Generating survival data with a clustered and multi-state structure is useful to study Multi-State Models, Competing Risks Models and Frailty Models. The simulation of such kind of data is not straightforward as one needs to introduce dependence between times of different transitions while taking under control the probability of each competing event, the median sojourn time in each state, the effect of covariates and the type and magnitude of heterogeneity. Here we propose a simulation procedure based on Clayton copulas for the joint distribution of times of each competing events block. It allows to specify the marginal distributions of time variables, while their dependence is induced by the copula. Furthermore, even though a dependence is obtained between all the time variables, only some joint distributions have to be handled. The choice of simulation parameters is done by numerical minimization of a criterion function based on the ratio of target and observed values of median times and of probabilities of competing events. The proposed method further allows to simulate discrete and continuous covariates and to specify their effect on each transition in a proportional hazards way. A frailty term can be added, too, in order to provide clustering. No particular restriction is needed on covariates distributions, frailty distribution, number and sizes of clusters. An example is provided simulating data mimicking those from an Italian multi-center study on head and neck cancer. The multi-state structure of these data arises from the interest in studying both time to local relapses and to distant metastases before death. We show that our proposed method reaches very good convergence to the target values.

1. A copula-based simulation method
for clustered multi-state survival data
F. Rotolo• , C. Legrand , I. Van Keilegom , M. Chiogna•
• Dipartmento di Scienze Statistiche Institut de Statistique, Biostatistique
et Sciences Actuarielles
Universit` degli Studi di Padova
a Universit´ Catholique de Louvain
e
September 23, 2011

2. Clustered Multi-State Survival Data F. Rotolo
Survival Data
Time since an origin event until an event of interest.
Example: from birth to death, since beginning of therapy until remission, etc.
Time
q q
T=5
0 1 2 3 4 5
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3. Clustered Multi-State Survival Data F. Rotolo
Survival Data
Time since an origin event until an event of interest.
Example: from birth to death, since beginning of therapy until remission, etc.
Time
q q
T=5
0 1 2 3 4 5
Censoring: some observations cannot be observed, the only
available information being a lower bound.
Example: migration, change of therapy, loss to follow-up, etc.
Time
q x q
T>3.25
0 1 2 3 4 5
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4. Clustered Multi-State Survival Data F. Rotolo
Modeling Survival Data
Because of this peculiarity, instead of modeling the density f (t) of
T , the hazard is considered
P[t ≤ T < t + ∆t|T ≥ t] f (t) d
h(t) = lim = = − log S(t),
∆t 0 ∆t S(t) dt
∞
with S(t) = t f (u)du = P[T > t].
t
Note: S(t) = exp{− 0 h(u)du}.
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5. Clustered Multi-State Survival Data F. Rotolo
Modeling Survival Data
Because of this peculiarity, instead of modeling the density f (t) of
T , the hazard is considered
P[t ≤ T < t + ∆t|T ≥ t] f (t) d
h(t) = lim = = − log S(t),
∆t 0 ∆t S(t) dt
∞
with S(t) = t f (u)du = P[T > t].
t
Note: S(t) = exp{− 0 h(u)du}.
The basic regression model for the hazard is the Proportional
Hazards (PH) Model (Cox, 1972)
h(t|X ) = h0 (t) exp{β X }.
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6. Clustered Multi-State Survival Data F. Rotolo
Survival Models
Complications of Cox models have been developed
Frailty Models (FMs)
account for overdispersion
or clustering by means
of random effects
h(t|Xij ) = h0 (t)Zi e β Xij ,
similar to GLMM
log[h(t|Xij )] = log[h0 (t)]+Wi +β Xij ,
with Zi = e Wi
(Duchateau & Janssen, 2008; Wienke, 2010)
A copula-based simulation method for clustered multi-state survival data 4/ 22

