One State at a Time, or the Whole Path?
A multi-state record can be viewed in two different ways. At a specified time, a person occupies one state: for example, remission, relapse, or death. Across follow-up, that same person may have visited several states, returned to a state, or followed a route that differs from other people’s routes. The fixed-time state and the sequence of transitions answer different questions. Confusing them can make a correct summary of one target seem to answer another.
For instance, the proportion of patients in remission at month 12 is a description of state occupation at that time. It does not tell us how many patients relapsed and later returned to remission. To answer the latter question, we need records of transitions, not just the month-12 state.
A history can be written as an initial state followed by a sequence of states and transition times, such as \(W \to I \to R\), with transition times \(t_1\) and \(t_2\). Its final state is \(X(t)\), provided the individual has had no further transition by time \(t\). The history includes more information than the final state: it preserves the route, the order of visits, and the timing of transitions. In a model that permits recurrent state changes, it can also record repeated visits.
What a Fixed-Time Summary Does and Does Not Preserve
For a population, the state-occupation probability for state \(r\) at time \(t\) is \(P\{X(t)=r\}\). In a fully observed sample of \(n\) individuals, the corresponding occupation proportion is the number in state \(r\) at time \(t\), divided by \(n\). These summaries classify each individual by one state at one specified time. They do not, on their own, show which route each individual followed.
A history-based summary uses a feature of \(H(t)\) instead. Examples include whether a person ever entered a particular state by time \(t\), how many transitions occurred, or whether a particular ordered route was followed. To calculate one of these summaries, the data must distinguish the relevant histories. A single state recorded at time \(t\) is generally insufficient.
The distinction is not that state occupation is incomplete or inferior. It is the appropriate summary when the question concerns who is in each state at the specified time. A history summary is appropriate when the question concerns prior events or routes. The estimand—the quantity the analysis is intended to describe—determines which record is needed.
Worked Example: Same Remission State, Different Histories
Consider three hypothetical patients observed through month 12. The allowed transitions in this example are well to illness, illness to remission, and remission to illness. Patient A moved from well to illness at month 2 and entered remission at month 5. Patient B followed the same route, relapsed at month 8, and returned to remission at month 10. Patient C moved from well to illness at month 4 and entered remission at month 9. Assume all three remain in remission through month 12.
At month 12, each patient occupies remission. Thus the sample occupation proportion for remission is
The histories are not identical. Patient B had a relapse and a return to remission; Patients A and C did not. The proportion of these three patients who had ever relapsed by month 12 is therefore
The occupation proportion is 1, while the proportion with a relapse history is about 0.333. Both calculations are correct, but they answer different questions. The month-12 state alone cannot identify Patient B’s earlier relapse.
A Many-to-One Mapping From Histories to States
The formal reason for the difference is that evaluating a history at time \(t\) keeps only its terminal state. Define the endpoint map \(\pi_t\) by \(\pi_t\{H(t)\}=X(t)\). Multiple distinct histories can have the same endpoint, so this map is generally many-to-one. Once only \(X(t)\) has been retained, the original history cannot in general be reconstructed.
For a discrete collection of possible histories, the occupation probability for state \(r\) can be written as the sum of the probabilities of all histories ending in \(r\). This equation shows exactly what an occupation summary combines:
The sum groups histories according to their endpoint. It does not show the separate probabilities of the histories within that group. In settings with continuously varying transition times, the corresponding expression uses an integral over histories rather than a sum; the same grouping principle applies.
Worked Example: Snapshot Occupation Versus Ever Entering Illness
Suppose a hypothetical cohort of 10 people is fully observed through month 6. Their states at month 6 are: 3 well, 2 currently ill, 4 in remission, and 1 dead. The study records that the 4 people in remission and the person who died had previously entered illness; the 2 currently ill people have also entered illness. The 3 people who are well have never entered illness.
The month-6 occupation proportion for illness is based only on those currently ill:
The proportion who ever entered illness by month 6 includes those currently ill, those now in remission, and the person who died. Therefore,
The 0.20 figure describes current illness occupation; the 0.70 figure describes whether illness appeared anywhere in the observed history. The latter cannot be obtained by counting only people currently ill. It requires historical information, including whether people who left illness had previously occupied it.
