Why the Time Origin Matters
A multi-state history records not only which state a person occupies, but also when the person entered that state. At a given observation time, the elapsed time since that entry is the person's sojourn time in the current state. This quantity has a different time origin from calendar time or time elapsed since study entry. Keeping the origins distinct is necessary whenever a question refers to how long someone has occupied a state.
For example, two people observed on the same calendar date may both be in state B, but one may have entered B yesterday and the other several months earlier. Their current states match; their sojourn times do not. Similarly, a person may have been enrolled in a study for a year but entered the current state only a week ago. The state history determines the relevant entry time.
This definition concerns a state-occupation history, not a particular probability model. It requires that the relevant entry time be known or recoverable from the available history. It does not, by itself, assume that future transitions depend on sojourn time, or that the process has any specific transition distribution. The Markov property and state sufficiency concern whether the current state alone is enough to characterize the future; here the immediate task is to calculate a time coordinate from a recorded path.
Three Time Coordinates
Use \(t\) for the time at which the person is observed. Calendar time refers to a fixed external origin, such as a date on a calendar. Time since study entry uses that person's enrollment time as its origin. Sojourn time uses the entry time into the current state as its origin. These coordinates can all be measured in days, but equal units do not make their meanings interchangeable.
| Time coordinate | Origin | What it answers |
|---|---|---|
| Calendar time | A fixed external date or time | When did the observation occur? |
| Time since study entry | The person's study-entry time | How long has the person been followed? |
| Sojourn time | The person's most recent entry into the current state | How long has the person occupied this state on the current visit? |
For a person whose study-entry time is \(S\) and current-state entry time is \(E\), time since study entry at \(t\) is \(t-S\), while sojourn time is \(t-E\). If the current state began after study entry, then $$ t-E=(t-S)-(E-S). $$ Thus the two elapsed times differ by the time from study entry to the current state's entry. If the state began before study entry, the same subtraction remains valid when the earlier entry time is known. If it is unknown, the sojourn time cannot be recovered simply by using study entry as a substitute.
Worked Example: Repeated Visits to the Same State
A hypothetical participant's observed history, with times measured in days from a defined time origin, is: state A from day 0 to day 4, state B from day 4 to day 9, state A from day 9 to day 12, and state C from day 12 onward. Calculate the current-state sojourn times at days 3, 7, 10, and 14.
Day 3: The participant is in the first visit to A, which began on day 0. The sojourn time is \(3-0=3\) days.
Day 7: The participant is in B, entered on day 4. The sojourn time is \(7-4=3\) days.
Day 10: The participant has returned to A, but this is a new visit. The most recent entry into A was day 9, so the sojourn time is \(10-9=1\) day, not \(10-0=10\) days.
Day 14: The current state is C, entered on day 12. The sojourn time is \(14-12=2\) days.
The defining feature is the most recent entry into the current state. A return to a previously visited state starts a new state spell for this calculation.
Calculating from an Event History
Represent a fully observed path by successive transition times and the states occupied between them. For instance, if a transition into state \(j\) occurs at time \(T_k\), then \(T_k\) is the start of the current spell in \(j\). At a later time \(t\) before the next transition, subtract \(T_k\) from \(t\). At the next transition, the old spell ends and the destination state's spell starts. The calculation is therefore a subtraction, but identifying the correct starting time requires following the path in order.
Identify the date or elapsed time \(t\) at which the state duration is requested.
Use the event history to determine the state occupied at \(t\).
Scan backward from \(t\) to the most recent transition into that state. Do not use an earlier visit to the same state.
Calculate \(t-E_i(t)\), and state the result in the units used for the event times.
The subtraction also gives a simple consistency check. If no transition occurs between \(t_1\) and \(t_2\), with \(t_2\geq t_1\), the entry time is unchanged, so $$ A(t_2)-A(t_1)=(t_2-E)-(t_1-E)=t_2-t_1. $$ The sojourn time increases by exactly the elapsed time. At a transition into a new state, its value resets to zero. If the person later returns to a state occupied before, the duration for the new visit starts again at zero.
Worked Example: Calendar Date, Follow-Up Time, and State Duration
In a hypothetical study, a participant enters the study on March 1. The participant enters state B on March 8, remains there through March 18, and is observed in B on March 18. Use elapsed calendar days, with March 1 as day 0. Calculate the calendar-time coordinate, time since study entry, and sojourn time on March 18.
March 18 is 17 days after March 1, so the calendar-time coordinate under this chosen origin is day 17. Time since study entry is also \(17-0=17\) days because study entry is day 0.