7. Clustered Multi-State Survival Data F. Rotolo
Survival Models
Complications of Cox models have been developed
Frailty Models (FMs) Multi-State Models (MSMs)
account for overdispersion consider several events
or clustering by means and their interactions
of random effects LR
T1 T4
h(t|Xij ) = h0 (t)Zi e β Xij ,
T3
NED De
similar to GLMM
log[h(t|Xij )] = log[h0 (t)]+Wi +β Xij ,
T2 T5
Wi
with Zi = e DM
(Duchateau & Janssen, 2008; Wienke, 2010)
(Putter et al., 2007; de Wreede et al., 2010)
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8. Clustered Multi-State Survival Data F. Rotolo
Survival Models
Complications of Cox models have been developed
Frailty Models (FMs) Multi-State Models (MSMs)
account for overdispersion consider several events
or clustering by means and their interactions
of random effects LR
T1 T4
h(t|Xij ) = h0 (t)Zi e β Xij ,
T3
NED De
similar to GLMM
log[h(t|Xij )] = log[h0 (t)]+Wi +β Xij ,
T2 T5
Wi
with Zi = e DM
(Duchateau & Janssen, 2008; Wienke, 2010)
(Putter et al., 2007; de Wreede et al., 2010)
Possible integration?
A copula-based simulation method for clustered multi-state survival data 4/ 22

