What a Panel-Observed History Records
In a continuously observed multi-state process, changes of state might be recorded at the times they occur. In panel observation, a person’s state is checked only at selected visits. The data therefore tell us which state was seen at each visit, not generally the exact times or complete sequence of changes between visits. Representing that distinction accurately is essential: an observation schedule is not a transition record.
Let \(X_i(t)\) denote the state of person \(i\) at time \(t\), with time measured from a defined origin such as enrollment. Let \(\mathcal{S}\) be the set of possible states. Person \(i\) has actual observation times \(t_{i0}<t_{i1}<\cdots<t_{im_i}\), and the recorded state at visit \(k\) is \(Y_{ik}=X_i(t_{ik})\). The visit times need not be equally spaced or the same across people.
An allowed state graph specifies which transitions are possible in the model or study description. For example, an illness-death process might have states 0 (healthy), 1 (ill), and 2 (dead), with allowed transitions \(0\to1\), \(0\to2\), and \(1\to2\). A proposed history between visits must respect those allowed transitions. But even when the graph is known, endpoint observations do not necessarily determine the path between them.
Represent Visits Before Interpreting Transitions
A reliable first representation is one row per person per actual visit. It keeps the observed information visible and avoids assigning an invented event date. In an analysis file, a person identifier, visit index, time, and observed state are usually sufficient for this basic record; other variables can be added with their own clear timing conventions.
| Person | Visit index | Time | Observed state |
|---|---|---|---|
| A | 0 | 0 | 0 |
| A | 1 | 2 | 1 |
| A | 2 | 5 | 2 |
The rows say that person A was observed in state 0 at time 0, state 1 at time 2, and state 2 at time 5. They do not say that the person entered state 1 at time 2 or state 2 at time 5. Those are visit times, and a state seen at a visit may have been entered earlier.
If a scheduled visit is missed, do not create a state observation for that time. Keep the actual observed visits and their times. A planned schedule may be useful metadata, but a scheduled time with no measurement is not an observed state. Likewise, distinguish a last observed state from an outcome known to occur after that visit; the latter requires separate information.
Worked Example: Bracketing Two Changes in an Illness-Death History
Suppose a hypothetical participant is observed at times 0, 2, and 5 years, with recorded states 0 (healthy), 1 (ill), and 2 (dead), respectively. Assume the allowed transitions are \(0\to1\), \(0\to2\), and \(1\to2\), and that the participant’s states are observed without error at the visits.
The visit record is \((0,0),(2,1),(5,2)\), with each ordered pair written as (time, observed state). The first and second observed states differ, so the change from healthy to ill must have occurred after the time-0 observation and no later than the time-2 observation. Under the convention that a transition at the right endpoint is reflected in the recorded state there, its possible time is \((0,2]\) years. The interval width is \(2-0=2\) years.
The participant is observed ill at year 2 and dead at year 5. Under the stated state graph, the transition from illness to death occurred in \((2,5]\) years, an interval of width \(5-2=3\) years. The observations support these brackets, not a particular onset or death date. In particular, it would be incorrect to record the illness onset as year 2 simply because illness was first observed then.
What an Interval Between Visits Can Mean
For an absorbing or otherwise persistent change that occurs once, two observations can bracket its time. Suppose a state change from \(a\) to \(b\) happens at time \(\tau\), the person is in \(a\) at an observation time \(L\), and in \(b\) at a later observation time \(R\). If the process is right-continuous and the state recorded at an observation is the state at that time, then
The strict inequality at \(L\) follows because a transition at \(L\) would already have placed the person in state \(b\) at that observation. Equality at \(R\) is possible because a transition at \(R\) is recorded as the state at \(R\). This endpoint convention must be stated and used consistently. The interval is \((L,R]\), not an assertion that the transition was uniformly distributed within it or most likely at its midpoint.
That simple bracket requires a simple path. If repeated transitions or reversals are allowed, equal states at two visits do not establish that no transition occurred. For example, a person could move from state 0 to state 1 and back to state 0 between visits. Nor does a pair of different endpoint states always identify one transition: the path might contain several allowed transitions. In those cases, the observation gap constrains the possible intervening paths, but does not necessarily provide a separate bracket for every transition.
Worked Example: Same State at Both Visits
In a hypothetical study, state 0 means disease-free and state 1 means diseased. A participant is observed in state 0 at year 1 and again in state 0 at year 4. The allowed graph permits both \(0\to1\) and \(1\to0\).
The visit record is \((1,0),(4,0)\). One compatible path is that the participant remained in state 0 throughout the three-year gap. Another is a transition \(0\to1\) at year 2 followed by \(1\to0\) at year 3. Those times are compatible because \(1<2<3\leq4\), and the state at the year-4 visit is again 0.
