A copula-based Simulation Method for Clustered Multi-State Survival Data
55
A copula-based simulation method for clustered multi-state survival data F. Rotolo •? , C. Legrand ? , I. Van Keilegom ? , M. Chiogna • • Dipartmento di Scienze Statistiche Universit` a degli Studi di Padova ? Institut de Statistique, Biostatistique et Sciences Actuarielles Universit´ e Catholique de Louvain September 23, 2011
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Generating survival data with a clustered and multi-state structure is useful to study Multi-StateModels, Competing Risks Models and Frailty Models. The simulation of such kind of datais not straightforward as one needs to introduce dependence between times of different transitionswhile taking under control the probability of each competing event, the median sojourn time ineach 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 distributionof times of each competing events block. It allows to specify the marginal distributions of timevariables, while their dependence is induced by the copula. Furthermore, even though a dependenceis 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 functionbased on the ratio of target and observed values of median times and of probabilities of competingevents.The proposed method further allows to simulate discrete and continuous covariates and tospecify 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 studyon head and neck cancer. The multi-state structure of these data arises from the interest instudying 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.
Transcript of A copula-based Simulation Method for Clustered Multi-State Survival Data
A copula-based simulation methodfor clustered multi-state survival data
F. Rotolo•?, C. Legrand?, I. Van Keilegom?, M. Chiogna•
•Dipartmento di Scienze Statistiche
Universita degli Studi di Padova
?Institut de Statistique, Biostatistiqueet Sciences Actuarielles
Universite Catholique de Louvain
September 23, 2011
Clustered Multi-State Survival Data F. Rotolo
Survival DataTime since an origin event until an event of interest.Example: from birth to death, since beginning of therapy until remission, etc.
●
0 1 2 3 4 5
●
Time
T=5
Censoring: some observations cannot be observed, the onlyavailable information being a lower bound.Example: migration, change of therapy, loss to follow-up, etc.
●
0 1 2 3 4 5
●xTime
T>3.25
A copula-based simulation method for clustered multi-state survival data 2/ 22
Clustered Multi-State Survival Data F. Rotolo
Survival DataTime since an origin event until an event of interest.Example: from birth to death, since beginning of therapy until remission, etc.
●
0 1 2 3 4 5
●
Time
T=5
Censoring: some observations cannot be observed, the onlyavailable information being a lower bound.Example: migration, change of therapy, loss to follow-up, etc.
●
0 1 2 3 4 5
●xTime
T>3.25
A copula-based simulation method for clustered multi-state survival data 2/ 22
Clustered Multi-State Survival Data F. Rotolo
Modeling Survival Data
Because of this peculiarity, instead of modeling the density f (t) ofT , the hazard is considered
h(t) = lim∆t↘0
P[t ≤ T < t + ∆t|T ≥ t]
∆t=
f (t)
S(t)= − d
dtlog S(t),
with S(t) =∫∞t f (u)du = P[T > t].
Note: S(t) = exp{−∫ t
0 h(u)du}.
The basic regression model for the hazard is the ProportionalHazards (PH) Model (Cox, 1972)
h(t|X ) = h0(t) exp{β′X}.
A copula-based simulation method for clustered multi-state survival data 3/ 22
Clustered Multi-State Survival Data F. Rotolo
Modeling Survival Data
Because of this peculiarity, instead of modeling the density f (t) ofT , the hazard is considered
h(t) = lim∆t↘0
P[t ≤ T < t + ∆t|T ≥ t]
∆t=
f (t)
S(t)= − d
dtlog S(t),
with S(t) =∫∞t f (u)du = P[T > t].
Note: S(t) = exp{−∫ t
0 h(u)du}.
The basic regression model for the hazard is the ProportionalHazards (PH) Model (Cox, 1972)
h(t|X ) = h0(t) exp{β′X}.