9. Clustered Multi-State Survival Data F. Rotolo
Survival Models
Complications of Cox models have been developed
Frailty Models (FMs) Multi-State Models (MSMs)
account for overdispersion consider several events
or clustering by means and their interactions
of random effects LR
T1 T4
h(t|Xij ) = h0 (t)Zi e β Xij ,
T3
NED De
similar to GLMM
log[h(t|Xij )] = log[h0 (t)]+Wi +β Xij ,
T2 T5
Wi
with Zi = e DM
(Duchateau & Janssen, 2008; Wienke, 2010)
(Putter et al., 2007; de Wreede et al., 2010)
Possible integration? Simulation studies
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10. Clustered Multi-State Survival Data F. Rotolo
Simulation of data
A simulation method should be able to generate
the dependence of times of
LR
competing events
NED De
DM
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11. Clustered Multi-State Survival Data F. Rotolo
Simulation of data
A simulation method should be able to generate
the dependence of times of
LR
competing events
the dependence of times of
subsequent events
NED De
DM
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12. Clustered Multi-State Survival Data F. Rotolo
Simulation of data
A simulation method should be able to generate
the dependence of times of
LR LR LR LR competing events
NED De NED De NED De NED De
LR LR
the dependence of times of
DM DM DM DM
NED De NED De subsequent events
LR LR LR LR
DM DM
NED De NED De NED De NED De
the dependence between clustered
DM DM DM DM observations
LR LR LR LR
NED De NED De NED De NED De
LR LR
DM DM DM DM
NED De NED De
LR LR LR LR
DM DM
NED De NED De NED De NED De
DM DM DM DM
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13. Clustered Multi-State Survival Data F. Rotolo
Simulation of data
A simulation method should be able to generate
the dependence of times of
LR
competing events
the dependence of times of
subsequent events
the dependence between clustered
observations
NED x De
the censoring due to competing
events occurrence
x
DM
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14. Clustered Multi-State Survival Data F. Rotolo
Simulation of data
A simulation method should be able to generate
the dependence of times of
LR
competing events
the dependence of times of
x subsequent events
the dependence between clustered
observations
NED x De
the censoring due to competing
events occurrence
x the censoring due to end of the
study or loss to follow up
DM
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15. Clustered Multi-State Survival Data F. Rotolo
Simulation of data
A simulation method should be able to generate
the dependence of times of
LR
competing events
the dependence of times of
T1 T4
subsequent events
the dependence between clustered
T3
observations
NED De
the censoring due to competing
events occurrence
the censoring due to end of the
T2 T5
study or loss to follow up
DM the event-specific covariates effect
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16. Simulation Algorithm F. Rotolo
Outline
Clustered Multi-State Survival Data
Simulation Algorithm
Clustering
Choice of Parameters
Example
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17. Simulation Algorithm F. Rotolo
Copula Model
LR
Marginal survival functions freely chosen
T1
S1 (t), S2 (t) and S3 (t)
T3
NED De
T2
DM
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18. Simulation Algorithm F. Rotolo
Copula Model
LR
Marginal survival functions freely chosen
T1
S1 (t), S2 (t) and S3 (t)
Joint survival function by Clayton Copula
3 −θ −1/θ
S123 (t) = i=1 Si (ti ) −2
T3
NED De
T2
DM
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19. Simulation Algorithm F. Rotolo
Copula Model
LR
Marginal survival functions freely chosen
T1 S1 (t), S2 (t) and S3 (t)
Joint survival function by Clayton Copula
3 −θ −1/θ
S123 (t) = i=1 Si (ti ) −2
NED
T3
De
Conditional survivals from the joint
θ −1/θ−1
S1 (t1 )
S2|1 (t2 |t1 ) = 1 + S2 (t2 )
− S1 (t1 )θ
T2
DM
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20. Simulation Algorithm F. Rotolo
Copula Model
LR
Marginal survival functions freely chosen
T1 S1 (t), S2 (t) and S3 (t)
Joint survival function by Clayton Copula
3 −θ −1/θ
S123 (t) = i=1 Si (ti ) −2
NED
T3
De
Conditional survivals from the joint
θ −1/θ−1
S1 (t1 )
S2|1 (t2 |t1 ) = 1 + S2 (t2 )
− S1 (t1 )θ
−1/θ−2
S3 (t3 )−θ −1
S3|12 (t3 |t1 , t2 ) = 1 + S1 (t1 )−θ +S2 (t2 )−θ −1
T2
DM
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21. Simulation Algorithm F. Rotolo
Algorithm
Data from the copula model (Kpanzou, 2007) are simulated as
follows
−1
1 T1 = S1 (U1 )
with U1 , U2 , U3 , UC i.i.d. U(0, 1)
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22. Simulation Algorithm F. Rotolo
Algorithm
Data from the copula model (Kpanzou, 2007) are simulated as
follows
−1
1 T1 = S1 (U1 )
−1
2 T2 |t1 = S2|1 (U2 |t1 ) =
θ −1/θ
−1 − 1+θ
S2 U2 − 1 S1 (t1 )−θ + 1
with U1 , U2 , U3 , UC i.i.d. U(0, 1)
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23. Simulation Algorithm F. Rotolo
Algorithm
Data from the copula model (Kpanzou, 2007) are simulated as
follows
−1
1 T1 = S1 (U1 )
−1
2 T2 |t1 = S2|1 (U2 |t1 ) =
θ −1/θ
−1 − 1+θ
S2 U2 − 1 S1 (t1 )−θ + 1
−1
3 T3 |t1 , t2 = S3|12 (U3 |t1 , t2 ) =
θ −1/θ
−1 − 1+2θ
S3 U3 −1 S1 (t1 )−θ + S2 (t2 )−θ − 1 + 1
with U1 , U2 , U3 , UC i.i.d. U(0, 1)
A copula-based simulation method for clustered multi-state survival data 8/ 22

