Why Structural Checks Matter
A multi-state model is built around a state space and the transitions permitted between its states. If the states overlap, leave some subjects unclassified, or misrepresent what can happen, later calculations may be precise but answer the wrong question. A diagram is therefore not just a picture of clinical events: its state definitions and arrows make claims about the process being studied.
The checks in this tutorial concern the structure of those definitions and arrows. They are separate from assessing whether the current state adequately summarizes relevant history, a question addressed by the Markov property and related material. Even when the Markov assumption is not at issue, however, the state space must describe a coherent process.
Mutually Exclusive and Complete States
A state definition must pass two distinct checks. Mutual exclusivity means a subject cannot satisfy two different state definitions at the same time. Completeness means every subject within the modeled population has a state at each time at which the process is defined. Together, these properties make the state assignment unambiguous.
For a finite set of states \(\mathcal{S}=\{1,\ldots,K\}\), define \(I_k(t)=1\) if a subject is in state \(k\) at time \(t\), and \(I_k(t)=0\) otherwise. The two requirements can be summarized as:
If the sum is greater than 1, the definitions overlap. If it is 0, the subject has no defined state. A state space may be defined for a narrower population or time interval, but that scope must be explicit; silently leaving some subjects out is not a valid classification.
Worked Example: Correcting Overlapping States
In a hypothetical follow-up of 20 patients, a day-30 report gives 6 patients as “hospitalized,” 17 as “alive,” and 3 as “dead.” The labels are intended to classify each patient at day 30. Can they serve as three mutually exclusive states?
No. “Hospitalized” is not disjoint from “alive”: a hospitalized patient may also be alive. In fact, the reported counts sum to \(6+17+3=26\), although there are only 20 patients. If the 6 hospitalized patients are among the 17 alive patients, the total counts state memberships, not distinct people.
A correction is to define “alive in hospital,” “alive outside hospital,” and “dead.” Suppose the same hypothetical records show 6 alive in hospital, 11 alive outside hospital, and 3 dead. Then \(6+11+3=20\), and every patient belongs to exactly one category. The revised labels distinguish location among living patients without counting anyone twice.
This correction is justified only if the data actually identify those categories at day 30. If hospital status is not known for some living patients, the classifications need to be refined or the missing status handled explicitly; it should not be assumed away.
Omitted Destinations and Unobserved Outcomes
A diagram is incomplete if it omits a possible destination that belongs to the process being modeled. This does not mean every imaginable event needs an arrow. Rather, all destinations relevant to the stated clinical question and population must be represented. An omitted event can make subjects appear to vanish from the process or can wrongly combine distinct outcomes.
Censoring is different from a clinical state. A subject who is lost to follow-up or reaches the planned end of observation has not necessarily experienced a clinical transition. Censoring records that the subject’s later state is not observed; it is not automatically a destination such as “dead” or “recovered.” A protocol-defined withdrawal may be modeled as a state if withdrawal itself is part of the scientific process, but that choice must be explicit and consistent with the question.
Worked Example: Finding an Omitted Clinical Destination
In a hypothetical cohort of 50 patients starting treatment, the one-year records show 38 still on treatment without relapse, 7 with relapse, 2 dead, and 3 lost to follow-up before one year. A proposed diagram has only “on treatment without relapse” and “relapse.” Is it adequate for a process that includes death?
No. The counts show two deaths, so death is a clinical destination relevant to the stated process. The two listed clinical categories account for only \(38+7=45\) patients, leaving \(50-45=5\). Of these, 2 died and 3 were censored before one year. A suitable model that includes death should represent death separately; the 3 lost patients should be recorded as censored, not silently assigned to a clinical state.
For example, if the initial state is “on treatment without relapse,” the diagram may include arrows to relapse and death when both are possible direct clinical transitions under the definitions. The hypothetical counts alone do not establish when or from which state each event occurred, so they cannot determine all arrows. That requires the individual event histories and the intended state definitions.
If the research question instead concerns relapse only among patients whose follow-up is available, the scope and handling of death must be stated. Simply leaving death out while describing the model as covering the full clinical process would be misleading.
Arrows Must Match the State Definitions
An arrow asserts that a direct transition is allowed within the process, not merely that two state labels can be written next to one another. An arrow is structurally impossible if it contradicts the defining meaning of a state or the time ordering of the modeled process. For example, if “dead” means death has occurred and the process follows an individual’s clinical status, an arrow from dead to alive contradicts that definition.
By contrast, an arrow should not be removed merely because it is rare, absent in a small data set, or not observed during a particular study. “Not observed” is evidence about the available records; “impossible” is a claim about the process as defined. The study protocol, clinical knowledge, and operational rules for recording states may inform that claim. When records appear to show a forbidden arrow, first check the definitions, event dates, and data coding rather than treating the contradiction as a genuine transition.
Self-transitions are usually not drawn in a continuous-time state diagram: remaining in a state is not a change from one state to another. In data recorded at discrete visits, however, consecutive observations may show the same state. That repeated observation is not itself a new state or necessarily a transition; the diagram convention and time scale should be made clear.
Worked Example: Distinguishing an Impossible Arrow From No Evidence
A hypothetical diagram for a process defined by “alive,” “hospitalized while alive,” and “dead” includes an arrow from dead to hospitalized. The records also contain no observed transitions from alive outside hospital to hospitalization during a small study. Which feature is structurally wrong?
The arrow from dead to hospitalized is inconsistent with the definitions: a person classified as dead cannot subsequently be hospitalized while alive in this process. It should be removed, or the state definitions and records should be investigated if the arrow arose from recorded data.
