Why the Clock Is Part of the Model
A multi-state model describes how a process moves among states, but it also needs a time scale. Time may mean years since enrollment, months since diagnosis, or age since birth. These choices are not merely labels: they determine what the model’s time index tells us about each individual at a given observation.
The Markov assumption says that, given the current state and the current time, the future does not additionally depend on the earlier path. Whether that is plausible can depend on the clock. If transition rates change with age, for example, age may be a more informative time index than time since study entry. If a transition rate depends on how long someone has already been ill, a clock that does not reveal that duration may leave relevant history out of the model.
The Markov property and time-homogeneity are distinct. A process can satisfy a Markov assumption while its transition law varies with time. Changing a clock can therefore affect the information used to condition on the future even when the model is allowed to be time-inhomogeneous.
In a continuous-time model, the same idea is often expressed through transition intensities. For distinct states \(r\) and \(s\), the intensity \(q_{rs}(t)\) is the instantaneous rate of moving from \(r\) to \(s\), conditional on being in \(r\) at time \(t\). Under a Markov model, this rate is determined by the current state and time, not by an unrecorded duration or earlier route.
Changing a Common Time Origin
A common, known shift of the time origin does not by itself change whether a process is Markov. Suppose every observation uses \(u=t-c\), where the same constant \(c\) applies to everyone. Knowing \(u\) is then equivalent to knowing \(t\), because \(t=u+c\). A transition law that depends on \(t\) can be written as a law depending on \(u\), with its time argument shifted.
This equivalence concerns a common shift of the time coordinate. It does not mean that every choice of origin is interchangeable in a study. If each person’s time zero is set at a different event, the resulting time index may not be recoverable from the original time index alone. Nor does changing the origin automatically encode time already spent in a state.
Worked Example: A Common Shift of the Time Origin
Suppose a two-state process has transition intensity \(q_{01}(t)=0.01+0.002t\) per year on the original clock, where time \(t\) is measured in years after a common study start. The analysis team instead sets time zero three years later, so \(u=t-3\). The process is assumed Markov on the original clock.
Since \(t=u+3\), the intensity on the new clock is $$ q^{*}_{01}(u)=q_{01}(u+3)=0.01+0.002(u+3). $$ At \(u=2\), the original time is \(t=5\), and the intensity is $$ q^{*}_{01}(2)=0.01+0.002(5)=0.0200 $$ per year.
The state and time on the new clock determine the same information as the state and time on the original clock. Thus the Markov description is preserved, although its intensity is now written as a function of \(u\). The intensity is not constant, so this example is time-inhomogeneous; the shift has not made it time-homogeneous.
When Time Since Enrollment Omits Duration
Suppose an individual enters a study in state 0 and later moves into state 1. If transition rates out of state 1 depend on time since entering state 1, then current state 1 and time since enrollment may not reveal the relevant duration. Two individuals can be in the same state at the same enrollment time but have entered that state at different times.
If their transition intensities differ because one has spent longer in state 1, then conditioning only on current state and enrollment time does not suffice. In that case, the state process indexed by enrollment time does not meet the stated Markov assumption, unless the omitted duration is irrelevant or the state representation is expanded.
Worked Example: Illness Duration Hidden by the Enrollment Clock
Consider a hypothetical study that follows participants from enrollment. At month 10, two participants are both in the illness state. One was diagnosed at month 2 and has been ill for 8 months; the other was diagnosed at month 8 and has been ill for 2 months. Assume, for illustration, that the intensity of recovery from illness is 0.020 per month during the first 3 months of illness and 0.005 per month thereafter.
At month 10, both participants have the same current state and the same time since enrollment. But their recovery intensities are different: $$ q_{\text{illness,recovery}}= \begin{cases} 0.005 & \text{per month for the participant ill for 8 months},\\ 0.020 & \text{per month for the participant ill for 2 months}. \end{cases} $$
By assumption, the intensity depends on duration of illness. The current state and enrollment time do not identify that duration, so the Markov condition using only those quantities fails for this model. This does not show that all enrollment-clock models fail: if the recovery intensity did not vary with illness duration, this particular source of history dependence would be absent. It shows why the time origin must be assessed against the mechanisms governing transitions.
Diagnosis Time and Birth Time as Alternative Clocks
Setting time zero at diagnosis can make duration since diagnosis explicit for people who enter follow-up at diagnosis. For a post-diagnosis model, time \(u\) then directly represents time since diagnosis. This may support a Markov description when the relevant transition intensities depend on that duration and the current state, but it is not a guarantee: rates may also vary with age or other omitted history.
The diagnosis clock is less informative if participants are enrolled at different points after diagnosis and the model records only time since enrollment. Their diagnosis durations then differ at the same enrollment time. A model can retain diagnosis time as a covariate or use an appropriate time scale, provided the needed information is observed and represented.
Similarly, age since birth can be useful when transition rates vary with age. If two people enter a study at different ages, equal times since enrollment do not mean equal ages. A model indexed by age can represent age-dependent transition rates directly. But age alone will not reveal time since disease onset when onset occurs at different ages.
Worked Example: Age Versus Time Since Enrollment
In a hypothetical cohort, suppose the transition intensity from a healthy state to illness depends on age: \(q(a)=0.010\) per year at age 50 and \(q(a)=0.040\) per year at age 70. Compare two participants, one enrolled at age 45 and the other at age 65. Five years after enrollment, both are healthy, but they are ages 50 and 70, respectively.
