Cause-Specific Hazards Across Different Time Scales

Why the Clock Changes the Question

A hazard is a rate for individuals who remain event-free just before a specified time. But “time” needs an origin. In a clinical study, it might mean years since enrollment, years since diagnosis, or attained age. These clocks can describe the same observed events while answering different questions: how does the event rate evolve during follow-up, or how does it vary with age or another prespecified origin?

The distinction matters in competing-risks analyses because a cause-specific hazard is conditional on still being free of every event type. Its interpretation therefore depends both on which cause is being considered and on which individuals are in the risk set at the clock time. The earlier material on cause-specific hazards establishes this event-free conditioning; here the focus is how the choice of clock changes the time at which that rate is evaluated.

Definition: Let $T$ be time from a prespecified origin to the first event of any type, and let $J$ label the event cause. On the origin-time scale, the cause-specific hazard for cause $k$ at time $t$ is $$ \lambda_k(t)=\lim_{\Delta t\downarrow 0} \frac{P(t\leq T<t+\Delta t,\ J=k\mid T\geq t)}{\Delta t}. $$ It is an instantaneous rate among individuals who have experienced no event before $t$, not a probability. If time is measured from study entry instead, define $S$ as elapsed time since entry and define the hazard by conditioning on being enrolled and still event-free at elapsed time $S$. The clock and the conditioning population should be stated when reporting either hazard.

Both hazards have units of inverse time, such as events per person-year. The number on the clock is not itself the rate: a duration of two years since entry may correspond to age 42 for one participant and age 72 for another. If entry occurs at origin time $A$, elapsed time $s$ corresponds to origin time $A+s$. That simple translation is exact for a fixed entry time, but a study with different entry times generally has a changing mixture of origin times in its risk set.

Fixed Entry Time: Translating the Clock

Suppose everyone enters at the same origin time $a$, and the origin-time hazard for cause $k$ is defined for the same underlying population being followed after entry. At elapsed time $s$, the corresponding origin time is $a+s$. The hazard on the elapsed-time scale is therefore the origin-time hazard evaluated at that translated time, within the group eligible and event-free at entry. This does not mean that the hazard at elapsed time $s$ equals the origin-time hazard at time $s$.

\lambda^{E}_k(s\mid A=a,\ \text{enrolled}) = \lambda_k(a+s\mid T\geq a,\ \text{enrolled},\ T\geq a+s).

Here $\lambda^{E}_k$ denotes a hazard indexed by elapsed time. The conditioning makes explicit that the entrants have reached time $a$ without an event. When the hazard depends only on origin time and the entry process does not otherwise select people with different hazards, this is the origin-time hazard at $a+s$, for those still event-free. If enrollment selects people according to characteristics related to their future event rates, the entrant-specific hazard need not equal the source-population hazard at that origin time.

Worked Example: Enrollment at Age 63

Consider a hypothetical cohort in which everyone enrolls at age 63. Suppose the cause-specific hazard for a particular event, conditional on being free of all event types, is $0.020$ per year at ages 60 through just before 65, and $0.060$ per year from age 65 through just before 70. Assume these hazards apply to the enrolled cohort and that age is the relevant origin-time clock.

At elapsed follow-up $s=1$ year, participants are age $63+1=64$. The hazard for the event is therefore $0.020$ per year. At elapsed follow-up $s=2$ years, they are age $63+2=65$, so the hazard is $0.060$ per year under the stated piecewise model.

The increase at two years is a change in the rate indexed by attained age, not evidence that two years of study participation itself caused the rate to rise. On the elapsed-time scale, the fitted hazard has a transition at $s=2$; on the age scale, it has a transition at age 65. Both descriptions refer to the same hypothetical pattern, but emphasize different clocks.

Different Entry Times: The Risk-Set Mixture

When people enter at different origin times, everyone at elapsed time $s$ has not reached the same origin time. A participant who entered at origin time $a$ is at origin time $a+s$, while another who entered later is at a later origin time. The elapsed-time hazard pooled across participants is consequently a weighted average of the origin-time hazards among people who remain at risk. The weights are not necessarily the enrollment proportions: participants can experience events at different rates and thereby leave the risk set at different speeds.