24. Simulation Algorithm F. Rotolo
Algorithm
Data from the copula model (Kpanzou, 2007) are simulated as
follows
−1
1 T1 = S1 (U1 )
−1
2 T2 |t1 = S2|1 (U2 |t1 ) =
θ −1/θ
−1 − 1+θ
S2 U2 − 1 S1 (t1 )−θ + 1
−1
3 T3 |t1 , t2 = S3|12 (U3 |t1 , t2 ) =
θ −1/θ
−1 − 1+2θ
S3 U3 −1 S1 (t1 )−θ + S2 (t2 )−θ − 1 + 1
−1
C TC = FC (UC )
with U1 , U2 , U3 , UC i.i.d. U(0, 1)
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25. Simulation Algorithm F. Rotolo
Algorithm
Data from the copula model (Kpanzou, 2007) are simulated as
follows
−1
1 T1 = S1 (U1 )
−1
2 T2 |t1 = S2|1 (U2 |t1 ) =
θ −1/θ
−1 − 1+θ
S2 U2 − 1 S1 (t1 )−θ + 1
−1
3 T3 |t1 , t2 = S3|12 (U3 |t1 , t2 ) =
θ −1/θ
−1 − 1+2θ
S3 U3 −1 S1 (t1 )−θ + S2 (t2 )−θ − 1 + 1
−1
C TC = FC (UC )
T min(TC , T1 , T2 , T3 )
with U1 , U2 , U3 , UC i.i.d. U(0, 1)
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26. Simulation Algorithm F. Rotolo
Second transitions
For patients with a transition into state LR or DM, an analogous
copula model is used for second transition to state De
LR
The following conditional survivals can be obtained
T1 T4 −1/θ−1
θ
S1 (t1 )
S4|1 (t4 |t1 ) = 1 + S4 (t4 )
− S1 (t1 )θ
θ −1/θ−1
S2 (t2 )
NED De
S5|2 (t5 |t2 ) = 1 + S5 (t5 )
− S2 (t2 )θ
and the same algorithm is used to simulate second
transition times, conditionally on first transition ones.
DM
A copula-based simulation method for clustered multi-state survival data 9/ 22

27. Simulation Algorithm F. Rotolo
Second transitions
For patients with a transition into state LR or DM, an analogous
copula model is used for second transition to state De
LR
The following conditional survivals can be obtained
θ −1/θ−1
S1 (t1 )
S4|1 (t4 |t1 ) = 1 + S4 (t4 )
− S1 (t1 )θ
θ −1/θ−1
S2 (t2 )
NED De
S5|2 (t5 |t2 ) = 1 + S5 (t5 )
− S2 (t2 )θ
and the same algorithm is used to simulate second
transition times, conditionally on first transition ones.
T2 T5
DM
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28. Simulation Algorithm F. Rotolo
Clustering
The algorithm allows to freely specify the marginal survivals Si (t).
How can we insert clustering?
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29. Simulation Algorithm F. Rotolo
Clustering
The algorithm allows to freely specify the marginal survivals Si (t).
How can we insert clustering?
In a PH way
hi (t|Z ) = Z h0i (t),
with h0i (t) the baseline hazard for transition i.
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30. Simulation Algorithm F. Rotolo
Clustering
The algorithm allows to freely specify the marginal survivals Si (t).
How can we insert clustering?
In a PH way
hi (t|Z ) = Z h0i (t),
with h0i (t) the baseline hazard for transition i.
t
Since S0i (t) = exp{− 0 h0i (u)du}, then
t
Si (t|Z ) = exp −Z h0i (u)du = [S0i (t)]Z
0
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31. Simulation Algorithm F. Rotolo
Clustering
The algorithm allows to freely specify the marginal survivals Si (t).
How can we insert clustering?
In a PH way
hi (t|Z ) = Z h0i (t),
with h0i (t) the baseline hazard for transition i.
t
Since S0i (t) = exp{− 0 h0i (u)du}, then
t
Si (t|Z ) = exp −Z h0i (u)du = [S0i (t)]Z
0
The copula model can be used for conditional survivals
{Si (t|Z )}i∈{1,2,3,4,5} and the same algorithm can be used,
conditionally on Z .
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32. Simulation Algorithm F. Rotolo
Clustering and covariates
The effect of covariates X can be inserted in an analogous way.
The marginals are then
βi X
Si (t|X , Z ) = S0i (t)Ze
and simulation via the copula model is done conditionally on
(X , Z ).
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33. Simulation Algorithm F. Rotolo
The Clayton–Weibull model
Despite the model is quite general, we consider in the following a
particular case:
Ti ∼ Wei(λi , ρi ), i ∈ {1, 2, 3, 4, 5}
TC ∼ Wei(λC , 1) ∼ Exp(λC )
72 months (6 years) of administrative censoring
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34. Simulation Algorithm F. Rotolo
The Clayton–Weibull model
Despite the model is quite general, we consider in the following a
particular case:
Ti ∼ Wei(λi , ρi ), i ∈ {1, 2, 3, 4, 5}
TC ∼ Wei(λC , 1) ∼ Exp(λC )
72 months (6 years) of administrative censoring
This model
1. gives simple forms of conditional distributions
T
2. implies that Si|X ,Z (t|x, z) = exp{−λi ze βi x t ρi },
T
i.e. Ti |X , Z ∼ Wei(λi ze βi x , ρi ) is still a Weibull r.v.
A copula-based simulation method for clustered multi-state survival data 12/ 22

