Marginal models for clustered time-to-event data with competing risks using pseudovalues

Abstract

Many time-to-event studies are complicated by the presence of competing risks and by nesting of individuals within a cluster, such as patients in the same center in a multicenter study. Several methods have been proposed for modeling the cumulative incidence function with independent observations. However, when subjects are clustered, one needs to account for the presence of a cluster effect either through frailty modeling of the hazard or subdistribution hazard, or by adjusting for the within-cluster correlation in a marginal model. We propose a method for modeling the marginal cumulative incidence function directly. We compute leave-one-out pseudo-observations from the cumulative incidence function at several time points. These are used in a generalized estimating equation to model the marginal cumulative incidence curve, and obtain consistent estimates of the model parameters. A sandwich variance estimator is derived to adjust for the within-cluster correlation. The method is easy to implement using standard software once the pseudovalues are obtained, and is a generalization of several existing models. Simulation studies show that the method works well to adjust the SE for the within-cluster correlation. We illustrate the method on a dataset looking at outcomes after bone marrow transplantation.

Figure 1

Boxplot of relative bias (Bias/β) for a partitioning of simulation settings, by number of clusters m, observations per cluster n_i, % type I events, and censoring percentage. Boxplots represent the range of values across all other simulation parameters, i.e. across α and the covariate number.

Figure 2

Boxplot of ratio of square root of average variances of parameter estimates to the Monte Carlo estimated SD, for covariate Z₁. Boxplots are for ratios across all simulation scenarios, and are shown separately by the variance estimation method (Unadjusted for clustering, Adjusted for clustering, and Bootstrap estimate), by number of clusters m, and across 3 panels for varying levels of association, α. A reference line at the target ratio of 1.0 is also provided.

Figure 3

Boxplot of ratio of square root of average variances of parameter estimates to the Monte Carlo estimated SD, for covariate Z₂. Boxplots are for ratios across all simulation scenarios, and are shown separately by the variance estimation method (Unadjusted for clustering, Adjusted for clustering, and Bootstrap estimate), by number of clusters m, and across 3 panels for varying levels of association, α. A reference line at the target ratio of 1.0 is also provided.

Figure 4

Boxplot of ratio of square root of average variances of parameter estimates to the Monte Carlo estimated SD, for covariate Z₃. Boxplots are for ratios across all simulation scenarios, and are shown separately by the variance estimation method (Unadjusted for clustering, Adjusted for clustering, and Bootstrap estimate), by number of clusters m, and across 3 panels for varying levels of association, α. A reference line at the target ratio of 1.0 is also provided.