Why a Constant-Rate Model Is Useful
A cause-specific hazard describes the instantaneous rate of a particular event among people who have not yet experienced any event. A subdistribution hazard describes the focal event rate using a different denominator: people with the focal event are removed, but those who have already had a competing event remain represented in the risk set. With constant cause-specific hazards, these two descriptions can be connected by explicit formulas. The example makes visible an important point: a competing cause can change the focal cause’s cumulative incidence and subdistribution hazard even when the focal cause-specific hazard stays fixed.
Consider two mutually exclusive first-event causes, labelled 1 and 2. The rates below are population-level model quantities, not observed estimates. Assume the event process has no other causes, the cause labels are correctly recorded, and simultaneous events have probability zero. There are no covariates in this simple model, and each cause-specific hazard is constant over time. These assumptions define a convenient model; they do not assert that latent event times for the two causes are independent.
Deriving Incidence and the Subdistribution Hazard
While a person remains event-free, the instantaneous rate of either event is the sum \(r\). Thus event-free survival \(S(t)\) satisfies \(S'(t)=-rS(t)\), with \(S(0)=1\). Solving gives \(S(t)=e^{-rt}\). Cause 1 can occur at time \(t\) only in someone who is event-free just before \(t\); the cause-1 event density is therefore \(f_1(t)=S(t)\lambda_1=\lambda_1e^{-rt}\). Integrating this density gives the cumulative incidence.
If \(r=0\), nonnegativity of the rates implies that both rates are zero, so no event occurs and \(F_1(t)=0\). For \(r>0\), substituting the density and cumulative incidence into the definition of the subdistribution hazard yields a useful closed form.
The exponent in the final expression is positive. To see the algebra, the denominator of the first ratio is \[ 1-\frac{\lambda_1}{r}+\frac{\lambda_1}{r}e^{-rt} = \frac{\lambda_2+\lambda_1e^{-rt}}{r}. \] Multiplying numerator and denominator of the ratio by \(e^{rt}\) gives the displayed form. When \(r=0\), the density is zero and \(1-F_1(t)=1\), so define \(\alpha_1(t)=0\). This separate definition avoids an indeterminate \(0/0\) in the closed-form expression.
The denominator \(1-F_1(t)\) includes both people who remain event-free and people who experienced cause 2 first. Consequently, \(\alpha_1(t)\) is a rate, not the probability of cause 1 in a time interval. Its units are inverse time, the same as those of \(\lambda_1\) and \(\lambda_2\).
Worked Example: Calculating Incidence and the Subdistribution Hazard
Suppose, hypothetically, that \(\lambda_1=0.08\) per year and \(\lambda_2=0.12\) per year. Find the cause-1 cumulative incidence and subdistribution hazard at five years.
Here \(r=0.08+0.12=0.20\) per year. Event-free survival at five years is \(e^{-0.20(5)}=e^{-1}=0.3679\). Hence \[ F_1(5)=\frac{0.08}{0.20}(1-e^{-1}) =0.4(0.6321)=0.2528. \] The probability of cause 1 as the first event by five years is about \(0.253\).
The cause-1 density at five years is \(f_1(5)=0.08e^{-1}=0.02943\) per year. Since \(1-F_1(5)=0.7472\), the defining ratio gives \[ \alpha_1(5)=\frac{0.02943}{0.7472}=0.03939 \quad\text{per year}. \] Using the closed form gives the same result: \[ \alpha_1(5)=\frac{0.08(0.20)}{0.08+0.12e^{1}} =\frac{0.016}{0.4062}=0.03939 \quad\text{per year}. \] As a probability check, the cause-2 cumulative incidence is \(0.12/0.20(1-e^{-1})=0.3793\). Together with event-free survival, the probabilities sum to \(0.2528+0.3793+0.3679=1.0000\), up to rounding.
What Changes When the Competing Rate Changes?
Hold the focal rate \(\lambda_1\) fixed and increase \(\lambda_2\). For \(t>0\) and \(\lambda_1>0\), the focal cumulative incidence can be written as an integral over the event-free probability at each time. Differentiating with respect to the competing rate shows why the incidence falls.
The competing event makes fewer people available to experience cause 1 later, so the probability of cause 1 occurring first declines. The subdistribution hazard also declines at positive times. Differentiating its closed form, for \(r>0\), gives
At \(t=0\), the subdistribution hazard equals \(\lambda_1\), regardless of \(\lambda_2\): no time has yet elapsed for the competing event to alter the denominator. If \(\lambda_2=0\) and \(\lambda_1>0\), then \(\alpha_1(t)=\lambda_1\) at every finite time, because the only possible event is the focal one. If \(\lambda_2>0\), the formula shows that \(\alpha_1(t)\) tends to zero as \(t\) grows without bound. This is a feature of the subdistribution denominator and does not mean the focal cause-specific hazard has declined: that hazard remains \(\lambda_1\) in the model.
Worked Example: Comparing Two Competing Rates
Consider two hypothetical populations with the same focal cause-specific hazard, \(\lambda_1=0.08\) per year. In the first, \(\lambda_2=0.04\) per year; in the second, \(\lambda_2=0.16\) per year. Compare cause-1 incidence and subdistribution hazard at five years.
For the first population, \(r=0.12\) per year: \[ F_1(5)=\frac{0.08}{0.12}(1-e^{-0.60}) =0.6667(0.4512)=0.3008. \] Its subdistribution hazard is \[ \alpha_1(5)=\frac{0.08(0.12)}{0.08+0.04e^{0.60}} =\frac{0.0096}{0.1529} =0.06279\quad\text{per year}. \]
For the second population, \(r=0.24\) per year: \[ F_1(5)=\frac{0.08}{0.24}(1-e^{-1.20}) =0.3333(0.6988)=0.2329. \] Its subdistribution hazard is \[ \alpha_1(5)=\frac{0.08(0.24)}{0.08+0.16e^{1.20}} =\frac{0.0192}{0.6112} =0.03141\quad\text{per year}. \] Thus the higher competing rate corresponds here to lower cause-1 incidence and a lower cause-1 subdistribution hazard at five years, while the cause-1 cause-specific hazard is \(0.08\) per year in both populations. These comparisons describe model-based associations between the rates and the resulting quantities; they do not by themselves establish a causal effect of changing the competing rate.