To make this precise, let $A$ denote an individual's entry time on the origin-time clock. For a group enrolled at time $a$, write $\lambda_k(a+s\mid A=a,\text{enrolled},T\geq a+s)$ for its cause-specific hazard at elapsed time $s$, conditional on still being event-free. Let $w(a,s)$ be the proportion of the elapsed-time risk set at $s$ that entered at $a$. For discrete entry-time groups, the pooled hazard is

\lambda^E_k(s) = \sum_a w(a,s)\, \lambda_k(a+s\mid A=a,\text{enrolled},T\geq a+s), \qquad \sum_a w(a,s)=1.

This identity follows by conditioning on entry time: over a very short interval, each entry-time group contributes its current risk-set size multiplied by its cause-specific rate. Dividing the sum of those contributions by the total risk-set size gives the weighted average. Because both the evaluated origin time $a+s$ and the weights $w(a,s)$ can vary, a pooled elapsed-time hazard need not equal the origin-time hazard at any single time.

Worked Example: A Pooled Rate at the Start of Follow-Up

Suppose a hypothetical study enrolls 100 event-free people. Eighty enter at age 50, and 20 enter at age 70. At the instant of entry, assume the cause-specific hazard for a specified cause is $0.010$ per year in the age-50 group and $0.050$ per year in the age-70 group. These rates are defined among people free of all event types, and all 100 are event-free and at risk at elapsed time zero.

The risk-set weights at $s=0$ are $80/100=0.80$ and $20/100=0.20$. Thus the pooled elapsed-time hazard at entry is $$ \lambda^E_k(0) =(0.80)(0.010)+(0.20)(0.050) =0.008+0.010 =0.018\ \text{per year}. $$

The result is an average rate for this particular entrant risk set at its start. It is not the age-50 hazard, the age-70 hazard, or a rate at one common attained age. In a short interval of length $\Delta$ years, the expected number of events of this cause is approximately $100(0.018)\Delta$, provided the rates are approximately constant over that short interval. For longer follow-up, the weights can change as people experience events, so the initial calculation cannot simply be carried forward.

A Short Justification for the Mixture Formula

The weighted-average result is a direct consequence of the definition of a cause-specific hazard. Consider the risk set at elapsed time $s$, divided into entry-time groups. For a small interval of length $\Delta$, the conditional probability of a cause-$k$ event in group $a$ is approximately $\lambda_{k,a}^E(s)\Delta$. If $Y_a(s)$ is the number at risk in that group, the expected number of such events across groups is approximately $\sum_a Y_a(s)\lambda_{k,a}^E(s)\Delta$. Dividing by the total risk set $\sum_a Y_a(s)$ and by $\Delta$ yields the pooled rate as the risk-set-weighted average. Taking the limit as $\Delta$ approaches zero gives the identity.

\lim_{\Delta\downarrow 0} \frac{\sum_a Y_a(s)\lambda_{k,a}^E(s)\Delta} {\left(\sum_a Y_a(s)\right)\Delta} = \sum_a \frac{Y_a(s)}{\sum_b Y_b(s)}\lambda_{k,a}^E(s).

This argument describes a pooled hazard; it does not say that the mixture can be interpreted as a hazard for a person with an “average” entry age. Nor does it imply that the weights remain fixed. The risk-set counts are those still event-free at $s$, so competing events as well as cause $k$ affect who contributes to later risk sets.

Worked Example: A Follow-Up Pattern That Reflects Age

Suppose a hypothetical disease process has a cause-specific hazard of $0.015$ per year before age 60 and $0.045$ per year from age 60 onward. Assume people enter event-free at age 58, and these rates describe their hazards after enrollment. Consider only follow-up through age 61.

For the first two years, elapsed times $0\leq s<2$ correspond to ages $58\leq 58+s<60$. The cause-specific hazard over that period is $0.015$ per year. At $s=2$, participants reach age 60; for $2\leq s\leq 3$, the hazard is $0.045$ per year, conditional on still being event-free.

