Why Define Another Hazard?
In competing-risks data, the outcome is the time and cause of the first event. A cause-specific hazard describes the instantaneous rate of cause \(k\) among individuals who are still event-free. The subdistribution hazard is another rate, defined using the cumulative incidence of cause \(k\). It is useful to define it precisely before considering models or interpretations built from it.
The definition involves two quantities: how quickly the cumulative incidence for cause \(k\) is changing at time \(t\), and the fraction of the population that has not yet experienced cause \(k\) by that time. The resulting ratio is a rate. It is not, by itself, a probability of an event during an interval.
The time scale must be specified: if \(t\) is measured in years, \(\lambda_k^*(t)\) has units of inverse years. Its numerator is a rate of change in a dimensionless probability, and its denominator is also dimensionless. The definition applies at times where the derivative exists and the denominator is positive. It does not require the cumulative incidence to be linear or the subdistribution hazard to be constant.
What the Formula Measures
The cumulative incidence \(F_k(t)\) is the probability that cause \(k\) has occurred as the first event by time \(t\). Its derivative \(f_k(t)\) describes the local rate of increase of that probability. Dividing by \(1-F_k(t)\) scales this rate by the complement of the cause-\(k\) cumulative incidence.
A short-interval calculation makes the rate interpretation explicit. If \(F_k\) is differentiable at \(t\), then for a small positive interval length \(\Delta t\), $$ F_k(t+\Delta t)-F_k(t)=f_k(t)\Delta t+o(\Delta t). $$ Therefore, $$ \frac{F_k(t+\Delta t)-F_k(t)}{1-F_k(t)} =\lambda_k^*(t)\Delta t+o(\Delta t). $$ The left side is a normalized increase in the cumulative incidence over the interval; for a sufficiently small interval, its leading term is the subdistribution hazard multiplied by the interval length. This calculation explains why the hazard has rate units. It does not turn the ratio into a conditional probability among people who are event-free at \(t\).
In particular, the denominator \(1-F_k(t)\) is not generally the probability of remaining free of every event. It is the complement of the cumulative incidence for cause \(k\) alone. The distinction matters because a competing cause may already have occurred even though cause \(k\) has not. The exact interpretation of that denominator and its implications for risk sets are separate questions; for now, the definition is the ratio shown above.
Worked Example: Calculating the Rate From a Smooth Curve
Suppose a hypothetical population has a cause-\(k\) cumulative incidence curve \(F_k(t)=0.30(1-e^{-0.4t})\), where \(t\) is measured in years. This is a valid increasing cause-specific cumulative incidence: it starts at zero, increases, and approaches \(0.30\), so it remains below one. Find the subdistribution hazard at \(t=2\) years.
Differentiate the curve: $$ f_k(t)=\frac{d}{dt}\{0.30(1-e^{-0.4t})\}=0.12e^{-0.4t}. $$ At \(t=2\), \(e^{-0.8}\approx0.4493\), so $$ F_k(2)=0.30(1-0.4493)\approx0.1652,\qquad f_k(2)=0.12(0.4493)\approx0.05391\text{ per year}. $$
Substitute into the definition: $$ \lambda_k^*(2)=\frac{0.05391}{1-0.1652} =\frac{0.05391}{0.8348}\approx0.06458\text{ per year}. $$ Thus the subdistribution hazard at two years is approximately \(0.06458\) per year. The value is a rate calculated from the local slope and the cause-\(k\) cumulative incidence, not a \(6.458\%\) probability of the event in a year.
Local Slope and the Denominator
The formula can vary over time because either part of the ratio can vary. A changing slope \(f_k(t)\) changes the numerator. Even with a constant slope, the denominator can change as \(F_k(t)\) increases. The next example isolates that arithmetic feature; it does not assume that a constant slope is a general model for competing-risks data.
Worked Example: A Constant Slope Over an Interval
Consider a hypothetical cause-\(k\) cumulative incidence that follows \(F_k(t)=0.05+0.04t\) for \(0\leq t\leq2\), with \(t\) in years. This segment is nondecreasing and ranges from \(0.05\) to \(0.13\), so it is a possible segment of a cumulative incidence curve. Within the interval, \(f_k(t)=0.04\) per year. Calculate the subdistribution hazard at \(t=0.5\) and \(t=1.5\) years.
At \(t=0.5\), \(F_k(0.5)=0.05+0.04(0.5)=0.07\). Therefore, $$ \lambda_k^*(0.5)=\frac{0.04}{1-0.07} =\frac{0.04}{0.93}\approx0.04301\text{ per year}. $$ At \(t=1.5\), \(F_k(1.5)=0.05+0.04(1.5)=0.11\), giving $$ \lambda_k^*(1.5)=\frac{0.04}{1-0.11} =\frac{0.04}{0.89}\approx0.04494\text{ per year}. $$
The slope is unchanged, but the calculated subdistribution hazard is slightly larger at \(1.5\) years because the denominator is smaller. This illustrates why a flat or changing cumulative-incidence slope alone does not determine the behavior of the ratio.
