A copula-based simulation method      for clustered multi-state survival data    F. Rotolo• , C. Legrand , I. Van Keilegom...
Clustered Multi-State Survival Data                                                    F. Rotolo                          ...
Clustered Multi-State Survival Data                                                    F. Rotolo                          ...
Clustered Multi-State Survival Data                                              F. Rotolo                                ...
Clustered Multi-State Survival Data                                              F. Rotolo                                ...
Clustered Multi-State Survival Data                                        F. Rotolo                                      ...
Clustered Multi-State Survival Data                                                                              F. Rotolo...
Clustered Multi-State Survival Data                                                                              F. Rotolo...
Clustered Multi-State Survival Data                                                                              F. Rotolo...
Clustered Multi-State Survival Data                                                                     F. Rotolo         ...
Clustered Multi-State Survival Data                                                                     F. Rotolo         ...
Clustered Multi-State Survival Data                                                                               F. Rotol...
Clustered Multi-State Survival Data                                                                     F. Rotolo         ...
Clustered Multi-State Survival Data                                                                      F. Rotolo        ...
Clustered Multi-State Survival Data                                                                       F. Rotolo       ...
Simulation Algorithm                                                       F. Rotolo                                      ...
Simulation Algorithm                                                                           F. Rotolo                  ...
Simulation Algorithm                                                                           F. Rotolo                  ...
Simulation Algorithm                                                                                             F. Rotolo...
Simulation Algorithm                                                                                             F. Rotolo...
Simulation Algorithm                                                       F. Rotolo                                      ...
Simulation Algorithm                                                              F. Rotolo                               ...
Simulation Algorithm                                                                          F. Rotolo                   ...
Simulation Algorithm                                                                          F. Rotolo                   ...
Simulation Algorithm                                                                          F. Rotolo                   ...
Simulation Algorithm                                                                                             F. Rotolo...
Simulation Algorithm                                                                                             F. Rotolo...
Simulation Algorithm                                                          F. Rotolo                                   ...
Simulation Algorithm                                                          F. Rotolo                                   ...
Simulation Algorithm                                                                          F. Rotolo                   ...
Simulation Algorithm                                                                          F. Rotolo                   ...
Simulation Algorithm                                                              F. Rotolo                               ...
Simulation Algorithm                                                       F. Rotolo                                The Cl...
Simulation Algorithm                                                           F. Rotolo                                Th...
Choice of Parameters                                                       F. Rotolo                                      ...
Choice of Parameters                                                                             F. Rotolo                ...
Choice of Parameters                                                                             F. Rotolo                ...
Choice of Parameters                                                                               F. Rotolo              ...
Choice of Parameters                                                                               F. Rotolo              ...
Choice of Parameters                                                                                F. Rotolo             ...
Choice of Parameters                                                                                F. Rotolo             ...
Choice of Parameters                                                                                F. Rotolo             ...
Choice of Parameters                                                       F. Rotolo                          Minimization...
Choice of Parameters                                                        F. Rotolo                          Minimizatio...
Example                                                                                          F. Rotolo                ...
Example                                                                                          F. Rotolo                ...
Example                                                                                                          F. Rotolo...
Example                                                                                    F. Rotolo                      ...
Example                                                                                          F. Rotolo                ...
Example                                                                                   F. Rotolo                       ...
Example                                                                                         F. Rotolo                 ...
Conclusion                                                                               F. Rotolo                        ...
Conclusion                                                                               F. Rotolo                        ...
References                                                                    F. Rotolo                                   ...
F. Rotolo [federico.rotolo@stat.unipd.it – federico.rotolo@uclouvain.be]PhD Student at University of Padova and Visiting P...
Upcoming SlideShare
Loading in …5
×

A copula-based Simulation Method for Clustered Multi-State Survival Data

1,173 views

Published on

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.

0 Comments
0 Likes
Statistics
Notes
  • Be the first to comment

  • Be the first to like this

No Downloads
Views
Total views
1,173
On SlideShare
0
From Embeds
0
Number of Embeds
76
Actions
Shares
0
Downloads
7
Comments
0
Likes
0
Embeds 0
No embeds

