Safety data are an essential part of the clinical evaluation of new medicinal products and regulatory submissions. However, their analysis might be challenged by the existence of competing risks. These are intercurrent events, defined as mutually exclusive events (death, other adverse events, change of treatment, noncompliance, end of study, etc.) whose occurrence precludes the event of interest from happening (Allignol, Beyersmann, & Schmoor, 2016). Competing risks are common. They are present in the vast majority of clinical trials (Koller, Raatz, Steyerberg, & Wolbers, 2012; van Walraven & McAlister, 2016) and might bias the results (Schuster, Hoogendijk, Kok, Twisk, & Heymans, 2020; van Walraven & McAlister, 2016). They represent a well-recognized problem in the analysis of adverse events (Stegherr, Beyersmann, et al., 2021; Stegherr, Schmoor, et al., 2021) and general recommendations urge the use of survival techniques that methodically account for the presence of competing risks (European Medicines Agency, 2020; Koller et al., 2012; Stegherr, Beyersmann, et al., 2021; Stegherr, Schmoor, et al., 2021). These techniques acknowledge that for a given adverse event there are other types of risks that occur at the same time.
The standard survival data situation corresponds to a Markov process with the two states: “event-free” and “event”. Splitting the “event” state into more states corresponding to different causes (“event 1”, “event 2”, “dead”, etc.) results in a Markov model for competing risks (Borgan, 1997). The analytical object in the presence of competing risks is the same as in standard marginal survival analysis: to estimate the probabilities, also named risks, and hazard rates of the event of interest over time and, if relevant, to assess whether there are differences between groups. However, a competing risks setting that extends the capabilities of analysis of two state survival models to deal with multi-state models (cf. Figure 1) is required, when subjects can experience more than one event (Therneau, Crowson, & Atkinson, 2020). The risk of the event of interest over time is estimated among the risk of other competing events whose occurrence precludes it from happening. The concepts of risks and rates generalize easily to the competing risk situation: hazard rates become cause-specific hazard rates and risks become cumulative incidences (Andersen, Geskus, de Witte, & Putter, 2012).
Two statistical frameworks exist to perform survival analysis in the presence of competing risks: the cause-specific and the subdistribution settings. All standard methods for survival data apply to the cause-specific setting (Geskus, 2016; Putter, Fiocco, & Geskus, 2007) which focuses on the cause-specific hazard function. This function estimates the probability of each type of event separately, right-censoring individuals at the time of the competing event, as well as for loss of follow-up, withdrawal, or at the end of the observation time. For the subdistribution setting, specific approaches were developed that based on the cumulative incidence function (Fine & Gray, 1999; R. J. Gray, 1988). This function focuses on the cumulative incidence (or “subdistribution”) from a particular cause and does not treat competing events as censored observations.
These settings differ in their definitions. The aim of this study is to compare their properties and to recommend how to perform safety analyses in clinical research and regulatory submissions. We investigate whether systematic differences exist between the estimates obtained with each approach and define to what extent the interpretation of the results of survival analysis depends on the choice of one or the other setting. For both settings non-parametric approaches (Borgan, 1997; Edwards, Hester, Gokhale, & Lesko, 2016; R. J. Gray, 1988; Schuster et al., 2020) as well as regression models (Fine & Gray, 1999; Klein, van Houwelingen, Ibrahim, & Scheike, 2013) exist. Classical hazard-based methods for survival data apply when analyzing cause-specific hazards: Kaplan-Meier and Nelson-Aalen estimators as well as the Cox proportional hazards regression model. These methods, however, do not allow to draw inference for subdistribution functions of competing risks. Specific approaches were developed: the Aalen-Johansen estimator and the Fine and Grey model. This paper focuses on (semi-) parametric approaches: cause-specific (Cox regression) and subdistribution hazard regression (Fine & Gray model). Both offer two major advantages in comparison to the non-parametric approaches. First, they allow to adjust for covariates when assessing and comparing event probabilities over time and thus provide more insight into the mechanisms that lead to the occurrence of an event. Second, they allow to use a fitted model to make predictions (e.g., for certain attributes of the population under study).
In the methods section, we provide a brief, nontechnical description of the cause-specific and subdistribution settings in survival analysis. Detailed technical descriptions can be found elsewhere (Borgan, 1997; Fine & Gray, 1999; Klein et al., 2013; Therneau & Grambsch, 2000). A short introduction to the non-parametric estimators can be found in the additional file 1. To examine and to compare the properties of the cause-specific and subdistribution settings in survival analysis, a simulation study was conducted. It covers all possible practical outcomes: from superiority to inferiority of the medical intervention and from small to large effect sizes. In the results section, we report the results of safety analyses performed on each simulated dataset with a cause-specific and a subdistribution setting. Finally, the results are discussed in terms of their practical implications and relevance for safety analyses of new medicinal products and regulatory submissions.