5.1 Data-generating mechanisms

I generated survival data from a Weibull distribution with shape parameter \(\lambda\) = 0.5, scale parameter \(p\) = 0.6, and parametrised as \(h(t) = \lambda p t ^ {p - 1}\) using the method of Bender, Augustin, and Blettner (2005), and applying administrative censoring at time \(t\) = 5. I included a binary covariate simulated by drawing from a Bernoulli random variable with parameter \(\pi\) = 0.5, and a frailty term shared between individuals in a cluster by drawing first from a Gamma distribution with shape parameter \(1 / \theta\) and scale parameter \(\theta\) (for identifiability purposes) and then by drawing from a normal distribution with mean \(\mu\) = 0 and standard deviation \(\sigma = \sqrt{\theta}\). I varied \(\theta\): \(\theta\) = {0.25, 0.75, 1.25}. I also varied the regression coefficient \(\beta\) associated with the binary covariate: \(\beta\) = {-0.50, 0.00, 0.50}. I simulated data for six different sample sizes: 15 clusters of 30, 100, or 500 individuals each; 50 clusters of 30 or 100 individuals; 1000 clusters of 2 individuals. Sample size varied between 450 and 7500 individuals. Applying a fully factorial design, it resulted in 3 times 2 times 3 times 6 = 108 different data-generating mechanisms, and for each of them I generated 1000 datasets.

The 54 simulated scenarios with a Gamma frailty will be used to answer the first aim of the simulation study, as it is possible to derive an analytical formulation of the likelihood. The remaining 54 scenarios with a log-normal frailty will be used to answer the second aim.

References

Bender, Ralf, Thomas Augustin, and Maria Blettner. 2005. “Generating Survival Times to Simulate Cox Proportional Hazards Models.” Statistics in Medicine 24: 1713–23. doi:10.1002/sim.2059.