Thus, the observations do not establish that the participant stayed disease-free between years 1 and 4. They also do not establish that a disease episode occurred. The defensible record is the two observed states and their times, with the intervening path unresolved. The gap length is \(4-1=3\) years; it is not a three-year interval-censored disease onset unless additional assumptions rule out return to state 0.
A Practical Representation Workflow
Separate the observed record from any derived description of possible transition intervals. This prevents a derived bracket from being mistaken for a directly measured event time.
For each person, record the ordered times at which a state was actually assessed. Do not substitute planned times for missed visits.
At each visit time, enter the state observed then. Keep unknown or unrecorded states distinct from any state category.
Compare each observed pair with the allowed transitions. A direct change between observations may require an intermediate state or several transitions.
For a single persistent change, use the interval between the last observation in the earlier state and the first observation in the later state. If reversals or multiple changes are possible, retain the visit record and describe the path as unresolved rather than forcing one event interval.
The state graph is a constraint, not a substitute for observation. If a person is seen in state 0 at one visit and state 2 at the next, and the only allowed route is \(0\to1\to2\), the record does not show when state 1 began or ended. It only says the observed endpoints are consistent with a route that includes state 1. If another route is allowed, it must also be considered rather than silently excluded.
Worked Example: Different Visit Schedules and an Unobserved Intermediate State
Consider a hypothetical process with states 0 (healthy), 1 (ill), and 2 (dead), and allowed transitions \(0\to1\), \(0\to2\), and \(1\to2\). Person B is observed in state 0 at time 0 and state 2 at time 6 months. Person C is observed in state 0 at time 0, state 1 at month 2, and state 2 at month 6.
For B, the visit record is \((0,0),(6,2)\). The observed endpoints do not reveal whether B moved directly from 0 to 2 or passed through state 1 before death. Under the allowed graph, both routes are possible. Consequently, the visit gap \((0,6]\) months bounds the overall change from the recorded initial state to the final state, but does not identify a separate interval for illness onset.
For C, the record is \((0,0),(2,1),(6,2)\). It supports a bracket of \((0,2]\) months for the change from healthy to ill and \((2,6]\) months for the change from ill to dead, given the stated graph and endpoint convention. The first bracket has width \(2-0=2\) months; the second has width \(6-2=4\) months. C’s intermediate visit narrows the description of the observed path compared with B’s record, but neither transition time is known exactly.
Diagnostics, Alternatives, and Common Mistakes
Before using a panel history, check that visit times are ordered, state labels belong to the declared state set, and each person’s records are linked to the correct person. Review repeated times, duplicate records, impossible state sequences, and missing assessments. An apparent impossible sequence may indicate a data error, a state graph that is too restrictive, or a coding convention that has not been clarified; it should not be repaired by guessing a transition time.
A visit-time record is the direct representation of panel observations. An interval record is a derived summary appropriate when a particular event is known to have occurred once and the observations bracket it. An exact transition-time record is appropriate only when the transition time was actually observed or otherwise known with the required precision. These are not interchangeable formats: replacing an interval by its midpoint discards the distinction between observed and assumed information.
- Pitfall: treating first detection as onset. If a state is absent at year 1 and present at year 4, setting onset to year 4 confuses the first observation in that state with its unknown entry time. Under the single-change convention, the supported interval is \((1,4]\).
- Pitfall: inferring stability from equal endpoints. Seeing the same state at both visits does not rule out departure and return when reversals are allowed. Record the endpoints; do not claim continuous occupancy without additional information or assumptions.
- Pitfall: forcing one transition into every gap. A gap with different endpoint states may contain multiple transitions, depending on the state graph. Check possible paths before assigning a single event interval.
- Pitfall: confusing missing with a state. A missed assessment is not a health state and does not provide an observed endpoint for a transition bracket.
Panel representation matters because the precision of the data limits the precision of any later analysis. Visit records preserve the schedule and observed states; interval descriptions communicate coarsening without pretending to know more than was measured. Detailed likelihood construction is a separate step and depends on the model for transitions as well as on these observation records.
Check Your Understanding
Answer each question using the stated observation and transition conventions.
- A person is in state A at time 3 and state B at time 8. If a single persistent change \(A\to B\) is assumed, give the interval for the change and its width.
- A person is observed in state 0 at months 1 and 7. The graph permits \(0\to1\) and \(1\to0\). Does this establish continuous occupancy of state 0?
- Why should a missed scheduled visit not be entered as a visit with the previous state?
- In an illness-death graph allowing \(0\to1\), \(0\to2\), and \(1\to2\), a person is observed in state 0 and later in state 2. Does this identify whether state 1 was occupied?
Solutions. (1) The change is in \((3,8]\), of width \(8-3=5\) time units. (2) No. A \(0\to1\to0\) path between visits is compatible with the equal observed states. (3) The state at a missed visit was not measured; carrying forward the previous state invents an observation and may hide a change. (4) No. A direct \(0\to2\) path and a path through state 1 are both compatible with those endpoints under the stated graph.