A copula-based simulation method for clustered multi-state survival data 3/ 22
Clustered Multi-State Survival Data F. Rotolo
Survival ModelsComplications of Cox models have been developed
Frailty Models (FMs)
account for overdispersionor clustering by means
of random effects
h(t|Xij) = h0(t)Zieβ′Xij ,
similar to GLMMlog[h(t|Xij)] = log[h0(t)]+Wi+β
′Xij ,
with Zi = eWi
(Duchateau & Janssen, 2008; Wienke, 2010)
Multi-State Models (MSMs)
consider several eventsand their interactions
NED
LR
DM
De
T1
T3
T2
T4
T5
(Putter et al., 2007; de Wreede et al., 2010)
Possible integration? I Simulation studies
A copula-based simulation method for clustered multi-state survival data 4/ 22
Clustered Multi-State Survival Data F. Rotolo
Survival ModelsComplications of Cox models have been developed
Frailty Models (FMs)
account for overdispersionor clustering by means
of random effects
h(t|Xij) = h0(t)Zieβ′Xij ,
similar to GLMMlog[h(t|Xij)] = log[h0(t)]+Wi+β
′Xij ,
with Zi = eWi
(Duchateau & Janssen, 2008; Wienke, 2010)
Multi-State Models (MSMs)
consider several eventsand their interactions
NED
LR
DM
De
T1
T3
T2
T4
T5
(Putter et al., 2007; de Wreede et al., 2010)
Possible integration? I Simulation studies
A copula-based simulation method for clustered multi-state survival data 4/ 22
Clustered Multi-State Survival Data F. Rotolo
Survival ModelsComplications of Cox models have been developed
Frailty Models (FMs)
account for overdispersionor clustering by means
of random effects
h(t|Xij) = h0(t)Zieβ′Xij ,
similar to GLMMlog[h(t|Xij)] = log[h0(t)]+Wi+β
′Xij ,
with Zi = eWi
(Duchateau & Janssen, 2008; Wienke, 2010)
Multi-State Models (MSMs)
consider several eventsand their interactions
NED
LR
DM
De
T1
T3
T2
T4
T5
(Putter et al., 2007; de Wreede et al., 2010)
Possible integration?
I Simulation studies
A copula-based simulation method for clustered multi-state survival data 4/ 22
Clustered Multi-State Survival Data F. Rotolo
Survival ModelsComplications of Cox models have been developed
Frailty Models (FMs)
account for overdispersionor clustering by means
of random effects
h(t|Xij) = h0(t)Zieβ′Xij ,
similar to GLMMlog[h(t|Xij)] = log[h0(t)]+Wi+β
′Xij ,
with Zi = eWi
(Duchateau & Janssen, 2008; Wienke, 2010)
Multi-State Models (MSMs)
consider several eventsand their interactions
NED
LR
DM
De
T1
T3
T2
T4
T5
(Putter et al., 2007; de Wreede et al., 2010)
Possible integration? I Simulation studiesA copula-based simulation method for clustered multi-state survival data 4/ 22
Clustered Multi-State Survival Data F. Rotolo
Simulation of data
A simulation method should be able to generate
NED
LR
DM
De
I the dependence of times ofcompeting events
I the dependence of times ofsubsequent events
I the dependence between clusteredobservations
I the censoring due to competingevents occurrence
I the censoring due to end of thestudy or loss to follow up
I the event-specific covariates effect
A copula-based simulation method for clustered multi-state survival data 5/ 22
Clustered Multi-State Survival Data F. Rotolo
Simulation of data
A simulation method should be able to generate
NED
LR
DM
De
I the dependence of times ofcompeting events
I the dependence of times ofsubsequent events
I the dependence between clusteredobservations
I the censoring due to competingevents occurrence
I the censoring due to end of thestudy or loss to follow up
I the event-specific covariates effect
A copula-based simulation method for clustered multi-state survival data 5/ 22
