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survival analysis prop. hazard model shared frailty model Simulation discussion
Procedures for analyzing Frailty-Models in SAS and R
Katharina Hirsch
Martin-Luther-Universitat Halle-WittenbergInstitut fur Medizinische Epidemiologie, Biometrie und Informatik
20.11.2009
Katharina Hirsch Frailty-Models 20.11.2009 1 / 23
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survival analysis prop. hazard model shared frailty model Simulation discussion
outline
1 survival analysis
2 proportional hazard model
3 shared frailty model
4 Simulation
5 discussion
Katharina Hirsch Frailty-Models 20.11.2009 2 / 23
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survival analysis prop. hazard model shared frailty model Simulation discussion
survival data
observation of single events
discharge from the hospitaldisruption of materialonset of a disease
analyzing the event time
estimation of the effect of prognostic factors
often censored data:
end of the studylost to follow-upcompeting risk
Katharina Hirsch Frailty-Models 20.11.2009 3 / 23
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survival analysis prop. hazard model shared frailty model Simulation discussion
proportional hazard model (Cox 1972)
µ(t | X ) = µ0(t)eβ′X
µ0(t) baseline hazard function
β′ = (β1, . . . , βk) vector of regression coefficients
X ′ = (X1, . . . ,Xk) vector of covariables
effect of covariables on the time until the occurrence of an event(regression model)
requirements:
proportional hazardsindependent life times
semiparametric model
Katharina Hirsch Frailty-Models 20.11.2009 4 / 23
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survival analysis prop. hazard model shared frailty model Simulation discussion
shared frailty model
clustered data
basic idea: extension of the Cox model
event times conditionally independent
Zi = frailty termdescribed unobserved heterogeneity
µij(t,Xij ,Zi ) = Ziµ0(t)eβ′Xij
Katharina Hirsch Frailty-Models 20.11.2009 5 / 23
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survival analysis prop. hazard model shared frailty model Simulation discussion
shared gamma frailty model
f (z) =bpzp−1e−bz
Γ(p)Z ∼ Γ(p, b)
hazard function
µ(t) =µ0(t)
1 + σ2M0(t)
assumption p = b
EZ = 1, σ2 =1
b
Z ∼ Γ(b, b)
typical assumptions for the parameters
one parametric gamma distribution
Katharina Hirsch Frailty-Models 20.11.2009 6 / 23
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survival analysis prop. hazard model shared frailty model Simulation discussion
shared log-normal frailty model
Ziµ0(t)eβ′Xij = µ0(t)eβ′Xij+Wi
W ∼ N(0, σ2)
assume a normal distributed random effect W
very flexible
assumption EW = 0
no explicit form of the Likelihood function
numerical methods have to be used
Katharina Hirsch Frailty-Models 20.11.2009 7 / 23
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survival analysis prop. hazard model shared frailty model Simulation discussion
Procedures for the shared gamma frailty model
name SPGAM1 frailtypenal coxph
software SAS R R
library - Frailtypack Survival
algorithm ML-EM PPL PPL
author Hien Vu Juan R. Gonzalez T. TherneauVirginie Rondeau
Katharina Hirsch Frailty-Models 20.11.2009 8 / 23
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survival analysis prop. hazard model shared frailty model Simulation discussion
Procedures for the shared log normal frailty model
name SPNL3 coxph coxme phmm
software SAS R R R
library - Survival Kinship Phmm
algorithm ML-EM PPL PPL EMlogN/reml MCMC
author Hien Vu T. Therneau, T. Therneau M. Donohue,C. McGilchrist R. Xu
Katharina Hirsch Frailty-Models 20.11.2009 9 / 23
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survival analysis prop. hazard model shared frailty model Simulation discussion
Simulation
3 different cluster sizes (10x100, 50x20, 100x10)
total sample size: 1000
3 covariables: x1 (uniform), x2 (normal), x3 (binomial)
β1 = 1β2 = −1β3 = 0.5σ2 = 0.5
gamma and log-normal frailty
3 different baseline hazards
Weibull-distributedGompertz-distributedExponential-distributed
50% and 80% censoring
Katharina Hirsch Frailty-Models 20.11.2009 10 / 23
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survival analysis prop. hazard model shared frailty model Simulation discussion
analysis of the simulated data
β estimators for Γ distributed frailty with 50% censoring
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coxphfrailtyPenalSPGAM1
ββ1
i ii iii
ββ2
i ii iii
ββ3
i ii iii
Katharina Hirsch Frailty-Models 20.11.2009 11 / 23
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survival analysis prop. hazard model shared frailty model Simulation discussion
analysis of the simulated data
σ2 estimators for Γ distributed frailty with 50% censoring
