4.4 Simulation
4.4.1 Simulation Set up
A simulation has been conducted to test the performance of the Gaussian
copula model, the Clayton copula model, the multi-state model based approach
and the Cox model approach. We simulate 1000 datasets with a sample size of
1000 for each data, with 500 patients in each of the control and treatment groups.
No other explanatory variables apart from treatment are considered in the sim-
ulations. As the illness-death model reflects the underlying survival process of
patients with cancer, it is assumed to be the true model. In all four simulation
scenarios, we assume the data are generated from a homogeneous semi-Markov
model with Weibull transition intensity functions. Let the scale parameter be λ01, λ02and λ12as well as the shape parameter be α01, α02and α12for each tran-
sition then the transition intensities are given by π01(t) = α01 1 λ01 α01 tα01−1, π02(t) = α02 1 λ02 α02 tα02−1, π12(s) = α12 1 λ12 α12 sα12−1,
where t and s refer to time since randomization and to time since progression,
respectively.
The simulation scenarios mainly differ with respect to which transitions are
assumed to be affected by the treatment. As there is not one rule of pattern how
the treatment affects the transitions, we choose different patterns to be the true
model. In scenario 1, we choose values of the shape and scale parameters such
that we have a situation with a treatment effect both on transition from study
entry to progression and from progression to death prior to progression. In sce-
nario 2, we consider a situation where treatment affects the transition between
state 0 and 1, while death given progression is not affected by the treatment.
In scenario 3, we assume a treatment effect on all three transitions and test for
example how the parametric multi-state model-based approach assuming only
treatment effects on transitions 0 → 1 and 1 → 2 performs. Scenario 4 represents
a situation where the treatment affects transitions π01and π12such that the true
hazard ratio decreases over time much faster than in the first scenario in order
to see whether that setting makes a difference in terms of the performance of the
methods.
In Table 4.1, the setting values of the scale parameters and shape parame-
ters of the Weibull distributions used in each scenario are summarized. Graphics
of the hazard function, survival function and cumulative hazard function of OS
Table 4.1: Parameter values for the simulation scenarios 1-4
Scenario
Parameter 1.Scenario 2.Scenario 3.Scenario 4.Scenario
(λ01, λ02, λ12)control (2.5, 9, 2.3) (2.5, 9, 2.5) (2.5, 7, 2.1) (1.5, 9, 2.3)
(λ01, λ02, λ12)treat (5, 9, 2.6) (5, 9, 2.5) (3.1, 10, 2.5) (5, 9, 6)
(α01, α02, α12)control (0.8, 1, 1.2) (0.8, 1, 1.2) (0.9, 1, 1) (0.7, 1, 1.2)
(α01, α02, α12)treat (0.8, 1, 1.2) (0.8, 1, 1.2) (0.9, 1, 1) (0.7, 1, 1.2)
low censoring U[4, 10] U[4, 10] U[3, 10] U[6, 10]
high censoring U[2, 7] U[3.5, 5] U[2, 6] U[2.5, 7]
As the rate of censoring might influence the performance of the methods, we
consider two censoring cases for each simulation scenario. In case 1, 20% of
patients are censored with respect to OS while in case 2 around 40% of OS times
are censored. The censoring times arise from a uniform distribution.
The following section addresses the setting about the prior distribution
for each considered method. The program JAGS (see Section 4.3 for more de-
tails on computation of the posterior distribution in Bayesian analysis) was used
for the Clayton-based model to obtain the posterior distribution of the baseline
scale parameters λP F S and λOS, the treatment effects θOS and θP F S and the de-
pendence parameter ξ. Furthermore, the posterior distribution of the shape pa-
rameters αP F Scontrol, αOScontrol, αP F Streat and αOStreat for the control and treatment
group will be generated. The initial values of all parameters were set to zero.
Like in the Clayton-based model approach, the posterior distribution of the pa-
rameters to be estimated were generated by JAGS within the framework of the
multi-state model-based approach. The various parameters include scale pa-
the treatment effects θ01, θ02and θ12. In both approaches, the prior density dis-
tribution of all parameters were set to be weakly informative. The initial values
for all parameters were all set to zero prior density. For the Clayton-based ap-
proach and the multi-state model based approach, the output of the posterior
results were based on 10000 iterations after a burn-in period of 5000.
The summary of the setting for the prior distributions for all parameters
and the two approaches is shown in Table 4.2. Following the standard recom-
mendation in the Clayton copula-models we parameterize the model on the log
scale. Regarding the scale parameter for each transition in the multi-state model-
based method, we used a half normal distribution truncated at 0.05 rather than
a log-normal distribution because of the potential for the scale parameter λ02to
get close to 0.
Table 4.2: Setting of prior distribution of the parameter values for Clayton copula-based and multi-sate model-based approach
Approach
Clayton copula-based
multi-state model-based
(in JAGS)
(in JAGS)
log(λ
P F S) ∼ N (0.2, 3
−2)
λ
01∼ N (0, 0.1)I[0.05, ]
log(λ
OS) ∼ N (0.2, 3
−2)
λ
02∼ N (0, 0.1)I[0.05, ]
θ
P F S∼ N (0.1, 3
−2)
λ
12∼ N (0, 0.1)I[0.05, ]
θ
OS∼ N (0.1, 3
−2)
θ
01∼ N (0, 0.1)
log(α
P F Scontrol) ∼ N (0.2, 3
−2)
θ
02∼ N (0, 0.1)
log(α
OScontrol) ∼ N (0.2, 3
−2)
θ
12∼ N (0, 0.1)
log(α
P F Streat) ∼ N (0.2, 3
−2)
α
01∼ N (0, 0.01)I[0.05, ]
log(α
OStreat) ∼ N (0.2, 3
−2)
α
02∼ N (0, 0.01)I[0.05, ]
ξ ∼ uni(−1, 50)
α
12∼ N (0, 0.01)I[0.05, ]
Instead of using the JAGS program to generate the posterior samples, the
Metropolis–Hastings algorithm was used for the Gaussian copula-based approach
order to perform the NICE model estimation, we adapted the code by Fu et al.32
Therefore, we will use the posterior mode as the estimate for each iteration
while assuming a non-informative prior in terms of improper uniform priors.
Because the Bayesian estimate uses a non-informative prior, it will be equiva-
lent to the maximum likelihood estimate (MLE) of the NICE model. Compared
to the NICE model in the paper, where exponential hazards for the transitions
are assumed, we instead use Weibull hazards. Therefore we also need posterior
estimates for the shape parameters. The scale and shape parameters λT T P, λOS,
αT T P and αOS are required for both control and treatment group. In addition,
the posterior density for the dependence parameter ρ of the Gaussian copula
will be generated. The initial values for were all set to be λT T P = 0.1, λOS = 0.1, αT T P = 0.1, αOS = 0.1 and ρ = 0.4 for both treatment and control group. The outcome of all posterior distributions of the parameters is the result from
20000 MCMC simulations with a burn-in period of 1000 iterations.