Clock-Reset And Clock-Forward Time

Why the Time Origin Matters

In a multi-state model, a transition hazard is measured at a particular time scale. That choice determines which histories are treated as comparable. For a person who becomes ill, should the death hazard be indexed by time since the study began, or by time since illness began? These are different questions: the first compares people at the same study time, while the second compares people at the same duration of illness.

Consider an illness-death process with state 0 for healthy, state 1 for ill, and state 2 for dead. The allowed transitions are \(0\to1\), \(0\to2\), and \(1\to2\); state 2 is absorbing. Let \(t\) denote time since a fixed analysis origin, such as study entry. If an individual enters state 1 at time \(a\), their duration in that state at time \(t\) is \(u=t-a\). The distinction between \(t\) and \(u\) is central: clock-forward uses \(t\), and clock-reset uses \(u\) for transitions out of the current state.

Definition: Under a clock-forward specification, transition intensities are indexed by time \(t\) from a fixed origin; entering a new state does not restart the clock. Under a clock-reset specification, the clock for transitions out of a state starts at zero upon entry to that state, so the relevant time is the current sojourn duration \(u=t-a\). These terms describe the time argument used in the model, not different definitions of a transition intensity.

For example, let \(\lambda_{12}(t)\) denote the instantaneous hazard of death while ill under a clock-forward model, and let \(\lambda_{12}(u)\) denote the corresponding hazard under a clock-reset model. The subscript \(12\) indicates a transition from state 1 to state 2. In either case, an intensity is a rate per unit time, conditional on currently being in state 1 and on the relevant time history. It is not itself a probability.

Sojourn Survival on Each Clock

In state 1, death is the only allowed exit in this three-state illness-death model. More generally, write \(\lambda_{1\cdot}\) for the sum of all intensities of transitions out of state 1. Conditional on entering state 1 at time \(a\), the probability of remaining there for at least another \(v\) time units is obtained by accumulating that total exit hazard over the interval.

S_1(v\mid a)= \begin{cases} \exp\left\{-\displaystyle\int_a^{a+v}\lambda_{1\cdot}(r)\,dr\right\}, & \text{clock-forward},\\[1.2em] \exp\left\{-\displaystyle\int_0^v\lambda_{1\cdot}(w)\,dw\right\}, & \text{clock-reset}. \end{cases}

The reset expression assumes the intensity after entry is a function of elapsed duration, with the duration clock set to zero at entry. For the forward expression, the same duration \(v\) covers calendar times from \(a\) through \(a+v\), so the entry time can affect the resulting survival probability. The expressions follow from the survival differential equation: if \(S(v)\) is survival in the state, then \(dS(v)/dv=-S(v)\lambda_{1\cdot}\) evaluated at the appropriate clock time. Dividing by \(S(v)\) and integrating gives the displayed exponential.

If state 1 has more than one possible exit, the total hazard is the sum of the cause-specific transition intensities. The probability of remaining in state 1 depends on that total, while the particular destination at exit depends on the transition-specific intensities. In the illness-death process here, death is the sole exit from state 1, so \(\lambda_{1\cdot}=\lambda_{12}\).

Worked Example: Comparing Hazards at the Same Study Time

Suppose a hypothetical model describes death while ill using either a clock-forward intensity \(\lambda_{12}(t)=0.04+0.01t\) or a clock-reset intensity \(\lambda_{12}(u)=0.04+0.02u\), with time measured in years. An individual entered the ill state at study time \(a=4\) years. Compare the two instantaneous hazards at study time \(t=7\) years.

Under the forward model, the clock reads \(t=7\):

\lambda_{12}(7)=0.04+0.01(7)=0.11\ \text{per year}.

Under the reset model, the duration in the ill state is \(u=t-a=7-4=3\) years:

\lambda_{12}(3)=0.04+0.02(3)=0.10\ \text{per year}.

At that instant, the specified forward model gives a death hazard of \(0.11\) per year and the specified reset model gives \(0.10\) per year. These are model-based instantaneous rates, not probabilities of dying during a year. The two numerical specifications are illustrative alternatives; the difference does not by itself show which model describes a real process better.

Entry Time Can Change Predicted Risk

A direct implication of the clock choice is whether two people with equal time in the ill state are assigned equal survival probabilities. Under a pure clock-reset specification, their predicted sojourn survival is the same, provided the model gives them the same relevant covariates and history. Under a clock-forward specification, their predictions can differ if they entered the ill state at different study times, because the hazard is accumulated over different calendar-time intervals.