Choose the Summary That Matches the Question
An occupation summary is a cross-section: it partitions individuals by their state at the selected time. A history summary instead classifies individuals by a property of their path. The table highlights the distinction.
| Scientific question | Information needed | Example summary |
|---|---|---|
| How many people are in remission at month 12? | State occupied at month 12 | Proportion in remission at month 12 |
| How many people have ever relapsed by month 12? | Whether relapse occurred anywhere in the history by month 12 | Proportion with at least one relapse |
| How many transitions did people experience? | Recorded transition sequence through month 12 | Distribution of transition counts |
| How many followed a specified route? | Order of states visited, and possibly transition times | Proportion whose history includes that route |
Worked Example: Counting Different History Types Within One Occupied State
In a hypothetical group of 8 patients, everyone begins in remission. The allowed transitions are remission to relapse, relapse to remission, and transition from remission or relapse to death. At month 10, Patients A, B, C, and D are in remission; E and F are in relapse; G and H are dead. Their histories are as follows: A and D remained in remission; B moved from remission to relapse and back to remission; C made that same relapse-and-return sequence twice; E moved from remission to relapse and remained there; F moved from remission to relapse, returned to remission, then relapsed again; G moved from remission to relapse and then to death; H moved directly from remission to death.
The occupation proportion for remission at month 10 is
Among these 4 patients in remission, 2 have a recorded relapse history (B and C), and 2 do not (A and D). Thus the proportion of the full group who are both in remission at month 10 and have previously relapsed is
The proportion with at least one relapse at any time by month 10 is larger: B, C, E, F, and G have all entered relapse, giving \(5/8=0.625\). These three summaries—current remission, remission with a prior relapse, and ever relapsed—are different because they apply different rules to the same set of histories. The current-state count alone gives neither of the history-based counts.
Assumptions, Diagnostics, and Limits
The arithmetic in the examples relies on clearly specified state definitions, a common time origin, and consistent observation through the stated time. A fixed-time occupation proportion also requires knowing each included individual’s state at that time. A history-based count requires enough observation to determine the relevant event history over the interval of interest. These are data and definition requirements; they do not make the two targets interchangeable.
Before calculating a summary, check the following:
- Is the target a snapshot or a path feature? “In state \(r\) at month 12” is an occupation question. “Ever entered \(r\) by month 12” is a history question.
- Are states defined consistently? If one record labels a person as recovered while another uses a different definition of recovery, their state counts may not be comparable.
- Are transition times and order recorded when needed? If a person can leave and re-enter a state, a single observation at the start and end of follow-up may miss intermediate transitions.
- Is the observation window adequate for the claim? If follow-up ends before the target time, an observed state before that time is not the state at the target time. Likewise, a history only supports claims through the time it was observed.
A common failure mode is to treat “not currently in a state” as equivalent to “never entered that state.” The first statement concerns occupation at one time; the second concerns the entire observed history. Another is to infer a route from the initial and final states. If several allowed routes connect those endpoints, they do not reveal which route an individual actually followed. When visits can recur, even knowing that a state was visited does not give the number or order of visits.
The main alternative to a snapshot summary is not one particular statistic; it is a history-based description selected to match the question. A snapshot is compact and directly answers who occupies each state at a fixed time. A path description retains transitions, ordering, and potentially timing, but requires more detailed records and can be harder to summarize. Neither view replaces the other. Also, a recorded history may be incomplete if transitions occur between observation visits and are not captured. In that case, a history-based measure can miss events or misstate their timing; this limitation should be described rather than silently treating the recorded sequence as complete.
Check Your Understanding
Answer each question, then compare your reasoning with the solutions.
- At month 8, 12 of 20 people occupy state \(R\). What occupation proportion does this give, and does it tell us how many people ever left and returned to \(R\)?
- In a cohort of 15 people, 3 are currently in state \(I\), 4 are in state \(R\) after previously entering \(I\), and 2 have died after entering \(I\). The remaining 6 never entered \(I\). What proportion is currently in \(I\), and what proportion ever entered \(I\)?
- Two people are both in remission at month 12. One remained in remission throughout; the other relapsed and returned to remission. Are their month-12 occupation states the same? Are their histories the same?
- Why can endpoint states alone fail to reveal the number of transitions made when cycles are allowed?
Solutions. (1) The occupation proportion is \(12/20=0.60\). It does not reveal how many people left and returned to \(R\), because those people may have the same month-8 state as people with different histories. (2) The current-\(I\) proportion is \(3/15=0.20\). The number who ever entered \(I\) is \(3+4+2=9\), so the ever-entered proportion is \(9/15=0.60\). (3) Yes, both occupy remission at month 12. No, their histories differ because only one includes a relapse and return. (4) A path can revisit states. Different sequences with different numbers of transitions can therefore have the same initial and final states; the endpoints discard the intervening visits.