Using Tabulated Values Carefully
In practice, one may have estimated cumulative incidence values at discrete time points rather than a formula that can be differentiated. A finite difference can approximate the derivative over a short interval. The resulting hazard is an approximation, not an exact pointwise derivative. Its quality depends on the interval width and on how smoothly the underlying curve changes.
Worked Example: A Finite-Difference Approximation
Suppose hypothetical estimated values are \(F_k(2.0)=0.120\) and \(F_k(2.1)=0.126\), where time is in years. Approximate the slope over this interval by the difference quotient: $$ \frac{F_k(2.1)-F_k(2.0)}{2.1-2.0} =\frac{0.126-0.120}{0.1} =0.060\text{ per year}. $$
Use the midpoint value \(F_k(2.05)\approx(0.120+0.126)/2=0.123\) to approximate the denominator. The subdistribution hazard near the midpoint is then $$ \lambda_k^*(2.05)\approx\frac{0.060}{1-0.123} =\frac{0.060}{0.877}\approx0.06842\text{ per year}. $$
This is a local approximation based on two tabulated values. It would be less credible if the interval were wide and the curve sharply bent within it, or if the estimated values were too noisy to give a stable slope. Reporting the interval used makes the approximation more transparent.
Assumptions, Scope, and Comparison
The definition is a population-level quantity for a specified first-event outcome. To use it as written, the event causes must be defined as mutually exclusive first-event types, \(F_k(t)\) must be a valid cumulative incidence function, and it must be differentiable at the time being considered. The denominator must satisfy \(1-F_k(t)>0\). No assumption of independence between hypothetical latent event times is needed for this definition: it is formed from the distribution of the observed first event and its cause.
The subdistribution hazard is distinct from the cause-specific hazard, which uses the event-free population as its conditioning group. Their denominators encode different quantities, so their values answer different questions and need not coincide. The distinction can be summarized without treating either rate as a probability:
| Quantity | Numerator or local event information | Denominator |
|---|---|---|
| Cause-specific hazard for cause \(k\) | Instantaneous rate of cause \(k\) | Probability of remaining free of every first event through \(t\) |
| Subdistribution hazard for cause \(k\) | Derivative \(f_k(t)\) of the cause-\(k\) cumulative incidence | \(1-F_k(t)\), the complement of that cause’s cumulative incidence |
This comparison is a reminder about definitions, not a rule for converting one hazard into the other. The subdistribution hazard can be defined even when no proportional-hazards model is being assumed. A particular model may impose additional assumptions, but they do not follow from the definition itself.
Diagnostics and Common Pitfalls
Before interpreting or calculating a value, check that the inputs behave like a cumulative incidence curve: \(F_k(t)\) should be between zero and one and should not decrease as time increases. At the time of interest, verify that the derivative or its approximation is meaningful and that \(1-F_k(t)\) is positive. A negative slope, a cumulative incidence above one, or a zero denominator signals an invalid input or a calculation outside the definition’s domain.
A further issue is the time structure of the data. If the cumulative incidence has jumps, as an empirical step curve does, an ordinary derivative may be zero between jumps and undefined at a jump. The derivative-based definition therefore cannot be applied mechanically at every time point to a step function. A suitable estimation or smoothing approach must address the time structure; a finite difference is only an approximation and can depend strongly on the chosen interval.
The definition is useful because it provides a rate-scale summary tied directly to the cumulative incidence for a specified cause. Its limitations follow from the same construction: the value depends on the local derivative, can be sensitive to how that derivative is estimated, and does not by itself supply a probability over a finite follow-up interval. For absolute risk, the cumulative incidence remains the direct probability quantity.
Check Your Understanding
Answer each question, then compare your reasoning with the solutions.
- If \(F_k(t)=0.20\) and \(f_k(t)=0.05\) per year at a particular time, calculate the subdistribution hazard and state its units.
- A cause-\(k\) cumulative incidence has derivative \(0.03\) per month and value \(0.25\). What is the subdistribution hazard at that time?
- Why is a finite-difference calculation from two cumulative incidence estimates an approximation rather than necessarily an exact pointwise hazard?
- A learner says a subdistribution hazard of \(0.08\) per year means an 8% probability of cause \(k\) during the next year. Explain the error.
Solutions. (1) \(\lambda_k^*(t)=0.05/(1-0.20)=0.0625\) per year. (2) \(\lambda_k^*(t)=0.03/(1-0.25)=0.04\) per month. (3) The difference quotient estimates a local slope over an interval; it equals the derivative at a particular time only under additional conditions, such as a linear curve over that interval. (4) A hazard has units of inverse time and is not a finite-interval probability; the 8% statement cannot be read directly from the rate without further assumptions and calculation.