Simulating time-to-event data subject to competing risks and clustering: A review and synthesis

Abstract

Simulation studies play an important role in evaluating the performance of statistical models developed for analyzing complex survival data such as those with competing risks and clustering. This article aims to provide researchers with a basic understanding of competing risks data generation, techniques for inducing cluster-level correlation, and ways to combine them together in simulation studies, in the context of randomized clinical trials with a binary exposure or treatment. We review data generation with competing and semi-competing risks and three approaches of inducing cluster-level correlation for time-to-event data: the frailty model framework, the probability transform, and Moran's algorithm. Using exponentially distributed event times as an example, we discuss how to introduce cluster-level correlation into generating complex survival outcomes, and illustrate multiple ways of combining these methods to simulate clustered, competing and semi-competing risks data with pre-specified correlation values or degree of clustering.

Keywords: Simulation study; cluster randomized trials; clustering; competing risks; semi-competing risks; time-to-event data.

Figure 1.

Scatter plot for Calculated ICC and pre-specified Kendall’s $\tau$ varying from 0.000 I to 0.99 in increments of 0.0002 under the Gamma frailty model with hazard rate $\lambda$ varied from 0.05 to 0.95 in increments of 0.1.

Figure 2.

Scatter plot for empirical correlation parameters (ICC and Kendall’s $\tau$) vs pre-specified Kendall’s $\tau$ under Gamma frailty model with 100 clusters, two subjects per cluster, and hazard rate $\lambda$ varied from 0.01 to 0.99 in increments of 0.01; the vertical axis represents empirical Kendall’s $\tau$ (left panel) and empirical ICC (right panel), and the horizontal axis represents pre-specified Kendall’s $\tau$ (both panels).

Figure 3.

Scatter plot for empirical correlation parameters (ICC and Kendall’s $\tau$) vs pre-specified Kendall’s $\tau$ under probability transform method with 100 clusters, two subjects per cluster, and hazard rate $\lambda$ varied from 0.01 to 0.99 in increments of 0.01; the vertical axis represents empirical Kendall’s $\tau$ (left panel) and empirical ICC (right panel), and the horizontal axis represents pre-specified Kendall’s $\tau$ (both panels).

Figure 4.

Scatter plot for empirical correlation parameters (ICC and Kendall’s $\tau$) vs pre-specified ICC under Moran’s algorithm with 100 clusters, two subjects per cluster, and hazard rate $\lambda$ varied from 0.01 to 0.99 in increments of 0.01; the vertical axis represents empirical ICC (left panel) and empirical Kendall’s $\tau$ (right panel), and the horizontal axis represents pre-specified ICC (both panels).