Competing Risks in Safety Analyses

Why Competing Events Change Safety Interpretation

A time-to-first adverse-event analysis often focuses on the time until a particular safety event occurs. But a patient may first experience another event that makes the safety event of interest impossible to observe. Death, for example, prevents a later nonfatal adverse event. When such events occur, the fraction who experience the safety event depends both on its own event rate and on how many people remain alive and free of every first event.

The distinction matters in exposure-response analyses. An exposure group may have a similar rate of a nonfatal event among people still at risk, yet a different observed probability of that event because its patients experience more competing events. Conversely, treating a competing event as ordinary censoring can answer a hypothetical question about risk if that competing event were removed, rather than the practical question of how often the safety event occurs in the population observed.

Let \(T\) be the time to the first event, and let \(J\) identify its type, such as \(J=1\) for the adverse event of interest and \(J=2\) for death. Write \(\lambda_k(t)\) for the cause-specific hazard of type \(k\). The event-free survival function, \(S(t)=P(T>t)\), is the probability of remaining free of all event types through time \(t\). With two causes, a patient who experiences either event leaves the event-free risk set.

From Cause-Specific Hazards to Cumulative Incidence

A cause-specific hazard is conditional on remaining event-free. It is not the probability of the event over a follow-up period. The probability of the event of interest depends on its hazard and on the probability of still being event-free at each moment. For continuous event times and cause-specific hazards \(\lambda_k(t)\), the relationships are:

The sum includes all mutually exclusive first-event types. The CIF accumulates the probability of event type \(k\) over time: at each time \(u\), the chance of still being event-free, \(S(u)\), is multiplied by the cause-specific event rate, \(\lambda_k(u)\). This is why the competing-event hazards affect the CIF even when the cause-specific hazard of the event of interest is unchanged.

A short derivation makes the distinction explicit. Over a small interval of length \(dt\), a patient who is event-free just before \(t\) has probability approximately \(\lambda_k(t)dt\) of experiencing cause \(k\). The probability of being event-free just before then is \(S(t)\). Thus the probability of a first event of type \(k\) in that interval is approximately \(S(t)\lambda_k(t)dt\). Summing these increments from zero to \(t\) gives the integral for \(F_k(t)\). Similarly, the probability of any first event in the interval is approximately \(S(t)\sum_j\lambda_j(t)dt\), so \(dS(t)/dt=-S(t)\sum_j\lambda_j(t)\); solving this equation with \(S(0)=1\) gives the expression above.

Estimating the CIF From Event-Time Data

When event times are observed at distinct or grouped times, a standard nonparametric estimator is the Aalen–Johansen estimator. At each event time, it updates the probability of remaining free of every event type, then assigns each cause its share of the remaining event-free probability. If \(Y(t)\) is the number at risk just before time \(t\), and \(d_k(t)\) is the number of type-\(k\) events at that time, the estimated CIF increment is:

Here \(\widehat{S}(u-)\) is the estimated probability of still being event-free immediately before \(u\). If there is right censoring, the person contributes to the risk set until censoring. Estimation relies on censoring being independent of future event times, at least conditional on the variables used in the analysis.

Choosing and Interpreting an Analysis

The appropriate quantity depends on the question. A cause-specific hazard model describes the instantaneous rate of a particular first event among those who are currently alive and free of all first events. It is often useful for assessing how exposure relates to that event-specific rate. A CIF describes the probability of experiencing that event first by a specified time in the presence of the other observed event types. It is often more directly aligned with questions such as, “What proportion of patients is expected to experience this adverse event by one year?”

For the Kaplan–Meier calculation, censoring a competing event treats that patient as though their subsequent event experience could stand in for those still followed. The resulting \(1-\widehat{S}(t)\) is not the observed probability of the event of interest when competing events occur. A subdistribution hazard model, often used to model CIF patterns directly, answers a different modeling question from a cause-specific hazard model. Neither model’s coefficient alone is a substitute for reporting the CIF when the objective is absolute event probability.

In exposure-response work, an exposure association is not automatically causal. Patient characteristics and clinical factors may be related to both exposure and event risk. Also, exposure can vary over follow-up. The exposure definition and time window should match the safety question; a model using a single baseline exposure should not be interpreted as though it represented later exposure changes. Cause-specific and cumulative-incidence results are complementary: one describes event rates among those still at risk, while the other describes realized event probabilities under the competing events observed.

Assumptions, Diagnostics, and Common Pitfalls

Before estimating or modeling competing risks, specify the event of interest, the competing event types, the time origin, and the follow-up horizon. The first-event types should be mutually exclusive and consistently classified. Check that a proposed competing event really prevents the endpoint as defined. Death generally prevents a later nonfatal event; treatment discontinuation is not automatically competing. If safety events continue to be measured after discontinuation, discontinuation does not itself make them unobservable. If follow-up stops at discontinuation, it may instead create informative censoring unless the analysis and assumptions account for that process.

• Tabulate numbers and timing of each first-event type by exposure group. Sparse events or very different follow-up can make estimates unstable.
• Check event coding, time origin, and risk windows. Misclassifying a competing event or recording it after the event of interest can distort both hazards and CIFs.
• Examine censoring patterns and reasons. If censoring depends on future event risk after accounting for included variables, standard estimates may be biased.
• For a cause-specific proportional-hazards model, assess whether the exposure hazard ratio is reasonably constant over time; a single summary can be misleading when effects vary.
• Compare model-predicted CIFs with nonparametric CIF estimates at clinically relevant times. Good cause-specific hazard fit does not by itself guarantee well-calibrated absolute risks.

Common mistake: “The adverse-event hazard ratio is near 1, so the two exposure groups must have the same probability of the event.” This is incorrect: the CIF also depends on the hazards of competing events and the event-free survival they produce. Report both the cause-specific hazard contrast and time-specific CIFs when both event rate and absolute probability matter, and interpret each in its own terms.