Clustered Multi-State Survival Data F. Rotolo
Simulation of data
A simulation method should be able to generate
NED
LR
DM
De NED
LR
DM
De
NED
LR
DM
De
NED
LR
DM
De NED
LR
DM
De
NED
LR
DM
De NED
LR
DM
De
NED
LR
DM
De
NED
LR
DM
De NED
LR
DM
De
NED
LR
DM
De NED
LR
DM
De
NED
LR
DM
De
NED
LR
DM
De NED
LR
DM
De
NED
LR
DM
De NED
LR
DM
De
NED
LR
DM
De
NED
LR
DM
De NED
LR
DM
De
I the dependence of times ofcompeting events
I the dependence of times ofsubsequent events
I the dependence between clusteredobservations
I the censoring due to competingevents occurrence
I the censoring due to end of thestudy or loss to follow up
I the event-specific covariates effect
A copula-based simulation method for clustered multi-state survival data 5/ 22
Clustered Multi-State Survival Data F. Rotolo
Simulation of data
A simulation method should be able to generate
NED
LR
DM
De
x
x
I the dependence of times ofcompeting events
I the dependence of times ofsubsequent events
I the dependence between clusteredobservations
I the censoring due to competingevents occurrence
I the censoring due to end of thestudy or loss to follow up
I the event-specific covariates effect
A copula-based simulation method for clustered multi-state survival data 5/ 22
Clustered Multi-State Survival Data F. Rotolo
Simulation of data
A simulation method should be able to generate
NED
LR
DM
De
x
x
x
I the dependence of times ofcompeting events
I the dependence of times ofsubsequent events
I the dependence between clusteredobservations
I the censoring due to competingevents occurrence
I the censoring due to end of thestudy or loss to follow up
I the event-specific covariates effect
A copula-based simulation method for clustered multi-state survival data 5/ 22
Clustered Multi-State Survival Data F. Rotolo
Simulation of data
A simulation method should be able to generate
NED
LR
DM
De
T1
T3
T2
T4
T5
I the dependence of times ofcompeting events
I the dependence of times ofsubsequent events
I the dependence between clusteredobservations
I the censoring due to competingevents occurrence
I the censoring due to end of thestudy or loss to follow up
I the event-specific covariates effect
A copula-based simulation method for clustered multi-state survival data 5/ 22
Simulation Algorithm F. Rotolo
Outline
Clustered Multi-State Survival Data
Simulation Algorithm
Clustering
Choice of Parameters
Example
A copula-based simulation method for clustered multi-state survival data 6/ 22
Simulation Algorithm F. Rotolo
Copula Model
NED
LR
DM
De
T1
T3
T2
I Marginal survival functions freely chosenS1(t), S2(t) and S3(t)
I Joint survival function by Clayton Copula
S123(t) =(∑3
i=1 Si (ti )−θ − 2
)−1/θ
I Conditional survivals from the joint
S2|1(t2|t1) =
[1 +
(S1(t1)S2(t2)
)θ− S1(t1)θ
]−1/θ−1
S3|12(t3|t1, t2) =(
1 + S3(t3)−θ−1
S1(t1)−θ+S2(t2)−θ−1
)−1/θ−2
A copula-based simulation method for clustered multi-state survival data 7/ 22
Simulation Algorithm F. Rotolo
Copula Model
NED
LR
DM
De
T1
T3
T2
I Marginal survival functions freely chosenS1(t), S2(t) and S3(t)
I Joint survival function by Clayton Copula
S123(t) =(∑3
i=1 Si (ti )−θ − 2
)−1/θ
I Conditional survivals from the joint
S2|1(t2|t1) =
[1 +
(S1(t1)S2(t2)
)θ− S1(t1)θ
]−1/θ−1
S3|12(t3|t1, t2) =(
1 + S3(t3)−θ−1
S1(t1)−θ+S2(t2)−θ−1
)−1/θ−2
A copula-based simulation method for clustered multi-state survival data 7/ 22
Simulation Algorithm F. Rotolo
Copula Model
NED
LR
DM
De
T1
T3
T2
I Marginal survival functions freely chosenS1(t), S2(t) and S3(t)
