Competing risk regression models for epidemiologic data

Abstract

Competing events can preclude the event of interest from occurring in epidemiologic data and can be analyzed by using extensions of survival analysis methods. In this paper, the authors outline 3 regression approaches for estimating 2 key quantities in competing risks analysis: the cause-specific relative hazard ((cs)RH) and the subdistribution relative hazard ((sd)RH). They compare and contrast the structure of the risk sets and the interpretation of parameters obtained with these methods. They also demonstrate the use of these methods with data from the Women's Interagency HIV Study established in 1993, treating time to initiation of highly active antiretroviral therapy or to clinical disease progression as competing events. In our example, women with an injection drug use history were less likely than those without a history of injection drug use to initiate therapy prior to progression to acquired immunodeficiency syndrome or death by both measures of association ((cs)RH = 0.67, 95% confidence interval: 0.57, 0.80 and (sd)RH = 0.60, 95% confidence interval: 0.50, 0.71). Moreover, the relative hazards for disease progression prior to treatment were elevated ((cs)RH = 1.71, 95% confidence interval: 1.37, 2.13 and (sd)RH = 2.01, 95% confidence interval: 1.62, 2.51). Methods for competing risks should be used by epidemiologists, with the choice of method guided by the scientific question.

Figure 1.

Cause-specific hazard schematic. The risk set starts with 30 individuals (solid circles). Over time, individuals have either event 1 (square) or event 2 (triangle). As individuals have either event, they are removed from the remaining risk sets. The calculation for the cause-specific hazard is given at the bottom of the figure.

Figure 2.

Subdistribution hazard schematic. The risk set starts with 30 individuals (solid circles). Over time, individuals have either event 1 (square) or event 2 (triangle). As individuals have the competing event (event 2, triangle), they are maintained in the risk set as triangles. Thus, over time, a greater proportion of the risk set becomes full of triangles that are individuals who have had the competing event prior to that time. The subdistribution hazard (SDH) for event 1 is given near the bottom of the figure along with the cause-specific hazard (CSH) for event 1 for comparison. Note that, because individuals are maintained in the risk set, the SDH tends to be lower than the CSH.

Figure 3.

Cumulative incidence of treatment initiation prior to acquired immunodeficiency syndrome (AIDS) or death (A and B) and the cumulative incidence of AIDS or death prior to treatment (C and D) by injection drug use status and type of cumulative incidence (cause-specific, A and C; subdistribution, B and D; _csPH, from proportional cause-specific hazards model; _sdPH, from proportional subdistribution hazards model). The mixture model comprised a lognormal distribution for initiation of treatment and a generalized-gamma distribution for the time to AIDS or death prior to treatment initiation. CI, confidence interval; _csRH, cause-specific relative hazard; HAART, highly active antiretroviral therapy; IDU, injection drug use; _sdRH, subdistribution relative hazard.