount of
information is needed to be collected
– Treatment
– Cardiac history
– Risk factors
• The presence of competing risks
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Outline
•
•
•
•
•
Case-cohort definition and analysis
Competing risks
PsL for case-cohort with competing risks
Prediction for a case-cohort study
Simulations and results
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Case-cohort
• Select randomly a subcohort from the
source cohort. This set contains some
cases
– 1/3 of the 3000 . Subcohort=1108, 105
cardiac events
• Add to this all the rest of the cases
– 204 cardiac events
• Take advantage of all 309 cardiac events,
but collect detail information only on
1108+204=1312 instead of ~3000.
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Analysis of case-cohort,
pseudolikelihood
j
The cases which are


n
not part of the
exp   x j 


PsL      

subcohort participate
I
exp

x


j 1  
rj
r 
 rR j

in the PsL only at the
1 if event was observed at t j
time of the event.
j  
0 if event was not observed at t j
1 observation r is in subcohort or

I rj  
r is a case not in the subcohort and tr  t j

0 r is a case not in the subcohort and tr  t j
Prentice RL. A Case-Cohort Design for Epidemiologic Cohort Studies and Disease
Prevention Trials. Biometrika 73: 1-11, 1986.
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Presence of competing risks
• Among the 1108 in the subcohort 198 patients died before having a
cardiac event. These are competing risks.
1 if the event of ineterest was observed at time t j


1 C j  1
C j  2 if the competing risk event was observed at time t j  j  
0 C j  0 or C j  2



0 no event was observed at time t j


n 
exp x j  

PL    
j 1   wrj exp  xr  
 rR~

 j

dj
wrj 
Gˆ  t j 
Gˆ  min(t j , tr ) 
R j  r ; tr  t j or Cr  2
Fine JP and Gray RJ. A proportional hazards model for the subdistribution of a
competing risk. Journal of the American Statistical Association 94: 496-509, 1999.
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Fine and Gray - modelling the
hazard of the subdistribution
j


r
 exp   x j  
PL(  )   

w
exp

x


j 1  
ji
i 
 iR j

w42
J=1
w32
R4
w65
w62
J=3
J=2
w32>w42>w62
R6
J=4 J=5
J=6
time
R3
R1
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SCT 2010
The analysis of case-cohort in the
presence of competing risks




exp x j 

PsL    
j 1   I rj wrj exp  xr  
 rR~

 j

dj
n
1 if the event of ineterest was observed at time t j

C j  2 if the competing risk event was observed at time t j

0 no event was observed at time t j
1

I rj  

0

1 C j  1
j  

 0 C j  0 or C j  2
observation r is in subcohort or
Gˆ  t j 
r is a case not in the subcohort and tr  t j w 
rj
ˆ min(t , t )
G
j r
r is a case not in the subcohort and tr  t j
R j  r ; tr  t j or Cr  2


Jackknife variance
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Prediction
Fˆ  t | x   1  e
 Hˆ  t|x 
Hˆ  t | x    hˆ  ti | x 
ti t
, where
ˆ x
ˆ
ˆ
h  t | x   h0  t  e
dj
h0 t j  
Using the
ˆx subcohort only
w
exp

 rj
r
~
rR j
 
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Simulations
• To verify that
– The estimated coefficient is correct
– The predicted probabilities are correct
– The type I error is correct
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Description of the simulations
The parameters
• The covariate
0
x
1
• The distribution of time
Tev , x 0 ~ Expev 0  0.03
Tev , x 1 ~ Expev1  0.07 
Tcr , x 0 /1 ~ Expcr  0.05
Tobs  min Tev , Tcr 
• The censor time
Tc ~ U 2,10
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Description of simulations
• Source cohort N=1200
• Subcohort = 1/3 of the source cohort,
n=400
• Case-cohort: 400+the rest of cases
• Obtain:
– Coefficients
– Predicted probabilities at 1,2,3,4,5 years
• Repeated 2000 times
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Coefficient obtained using the case-cohort dataset
Coefficients
1.4
1.2
1.0
0.8
0.6
0.4
0.2
0.2
0.4
0.6
0.8
1.0
Coefficient obtained using the source cohort
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1.2
1.4
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Predicted probabilities
Probability of event of interest
0.4
0.3
0.2
0.1
0.0
at 1 years
at 3 years
x=0
SCT 2010
at 5 years
at 2 years
at 4 years
x=1
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To verify the type I error
• Source cohort N=120
• Subcohort = 1/3 of the source cohort,
n=40
• Case-cohort: 40+the rest of cases
• Repeated 2000 times
• Type I error = 0.0375, close to 0.05
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Thank you
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References
1.Prentice RL. A Case-Cohort Design for Epidemiologic Cohort Studies
and Disease Prevention Trials. Biometrika 73: 1-11, 1986.
2.Fine JP and Gray RJ. A proportional hazards model for the
subdistribution of a competing risk. Journal of the American Statistical
Association 94: 496-509, 1999.
3.Barlow WE, Ichikawa L, Rosner D, and Izumi S. Analysis of casecohort designs. Journal of Clinical Epidemiology 52: 1165-1172, 1999.
4.Borgan O, Goldstein L, and Langholz B. Methods for the analysis of
sampled cohort data in the Cox proportional hazards model. Annals of
Statistics 23: 1749-1778, 1995.
5.Barlow WE. Robust Variance-Estimation for the Case-Cohort Design.
Biometrics 50: 1064-1072, 1994.
6.Self SG and Prentice RL. Asymptotic-Distribution Theory and Efficiency
Results for Case Cohort Studies. Annals of Statistics 16: 64-81, 1988.
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7. Therneau TM and Li HZ. Computing the Cox model for case cohort
designs. Lifetime Data Analysis 5: 99-112, 1999.
8. Sorensen P and Andersen PK. Competing risks analysis of the casecohort design. Biometrika 87: 49-59, 2000.
9. Crowder MJ. Classical competing risks. London: Chapman and
Hall/CRC Press, 2001.
10. Tibshirani RJ and Efron B. An Introduction to the Bootstrap. New
York: Chapman & Hall, 1993.
11. Lag R, Melbert D, Krapcho M, Stinchcomb DG, Howlader N,
Horner MJ, Mariotto A, Miller BA, Feuer EJ, Altekruse SF, Lewis
DR, Clegg L, Eisner MP, Reichman M, and Edwards BK. SEER
Cancer Statistics Review, 1975-2005. Bethesda, MD: National Cancer
Institute.
12. Myrehaug S, Pintilie M, Tsang R, Mackenzie R, Crump M, Chen ZL,
Sun A, and Hodgson DC. Cardiac morbidity following modern
treatment for Hodgkin lymphoma: Supra-additive cardiotoxicity of
doxorubicin and radiation therapy. Leukemia & Lymphoma 49: 14861493, 2008.
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Women, 45 years old, no risk factors
Probability for cardiac hospitalization
0.20
D+RT
RT
D
0.15
0.10
0.05
0.00
0
5
10
Time to cardiac hospitalization
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20
Men, 45 years old, no risk factors
Probability for cardiac hospitalization
0.20
D+RT
RT
D
0.15
0.10
0.05
0.00
0
5
10
Time to cardiac hospitalization
SCT 2010
15
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