I Joint survival function by Clayton Copula
S123(t) =(∑3
i=1 Si (ti )−θ − 2
)−1/θ
I Conditional survivals from the joint
S2|1(t2|t1) =
[1 +
(S1(t1)S2(t2)
)θ− S1(t1)θ
]−1/θ−1
S3|12(t3|t1, t2) =(
1 + S3(t3)−θ−1
S1(t1)−θ+S2(t2)−θ−1
)−1/θ−2
A copula-based simulation method for clustered multi-state survival data 7/ 22
Simulation Algorithm F. Rotolo
Copula Model
NED
LR
DM
De
T1
T3
T2
I Marginal survival functions freely chosenS1(t), S2(t) and S3(t)
I Joint survival function by Clayton Copula
S123(t) =(∑3
i=1 Si (ti )−θ − 2
)−1/θ
I Conditional survivals from the joint
S2|1(t2|t1) =
[1 +
(S1(t1)S2(t2)
)θ− S1(t1)θ
]−1/θ−1
S3|12(t3|t1, t2) =(
1 + S3(t3)−θ−1
S1(t1)−θ+S2(t2)−θ−1
)−1/θ−2
A copula-based simulation method for clustered multi-state survival data 7/ 22
Simulation Algorithm F. Rotolo
Algorithm
Data from the copula model (Kpanzou, 2007) are simulated asfollows
1 I T1 = S−11 (U1)
2 I T2|t1 = S−12|1 (U2|t1) =
S−12
({[U− θ
1+θ
2 − 1
]S1(t1)−θ + 1
}−1/θ)
3 I T3|t1, t2 = S−13|12(U3|t1, t2) =
S−13
({[U− θ
1+2θ
3 − 1
] [S1(t1)−θ + S2(t2)−θ − 1
]+ 1
}−1/θ)
C I TC = F−1C (UC )
T I min(TC ,T1,T2,T3)
with U1,U2,U3,UC i.i.d. U(0, 1)A copula-based simulation method for clustered multi-state survival data 8/ 22
Simulation Algorithm F. Rotolo
Algorithm
Data from the copula model (Kpanzou, 2007) are simulated asfollows
1 I T1 = S−11 (U1)
2 I T2|t1 = S−12|1 (U2|t1) =
S−12
({[U− θ
1+θ
2 − 1
]S1(t1)−θ + 1
}−1/θ)
3 I T3|t1, t2 = S−13|12(U3|t1, t2) =
S−13
({[U− θ
1+2θ
3 − 1
] [S1(t1)−θ + S2(t2)−θ − 1
]+ 1
}−1/θ)
C I TC = F−1C (UC )
T I min(TC ,T1,T2,T3)
with U1,U2,U3,UC i.i.d. U(0, 1)A copula-based simulation method for clustered multi-state survival data 8/ 22
Simulation Algorithm F. Rotolo
Algorithm
Data from the copula model (Kpanzou, 2007) are simulated asfollows
1 I T1 = S−11 (U1)
2 I T2|t1 = S−12|1 (U2|t1) =
S−12
({[U− θ
1+θ
2 − 1
]S1(t1)−θ + 1
}−1/θ)
3 I T3|t1, t2 = S−13|12(U3|t1, t2) =
S−13
({[U− θ
1+2θ
3 − 1
] [S1(t1)−θ + S2(t2)−θ − 1
]+ 1
}−1/θ)
C I TC = F−1C (UC )
T I min(TC ,T1,T2,T3)
with U1,U2,U3,UC i.i.d. U(0, 1)A copula-based simulation method for clustered multi-state survival data 8/ 22
Simulation Algorithm F. Rotolo
Algorithm
Data from the copula model (Kpanzou, 2007) are simulated asfollows
1 I T1 = S−11 (U1)
2 I T2|t1 = S−12|1 (U2|t1) =
S−12
({[U− θ
1+θ
2 − 1
]S1(t1)−θ + 1
}−1/θ)
3 I T3|t1, t2 = S−13|12(U3|t1, t2) =
S−13
({[U− θ
1+2θ
3 − 1
] [S1(t1)−θ + S2(t2)−θ − 1
]+ 1
}−1/θ)
C I TC = F−1C (UC )
T I min(TC ,T1,T2,T3)
with U1,U2,U3,UC i.i.d. U(0, 1)A copula-based simulation method for clustered multi-state survival data 8/ 22
Simulation Algorithm F. Rotolo
Algorithm
Data from the copula model (Kpanzou, 2007) are simulated asfollows
1 I T1 = S−11 (U1)
2 I T2|t1 = S−12|1 (U2|t1) =
S−12
({[U− θ
1+θ
2 − 1
]S1(t1)−θ + 1
}−1/θ)
3 I T3|t1, t2 = S−13|12(U3|t1, t2) =
S−13
({[U− θ
1+2θ
3 − 1
] [S1(t1)−θ + S2(t2)−θ − 1
]+ 1
}−1/θ)
C I TC = F−1C (UC )
T I min(TC ,T1,T2,T3)
with U1,U2,U3,UC i.i.d. U(0, 1)A copula-based simulation method for clustered multi-state survival data 8/ 22
Simulation Algorithm F. Rotolo
Second transitionsFor patients with a transition into state LR or DM, an analogouscopula model is used for second transition to state De
NED
LR
DM
De
T1 T4
The following conditional survivals can be obtained
I S4|1(t4|t1) =
[1 +
(S1(t1)S4(t4)
)θ− S1(t1)θ
]−1/θ−1
I S5|2(t5|t2) =
[1 +
(S2(t2)S5(t5)
)θ− S2(t2)θ
]−1/θ−1
and the same algorithm is used to simulate secondtransition times, conditionally on first transition ones.
