How Competing Events Shape The Subdistribution Hazard

Why the Same Focal Rate Can Yield Different Subdistribution Hazards

A cause-specific hazard describes the instantaneous rate of a focal event among people who have not yet experienced any event. A subdistribution hazard instead uses a risk-set construction that retains people after a competing event. These quantities answer different questions, so a change in the subdistribution hazard need not mean that the focal cause-specific hazard has changed.

The key connection is that the subdistribution hazard depends both on the focal cause-specific hazard at time \(t\) and on the event history accumulated by that time. In particular, competing events affect how much event-free probability remains available for a new focal event, while people who already had a competing event remain represented in the subdistribution-hazard denominator. The result is a rate shaped by the competing-event distribution, not just by the focal rate.

Definition: Let \(T\) be the time to the first event, and let \(J\) identify its cause. For cause \(j\), the cumulative incidence function is \(F_j(t)=P(T\leq t,J=j)\), and the event-free probability is \(S(t)=P(T>t)\). With continuous event times, the cause-specific hazard is \(\lambda_j(t)=f_j(t)/S(t)\), where \(f_j(t)\) is the density of a first event from cause \(j\). The subdistribution hazard is \(\tilde{\lambda}_j(t)=f_j(t)/[1-F_j(t)]\), whenever the denominator is positive.

The denominator \(1-F_j(t)\) includes both people who are still event-free and people who have already experienced a cause other than \(j\). That is why the subdistribution hazard is not the cause-specific hazard with a different label. The earlier material on the subdistribution risk set and the subdistribution hazard identity gives the background for this distinction; here the focus is on how other causes enter the identity.

The Identity and What the Competing Causes Contribute

Assume that the first-event causes are mutually exclusive, event times are continuous, and the cause-specific hazards and densities are well-defined. At time \(t\), a first event from cause \(j\) can occur only among people who remain event-free just before \(t\). Thus its density is the event-free probability multiplied by the cause-specific hazard:

f_j(t)=S(t)\lambda_j(t).

Substituting this expression into the definition of the subdistribution hazard gives the identity:

\tilde{\lambda}_j(t) = \frac{S(t)\lambda_j(t)}{1-F_j(t)}.

Because the first-event causes are mutually exclusive, the event-free probability is \(S(t)=1-\sum_k F_k(t)\), where the sum includes all causes. Therefore an equivalent expression is:

\tilde{\lambda}_j(t) = \lambda_j(t) \frac{1-\sum_k F_k(t)}{1-F_j(t)} = \lambda_j(t) \left(1-\frac{\sum_{k\ne j}F_k(t)}{1-F_j(t)}\right).

This decomposition isolates two parts. The focal cause-specific hazard is \(\lambda_j(t)\). The multiplier is the event-free probability divided by the probability of not yet having experienced the focal cause. It is at most one, since \(S(t)\leq 1-F_j(t)\). The multiplier reflects the competing-event incidence accumulated by time \(t\): if more people have had other first events, fewer are event-free relative to those counted in \(1-F_j(t)\).

For a fixed time, a fixed focal cumulative incidence, and a fixed focal cause-specific hazard, increasing the cumulative incidence of competing causes lowers this multiplier and therefore lowers the subdistribution hazard. This is a conditional comparison at the stated values. In a comparison of whole populations or processes, the focal cumulative incidence can itself change when competing hazards change. The identity shows the mechanism, but it does not justify claiming that every change in a competing hazard must produce the same direction or size of change in the subdistribution hazard at every time.

Worked Calculations

Worked Example: Reconstructing the Subdistribution Hazard

Consider a hypothetical population at a particular time \(t\). Suppose the focal cause-specific hazard is \(0.4004\) events per person-year and the event-free probability is \(S(t)=0.7400\). Suppose the focal cumulative incidence is \(F_j(t)=0.1728\). The remaining probability, \(0.0872\), has experienced competing events, since \(1-0.7400-0.1728=0.0872\).

First calculate the instantaneous focal-event density:

f_j(t)=S(t)\lambda_j(t) =(0.7400)(0.4004) =0.296296\ \text{per year}.

