Statistical Methods for Cohort Studies of CKD: Survival Analysis in the Setting of Competing Risks

Abstract

Survival analysis is commonly used to evaluate factors associated with time to an event of interest (e.g., ESRD, cardiovascular disease, and mortality) among CKD populations. Time to the event of interest is typically observed only for some participants. Other participants have their event time censored because of the end of the study, death, withdrawal from the study, or some other competing event. Classic survival analysis methods, such as Cox proportional hazards regression, rely on the assumption that any censoring is independent of the event of interest. However, in most clinical settings, such as in CKD populations, this assumption is unlikely to be true. For example, participants whose follow-up time is censored because of health-related death likely would have had a shorter time to ESRD, had they not died. These types of competing events that cause dependent censoring are referred to as competing risks. Here, we first describe common circumstances in clinical renal research where competing risks operate and then review statistical approaches for dealing with competing risks. We compare two of the most popular analytical methods used in settings of competing risks: cause-specific hazards models and the Fine and Gray approach (subdistribution hazards models). We also discuss practical recommendations for analysis and interpretation of survival data that incorporate competing risks. To demonstrate each of the analytical tools, we use a study of fibroblast growth factor 23 and risks of mortality and ESRD in participants with CKD from the Chronic Renal Insufficiency Cohort Study.

Keywords: Cardiovascular Diseases; Cause-specific; Chronic; Cox proportional hazards models; Cumulative incidence function; FGF-23; Fibroblast Growth Factors; Fibroblast growth factor 23; Follow-Up Studies; Kidney Failure; Proportional Hazards Models; Renal Insufficiency; Risk; Survival Analysis.

Figure 1.

In a hypothetical example of 25 subjects, the cause-specific hazard function for the event of interest at time t and the subdistribution hazard function for the event of interest at time t are calculated using different risk sets (i.e., differences in their denominators). Solid circles: subjects without experiencing any events; squares: subjects experiencing the event of interest; empty (orange) triangles: subjects experiencing the competing event; solid (gray) triangle: subjects experiencing the competing event in earlier time points and stayed in the risk set; circles: subjects being censored.

Figure 2.

A simple flow chart to illustrate analytical strategies on the basis of the goal of the analysis. CIF, cumulative incidence function; HR, hazard ratio.

Figure 3.

Estimated cause-specific hazard ratio (solid line) and subdistribution hazard ratio (dashed line) over time and their corresponding 95% confidence intervals (red area: cause-specific hazard ratio; blue area: subdistribution hazard ratio) from models including a time-dependent coefficient for levels of FGF-23 to allow hazard ratios to change over time. 95% CI, 95% confidence interval; FGF-23, fibroblast growth factor 23; HR, hazard ratio.

Figure 4.

Schoenfeld residuals plots for cause-specific hazards models and Schoenfeld-type residuals plots for subdistribution hazards models. Circles: Schoenfeld residuals; lines: smooth curve; dashed lines: confidence bands at two standard errors.

Figure 5.

Cumulative incidence functions for ESRD by FGF-23 groups on the basis of cause-specific hazards models with a time-dependent coefficient and subdistribution hazards models with a time-dependent coefficient. Note that the gray solid and dotted lines from sub-distribution hazards models are cumulative incidence functions for ESRD by FGF-23 groups on the basis of subdistribution hazards models without time-dependent coefficient. FGF-23, fibroblast growth factor 23.