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ercoxphfrailtyPenalSPGAM1
i ii iii
Katharina Hirsch Frailty-Models 20.11.2009 12 / 23
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survival analysis prop. hazard model shared frailty model Simulation discussion
analysis of the simulated data
β estimators for Γ distributed frailty with 80% censoring
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coxphfrailtyPenalSPGAM1
ββ1
i ii iiiββ2
i ii iii
ββ3
i ii iii
Katharina Hirsch Frailty-Models 20.11.2009 13 / 23
logo
survival analysis prop. hazard model shared frailty model Simulation discussion
analysis of the simulated data
σ2 estimators for Γ distributed frailty with 80% censoring
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ercoxphfrailtyPenalSPGAM1
i ii iii
Katharina Hirsch Frailty-Models 20.11.2009 14 / 23
logo
survival analysis prop. hazard model shared frailty model Simulation discussion
analysis of the simulated data
β estimators for lognormal distributed frailty with 50% censoring
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coxphcoxmephmmSPLN3
ββ1
i ii iii
ββ2
i ii iii
ββ3
i ii iii
Katharina Hirsch Frailty-Models 20.11.2009 15 / 23
logo
survival analysis prop. hazard model shared frailty model Simulation discussion
analysis of the simulated data
σ2 estimators for lognormal distributed frailty with 50% censoring
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Katharina Hirsch Frailty-Models 20.11.2009 16 / 23
logo
survival analysis prop. hazard model shared frailty model Simulation discussion
analysis of the simulated data
β estimators for lognormal distributed frailty with 80% censoring
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i ii iii
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i ii iii
Katharina Hirsch Frailty-Models 20.11.2009 17 / 23
logo
survival analysis prop. hazard model shared frailty model Simulation discussion
analysis of the simulated data
σ2 estimators for lognormal distributed frailty with 80% censoring
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Katharina Hirsch Frailty-Models 20.11.2009 18 / 23
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survival analysis prop. hazard model shared frailty model Simulation discussion
discussion
easy to handle
all procedures are suitable
close estimation for β
just the SAS macros provides an estimation for the SE for Var(Z)
Katharina Hirsch Frailty-Models 20.11.2009 19 / 23
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survival analysis prop. hazard model shared frailty model Simulation discussion
discussion
SPGAM1 frailtypenal coxph
one distribution three distributions
definition of control parameters
long runtime low runtime
alternative to coxph not adequate preferable
Katharina Hirsch Frailty-Models 20.11.2009 20 / 23
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survival analysis prop. hazard model shared frailty model Simulation discussion
discussion
SPNL3 coxph coxme phmm
one distribution three distributions one distribution
definition of control parameters
long runtime low runtime long runtime
less adequate less adequate preferable less adequate
Katharina Hirsch Frailty-Models 20.11.2009 21 / 23
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survival analysis prop. hazard model shared frailty model Simulation discussion
Literature
D.R. Cox: Regression models and life tables. Journal of the RoyalStatistical Society 34, 187 – 202, 1972.L. Duchateau, P. Janssen: The Frailty Model. Springer New York, 2008.H. T. V. Vu und M. W. Knuiman: A hybrid ML-EM algorithm forcalculation of maximum likelihood estimates in semiparametric shared frailtymodels. Computational Statistics & Data Analysis, 40(1), 173 – 187, 2002.H. Vu, M. Segal, M. Knuiman and I. James: Asymptotic and smallsample statistical properties of random frailty variance estimates for sharedgamma frailty models. Communication in Statistics: Simulation andComputation30(3), 581 – 595, 2001.V. Rondeau and J. R. Gonzalez: frailtypack:A computer program for theanalysis of correlated failure time data using penalized likelihood estimation.Computer Methods and Programs in Biomedicine 80, 154 – 164, 2005.G. Kauermann and R. Xu and F. Vaida: Stacked Laplace-EMalgorithm for duration models with time-varying and random effects.Computational Statistics and Data Analysis 52, 2514 – 2528, 2008.R Development Core Team: The R project for statistical computing.URL: http://www.r-project.org, 2008.
Katharina Hirsch Frailty-Models 20.11.2009 22 / 23
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survival analysis prop. hazard model shared frailty model Simulation discussion
Thank you for your attention!
Katharina Hirsch Frailty-Models 20.11.2009 23 / 23