Survival analysis—part 3: intermediate events and the importance of competing risks

Abstract

Learning objectives:

1. Understand what events can be labelled as intermediate events in survival analysis.

2. Understand why the Kaplan and Meier method cannot be used in the presence of competing events.

3. Regression analysis in the presence of competing events.

Introduction

Survival analysis—parts 1 and 2 covered an introduction to survival analysis, use of the Kaplan and Meier method (KM) for calculating survival estimates in the presence of censoring, and the Cox proportional hazards model (CPH). In this paper, we introduce and discuss a framework for the analysis of competing events. Our manuscript is formatted as follows: (1) the first section provides readers with examples of competing events encountered in cardiac surgery, (2) the next section outlines the preferred method to estimate event rates when competing events are present, and (3) lastly, we discuss regression methods in competing risk analysis.

We also provide readers a dummy dataset with code that outlines standard methods to analyze such data.

What are competing events?

The Kaplan and Meier method, which allows for right censoring, is often used to provide crude estimates of mortality at specific time points [1]. This method is useful when the event modeled is a terminal event. However, when we want to model a non-terminal event, then we encounter a situation of competing events. In Fig. 1, we present time line of 10 patients that had coronary artery bypass grafting (CABG) and are part of our study. While they had surgery on different dates, their time history is arranged from time 0, i.e., from the point that they underwent surgery. In this group, if we want to model death, then we are modeling a terminal event (Fig. 1a). No event can occur after death. In this situation, the Kaplan and Meier method will provide a reliable death rate at specific time points during follow-up. However, consider that we are interested in knowing the rate of myocardial infarction (MI) during follow-up. Here, we now have two events of interest—myocardial infarction (blue dot) and death (red dot) (Fig. 1b). The possible situation that may occur during follow-up are as follows: (1) Patient A died very early after CABG; hence, patient A cannot be at risk for MI after death. (2) Patient B, on the other hand, did not suffer MI and died during the study period. (3) Patient C had an MI, survived that, and then died later during the follow-up period, and lastly, (4) patient D did not suffer any event and is alive at the end of follow-up. Hence, here, suffering MI is our event of interest and death is the competing event (Fig. 2a).

Fig. 1.

Hypothetical example of 10 patients. a In this group, if we want to model death, then we are modeling a terminal event (depicted by a red dot). b This figure presents the situation where we are interesting in modeling an intermediate event, myocardial infarction (MI). Here, we now have two events of interest—myocardial infarction (blue dot) and death (red dot)

Fig. 2.

These two examples provide different situations where a competing risk analysis should be performed. a Here, the event of interest is myocardial infarction (MI) while death is a competing event. However, in this situation, these events are not mutually exclusive. b However, consider that we want to model two different causes of death (CV and non-CV mortality). Here, these two causes are mutually exclusive as a patient that has one event cannot suffer the other event. Here too, a competing risk analysis is recommended

However, we can also have a competing event’s situation even if the events of interest are terminal events. This occurs when we are interested in evaluating cause-specific mortality. If our event of interest is cardiovascular related mortality (CV mortality), then death due to non-cardiovascular causes (non-CV mortality) becomes a competing event (Fig. 2b).

Non-parametric estimation—the cumulative incidence function

The Kaplan-Meier method can estimate survival [S(t)] when modeling a terminal event. As discussed in earlier parts, the survival estimate [S(t)] = 1 when time t = 0. As the time increases, both events (observing the terminal event and censoring) influence estimated survival. In the absence of competing events, the cumulative failure [F(t)] of observing the event is:

$$F\left(t\right)=1–S\left(t\right).$$

However, in the competing risk framework, the cumulative incidence function (CIF) should be used to obtain reliable estimates of event occurrence. If we consider modeling two separate events (a and b), then:

• CIF_a = Pr(T > t, D = a)

• CIF_b = Pr(T > t, D = b) and

• CIF_total = CIF_a + CIF_b

If we consider two mutually exclusive events (A and B), Austin et al. recommend that non-parametric estimates should always be calculated using the CIF rather than the Kaplan and Meier method. In fact, authors demonstrate that, compared to the CIF, the KM method will always overestimate the cumulative incidence for each event [2, 3].