State B began on March 8, which is day 7. Therefore the sojourn time in B on March 18 is \(17-7=10\) days. The three coordinates are: calendar coordinate, day 17; time since study entry, 17 days; and sojourn time in B, 10 days.
The first two values happen to match because this example uses the study-entry date as the calendar origin. That numerical equality does not make the concepts identical. The state-duration calculation needs March 8, the entry date into B.
Initial States, Partial Histories, and Censoring
The first state recorded for a participant needs special attention. If observation begins at study entry and the participant enters that state at that same time, the sojourn time can be calculated from study entry. But study entry is not automatically state entry. A participant may already have been in the initial recorded state for some time before enrollment.
If the actual entry time into that initial state is known from earlier records, it can be used even if it precedes study entry. If it is unknown, the exact current sojourn time is unknown too. Replacing the unknown entry time with the enrollment date produces time since study entry, not the true duration in state. This distinction matters even when the state label at enrollment is recorded without error.
Worked Example: A State Spell Already Underway at Enrollment
A hypothetical participant enrolls on day 20 and is recorded in state B. The history establishes that the participant entered B on day 14, and the participant remains in B until the last observation on day 25, when follow-up ends without a transition. Calculate the sojourn time at the last observation and describe what is known about the completed spell.
The current-state entry time is day 14, not the enrollment time of day 20. At day 25, the duration in B so far is \(25-14=11\) days. The participant has been followed for \(25-20=5\) days, a different quantity.
Because observation ends while the participant remains in B, 11 days is the duration accumulated by the end of follow-up, not a known completed duration of the state spell. The spell's eventual length may be longer. The event history gives a right-censored spell: follow-up tells us the participant stayed in B for at least 11 days, but not when the transition out of B occurred.
If the day-14 entry time had not been known, the study records alone would not establish an 11-day state duration. They would establish only 5 days of observed follow-up in B.
Assumptions, Diagnostics, and Common Pitfalls
A numerical sojourn time is well-defined from an event history when the time origin and units are specified, the state occupied at the observation time is known, and the current spell's entry time can be identified. The arithmetic also presumes that the recorded path captures the relevant state changes. If transitions are missed or dates are inaccurate, the calculated duration may not describe the actual period in state.
Check a proposed calculation against the history. The entry time must not be later than the observation time, so \(A(t)\geq 0\). If the person has remained in the state continuously, a later observation should have a duration at least as large as an earlier one, increasing by the time between observations. A reported duration that drops without a transition may indicate a time-origin mix-up or a missed transition. A negative value usually signals reversed dates or inconsistent units.
- Using first-ever entry instead of current-spell entry. If a person returns to a state, use the latest entry. The first-ever entry would count time spent in other states.
- Using study follow-up as state duration. These coincide only when the current state began at study entry. Otherwise, calculate from the current state's entry time.
- Treating an observation endpoint as a transition. If follow-up ends in a state, the time to that endpoint is accumulated duration; it is not evidence of a transition out of the state.
- Ignoring unknown pre-enrollment history. When the current spell began before enrollment and its entry date is unavailable, report the duration as unknown rather than silently setting it to time since enrollment.
The main alternatives answer different questions. Calendar time is useful when the date itself matters; time since study entry describes follow-up; sojourn time describes the duration of the current state spell. None is universally preferable. A model or summary must use the time coordinate appropriate to its scientific question, and a number reported simply as “time” is ambiguous unless its origin is stated. A sojourn-time calculation alone does not specify how transition behavior depends on duration; that is a separate modeling choice.
Check Your Understanding
Answer each question, then compare your reasoning with the solutions below.
- A participant moves from A to B on day 6, from B to C on day 10, and from C back to B on day 15. What is the sojourn time in B on day 18?
- A participant enters the study on day 30, is already in state D, and is observed in D on day 34. The entry time into D is unknown. What duration is known, and can the exact sojourn time be calculated?
- A person remains in state E from day 4 through day 9 with no transition. By how much should sojourn time increase between those observations?
- At day 12, a person is in state F, entered on day 8. Follow-up ends that day with no exit from F. What can be said about the state-spell duration?
Solutions. (1) The most recent entry into B was day 15, so the sojourn time is \(18-15=3\) days. (2) Four days of follow-up in D are observed, from day 30 to day 34, but the exact sojourn time is unknown because D may have begun before enrollment. (3) It should increase by \(9-4=5\) days, provided the person stayed in E throughout. (4) The observed duration is \(12-8=4\) days; the spell is right-censored at follow-up, so it lasted at least 4 days, and its completed duration is not known.