A copula-based simulation method for clustered multi-state survival data 9/ 22
Simulation Algorithm F. Rotolo
Second transitionsFor patients with a transition into state LR or DM, an analogouscopula model is used for second transition to state De
NED
LR
DM
De
T2 T5
The following conditional survivals can be obtained
I S4|1(t4|t1) =
[1 +
(S1(t1)S4(t4)
)θ− S1(t1)θ
]−1/θ−1
I S5|2(t5|t2) =
[1 +
(S2(t2)S5(t5)
)θ− S2(t2)θ
]−1/θ−1
and the same algorithm is used to simulate secondtransition times, conditionally on first transition ones.
A copula-based simulation method for clustered multi-state survival data 9/ 22
Simulation Algorithm F. Rotolo
Clustering
The algorithm allows to freely specify the marginal survivals Si (t).How can we insert clustering?
In a PH wayhi (t|Z ) = Zh0i (t),
with h0i (t) the baseline hazard for transition i .
Since S0i (t) = exp{−∫ t
0 h0i (u)du}, then
Si (t|Z ) = exp
{−Z
∫ t
0h0i (u)du
}= [S0i (t)]Z
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 .
A copula-based simulation method for clustered multi-state survival data 10/ 22
Simulation Algorithm F. Rotolo
Clustering
The algorithm allows to freely specify the marginal survivals Si (t).How can we insert clustering?
In a PH wayhi (t|Z ) = Zh0i (t),
with h0i (t) the baseline hazard for transition i .
Since S0i (t) = exp{−∫ t
0 h0i (u)du}, then
Si (t|Z ) = exp
{−Z
∫ t
0h0i (u)du
}= [S0i (t)]Z
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 .
A copula-based simulation method for clustered multi-state survival data 10/ 22
Simulation Algorithm F. Rotolo
Clustering
The algorithm allows to freely specify the marginal survivals Si (t).How can we insert clustering?
In a PH wayhi (t|Z ) = Zh0i (t),
with h0i (t) the baseline hazard for transition i .
Since S0i (t) = exp{−∫ t
0 h0i (u)du}, then
Si (t|Z ) = exp
{−Z
∫ t
0h0i (u)du
}= [S0i (t)]Z
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 .
A copula-based simulation method for clustered multi-state survival data 10/ 22
Simulation Algorithm F. Rotolo
Clustering
The algorithm allows to freely specify the marginal survivals Si (t).How can we insert clustering?
In a PH wayhi (t|Z ) = Zh0i (t),
with h0i (t) the baseline hazard for transition i .
Since S0i (t) = exp{−∫ t
0 h0i (u)du}, then
Si (t|Z ) = exp
{−Z
∫ t
0h0i (u)du
}= [S0i (t)]Z
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 .
A copula-based simulation method for clustered multi-state survival data 10/ 22
Simulation Algorithm F. Rotolo
Clustering and covariates
The effect of covariates X can be inserted in an analogous way.The marginals are then
Si (t|X ,Z ) = S0i (t)Zeβ′i X
and simulation via the copula model is done conditionally on(X ,Z ).
