From a State at One Time to a History Over Time
A record that someone is in state B at time 13 answers a question about one instant. It does not, by itself, tell us which states that person occupied earlier or whether the person changed states and returned to B before time 13. Multi-state data can be represented at different levels of detail: as states observed at selected times, or as an ordered history that records each observed transition and its time. Keeping these representations distinct prevents stronger conclusions than the data support.
For an individual \(i\), let \(X_i(t)\) denote the state occupied at time \(t\) on the study time scale. The state labels, time origin, and observation end must be specified for a record to be interpretable. A state process describes occupation at each time; an event history describes the changes that lead from one state to another.
A useful representation assumes a piecewise-constant path: the state remains unchanged between transitions and changes at isolated transition times. Write \(m_i\) for the number of transitions by \(C_i\), \(t_{i0}=0\) for the start of observation, and \(t_{i1},\ldots,t_{im_i}\) for the transition times. Let \(s_{i0}\) be the initial state and \(s_{ij}\) the state entered at transition \(j\). The history is the ordered sequence
The times satisfy \(0<t_{i1}<\cdots<t_{im_i}\leq C_i\). When \(m_i=0\), the history consists of the initial state and observation end. For a complete history, the state at a time \(t\) is the state entered at the latest transition at or before \(t\), or the initial state if no transition has yet occurred. Thus, for \(t<C_i\),
At a transition time, this convention assigns the person to the state entered at that time. At \(C_i\), the state is the state after any transition recorded exactly at \(C_i\). This is a bookkeeping convention for representing the path; the data collection must still establish which state was occupied.
Reading a Complete Sequence
The ordered sequence preserves both the states visited and the times at which they were entered. It can therefore answer questions such as “What state was occupied at time \(t\)?” and “How long was the person recorded in state B?” A list of states without times does not answer the second question, while a list of times without destination states does not identify the path.
Worked Example: Recovering State and Duration From a History
Suppose, hypothetically, that every state change for one participant is ascertained from time 0 through time 12 days. The participant starts in A, enters B at day 3, and enters C at day 8. The history is \((A,3,B,8,C;12)\).
The state intervals are A from day 0 up to day 3, B from day 3 up to day 8, and C from day 8 through the observation end at day 12. Therefore \(X(2)=A\), \(X(3)=B\), \(X(7)=B\), and \(X(8)=C\). The transition-time convention explains why the state at day 3 is B rather than A.
The recorded time in A is \(3-0=3\) days; in B it is \(8-3=5\) days; and in C it is \(12-8=4\) days. The total is \(3+5+4=12\) days, matching the observation interval. These durations follow from the stated complete ascertainment; they would not be justified by visits only at days 0, 3, 8, and 12 unless the transitions were known to occur exactly at days 3 and 8.
More generally, if the full path is known, the time spent in a state \(r\) is the sum of the lengths of the intervals assigned to that state. Write \(I\{X_i(t)=r\}\) for 1 when the individual is in \(r\) at time \(t\), and 0 otherwise. Then the occupation time over the observed interval is
The sum works because each interval between consecutive recorded transition times has a single state, and its length is its right endpoint minus its left endpoint. Adding the lengths for intervals in state \(r\) gives the total time in that state. Summing over all states gives \(C_i\), provided the states cover the full observation interval without gaps.
Why Observation Times Are Not a Full History
In many studies, state is assessed at visits rather than continuously. A visit-based record may establish the state at each visit, but it generally does not reveal the transition times or all intermediate states. A complete sequence should not be constructed from such a record unless the study design or other measurement information justifies the transitions and their timing.
Worked Example: What Two Visits Establish
Suppose, hypothetically, that a participant is observed in B at day 5 and is again observed in B at day 13. The record contains no intermediate assessment. It establishes \(X(5)=B\) and \(X(13)=B\), assuming the assessments correctly classify state at those times.
It does not establish that \(X(t)=B\) for every \(t\) between days 5 and 13. The participant could have moved from B to another state and then returned to B before day 13. Alternatively, the participant might have remained in B throughout. Both paths agree with the two recorded states. Since the actual transition times and intervening states are not known, neither a complete ordered history nor the time spent in B over the entire eight-day interval can be recovered from these visits alone.
This distinction is not a technicality: an absence of recorded transitions between visits is not evidence that no transition occurred. A statement that B was occupied throughout \([5,13]\) would require complete ascertainment of state changes in that interval, or another justified source of continuous-path information.