35. Choice of Parameters F. Rotolo
Outline
Clustered Multi-State Survival Data
Simulation Algorithm
Clustering
Choice of Parameters
Example
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36. Choice of Parameters F. Rotolo
Choice of parameters
When simulating a dataset, one should be able to choose parameters in
order to obtain particular target values for
LR pi probabilities of LR, DM, De and
censoring from NED
T1 T4
T3
NED De
T2 T5
DM
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37. Choice of Parameters F. Rotolo
Choice of parameters
When simulating a dataset, one should be able to choose parameters in
order to obtain particular target values for
LR pi probabilities of LR, DM, De and
censoring from NED
T1 T4
mi median of uncensored LR, DM
and De times from NED
T3
NED De
T2 T5
DM
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38. Choice of Parameters F. Rotolo
Choice of parameters
When simulating a dataset, one should be able to choose parameters in
order to obtain particular target values for
LR pi probabilities of LR, DM, De and
censoring from NED
T1 T4
mi median of uncensored LR, DM
and De times from NED
T3
NED De
pi probabilities of De and censoring
from LR and from DM
T2 T5
DM
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39. Choice of Parameters F. Rotolo
Choice of parameters
When simulating a dataset, one should be able to choose parameters in
order to obtain particular target values for
LR pi probabilities of LR, DM, De and
censoring from NED
T1 T4
mi median of uncensored LR, DM
and De times from NED
T3
NED De
pi probabilities of De and censoring
from LR and from DM
T2 T5
mi median of uncensored De times
from LR and from DM
DM
It is not possible to analytically express these quantities as functions of
the parameters.
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40. Choice of Parameters F. Rotolo
Criterion function
In order to find appropriate parameters for given target values
{pi , mi }, we want to minimize the criterion function
2 2
pi mi
Υ(Π) = log + log
pi (Π)
ˆ mi (Π)
ˆ
i∈{1,2,3,4,5}
≥0
with
Π = {λi }i∈{1,2,3,C ,4,C 4,5,C 5} ∪ {ρi }i∈{1,2,3,4,5} ∈ R13
+
and {ˆi (Π), mi (Π)} the observed values in a simulated dataset with
p ˆ
parameters Π
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41. Choice of Parameters F. Rotolo
Criterion function
In order to find appropriate parameters for given target values
{pi , mi }, we want to minimize the criterion function
2 2
pi mi
Υ(Π) = log + log
pi (Π)
ˆ mi (Π)
ˆ
i∈{1,2,3,4,5}
= Υ123 (Π123 ) + Υ4 (Π4 ) + Υ5 (Π5 ) ≥ 0
with
Π = {λi }i∈{1,2,3,C ,4,C 4,5,C 5} ∪ {ρi }i∈{1,2,3,4,5} ∈ R13
+
Π = Π123 ∪ Π4 ∪ Π5 ∈ R7 × R3 × R3
+ + +
and {ˆi (Π), mi (Π)} the observed values in a simulated dataset with
p ˆ
parameters Π
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42. Choice of Parameters F. Rotolo
Criterion function
In order to find appropriate parameters for given target values
{pi , mi }, we want to minimize the criterion function
2 2
pi mi
Υ(Π) = log + log
pi (Π)
ˆ mi (Π)
ˆ
i∈{1,2,3,4,5}
= Υ123 (Π123 ) + Υ4 (Π4 ) + Υ5 (Π5 ) ≥ 0
with
Π = {λi }i∈{1,2,3,C ,4,C 4,5,C 5} ∪ {ρi }i∈{1,2,3,4,5} ∈ R13
+
Π = Π123 ∪ Π4 ∪ Π5 ∈ R7 × R3 × R3
+ + +
Further reduction of problem dimension...
Π = Π123 ∪ Π4 ∪ Π5 ∈ R4+3 × R2+1 × R2+1
+ + +
and {ˆi (Π), mi (Π)} the observed values in a simulated dataset with
p ˆ
parameters Π
A copula-based simulation method for clustered multi-state survival data 15/ 22