A copula-based simulation method for clustered multi-state survival data 11/ 22
Simulation Algorithm F. Rotolo
The Clayton–Weibull model
Despite the model is quite general, we consider in the following aparticular case:
I Ti ∼Wei(λi , ρi ), i ∈ {1, 2, 3, 4, 5}I TC ∼Wei(λC , 1) ∼ Exp(λC )
I 72 months (6 years) of administrative censoring
This model
1. gives simple forms of conditional distributions
2. implies that Si |X ,Z (t|x , z) = exp{−λizeβTi x tρi},
i.e. Ti |X ,Z ∼Wei(λizeβTi x , ρi ) is still a Weibull r.v.
A copula-based simulation method for clustered multi-state survival data 12/ 22
Simulation Algorithm F. Rotolo
The Clayton–Weibull model
Despite the model is quite general, we consider in the following aparticular case:
I Ti ∼Wei(λi , ρi ), i ∈ {1, 2, 3, 4, 5}I TC ∼Wei(λC , 1) ∼ Exp(λC )
I 72 months (6 years) of administrative censoring
This model
1. gives simple forms of conditional distributions
2. implies that Si |X ,Z (t|x , z) = exp{−λizeβTi x tρi},
i.e. Ti |X ,Z ∼Wei(λizeβTi x , ρi ) is still a Weibull r.v.
A copula-based simulation method for clustered multi-state survival data 12/ 22
Choice of Parameters F. Rotolo
Outline
Clustered Multi-State Survival Data
Simulation Algorithm
Clustering
Choice of Parameters
Example
A copula-based simulation method for clustered multi-state survival data 13/ 22
Choice of Parameters F. Rotolo
Choice of parameters
When simulating a dataset, one should be able to choose parameters inorder to obtain particular target values for
NED
LR
DM
De
T1
T3
T2
T4
T5
pi I probabilities of LR, DM, De andcensoring from NED
mi I median of uncensored LR, DMand De times from NED
pi I probabilities of De and censoringfrom LR and from DM
mi I median of uncensored De timesfrom LR and from DM
It is not possible to analytically express these quantities as functions of
the parameters.
A copula-based simulation method for clustered multi-state survival data 14/ 22
Choice of Parameters F. Rotolo
Choice of parameters
When simulating a dataset, one should be able to choose parameters inorder to obtain particular target values for
NED
LR
DM
De
T1
T3
T2
T4
T5
pi I probabilities of LR, DM, De andcensoring from NED
mi I median of uncensored LR, DMand De times from NED
pi I probabilities of De and censoringfrom LR and from DM
mi I median of uncensored De timesfrom LR and from DM
It is not possible to analytically express these quantities as functions of
the parameters.
A copula-based simulation method for clustered multi-state survival data 14/ 22
Choice of Parameters F. Rotolo
Choice of parameters
When simulating a dataset, one should be able to choose parameters inorder to obtain particular target values for
NED
LR
DM
De
T1
T3
T2
T4
T5
pi I probabilities of LR, DM, De andcensoring from NED
mi I median of uncensored LR, DMand De times from NED
pi I probabilities of De and censoringfrom LR and from DM
mi I median of uncensored De timesfrom LR and from DM
It is not possible to analytically express these quantities as functions of
the parameters.
A copula-based simulation method for clustered multi-state survival data 14/ 22
Choice of Parameters F. Rotolo
Choice of parameters
When simulating a dataset, one should be able to choose parameters inorder to obtain particular target values for
NED
LR
DM
De
T1
T3
T2
T4
T5
pi I probabilities of LR, DM, De andcensoring from NED
mi I median of uncensored LR, DMand De times from NED
pi I probabilities of De and censoringfrom LR and from DM
mi I median of uncensored De timesfrom LR and from DM
It is not possible to analytically express these quantities as functions of
the parameters.
