Survival analysis in the presence of competing risks

Abstract

Survival analysis in the presence of competing risks imposes additional challenges for clinical investigators in that hazard function (the rate) has no one-to-one link to the cumulative incidence function (CIF, the risk). CIF is of particular interest and can be estimated non-parametrically with the use cuminc() function. This function also allows for group comparison and visualization of estimated CIF. The effect of covariates on cause-specific hazard can be explored using conventional Cox proportional hazard model by treating competing events as censoring. However, the effect on hazard cannot be directly linked to the effect on CIF because there is no one-to-one correspondence between hazard and cumulative incidence. Fine-Gray model directly models the covariate effect on CIF and it reports subdistribution hazard ratio (SHR). However, SHR only provide information on the ordering of CIF curves at different levels of covariates, it has no practical interpretation as HR in the absence of competing risks. Fine-Gray model can be fit with crr() function shipped with the cmprsk package. Time-varying covariates are allowed in the crr() function, which is specified by cov2 and tf arguments. Predictions and visualization of CIF for subjects with given covariate values are allowed for crr object. Alternatively, competing risk models can be fit with riskRegression package by employing different link functions between covariates and outcomes. The assumption of proportionality can be checked by testing statistical significance of interaction terms involving failure time. Schoenfeld residuals provide another way to check model assumption.

Keywords: Competing risk; Fine-Gary model; cumulative incidence; hazard function.

Zhongheng Zhang, MMed.

Figure 1

Non-parametric estimates of cumulative incidence functions for deaths from melanoma (status==1) and other causes (status==2). Each outcome is compared between male and female patients.

Figure 2

Parametric estimates of cumulative incidence functions for three patients with given covariate values.

Figure 3

Cumulative incidence functions at different invasion levels, setting age to 52 years and sex to male. There is no evidence of violation to the proportionality assumption for the variable invasion.

Figure 4

Time-dependent effects in the Fine-Gray regression model for invasion. The nonparametric estimates are shown with 95% pointwise confidence intervals.

Figure 5

Schoenfeld residuals against failure time for each covariate. It is noted that the residuals follows a non-constant distribution across failure times, indicating a potential violation to the proportional assumption.