The subdistribution-hazard denominator is \(1-F_j(t)=1-0.1728=0.8272\). Using the rounded density \(0.29630\) per year for the division gives:

\tilde{\lambda}_j(t) = \frac{0.29630}{0.8272} \approx 0.3581963\ \text{per year}.

Using the unrounded product instead gives \(0.296296/0.8272\approx0.3581915\) per year. The slight difference is due to rounding the density before division. Either calculation shows the identity: the subdistribution hazard is smaller than the focal cause-specific hazard here because \(S(t)/(1-F_j(t))=0.7400/0.8272\approx0.8946\). These are instantaneous rates, not probabilities of an event during a year.

Worked Example: More Competing Incidence at the Same Snapshot

Compare two hypothetical snapshots with the same focal cause-specific hazard, \(0.3000\) per year, and the same focal cumulative incidence, \(F_j(t)=0.1000\). At the first snapshot, let the cumulative incidence of all competing causes be \(0.0500\); at the second, let it be \(0.2000\). These are valid probability configurations: the event-free probabilities are \(0.8500\) and \(0.7000\), respectively.

For the first snapshot:

\tilde{\lambda}_j(t) = 0.3000\frac{0.8500}{1-0.1000} = 0.3000\frac{0.8500}{0.9000} = 0.2833\ \text{per year}.

For the second snapshot:

\tilde{\lambda}_j(t) = 0.3000\frac{0.7000}{0.9000} = 0.2333\ \text{per year}.

At these specified snapshots, the focal cause-specific hazard and focal cumulative incidence are held fixed, while competing-event incidence is larger in the second configuration. The subdistribution hazard is consequently lower. This illustrates the identity’s direct effect; it is not, by itself, a claim that two complete populations with different competing hazards must have the same focal cumulative incidence at every time.

Worked Example: Accounting for More Than One Competing Cause

Suppose a hypothetical population has three possible first-event causes. At time \(t\), the focal cause has cumulative incidence \(F_j(t)=0.0800\), while the other two causes have cumulative incidences \(0.1200\) and \(0.2000\). The focal cause-specific hazard at this time is \(0.2500\) per year. The event-free probability is:

$$ S(t)=1-0.0800-0.1200-0.2000=0.6000. $$

The focal-event density is \(f_j(t)=(0.6000)(0.2500)=0.1500\) per year, and the subdistribution-hazard denominator is \(1-0.0800=0.9200\). Thus:

\tilde{\lambda}_j(t) = \frac{0.1500}{0.9200} = 0.1630\ \text{per year}.

Both competing causes must be included in the event-free probability. If one incorrectly omitted the cause with cumulative incidence \(0.1200\), one would use \(S(t)=0.7200\), calculate a density of \(0.1800\) per year, and obtain \(0.1800/0.9200=0.1957\) per year. That is not the subdistribution hazard for the stated three-cause system, because it treats people who experienced one competing cause as if they remained event-free.

Why the Identity Works

The identity follows directly from the meaning of the two rates, but writing out the steps prevents a common denominator error. The cause-specific hazard conditions on remaining free of every first event, whereas the subdistribution hazard divides the focal-event density by the probability of not yet having had the focal event. The numerator is the same focal-event density in both expressions; the denominators differ.

Express the focal density.
Under continuous event times, the focal cause-specific hazard satisfies \(f_j(t)=S(t)\lambda_j(t)\), because only event-free individuals can have a first event at time \(t\).

Use the subdistribution denominator.
By definition, \(\tilde{\lambda}_j(t)=f_j(t)/[1-F_j(t)]\). Substitute \(S(t)\lambda_j(t)\) for \(f_j(t)\) to obtain \(\tilde{\lambda}_j(t)=\lambda_j(t)S(t)/[1-F_j(t)]\).

Separate the competing incidence.
Since \(1-F_j(t)=S(t)+\sum_{k\ne j}F_k(t)\), the multiplier is \(S(t)/[S(t)+\sum_{k\ne j}F_k(t)]\). It is at most one and decreases as the competing cumulative incidence increases, if \(S(t)\) and the focal cumulative incidence are considered in this fixed-time comparison.