A copula-based simulation method for clustered multi-state survival data 14/ 22
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
Υ(Π) =∑
i∈{1,2,3,4,5}
{[log
pipi (Π)
]2
+
[log
mi
mi (Π)
]2}
≥ 0
with
Π = {λi}i∈{1,2,3,C ,4,C4,5,C5} ∪ {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 {pi (Π), mi (Π)} the observed values in a simulated dataset with
parameters Π
A copula-based simulation method for clustered multi-state survival data 15/ 22
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
Υ(Π) =∑
i∈{1,2,3,4,5}
{[log
pipi (Π)
]2
+
[log
mi
mi (Π)
]2}
= Υ123(Π123) + Υ4(Π4) + Υ5(Π5) ≥ 0
with
Π = {λi}i∈{1,2,3,C ,4,C4,5,C5} ∪ {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 {pi (Π), mi (Π)} the observed values in a simulated dataset with
parameters Π
A copula-based simulation method for clustered multi-state survival data 15/ 22
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
Υ(Π) =∑
i∈{1,2,3,4,5}
{[log
pipi (Π)
]2
+
[log
mi
mi (Π)
]2}
= Υ123(Π123) + Υ4(Π4) + Υ5(Π5) ≥ 0
with
Π = {λi}i∈{1,2,3,C ,4,C4,5,C5} ∪ {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 {pi (Π), mi (Π)} the observed values in a simulated dataset with
parameters Π
A copula-based simulation method for clustered multi-state survival data 15/ 22
Choice of Parameters F. Rotolo
Minimization of criterion function
In order to further reduce the dimension of the problem, each ofthe parameter sets ΠK ,K ∈ {{123}, {4}, {5}} is split into thescale {λi} and the shape parameters {ρi}. The optimization of thecriterion function ΥK (ΠK ) is iterated on each subset
Example: algorithm for K = {123}I Set J = 1λ(0) = {λ(0)
i }i∈{C ,1,2,3} = {1, 1, 1, 1}ρ(0) = {ρ(0)
i }i∈{1,2,3} = {1, 1, 1}
I Repeat until J = maxit or Υ123(λ(J−1), ρ(J−1)) < thI Obtain λ(J) by minimizing Υ123(λ, ρ(J−1)) over λI Obtain ρ(J) by minimizing Υ123(λ(J), ρ) over ρI 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
Choice of Parameters F. Rotolo
Minimization of criterion function
In order to further reduce the dimension of the problem, each ofthe parameter sets ΠK ,K ∈ {{123}, {4}, {5}} is split into thescale {λi} and the shape parameters {ρi}. The optimization of thecriterion function ΥK (ΠK ) is iterated on each subset
Example: algorithm for K = {123}I Set J = 1λ(0) = {λ(0)
i }i∈{C ,1,2,3} = {1, 1, 1, 1}ρ(0) = {ρ(0)
i }i∈{1,2,3} = {1, 1, 1}I Repeat until J = maxit or Υ123(λ(J−1), ρ(J−1)) < th
I Obtain λ(J) by minimizing Υ123(λ, ρ(J−1)) over λI Obtain ρ(J) by minimizing Υ123(λ(J), ρ) over ρI 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
Example F. Rotolo
An example
A dataset of size 44 is available from a multi-center study on head and neck cancer.
NED
LR
DM
De22
0
8
14
4
15
3
4
7
Tot: 44
Target values {pi} and {mi}
Frailty term
I 40 Hospitals
I random sizes
I Z ∼ Gam(1, 0.5)
Covariates
I Age ∼ N (60, 7)with
βi,Age =
log(0.8)/10 i = 1
log(0.9)/10 i = 2
log(1.2)/10 i = 3, 4, 5
I Treat ∼ Bin(0.5)with
βi,Treat =
log(1/3) i = 1
0 i = 2
log(1.2) i = 3, 4, 5
A copula-based simulation method for clustered multi-state survival data 17/ 22
Example F. Rotolo
An example
A dataset of size 44 is available from a multi-center study on head and neck cancer.
NED
LR
DM
De22
0
8
14
4
15
3
4
7
Tot: 44
Target values {pi} and {mi}
Frailty term
I 40 Hospitals
I random sizes
I Z ∼ Gam(1, 0.5)
Covariates
I Age ∼ N (60, 7)with
βi,Age =
log(0.8)/10 i = 1
log(0.9)/10 i = 2
log(1.2)/10 i = 3, 4, 5
I Treat ∼ Bin(0.5)with
βi,Treat =
log(1/3) i = 1
0 i = 2
log(1.2) i = 3, 4, 5
A copula-based simulation method for clustered multi-state survival data 17/ 22
Example F. Rotolo
An example
A dataset of size 44 is available from a multi-center study on head and neck cancer.
