Competing Risks Analysis

How long does it take for a kidney transplant patient’s graft to fail? If a patient dies while the graft is still working normally, how should that patient be analyzed? Could the likelihood of the two events differ depending on the treatment?

Competing risks analysis is a method for analyzing situations in which, among several types of events, the one that happens first prevents the others from happening. After a kidney transplant, “graft failure” and “death before graft failure” are classic competing events.

If you use the usual Kaplan–Meier approach, treat competing events as ordinary censoring, and calculate 1 − survival probability, you can overestimate the actual cumulative probability of the event of interest. Competing risks analysis uses a cumulative incidence function (CIF) for each event type and, depending on the research question, examines covariate relationships with a cause-specific hazard model or a Fine–Gray model.

This method is used for treatment outcomes in transplant patients, cancer recurrence versus death before recurrence, discharge versus in-hospital death, analysis of equipment failure causes, and more. In this article we use a kidney transplant example to cover the difference between competing events and censoring, cumulative incidence curves, interpreting hazard ratios, choosing a model, and points to watch out for when reporting results.

Competing risks analysis handles both the type of the first event each subject experiences during follow-up and the time until that event. For example, a kidney transplant patient may experience graft failure first, or may die while the graft is still functioning. The key point is that, depending on which event happens first, the other event can no longer be observed in the same way.

The analysis usually records the time to the event, the event type, and the covariates. The event type distinguishes the event of interest (e.g., graft failure), the competing event (death before graft failure), and censoring due to the end of observation.

In Kaplan–Meier analysis, a censored observation is treated as carrying only the information that no event had occurred up to that time. But a patient who died before graft failure can never experience graft failure afterward. If you treat death the same as simply losing track of a patient and calculate 1 − Kaplan–Meier, the result differs from the real-world cumulative probability of graft failure.

The same patient can die after graft failure, but here we analyze graft failure as the first event and death before graft failure. To also analyze death after graft failure, you need a different study design, such as a multistate model.

Suppose 400 kidney transplant patients were followed from the time of transplant for up to 4 years. The data record time since transplant, type of first event, age, donor type, and key clinical characteristics. The numbers below are a hypothetical example to illustrate interpretation.

The shares in this table simply divide each patient’s observed outcome by the total, and they are not the same as the estimated cumulative incidence at 4 years. Because follow-up time can differ from patient to patient, the CIF must be estimated using the time information.

The cumulative incidence function (CIF) takes the possibility of competing events into account and estimates the probability that a particular event occurs first by time t. You can plot a CIF for graft failure and a CIF for death before graft failure separately.

At any given time, the sum of the event-specific CIFs is the cumulative probability of a first event from any cause. Adding the probability that no event has happened yet gives 1.

For example, suppose that in a hypothetical analysis the CIF of graft failure at 4 years is 0.24 and the CIF of death before graft failure is 0.13. This means the cumulative probability that each event occurs before the other within 4 years is estimated at about 24% and 13%, respectively.

Regression models for competing risks fall into two main types: the cause-specific hazard model and the Fine–Gray model. The former describes the instantaneous hazard of a particular event among people who have not yet experienced any event, and the latter describes the subdistribution hazard, which is linked to the CIF of a particular event.

If, in hypothetical results, the csHR for graft failure in older patients is 1.35, that means that among patients who have not yet had either event, the cause-specific instantaneous hazard under the same conditions is higher in the older group than in the comparison group. An sHR of 1.20 means the older group is higher on the subdistribution hazard scale of the Fine–Gray model; it does not mean the 4-year probability of graft failure is exactly 20% higher.

The numbers in this results table are an example made to explain how the measures differ; they are not model fits to the actual 400-patient data. The confidence intervals and p-values should not be used as real estimates either.

Plotting the CIF of graft failure by donor type shows how cumulative incidence evolves over time in the two groups. Gray’s test is a way to check whether the CIFs differ between groups while accounting for competing risks. However, this test alone does not adjust for patient age or comorbidities.

If you need to adjust for covariates as well, consider Fine–Gray regression or a cause-specific hazard model suited to your purpose. If the curves cross during follow-up, it may be hard to summarize the relationship with a single hazard ratio.

Event definition: Be clear about whether graft failure was defined as the start of dialysis again, re-transplantation, or a particular laboratory criterion. You also need to distinguish whether a death occurred before or after graft failure.

Censoring: If loss to follow-up is related to patient prognosis, the assumption of noninformative censoring may not hold. Check each patient’s follow-up time and the reason follow-up stopped.

Proportional hazards assumption: For the cause-specific Cox model and the Fine–Gray model, check the form of proportional hazards each one uses. If an effect changes over time, you can consider a time-dependent effect.

Causal interpretation: Even if a difference in cumulative incidence by donor type is observed, you cannot conclude it is a causal effect of donor type unless confounders such as patient selection and underlying conditions are ruled out.

A research report should give the total number of patients, the starting point of observation after transplant and the follow-up period, the definition and number of each competing event, the number censored, the CIF and confidence interval at the time points of interest, and the regression model and covariates used. Report the results of Gray’s test and Fine–Gray regression separately without mixing them up.

The text above is a hypothetical example to show the format. In an actual paper, fill in these estimates only after checking the data that produced them, how the confidence intervals were calculated, and the model diagnostics.

In the kidney transplant example, graft failure and death before graft failure must be distinguished as separate competing events. Looking at the cumulative incidence function (CIF) of each event shows the probability of which event appears first by a given time, and the cause-specific hazard model and the Fine–Gray model describe relationships with covariates from different perspectives.

The most important thing is to choose the measure that fits the purpose of the analysis. Do not confuse cause-specific instantaneous hazards with cumulative probabilities, and also check how censoring arose and the assumptions the hazard ratios require. In particular, the starting point for interpreting results correctly is not to treat a hazard ratio as a change in absolute probability over a specific period or as a causal effect.