No notes for slide

A copula-based Simulation Method for Clustered Multi-State Survival Data

  1. 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. 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 5A copula-based simulation method for clustered multi-state survival data 2/ 22
  3. 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 5A copula-based simulation method for clustered multi-state survival data 2/ 22
  4. 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}.A copula-based simulation method for clustered multi-state survival data 3/ 22
  5. 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 }.A copula-based simulation method for clustered multi-state survival data 3/ 22
  6. 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. 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)A copula-based simulation method for clustered multi-state survival data 4/ 22
  8. 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. 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 studiesA copula-based simulation method for clustered multi-state survival data 4/ 22
  10. 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 DMA copula-based simulation method for clustered multi-state survival data 5/ 22
  11. 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 DMA copula-based simulation method for clustered multi-state survival data 5/ 22
  12. 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 DMA copula-based simulation method for clustered multi-state survival data 5/ 22
  13. 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 DMA copula-based simulation method for clustered multi-state survival data 5/ 22
  14. 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 DMA copula-based simulation method for clustered multi-state survival data 5/ 22
  15. 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 effectA copula-based simulation method for clustered multi-state survival data 5/ 22
  16. 16. Simulation Algorithm F. Rotolo Outline Clustered Multi-State Survival Data Simulation Algorithm Clustering Choice of Parameters ExampleA copula-based simulation method for clustered multi-state survival data 6/ 22
  17. 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 DMA copula-based simulation method for clustered multi-state survival data 7/ 22
  18. 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 DMA copula-based simulation method for clustered multi-state survival data 7/ 22
  19. 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 DMA copula-based simulation method for clustered multi-state survival data 7/ 22
  20. 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 DMA copula-based simulation method for clustered multi-state survival data 7/ 22
  21. 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)A copula-based simulation method for clustered multi-state survival data 8/ 22
  22. 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)A copula-based simulation method for clustered multi-state survival data 8/ 22
  23. 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. 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)A copula-based simulation method for clustered multi-state survival data 8/ 22
  25. 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)A copula-based simulation method for clustered multi-state survival data 8/ 22
  26. 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. DMA copula-based simulation method for clustered multi-state survival data 9/ 22
  27. 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 DMA copula-based simulation method for clustered multi-state survival data 9/ 22
  28. 28. Simulation Algorithm F. Rotolo Clustering The algorithm allows to freely specify the marginal survivals Si (t). How can we insert clustering?A copula-based simulation method for clustered multi-state survival data 10/ 22
  29. 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.A copula-based simulation method for clustered multi-state survival data 10/ 22
  30. 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 0A copula-based simulation method for clustered multi-state survival data 10/ 22
  31. 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 .A copula-based simulation method for clustered multi-state survival data 10/ 22
  32. 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 ).A copula-based simulation method for clustered multi-state survival data 11/ 22
  33. 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 censoringA copula-based simulation method for clustered multi-state survival data 12/ 22
  34. 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. 35. Choice of Parameters F. Rotolo Outline Clustered Multi-State Survival Data Simulation Algorithm Clustering Choice of Parameters ExampleA copula-based simulation method for clustered multi-state survival data 13/ 22
  36. 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 DMA copula-based simulation method for clustered multi-state survival data 14/ 22
  37. 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 DMA copula-based simulation method for clustered multi-state survival data 14/ 22
  38. 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 DMA copula-based simulation method for clustered multi-state survival data 14/ 22
  39. 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.A copula-based simulation method for clustered multi-state survival data 14/ 22
  40. 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 ΠA copula-based simulation method for clustered multi-state survival data 15/ 22
  41. 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 ΠA copula-based simulation method for clustered multi-state survival data 15/ 22
  42. 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 ΠA copula-based simulation method for clustered multi-state survival data 15/ 22
  43. 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}A copula-based simulation method for clustered multi-state survival data 16/ 22
  44. 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. 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 0A copula-based simulation method for clustered multi-state survival data 17/ 22
  46. 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 0A copula-based simulation method for clustered multi-state survival data 17/ 22
  47. 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. 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 0.569A copula-based simulation method for clustered multi-state survival data 18/ 22
  49. 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 ) = 0.24A copula-based simulation method for clustered multi-state survival data 18/ 22
  50. 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 1.000A copula-based simulation method for clustered multi-state survival data 19/ 22
  51. 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 ) = 0.0064A copula-based simulation method for clustered multi-state survival data 19/ 22
  52. 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 hazardsparMod choose marginal distributions of time variablesA copula-based simulation method for clustered multi-state survival data 20/ 22
  53. 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 hazardsparMod 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 administrativeA copula-based simulation method for clustered multi-state survival data 20/ 22
  54. 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.A copula-based simulation method for clustered multi-state survival data 21/ 22
  55. 55. F. Rotolo [federico.rotolo@stat.unipd.it – federico.rotolo@uclouvain.be]PhD Student at University of Padova and Visiting PhD Student at UCLunder the supervision of prof. C. Legrand, UCL prof. I. Van Keilegom, UCL prof. M. Chiogna, UniPd

×