NED
LR
DM
De22
0
8
14
4
15
3
4
7
Tot: 44
Target values {pi} and {mi}
Frailty term
I 40 Hospitals
I random sizes
I Z ∼ Gam(1, 0.5)
Covariates
I Age ∼ N (60, 7)with
βi,Age =
log(0.8)/10 i = 1
log(0.9)/10 i = 2
log(1.2)/10 i = 3, 4, 5
I Treat ∼ Bin(0.5)with
βi,Treat =
log(1/3) i = 1
0 i = 2
log(1.2) i = 3, 4, 5
A copula-based simulation method for clustered multi-state survival data 17/ 22
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 3.00Simulated 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
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 3.00Simulated 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
Example F. Rotolo
ResultsSecond transitions. Conditionally on first transitions data, thealgorithm is run for second transitions from LR and DM withmaxit = 6 and th = 0.05. The times of execution were 4:31’ and3:57’ hours, respectively.
LR→De
λ4 λC4 ρ4
0.029 0.099 1.078
DM→De
λ5 λC5 ρ5
0.192 0.039 1.000
LR→De DM→Depi mi pi mi
De C De De C De
0.53 0.47 3.25 Target 0.95 0.05 0.500.50 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
Example F. Rotolo
ResultsSecond transitions. Conditionally on first transitions data, thealgorithm is run for second transitions from LR and DM withmaxit = 6 and th = 0.05. The times of execution were 4:31’ and3:57’ hours, respectively.
LR→De
λ4 λC4 ρ4
0.029 0.099 1.078
DM→De
λ5 λC5 ρ5
0.192 0.039 1.000
LR→De DM→Depi mi pi mi
De C De De C De
0.53 0.47 3.25 Target 0.95 0.05 0.500.50 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
Conclusion F. Rotolo
Conclusion
The proposed simulation procedure for clustered MS allows to
MSMsJ generate dependence between times of the same subject(between both competing and subsequent event times)
FMsJ generate dependence between times of clustered subjects(with arbitrary number and size of groups and free frailty distribution)
PHJ insert covariates via proportional hazards
parModJ choose marginal distributions of time variables
I automatically find appropriate parameters, given arbitrarytarget values for probabilities of censoring, of competingevents and for medians of uncensored times
I generate censoring, both random and administrative
A copula-based simulation method for clustered multi-state survival data 20/ 22
Conclusion F. Rotolo
Conclusion
The proposed simulation procedure for clustered MS allows to
MSMsJ generate dependence between times of the same subject(between both competing and subsequent event times)
FMsJ generate dependence between times of clustered subjects(with arbitrary number and size of groups and free frailty distribution)
PHJ insert covariates via proportional hazards
parModJ choose marginal distributions of time variables
I automatically find appropriate parameters, given arbitrarytarget values for probabilities of censoring, of competingevents and for medians of uncensored times
I generate censoring, both random and administrative
A copula-based simulation method for clustered multi-state survival data 20/ 22
References F. Rotolo
References
Cox, D. R. (1972). Regression models and life-tables. Journal of theRoyal Statistical Society. Series B (Methodological) 34, 187–220.
de Wreede, L. C., Fiocco, M. & Putter, H. (2010). The mstatepackage for estimation and prediction in non- and semi-parametricmulti-state and competing risks models. Comput Methods ProgramsBiomed 99, 261–74.
Duchateau, L. & Janssen, P. (2008). The frailty model. Springer.
Kpanzou, T. A. (2007). Copulas in statistics. African Institute forMathematical Sciences (AIMS) .
Putter, H., Fiocco, M. & Geskus, R. B. (2007). Tutorial inbiostatistics: 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.
A copula-based simulation method for clustered multi-state survival data 21/ 22
F. Rotolo [[email protected] – [email protected]]
PhD Student at University of Padova and Visiting PhD Student at UCL
under the supervision of
prof. C. Legrand, UCLprof. I. Van Keilegom, UCLprof. M. Chiogna, UniPd
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