Multi-variable regression methods for competing events

Regression methods allow researchers to understand and interpret the association of clinical factors with the observed end-point. They provide an estimate for each covariate adjusting for differences in other factors included in the model. The hazard function is the scale used most often to model time-to-event data in the presence of censoring. In fact, with the most widely used survival model, the CPH model, results are presented as a hazard ratio for each included covariate. Readers can review part 2 for a more detailed understanding of the CPH model. In the situation that only one event is possible, the CPH can be implemented without concern. However, in the presence of competing risks, the hazard rates obtained from the CPH model do not correspond to the cumulative incidence observed for each competing event. This occurs because CPH model does not account for the fact that a second event can also occur during follow-up. Hence, in a competing risk situation, while the CPH model provides a reliable risk estimate, it does not provide an accurate estimate of the incidence of an event.

Thus, to model the cumulative incidence, Fine and Gray introduced the sub-distribution hazard model (FG model). We now attempt to explain this in more detail without confusing the reader with complicated mathematical equations. The FG model, like the CPH model, is also based on the hazard scale. However, it takes into account occurrence of the second competing event. Hence, the hazard ratio obtained from the FG model is the risk of observing the first event provided the competing event has not yet occurred. The hazard ratio obtained from the FG model is denoted as the sub-distribution hazard ratio (sub-HR). By accounting for both events, the sub-HR obtained from the FG model directly correlates with the cumulative incidence calculated using non-parametric methods.

Which regression method is appropriate for competing events?

Austin et al. recommend that both regression methods are appropriate in analyzing data with competing events [2]. They recommend choosing the model depending upon the research question. Thus, to understand the association of clinical factors with a particular end-point, Lau et al. recommend the CPH model [4]. However, when the aim is to create a risk model for a specific secondary end-point, the sub-HR values should be calculated.

A recent study demonstrates that in spite of wide availability of these statistical tools in contemporary statistical packages, competing risk analysis is underutilized [5]. Given the erroneous estimates obtained when a competing risk model is not utilized, readers should carefully study the methods section of papers to obtain a deeper understanding of presented results.

Competing risk analysis has been implemented in recent peer-reviewed studies. Deo et al. conducted an observational study analyzing the clinical outcome of patients with metabolic syndrome undergoing coronary artery bypass grafting [6]. They studied all-cause mortality, myocardial infarction, stroke, and heart failure hospitalization as the end-points for analysis. Here, MI and stroke were modeled within a competing risk framework. Thus, they report unadjusted event rates for MI and stroke using the cumulative incidence function. The Fine and Gray sub-distribution hazard model is used to identify the association between clinical covariates and the studied end-point.

A review of statistical software routinely used in bio-statistics was performed to identify programs that allow for modeling competing risk events. Among the many software programs available for bio-statistical analysis, we can say with certainty that R (The R Foundation for Statistical Computing, Austria), STATA (StataCorp, College Station, TX) and SAS (SAS Corporation, Cary, NC) have functions and programs that allow for modeling competing risk data. At the time of reviewing this topic, we were unable to find any source that would allow researchers to perform cumulative incidence function and the Fine and Gray model in SPSS (IBM-SPSS, Armonk, NY).

Suggested reading

1. Austin PC, Fine JP. Practical recommendations for reporting Fine-Gray model analyses for competing risk data. Stat Med. 2017;36:4391–4400. 10.1002/sim.7501. Epub 2017 Sep 15. PMID: 28913837; PMCID: PMC5698744.

2. Austin PC, Lee DS, Fine JP. Introduction to the analysis of survival data in the presence of competing risks. Circulation. 2016;133:601–9. 10.1161/ CIRCULATIONAHA.115.017719. PMID: 26858290; PMCID: PMC4741409.

These articles (referenced throughout this paper) provide excellent understanding of competing risk models. In fact, in all of his papers, he beautifully explains bio-statistical techniques in a reasonably non-mathematical format. They are a “must-read” for all clinical researchers.

• 3.Data Analysis with competing risks and Intermediate states. Ronald B Geskus. CRC Press. © 2016 by Taylor & Francis Group. ISBN 13: 978-1-4665-7035-1

This book (dedicated to modeling competing risks and multistate models) is an excellent reference for using R in modeling multi-state and competing risk data.

Funding

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Conflict of interest

The authors declare that they have no conflict of interest.

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