43. Choice of Parameters F. Rotolo
Minimization of criterion function
In order to further reduce the dimension of the problem, each of
the parameter sets ΠK , K ∈ {{123}, {4}, {5}} is split into the
scale {λi } and the shape parameters {ρi }. The optimization of the
criterion function ΥK (ΠK ) is iterated on each subset
Example: algorithm for K = {123}
Set J = 1
(0)
λ(0) = {λi }i∈{C ,1,2,3} = {1, 1, 1, 1}
(0)
ρ(0) = {ρi }i∈{1,2,3} = {1, 1, 1}
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44. Choice of Parameters F. Rotolo
Minimization of criterion function
In order to further reduce the dimension of the problem, each of
the parameter sets ΠK , K ∈ {{123}, {4}, {5}} is split into the
scale {λi } and the shape parameters {ρi }. The optimization of the
criterion function ΥK (ΠK ) is iterated on each subset
Example: algorithm for K = {123}
Set J = 1
(0)
λ(0) = {λi }i∈{C ,1,2,3} = {1, 1, 1, 1}
(0)
ρ(0) = {ρi }i∈{1,2,3} = {1, 1, 1}
Repeat until J = maxit or Υ123 (λ(J−1) , ρ(J−1) ) < th
Obtain λ(J) by minimizing Υ123 (λ, ρ(J−1) ) over λ
Obtain ρ(J) by minimizing Υ123 (λ(J) , ρ) over ρ
Set J = J + 1
where maxit and th are arbitrary termination parameters.
A copula-based simulation method for clustered multi-state survival data 16/ 22

45. Example F. Rotolo
An example
A dataset of size 44 is available from a multi-center study on head and neck cancer.
Target values {pi } and {mi }
LR
8
15 7
3
NED De
22 14
4 4
DM
Tot: 44 0
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46. Example F. Rotolo
An example
A dataset of size 44 is available from a multi-center study on head and neck cancer.
Target values {pi } and {mi }
Frailty term
40 Hospitals
LR
8
random sizes
15 7 Z ∼ Gam(1, 0.5)
3
NED De
22 14
4 4
DM
Tot: 44 0
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47. Example F. Rotolo
An example
A dataset of size 44 is available from a multi-center study on head and neck cancer.
Target values {pi } and {mi }
Frailty term
40 Hospitals
LR
8
random sizes
15 7 Z ∼ Gam(1, 0.5)
Covariates
Age ∼ N (60, 7)
3
NED De with 
22 14 log(0.8)/10
 i =1
βi,Age = log(0.9)/10 i =2

log(1.2)/10 i = 3, 4, 5

4 4
Treat ∼ Bin(0.5)
with 
DM log(1/3) i =1
0

Tot: 44
βi,Treat = 0 i =2

log(1.2) i = 3, 4, 5

A copula-based simulation method for clustered multi-state survival data 17/ 22

48. Example F. Rotolo
Results
First transitions. The algorithm is run with datasets of size 104 ,
maxit = 10 and th = 0.1. The time of execution was 11:57’ hours
NED→ {LR,DM,De}
λ1 λ2 λ3 λC ρ1 ρ2 ρ3
0.276 0.019 0.013 0.031 0.851 1.076 0.569
A copula-based simulation method for clustered multi-state survival data 18/ 22

