Cycles and Recurrent State Changes

Why Cycles Need Careful Representation

Some clinical histories do not move steadily toward one final state. A patient may enter remission, relapse, and later enter remission again. A diagram for such a process must allow the path to return to a state already visited. A single arrow from active disease to remission is not enough to express that possibility: the diagram must also allow a return transition from remission to active disease.

A cycle in a state diagram describes which state changes are possible; a particular patient’s history records which changes occurred, and in what order. Repeated movement around a cycle can therefore produce repeated transitions of the same type. But this does not automatically make every transition a recurrent clinical event. The event definition and the state definition determine whether the counts coincide.

Definition: A directed state diagram has a set of states and directed arrows representing allowed transitions. A cycle is a directed route that returns to a state already visited. A state path is an ordered sequence of visited states together with the times at which successive transitions occur. If the path is \(x_0,x_1,\ldots,x_m\), then each consecutive pair \(x_{k-1}\to x_k\) must be an allowed transition, and its transition time \(\tau_k\) must follow the preceding transition time.

For a finite observed history, write the transitions as occurring at \(0<\tau_1<\cdots<\tau_m\). The patient occupies \(x_0\) before \(\tau_1\), then \(x_1\) from \(\tau_1\) until \(\tau_2\), and so on, ending in \(x_m\) after \(\tau_m\) through the observed follow-up. A state can occur more than once in this sequence. Each occurrence after the first is a return to that state, not an error in the path.

The sequence of states and times is meaningful only relative to a stated set of states and allowed arrows. In particular, when a path is intended to give the exact observed history, the states must classify the patient’s status at each time, the recorded changes must be ordered, and the data must capture the relevant changes. If an intermediate change was missed, the recorded route may be incomplete.

Counting Transitions Along a Cycle

For a specified arrow \(i\to j\), its transition count over the observed path is the number of times that arrow is traversed. In a multi-state history this count can be calculated directly from the consecutive pairs in the state sequence. A patient may traverse the same arrow multiple times if the diagram permits a return route.

Definition: Let \(N_{ij}(t)\) be the number of observed transitions from state \(i\) to state \(j\) by time \(t\). This is a transition-specific count defined from the state path. A recurrent-event count \(N_E(t)\), by contrast, counts occurrences of an event \(E\) as defined for the scientific question. The two counts agree only when the event definition and observation rules make each occurrence correspond to exactly one \(i\to j\) transition, and every such transition correspond to exactly one occurrence.

Worked Example: Counting Relapse Transitions in a Cyclic History

Consider a hypothetical follow-up with three states: active disease (\(A\)), remission (\(R\)), and death (\(D\)). The allowed arrows are \(A\to R\), \(R\to A\), \(A\to D\), and \(R\to D\); death is absorbing. One patient is in \(A\) at day 0, enters \(R\) on day 3, returns to \(A\) on day 8, enters \(R\) on day 10, and dies on day 15. Count the transitions and determine how many relapse transitions occurred.

The state path, including transition times, is \(A \xrightarrow{3} R \xrightarrow{8} A \xrightarrow{10} R \xrightarrow{15} D\). Each arrow is allowed, and the times are strictly increasing, so this is a valid path under the specified diagram.

There are four transitions in total. The \(A\to R\) arrow is traversed twice, \(R\to A\) once, and \(R\to D\) once; \(A\to D\) is not traversed. If relapse is defined here as a transition from remission to active disease, the relapse-transition count is

N_{RA}(15)=1.

The two entries into remission are not two relapses. The count follows the chosen relapse definition, not the number of times the path changes state or revisits remission.

A Consistency Check for Two-State Cycles

A two-state diagram with arrows in both directions imposes a simple relationship between the starting state, ending state, and transition counts. This provides a useful way to check whether a claimed path or count is possible.

Result: For a path involving only states \(A\) and \(B\), with both \(A\to B\) and \(B\to A\) allowed, let \(n_{AB}\) and \(n_{BA}\) be the numbers of transitions in each direction. Then

n_{AB}-n_{BA} = \mathbf{1}(x_m=B)-\mathbf{1}(x_0=B),

where \(\mathbf{1}(\cdot)\) equals 1 when its condition is true and 0 otherwise. To justify the identity, assign the value 0 to state \(A\) and 1 to state \(B\). At each \(A\to B\) transition that value increases by 1; at each \(B\to A\) transition it decreases by 1. Adding all changes along the path cancels the intermediate values and leaves only the final value minus the initial value. That total is exactly \(n_{AB}-n_{BA}\).

Consequently, a path that starts and ends in \(A\) has the same number of transitions in both directions. A path starting in \(A\) and ending in \(B\) has one more \(A\to B\) transition than \(B\to A\) transitions. The check does not determine the order of the transitions, but it can expose an impossible summary.

Worked Example: Checking the Endpoint of a Two-State Path

Suppose a hypothetical patient’s states alternate between \(A\) and \(B\), and the recorded path is \(A\to B\to A\to B\to A\). Check the number of transitions in each direction and the endpoint.

Reading each consecutive pair gives \(A\to B\) twice and \(B\to A\) twice. Thus \(n_{AB}=2\), \(n_{BA}=2\), and the path has four transitions. The path starts in \(A\) and ends in \(A\), so the identity gives

n_{AB}-n_{BA}=2-2=0 =\mathbf{1}(A=B)-\mathbf{1}(A=B).

The endpoint check is consistent. In particular, this is a four-transition path starting in \(A\), not a path from \(A\) to \(B\): after an even number of changes between two states, it ends where it started.