When observations are intermittent, a transparent representation preserves what was measured: the observed states and their assessment times, with uncertainty about what occurred between them left explicit. If a transition is known only to have occurred between two visits, its time is interval-censored rather than exactly observed. If the state at the next visit differs from the state at the previous visit, the observations establish that at least one change occurred under the assumption that both assessments are correct, but they do not necessarily reveal the number or order of changes between those visits.
Worked Example: Distinct Histories With the Same Current State
Consider two hypothetical participants observed completely through day 10. Both start in A and are in C at day 10. Participant 1 enters C directly at day 6, giving \((A,6,C;10)\). Participant 2 enters B at day 2 and then C at day 6, giving \((A,2,B,6,C;10)\).
For both participants, \(X(10)=C\). Their histories are nevertheless different: participant 1 has one recorded transition, and participant 2 has two. Their time in C is \(10-6=4\) days in both cases, but participant 1 spends \(6-0=6\) days in A, whereas participant 2 spends \(2-0=2\) days in A and \(6-2=4\) days in B. Each total is 10 days: \(6+4=10\) for participant 1 and \(2+4+4=10\) for participant 2.
Thus, a current-state summary can be adequate for a question about state at day 10 but cannot distinguish these histories. The appropriate representation depends on whether the question concerns occupation at a particular time, the sequence of states, transition timing, or time spent in states.
Assumptions, Checks, and Information Lost
Before interpreting a sequence as a complete path, check what the observation process supports. The sequence representation relies on a specified time origin and observation end, correctly labelled states, correctly ordered transition times, and complete ascertainment of changes if the claim is that the entire intervening path is known. A piecewise-constant path with isolated changes is the representation used here. If time is recorded only to a coarse unit, two transitions may appear tied even when they occurred in a known order; if their order cannot be recovered, the record should show that ambiguity rather than impose an unsupported ordering.
A practical construction and verification procedure is:
Record the time origin, the time scale, and the end \(C_i\). Distinguish a study end or loss of observation from a transition to a state.
List states and times that were actually ascertained. Mark transition times as unknown or interval-located when exact timing was not observed.
Verify that transition times are ordered, that each transition has a destination state, and that each segment has a coherent state assignment.
Confirm that the final recorded state agrees with the last observed state and, for a fully observed path, that state-specific durations sum to \(C_i\).
Several approaches can summarize a history, but they answer different questions:
| Representation | What it retains | Main limitation |
|---|---|---|
| State at one specified time | The state at that instant | Does not reveal the route or time spent in earlier states |
| Ordered states and transition times | The complete ascertained path and its timing | Requires adequate transition ascertainment; may be unavailable with visit-only data |
| Transition count or endpoint alone | A compact summary for a targeted question | Discards the order, timing, or state labels needed for other questions |
A count can be useful when only the number of changes matters, and an endpoint can be useful when only the final state matters. Neither is a substitute for the full sequence when the order and timing of the path are important. Conversely, storing a complete sequence does not remove measurement error or recover transitions that were never observed. Conclusions must remain limited to the information captured by the observation process.
Common Mistakes
- Inferring a whole interval from its endpoints. Seeing B at two visits does not establish continuous occupation of B between them. It establishes B at the two assessment times.
- Treating “not recorded” as “did not happen.” No recorded change may reflect sparse observation rather than complete surveillance. State the ascertainment assumption before treating the sequence as complete.
- Confusing current state with the route taken. Two people can occupy the same state at a given time after different sequences of states and transitions.
- Omitting the observation end. A last state label without its time frame can be mistaken for a permanent or final state. A history describes the observed interval, not necessarily what happened afterward.
Check Your Understanding
Answer each question, then compare your reasoning with the solutions below.
- A complete history is \((A,4,B,9,C;12)\). What state is occupied at day 4, and how many days are spent in each state by day 12?
- A participant is observed in A at day 2 and A again at day 7. What can be concluded about the state between those times?
- Two fully observed histories have the same state at day 10 but different transition sequences. Is that a contradiction? Explain.
- A history contains transition times 3 and 8, but the time unit is not stated. What information is missing for interpreting durations?
- Why should a recorded absence of transitions between two visits not automatically be treated as a complete no-transition interval?
Solutions. (1) At day 4 the participant is in B, the state entered at that transition. The durations are \(4-0=4\) days in A, \(9-4=5\) days in B, and \(12-9=3\) days in C; \(4+5+3=12\) days. (2) The record establishes A at days 2 and 7 only. Without complete ascertainment, the participant may have left A and returned between visits. (3) No. The state at one instant does not uniquely determine the sequence leading to it; both histories can end in the same state. (4) The time unit and time origin are needed to interpret the transition times as durations in context. (5) Because changes may have occurred without being assessed or recorded; non-recording does not establish non-occurrence.