Worked Example: Two Illness-Entry Times

Suppose a hypothetical forward model has \(\lambda_{12}(t)=0.02+0.01t\) per year. Compare the probability of death within the next two years for someone entering state 1 at \(a=2\) years and someone entering it at \(a=6\) years. Assume the person remains under observation and death is the only possible exit from state 1. Also calculate the corresponding probability under a reset model with \(\lambda_{12}(u)=0.02+0.01u\).

For entry at \(a=2\), the integrated forward hazard over the next two years is:

\int_2^4(0.02+0.01t)\,dt =0.02(4-2)+0.005(4^2-2^2) =0.04+0.06=0.10.

The probability of death by two years after entry is one minus the probability of remaining alive and ill:

1-\exp(-0.10)=0.09516.

Thus the forward model predicts about 9.52% death risk over this two-year interval. For entry at \(a=6\), the integrated hazard is:

\int_6^8(0.02+0.01t)\,dt =0.02(8-6)+0.005(8^2-6^2) =0.04+0.14=0.18,

so the predicted two-year death probability is \(1-\exp(-0.18)=0.1647\), or about 16.5%. The difference arises because the two people traverse different study-time intervals under the forward model.

Under the reset model, both people have the same duration interval, from \(u=0\) to \(u=2\):

\int_0^2(0.02+0.01u)\,du=0.04+0.02=0.06, \qquad 1-\exp(-0.06)=0.05824.

The reset model therefore predicts about 5.82% risk for either entry time. Each calculation assumes the displayed intensity is the relevant hazard throughout the interval and that no other transition can remove someone from state 1. In a model with multiple exits, the probability of death by the interval end would require accounting for all exits, not just integrating the death intensity as if it were the sole exit.

Path Likelihoods Show the Clock Explicitly

The time scale also changes the contribution of an observed path to a likelihood. For a path that remains healthy until an illness transition at time \(a\), then dies at time \(d>a\), the likelihood contains (1) the probability of no transition out of state 0 before \(a\), (2) the intensity of the observed \(0\to1\) transition at \(a\), (3) the probability of no transition out of state 1 before \(d\), and (4) the intensity of the observed \(1\to2\) transition at \(d\). The no-transition terms are exponentials of integrated total outgoing intensities. A transition intensity evaluated at an observed event time is a density contribution, not a probability mass.

Worked Example: Likelihood for an Illness-Then-Death Path

Consider a hypothetical individual who is healthy from \(t=0\) until becoming ill at \(a=2\), then dies at \(d=5\). Suppose the healthy-state intensities are constant: \(\lambda_{01}=0.10\) and \(\lambda_{02}=0.05\) per year. Compare two specifications for the ill-to-death intensity: a forward model \(\lambda_{12}(t)=0.02+0.01t\), and a reset model \(\lambda_{12}(u)=0.02+0.01u\). Assume the stated intensities fully specify the outgoing transitions, and there is no censoring before death.

While healthy, the total outgoing intensity is \(0.10+0.05=0.15\) per year. The likelihood contribution for remaining healthy for two years and then transitioning to illness is:

\exp\{-0.15(2)\}\lambda_{01}(2)=\exp(-0.30)(0.10).

In state 1, the individual is observed for three years before dying. Under the forward model, the integrated ill-state hazard is \(\int_2^5(0.02+0.01t)\,dt=0.06+0.105=0.165\), and the intensity at death time \(t=5\) is \(0.02+0.01(5)=0.07\) per year. The full path likelihood contribution is therefore:

L_F=\exp(-0.30)(0.10)\exp(-0.165)(0.07) =0.004397.

Under the reset model, the three-year sojourn corresponds to \(u=0\) through \(u=3\). Its integrated hazard is \(\int_0^3(0.02+0.01u)\,du=0.06+0.045=0.105\), and the intensity at the observed death is \(0.02+0.01(3)=0.05\) per year. Thus:

L_R=\exp(-0.30)(0.10)\exp(-0.105)(0.05) =0.003337.

These are likelihood contributions under two different models for this observed path; they have units involving inverse time because the path includes transition-event densities. The larger contribution for this one path does not establish that the forward specification is preferable. Model assessment requires the full observed data and appropriate comparison, not a conclusion from one individual.