State Changes and Recurrent Events Are Not Interchangeable

A state path records the patient’s state over time. A recurrent-event count records occurrences of an event under a separate operational definition. For example, “relapse” might be defined as the transition \(R\to A\). Another study might define an event as a symptom episode, hospitalization, or treatment failure. These definitions need not correspond to the same observations.

If every event of interest is exactly one transition of a specified type, and the state data capture all such transitions, then the event count and transition count match. Otherwise, they may differ. A patient can experience multiple events without changing the modeled state, or a state transition may occur without meeting the event definition. Neither count is inherently more correct; each answers a different question.

Worked Example: Three Events Without a State Transition

In a hypothetical study, “active disease” is a broad state that does not change when a patient has a symptom episode. A patient remains in \(A\) throughout follow-up and has qualifying symptom episodes on days 2, 5, and 9. The event definition counts each qualifying episode separately. What are the state-transition count and the recurrent-event count?

The state path is \(A\) throughout. Since no state change occurs, the number of state transitions is 0. By the stated event definition, the recurrent-event count is \(3\), with events at days 2, 5, and 9. Thus the recurrent-event count cannot be recovered from this state path alone: the state description deliberately does not distinguish one symptom episode from another.

If the scientific aim were to represent symptom episodes as changes of status, the state definitions or the model representation would need to change. Simply counting every event as a transition would be misleading unless an allowed arrow and a state history were defined that captured those occurrences.

The distinction also works in the other direction. A cycle can contain repeated state changes that are not all occurrences of the event being counted. In the remission example, two \(A\to R\) transitions occurred, but under the stated definition only \(R\to A\) was a relapse. When reporting results, name the arrow or event definition and the counting interval rather than using an unqualified phrase such as “number of recurrences.”

How to Build and Check a Cyclic History

A reliable representation begins with the clinical meaning of the states and transitions, not with an assumption that every repeated clinical occurrence must be a new state. The following sequence keeps the diagram, path, and count aligned.

Specify the state meanings.
Define what each state means at a given time, including how remission, active disease, or other relevant statuses are classified.

List allowed arrows.
Decide which state changes are possible under the scientific definition. Include a return arrow when the process permits a return to an earlier state.

Write the observed path.
List states in time order and attach each observed transition time to the arrow that occurred. Check that every consecutive pair is allowed.

Define the count of interest.
State whether the target is a particular transition count or a separately defined recurrent-event count, and check whether the two definitions coincide.

A practical diagnostic is to check the path against the diagram before summarizing it. Look for an unallowed arrow, a transition recorded out of time order, duplicate or tied transition times that have not been resolved, and gaps in observation during which an intermediate change could have been missed. In a two-state path, also compare the transition counts with the starting and ending states using the identity above. Passing these checks does not prove that the state definitions are clinically adequate, but failing one signals that the path or its summary needs attention.

What Cyclic Representation Can and Cannot Do

A cyclic state diagram is useful when return transitions are clinically meaningful and the current status can change repeatedly. Compared with a recurrent-event count alone, a state path preserves the sequence of statuses and the transitions connecting them. Compared with a one-way progression diagram, it can represent remission followed by relapse and further remission. The appropriate representation depends on the question: an event-count analysis focuses on event occurrences and their timing, while a multi-state path describes state occupancy and movement between states.

A state path may still omit information. If two clinically distinct histories are assigned to the same state sequence, the path will not distinguish them. If events occur without changing state, their count requires separate event data or a more detailed state definition. Conversely, adding a new state for every event can create a needlessly complicated representation. The state definitions should preserve information relevant to the scientific question without claiming that a transition count captures every recurrent occurrence.

These representations describe histories; by themselves, they do not establish why a transition occurred or estimate a causal effect. Their usefulness also depends on observation quality. Infrequent assessments, ambiguous definitions of relapse, or missed transitions can make an observed path differ from the underlying clinical history. Report the observation schedule and the operational definitions that determine which changes and events are counted.

Common Mistakes

Forbidding a return to a previously visited state. A cycle is precisely a route that can revisit a state when the corresponding arrows are allowed.
Calling every state change a relapse. A relapse count requires an explicit event definition; it may correspond to one arrow, but not necessarily to all transitions.
Confusing visits with transitions. A state appearing repeatedly in a sequence indicates returns, while the number and type of transitions are determined by consecutive pairs.
Reporting counts without checking endpoints. For a two-state path, the difference in directional counts must agree with the starting and ending states.
Treating an unchanged state as evidence of no recurrent events. If events do not alter the defined state, a state path can remain constant while the event count increases.

Key takeaway: A cycle permits a path to revisit states and traverse the same arrow repeatedly. Count transitions from consecutive states, and count recurrent events from their own definition; equate the counts only when a one-to-one correspondence is justified.

Check Your Understanding

Answer each question, then compare with the brief solutions.

A two-state path starts in \(A\) and has five transitions, with both directions allowed. Which state must it end in?
A path starts and ends in \(A\) and traverses \(A\to B\) three times. How many \(B\to A\) transitions must it contain?
A patient has two \(R\to A\) transitions, and relapse is defined as \(R\to A\). What is the relapse-transition count?
A patient has four qualifying symptom events but remains in the same defined state throughout follow-up. What can be concluded about the state-transition count and event count?
Why is a recorded sequence \(A\to B\to C\) invalid if the diagram has no allowed arrow \(B\to C\)?

Solutions. (1) It ends in \(B\): each transition switches states, so an odd number of transitions ends in the other state. (2) It must contain three \(B\to A\) transitions, since a path starting and ending in \(A\) has equal counts in both directions. (3) The count is 2, by the stated definition. (4) The state-transition count is 0, while the recurrent-event count is 4; the events do not change the state as defined. (5) Every consecutive pair in a valid path must be an allowed arrow, and \(B\to C\) is not allowed in that diagram.