49. Example F. Rotolo
Results
First transitions. The algorithm is run with datasets of size 104 ,
maxit = 10 and th = 0.1. The time of execution was 11:57’ hours
NED→ {LR,DM,De}
λ1 λ2 λ3 λC ρ1 ρ2 ρ3
0.276 0.019 0.013 0.031 0.851 1.076 0.569
NED→ {LR,DM,De}
pi mi
LR DM De C LR DM De
Target 0.34 0.09 0.07 0.50 6.00 10.00 3.00
Simulated 0.33 0.12 0.09 0.46 5.41 9.33 2.29
Υ123 (Π123 ) = 0.24
A copula-based simulation method for clustered multi-state survival data 18/ 22

50. Example F. Rotolo
Results
Second transitions. Conditionally on first transitions data, the
algorithm is run for second transitions from LR and DM with
maxit = 6 and th = 0.05. The times of execution were 4:31’ and
3:57’ hours, respectively.
LR→De DM→De
λ4 λC 4 ρ4 λ5 λC 5 ρ5
0.029 0.099 1.078 0.192 0.039 1.000
A copula-based simulation method for clustered multi-state survival data 19/ 22

51. Example F. Rotolo
Results
Second transitions. Conditionally on first transitions data, the
algorithm is run for second transitions from LR and DM with
maxit = 6 and th = 0.05. The times of execution were 4:31’ and
3:57’ hours, respectively.
LR→De DM→De
λ4 λC 4 ρ4 λ5 λC 5 ρ5
0.029 0.099 1.078 0.192 0.039 1.000
LR→De DM→De
pi mi pi mi
De C De De C De
0.53 0.47 3.25 Target 0.95 0.05 0.50
0.50 0.50 3.32 Simulated 0.97 0.03 0.54
Υ4 (Π4 ) = 0.0043 Υ5 (Π5 ) = 0.0064
A copula-based simulation method for clustered multi-state survival data 19/ 22

52. Conclusion F. Rotolo
Conclusion
The proposed simulation procedure for clustered MS allows to
MSMs generate dependence between times of the same subject
(between both competing and subsequent event times)
FMs generate dependence between times of clustered subjects
(with arbitrary number and size of groups and free frailty distribution)
PH insert covariates via proportional hazards
parMod choose marginal distributions of time variables
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53. Conclusion F. Rotolo
Conclusion
The proposed simulation procedure for clustered MS allows to
MSMs generate dependence between times of the same subject
(between both competing and subsequent event times)
FMs generate dependence between times of clustered subjects
(with arbitrary number and size of groups and free frailty distribution)
PH insert covariates via proportional hazards
parMod choose marginal distributions of time variables
automatically find appropriate parameters, given arbitrary
target values for probabilities of censoring, of competing
events and for medians of uncensored times
generate censoring, both random and administrative
A copula-based simulation method for clustered multi-state survival data 20/ 22

54. References F. Rotolo
References
Cox, D. R. (1972). Regression models and life-tables. Journal of the
Royal Statistical Society. Series B (Methodological) 34, 187–220.
de Wreede, L. C., Fiocco, M. & Putter, H. (2010). The mstate
package for estimation and prediction in non- and semi-parametric
multi-state and competing risks models. Comput Methods Programs
Biomed 99, 261–74.
Duchateau, L. & Janssen, P. (2008). The frailty model. Springer.
Kpanzou, T. A. (2007). Copulas in statistics. African Institute for
Mathematical Sciences (AIMS) .
Putter, H., Fiocco, M. & Geskus, R. B. (2007). Tutorial in
biostatistics: competing risks and multi-state models. Stat Med 26,
2389–430.
Wienke, A. (2010). Frailty Models in Survival Analysis. Chapman &
Hall/CRC biostatistics series. Taylor and Francis.
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55. F. Rotolo [federico.rotolo@stat.unipd.it – federico.rotolo@uclouvain.be]
PhD Student at University of Padova and Visiting PhD Student at UCL
under the supervision of
prof. C. Legrand, UCL
prof. I. Van Keilegom, UCL
prof. M. Chiogna, UniPd