Choosing and Checking the Clock

The clock should correspond to the scientific mechanism and the intended prediction. A forward clock is natural when transition risk is expected to vary with study time or calendar time even after accounting for the current state. A reset clock is natural when the relevant feature is how long the individual has occupied the current state. Neither choice is universally correct. The state labels alone do not determine the time scale.

A useful way to diagnose a mismatch is to compare transition patterns after aligning observations in two ways: by time since the fixed origin and by time since entry into the current state. If patterns line up more consistently by duration, that supports considering a reset-time formulation; if they line up by study time, a forward formulation may be more suitable. Such patterns are diagnostic evidence, not proof of a model. Covariates, selection into a state, and other aspects of the observation process may also affect observed transition patterns.

Data must contain enough variation in entry times to separate the explanations. If every person enters state 1 at the same time \(a\), then forward time and reset time satisfy \(t=a+u\) for everyone in that state. Over the observed range, one clock’s intensity curve can be rewritten as a function of the other clock. The data then provide little or no information for distinguishing the time-scale interpretations without additional restrictions or information.

Question	Clock-forward specification	Clock-reset specification
Time argument for an exit from state 1	Time \(t\) from the fixed origin	Duration \(u=t-a\) since entry to state 1
Do equal state durations imply the same hazard?	Not necessarily; entry times may differ	Yes, under the same reset model inputs
Do equal study times imply the same hazard?	Yes, under the same forward model inputs	Not necessarily; durations may differ
Typical interpretation	Risk changes with position on the study-time scale	Risk changes with time spent in the current state

A common mistake is to calculate a clock-reset prediction using time since study entry, or to calculate a clock-forward prediction using only time since illness entry. The correction is to write down the time origin before evaluating an intensity or integrating it. For an individual entering state 1 at \(a\), explicitly check that the forward integral runs over \(t\in[a,a+v]\), while the reset integral runs over \(u\in[0,v]\). Another mistake is to interpret a hazard value, such as \(0.10\) per year, as a 10% probability of death in the next year. A hazard is an instantaneous rate; even with a constant hazard of \(0.10\) per year and no competing exits, the one-year event probability is \(1-\exp(-0.10)\), about 9.52%.

The clock choice is also distinct from the choice of state structure. Both formulations can describe the same healthy, ill, and dead states, but they can imply different transition probabilities because their hazard integrals differ. Transition-probability calculations must therefore use intensities evaluated on the clock specified by the model. The introductory material on semi-Markov transition kernels provides the broader context for representing transitions according to time spent in a state; here, the key practical issue is aligning the model’s time argument with the history that is scientifically relevant.

Key takeaway: For every transition, identify the fixed origin and the state-entry time before writing its intensity. Clock-forward time is \(t\); clock-reset time is \(u=t-a\). The choice changes both the integrated hazard and the predictions, and should be justified by the process and the information in the data.

Check Your Understanding

Use the illness-death setup above. Each question includes a solution so you can check the time scale and calculation.

An individual enters the ill state at \(t=3\). At \(t=8\), what are the forward time and reset duration?
Under a reset intensity \(\lambda_{12}(u)=0.03+0.01u\) per year, calculate the probability of remaining alive and ill for two years, assuming death is the only exit.
Under a forward intensity \(\lambda_{12}(t)=0.02\) per year, calculate the probability of death within one year after entering the ill state at \(a=4\), assuming death is the only exit.
Why can data in which every individual enters the ill state at the same study time be uninformative for distinguishing the two clocks?

Solutions. (1) The forward time is \(t=8\) years, while the reset duration is \(u=8-3=5\) years. (2) The integrated reset hazard is \(\int_0^2(0.03+0.01u)\,du=0.06+0.02=0.08\). Therefore, the probability of remaining alive and ill is \(\exp(-0.08)=0.9231\), or about 92.3%. (3) The forward hazard is constant, so the one-year integrated hazard is \(0.02(1)=0.02\). The death probability is \(1-\exp(-0.02)=0.01980\), or about 1.98%. (4) If all entry times equal \(a\), then \(t=a+u\) for every observation in state 1. On the observed range, a function of \(t\) can be expressed as a function of \(u\) by shifting its argument, so the observed data may not separate the clock interpretations without further assumptions or variation in entry times.