On the enrollment clock, both have the same time index, 5 years, and the same current state. Yet the assumed illness intensities differ by age. Under the stated model, their one-year probabilities of leaving the healthy state are $$ P(\text{leave healthy within 1 year}\mid a=50) =1-e^{-0.010(1)}\approx 0.009950, $$ and $$ P(\text{leave healthy within 1 year}\mid a=70) =1-e^{-0.040(1)}\approx 0.03921. $$
These calculations assume a constant intensity over the one-year interval at each stated age and no other outgoing transition from the healthy state. With an age clock, the two participants have different time indices, so the assumed age-specific rates are represented directly. With an enrollment clock, current state and elapsed enrollment time alone cannot distinguish the ages in this example; age must be included or otherwise represented for the assumed rates to be described.
A Short Justification: What a Clock Change Preserves
The common-shift result follows directly from conditioning. Let \(u=t-c\) for a fixed constant \(c\), and let \(X^*(u)=X(u+c)\) denote the same process expressed on the shifted clock. Because \(t=u+c\), the time \(u\) determines \(t\), and the state history up to \(u\) corresponds to the state history up to \(t\). If the process is Markov on the original clock, then $$ P\{X^*(u+h)=j\mid \mathcal{H}^*(u)\} = P\{X^*(u+h)=j\mid X^*(u)\}. $$
The transition probabilities are simply re-expressed at shifted times. For example, the probability of moving from \(r\) to \(s\) over a gap \(h\), conditional on being in \(r\), is evaluated at original time \(u+c\). Thus a common shift preserves the Markov property, while it may change how a time-varying transition law is written.
The argument relies on the shift being known and common. If individual \(i\) uses \(u_i=t-c_i\), the pooled time index \(u_i\) does not necessarily determine a common \(t\), and participants at the same \(u_i\) may be at different ages or calendar times. A model may still be Markov after accounting for those quantities, but the common-shift argument no longer establishes it by itself.
Choosing a Clock and Recognizing Failure
A useful choice of time origin follows the scientific question and the transition mechanisms. Before analysis, state exactly what time zero means, whether it is common to all individuals, and what duration the time index represents. Then ask whether the current state and that time index contain the information needed to specify transition behavior.
- Time since enrollment naturally describes follow-up, but may omit age or time since entry into a later state.
- Time since diagnosis makes diagnosis duration explicit for a diagnosis-entry cohort, but may omit age or other relevant durations.
- Age since birth makes age explicit, but does not generally identify time since onset or time since a treatment began.
Warning signs include transition patterns that differ for people in the same state and at the same analysis time but with different known durations in that state, or clear rate differences by age when age is not represented. These patterns suggest that the chosen state-and-clock description may be missing information. They are reasons to examine the model specification, not automatic proof of a particular alternative.
The main alternatives are to use a more suitable time scale, include relevant observed history in the model, or expand the state representation to record a duration or history category. When transition rates depend explicitly on time spent in the current state, a duration-dependent or semi-Markov model may be appropriate. Each choice has limitations: a richer model needs reliable history data and may require more parameters, while a simpler Markov model can be easier to interpret and estimate if its state and time variables are adequate.
| Modeling choice | What the clock or state captures | Potential limitation |
|---|---|---|
| Enrollment-time Markov model | Current state and follow-up time | May omit age or time since later state entry |
| Diagnosis-time model | Current state and duration from diagnosis | May not represent age or pre-diagnosis history |
| Age-time model | Current state and age | May omit time since onset or treatment |
| Duration-dependent or expanded-state model | Current state plus a relevant duration or history summary | Needs more information and a more complex model |
Common Mistake
A common mistake is to say, “The process is Markov because the rate is allowed to vary with time.” Allowing time-inhomogeneous transition rates accounts for variation with the specified time index; it does not account for a duration that varies among individuals at the same index time. In the illness example, both participants were at month 10 since enrollment, but their illness durations were different. If duration changes the rate, time-inhomogeneity alone does not resolve the omitted-history problem.
The correction is to identify what the time index tells us for each individual and what it leaves unknown. A common shift changes the coordinate without losing information. A clock reset to different individual events can change what is represented, and a clock that fails to encode a relevant duration may leave the Markov assumption implausible.
Check Your Understanding
Answer each question, then compare your reasoning with the solutions.
- A common study clock is shifted by 4 years for every participant. Does this shift alone alter whether the process is Markov?
- Two participants are in the same illness state at the same month since enrollment, but one has been ill longer. If recovery intensity depends on illness duration, what is the modeling concern?
- Why might age be a more useful clock than time since enrollment for a transition rate that depends on age?
- Does setting time zero at diagnosis guarantee that a post-diagnosis process is Markov? Give one reason it might not.
- At the same state and time on the enrollment clock, one participant is age 50 and another is age 70. If transition intensity depends on age, what information may need to be represented?
Solutions. (1) No. A common known shift preserves the information in the time index, since either time coordinate determines the other. (2) The enrollment clock and current state do not reveal illness duration, so they may not suffice to determine the recovery intensity. (3) Age-time directly represents age, whereas equal enrollment times can correspond to different ages. (4) No. Rates could also depend on age, prior treatment, or another history not captured by current state and diagnosis duration. (5) Age may need to enter as the time scale or as an additional modeled variable; otherwise the specified intensity difference is not represented by state and enrollment time alone.