A model using elapsed time would represent the change at two years after entry. A model using attained age would represent it at age 60. For this fixed-entry cohort, the two descriptions align through $t=58+s$. If entry ages varied, the same age-related change would occur at different elapsed follow-up times in different participants, and a pooled elapsed-time pattern could smooth or shift that transition.

Choosing and Checking the Time Scale

The choice of clock should follow the scientific question and the design, not whichever scale makes a plot look smoother. Time since entry is natural when the question concerns the rate over follow-up from a common study milestone. Attained age may be more informative when age is central to the process and participants enroll at different ages. Another prespecified origin, such as time since diagnosis, may be useful when the relevant process is tied to that event. These scales are not interchangeable labels: they define different ways of indexing the risk sets and comparing rates.

A comparison across groups also requires care. If one group enters at older ages than another, a difference in duration-scale hazards can reflect their different age composition, their different age-specific rates, or both. Conversely, comparing rates at the same attained age may involve different amounts of prior follow-up and different selected risk sets. State the clock, the event-free conditioning, the population, and the time units before interpreting a reported cause-specific hazard.

Useful checks include plotting or tabulating entry times, checking how entry-time composition changes among people still at risk, and comparing plausible time-scale specifications. If an elapsed-time pattern shows a sharp change at a common duration despite widely varying entry ages, ask whether a study milestone could explain it. If the scientific expectation is an age-related pattern, examine whether the change aligns more consistently on the attained-age scale. These are diagnostic clues, not proofs of a particular mechanism.

Clock	Time value for a participant	Question emphasized	Potential interpretive issue
Since study entry	Years since enrollment	How the rate changes during follow-up	Different entry ages can mix different origin times
Attained age	Age at the current time	How the rate varies with age	Entry at older ages selects people who reached those ages event-free
Since diagnosis	Time since diagnosis	How the rate varies over the post-diagnosis course	Diagnosis timing and study entry may differ

Common Mistake and Limitations

Wrong reasoning: “The cause-specific hazard is $0.04$ at two years, so it is the hazard at origin time two.” This treats elapsed time since entry as if it were time since the prespecified origin.
Correction: If entry is at origin time $a$, elapsed time $s$ corresponds to origin time $a+s$. With varying entry times, the pooled duration hazard averages hazards at different origin times using the current risk-set composition.
Wrong reasoning: “The enrollment proportions are the weights for the pooled hazard throughout follow-up.” Those are the weights at the start only if all participants begin at risk then. Later weights depend on who remains event-free.
Correction: Use the entry-time composition of the risk set at the time being interpreted. If entry times are continuous, the weighted sum is replaced by an average over the entry-time distribution among those still at risk.

A chosen clock does not by itself fix problems caused by selective enrollment, incomplete event observation, or differences in the populations being compared. In particular, a hazard among people enrolled at age 70 describes those who reached enrollment event-free and were selected into the study; it need not describe everyone in the source population at age 70. Estimating hazards from censored data also requires appropriate assumptions about censoring. These issues are distinct from the arithmetic translation between clocks.

Key takeaway: A cause-specific hazard is indexed by a chosen clock and conditioned on remaining event-free. With a fixed entry time, elapsed time translates to origin time by addition. With varying entry times, the pooled elapsed-time hazard is a risk-set-weighted mixture, so name the clock and the risk-set population whenever interpreting it.

Check Your Understanding

Answer each question, then compare with the brief solutions below.

A cohort enters at age 45. What age corresponds to three years since entry?
Why might a pooled hazard by time since entry fail to equal the origin-time hazard at one particular age?
In a risk set, 60 people have a cause-specific hazard of $0.020$ per year and 40 have a hazard of $0.050$ per year. Calculate the pooled hazard at that instant.
A hazard change occurs at the same attained age among participants enrolled at different ages, but at different durations of follow-up. Which clock gives the more direct description of this pattern?

Solutions. (1) Age $45+3=48$. (2) People at the same elapsed follow-up time can have different entry times and therefore different origin times; the pooled hazard averages their hazards with weights determined by the current event-free risk set. (3) The weights are $0.60$ and $0.40$, so the pooled rate is $(0.60)(0.020)+(0.40)(0.050)=0.032$ per year. (4) Attained age gives the more direct description, because the change aligns at a common age rather than a common duration since entry.