This derivation also explains an important boundary case. If no competing events have occurred by \(t\), then \(\sum_{k\ne j}F_k(t)=0\), so \(S(t)=1-F_j(t)\) and \(\tilde{\lambda}_j(t)=\lambda_j(t)\). Once competing events have occurred, the multiplier can be below one. This is a statement about the two hazard definitions at that time, not a general equivalence between their interpretations.

Interpretation, Alternatives, and Diagnostics

The cause-specific hazard is useful for describing the instantaneous focal-event rate among those still free of any event. The subdistribution hazard is tied to the focal cumulative incidence and its special denominator. If the scientific question concerns the actual probability of the focal event by a specified time, the cumulative incidence itself is the direct quantity to interpret. Neither hazard should be described as a probability.

A practical check is to verify the probability accounting at each time: \(S(t)+\sum_k F_k(t)=1\), up to rounding, and \(0\leq F_j(t)\leq1\). Then verify that \(f_j(t)=S(t)\lambda_j(t)\) and that \(1-F_j(t)\geq S(t)\). If the calculated subdistribution hazard exceeds the cause-specific hazard under these assumptions, inspect the cause totals, the denominator, and the units; the identity requires \(\tilde{\lambda}_j(t)\leq\lambda_j(t)\). A zero denominator means \(F_j(t)=1\), so the subdistribution hazard is not defined by this ratio at that time.

These checks concern the mathematical quantities. In data analysis, estimation additionally depends on how event times, causes, and censoring are observed and handled. Misclassified causes or incorrectly treating competing events as event-free follow-up can distort the inputs. The identity does not resolve those data-quality issues, nor does it establish a causal effect of a competing event.

Common Mistakes and Limitations

Calling the subdistribution hazard a probability. It is a rate, with units such as per year. It is not the probability of a focal event in the next year.
Using only the focal cause-specific hazard. The identity contains the multiplier \(S(t)/[1-F_j(t)]\), which depends on the incidence of all competing causes.
Putting \(S(t)\) in the subdistribution denominator. The denominator is \(1-F_j(t)\), not the probability of being event-free. Using \(S(t)\) instead gives the cause-specific hazard.
Claiming an unconditional direction from a snapshot comparison. A competing hazard can change the focal cumulative incidence as well as event-free probability. State what is held fixed before asserting that a subdistribution hazard is lower.

Key takeaway: At a given time, \(\tilde{\lambda}_j(t)=\lambda_j(t)S(t)/[1-F_j(t)]\). Competing-event incidence enters through the ratio of event-free probability to the probability of not yet having had the focal event. Thus the subdistribution hazard can change even when the focal cause-specific hazard is unchanged.

Check Your Understanding

Answer each question, then compare your reasoning with the solutions below.

At a given time, \(F_j(t)=0.10\), the total competing-cause incidence is \(0.15\), and \(\lambda_j(t)=0.40\) per year. Calculate \(S(t)\), \(f_j(t)\), and \(\tilde{\lambda}_j(t)\).
Why can the subdistribution hazard differ from the cause-specific hazard even though both use the focal cause?
If the competing-cause incidence increases while \(F_j(t)\) and \(\lambda_j(t)\) are fixed, what happens to the subdistribution hazard?
A calculation gives \(\tilde{\lambda}_j(t)>\lambda_j(t)\) under the continuous-time competing-risks definitions. Name a quantity or step to check.

Solutions. (1) \(S(t)=1-0.10-0.15=0.75\). Then \(f_j(t)=0.75(0.40)=0.30\) per year, and \(\tilde{\lambda}_j(t)=0.30/(1-0.10)=0.3333\) per year. (2) The cause-specific hazard divides the focal-event density by \(S(t)\), while the subdistribution hazard divides it by \(1-F_j(t)\), which also includes people who experienced competing causes. (3) It decreases in this fixed-\(F_j\), fixed-\(\lambda_j\) comparison, because \(S(t)=1-F_j(t)-\sum_{k\ne j}F_k(t)\) decreases. (4) Check whether competing cumulative incidences were omitted, whether \(S(t)\) and \(1-F_j(t)\) were confused, and whether the probabilities sum consistently; the identity requires \(S(t)\leq1-F_j(t)\).