Worked Example: Zero-Rate Boundary Cases
Suppose first that the hypothetical rates are \(\lambda_1=0\) and \(\lambda_2=0.09\) per year. The total rate is positive, so the closed forms apply. For every finite \(t\), \[ F_1(t)=\frac{0}{0.09}(1-e^{-0.09t})=0, \qquad \alpha_1(t)=\frac{0(0.09)}{0+0.09e^{0.09t}}=0. \] There is no cause-1 event under this model, though cause 2 can occur.
If instead both rates are zero, then \(r=0\), and the ratio formulas involving \(r\) are not used. No event occurs, so \(F_1(t)=0\), \(f_1(t)=0\), and \(\alpha_1(t)=f_1(t)/(1-F_1(t))=0/1=0\). Treating this boundary case directly is necessary; substituting zero rates into a formula with a zero denominator would be invalid.
Checks, Comparisons, and Limitations
Several checks help detect algebra or interpretation errors. The cause-specific hazards must be nonnegative and expressed in the same time units. For \(r>0\), the cause-1 incidence must be between zero and \(\lambda_1/r\), its long-run limit in this model. The event-free probability and both cause-specific cumulative incidences must sum to one. The subdistribution hazard must have units of inverse time and be nonnegative. When \(\lambda_2=0\), it must reduce to \(\lambda_1\); when \(\lambda_1=0\), it must equal zero. These checks also expose a copied formula with \(e^{-rt}\) rather than \(e^{rt}\) in the final expression for \(\alpha_1(t)\).
The main quantities answer different questions. The cause-specific hazard \(\lambda_1\) is the instantaneous cause-1 rate among those still event-free; under this model it is constant. The cumulative incidence \(F_1(t)\) is an absolute probability by a specified time and incorporates the competition between causes. The subdistribution hazard \(\alpha_1(t)\) is the density of cause-1 events divided by the probability of not yet having had cause 1; it incorporates the competing rate through that denominator. None of these quantities is interchangeable with the others.
| Quantity | Denominator or interpretation | Constant-rate expression |
|---|---|---|
| Cause-1 hazard | Event-free people; instantaneous cause-1 rate | \(\lambda_1\) |
| Cause-1 cumulative incidence | Probability cause 1 occurs first by \(t\) | \(\lambda_1(1-e^{-rt})/r\), if \(r>0\) |
| Cause-1 subdistribution hazard | Density divided by \(1-F_1(t)\) | \(\lambda_1r/(\lambda_1+\lambda_2e^{rt})\), if \(r>0\) |
A common mistake is to say that increasing \(\lambda_2\) must change \(\lambda_1\), because fewer cause-1 events are observed. That reasoning confuses the event-free risk set with the probability of eventually experiencing a particular cause first. Under the stated model, the cause-1 cause-specific hazard stays fixed at \(\lambda_1\); the competing rate changes event-free survival and therefore changes \(F_1(t)\) and \(\alpha_1(t)\). Another mistake is to call \(\alpha_1(t)\) a probability. It is a hazard rate; a short-interval probability is approximately \(\alpha_1(t)\,dt\) only under the corresponding small-interval approximation.
The formulas depend on constant cause-specific rates, correctly identified first-event causes, and the two-cause setup. Real rates may vary with time or covariates, and averaging heterogeneous rates need not produce this constant-rate model. The calculations describe the event process; analyses of censored observations additionally require appropriate handling of censoring. In practice, compare model implications with observed event patterns rather than assuming the constant-rate form is adequate.
Check that \(\lambda_1\) and \(\lambda_2\) are nonnegative, have the same time unit, and refer to the event-free population.
Set \(r=\lambda_1+\lambda_2\). If \(r=0\), both incidences and subdistribution hazards are zero.
For \(r>0\), use \(F_1(t)=\lambda_1(1-e^{-rt})/r\) and \(\alpha_1(t)=\lambda_1r/(\lambda_1+\lambda_2e^{rt})\).
Report incidence as a probability by time \(t\), report hazards in inverse-time units, and verify boundary cases and probability totals.
Check Your Understanding
Try the questions first, then compare your reasoning with the solutions.
- For \(\lambda_1=0.05\) and \(\lambda_2=0.05\) per month, what is \(F_1(4)\)?
- At \(t=0\), what is the subdistribution hazard, and why does it not depend on \(\lambda_2\)?
- If \(\lambda_1=0.06\) per year and \(\lambda_2=0\), what is \(\alpha_1(t)\) for finite \(t\)?
- Why is \(\alpha_1(t)\) not a probability, and what does it mean if it decreases as time passes?
Solutions. (1) Here \(r=0.10\) per month, so \(F_1(4)=0.05/0.10(1-e^{-0.4})=0.5(0.3297)=0.1648\), or about \(0.165\). (2) \(\alpha_1(0)=\lambda_1\), since \(F_1(0)=0\) and \(f_1(0)=\lambda_1\); no time has elapsed for cause 2 to affect the subdistribution denominator. (3) With no competing cause, \(\alpha_1(t)=\lambda_1=0.06\) per year. (4) It is a density-to-survival ratio with units of inverse time, not a probability. A decline can reflect the evolving subdistribution denominator and the competing-event process; it does not imply that the cause-specific hazard has declined.