The absence of observed transitions from alive outside hospital to hospitalization does not, by itself, prove that such a transition is impossible. If a living person can be admitted under the study’s clinical scope, the arrow may be allowed even if no one in this small hypothetical study made that transition. The diagram represents the permitted process, not only the transitions that happened to appear in one sample.
This judgment assumes “dead” is an absorbing clinical status and that hospitalization is defined only for living patients. If the labels instead refer to a different construct, such as a historical record or a posthumous administrative classification, their meanings must be clarified before judging the arrow.
Endpoints, Absorbing States, and Censoring
An endpoint in a diagram is not necessarily an absorbing state. In a time-invariant state model, a state is absorbing if, once entered, the process cannot leave it. A state is transient in the relevant structural sense if the process may leave it for another state. These properties depend on what the states mean and which transitions are allowed, not on whether a label sounds final or appears at the end of a diagram.
Death is commonly absorbing when the process describes an individual’s living clinical status. Remission is not necessarily absorbing if relapse can occur. Discharge from hospital is not necessarily absorbing if readmission is possible. Administrative study completion, in contrast, usually ends observation rather than changing the subject’s clinical state. It is therefore generally represented as the end of observation or censoring, not as an absorbing clinical state.
If a model treats a genuinely absorbing state as transient, it allows an incoherent exit and can misrepresent the process. If it treats a state as absorbing merely because it is a study endpoint, it can erase clinically possible later transitions. The right check is whether leaving that state is possible under the definitions and time scope—not where the box was drawn.
Worked Example: Is the Endpoint Really Absorbing?
In a hypothetical illness-death diagram, the states are healthy, ill, and dead. The allowed clinical transitions are healthy to ill, healthy to dead, and ill to dead. Is it correct to draw an outgoing arrow from dead to healthy because “dead” is the final study endpoint?
No. Under the stated definition of dead, the process cannot leave that state. “Final study endpoint” is not a reason to add an outgoing arrow; death is absorbing for this clinical-status process. Healthy and ill are transient in this diagram because each has at least one permitted exit, while dead has none.
Now change the endpoint to remission in a different hypothetical process where relapse is possible. A diagram with “remission” as a final box would then be structurally wrong if it excludes remission to relapse. The endpoint label does not make remission absorbing. A planned end-of-study visit is also not, by itself, a clinical state transition: observation may stop while the participant remains in whichever clinical state they occupy.
A Practical Structural Audit
A short audit before analysis can catch errors that are difficult to repair after transition records have been summarized. Check the state definitions and the arrows separately: a clear diagram cannot compensate for ambiguous definitions, and precise definitions do not guarantee that every relevant arrow has been included.
Specify the population, time origin, and period over which the state process is defined. Identify whether the model concerns clinical status, a protocol process, or another clearly stated construct.
At a fixed time, ask whether each eligible subject belongs to exactly one state. Look for overlapping labels and for people who fit none of them.
Ask what clinically or scientifically relevant outcomes can occur within scope. Distinguish those outcomes from loss to follow-up and administrative end of observation.
For every arrow, check whether it contradicts the source or destination definitions. Do not infer impossibility solely from a lack of observed transitions.
For every state with no outgoing arrows, ask whether departure is impossible in the modeled process or whether the diagram has mistaken an endpoint for an absorbing state.
These checks do not establish that a state process satisfies the Markov property, nor do they identify the best state space for every scientific question. A diagram can be structurally coherent while still failing to summarize relevant history or being too coarse for the intended analysis. Those are separate modeling concerns; structural coherence is necessary, not sufficient.
Common Mistakes and Their Corrections
- Wrong reasoning: “Alive” and “hospitalized” are different labels, so they are separate states. Correction: Check whether a hospitalized person can also be alive. If so, refine the labels into disjoint categories or define a different state space.
- Wrong reasoning: “The records do not show this event, so the arrow is impossible.” Correction: Lack of observations does not establish impossibility. Decide whether the transition is allowed under the process definitions and scope.
- Wrong reasoning: “The study ended, so study completion is an absorbing clinical state.” Correction: Study completion usually marks the end of observation, not a change in clinical status.
- Wrong reasoning: “Remission is the last box, so it cannot be left.” Correction: Check whether relapse or another exit is possible. A state is absorbing only when the process cannot leave it.
- Wrong reasoning: “Everyone missing from the listed outcome states can be counted as censored.” Correction: Determine whether the subject had a clinical event, was censored, or had an unknown classification. These are not interchangeable explanations.
Check Your Understanding
Answer each question using the definitions and distinctions above. Brief solutions follow the questions.
- A diagram has the states “alive,” “in hospital,” and “dead.” What structural problem may arise, and how could the states be revised?
- In a hypothetical cohort of 40, 30 are recorded in a clinical state, 6 have a second clinical outcome, and 4 are lost to follow-up. Should the four lost subjects automatically be added to the second outcome state?
- A transition is permitted by the clinical definitions but does not occur in a small sample. Does that absence alone justify removing the arrow?
- A model labels remission as its final box, but relapse can occur after remission. Should remission be absorbing?
- In a process defined by living clinical status, should death have an outgoing arrow to a living state? Explain.
Solutions. (1) “Alive” may overlap with “in hospital.” Use mutually exclusive categories such as alive in hospital, alive outside hospital, and dead, if these categories are observed and appropriate. (2) No. The four are censored if follow-up ended without observing their later clinical state; they should not be assigned the second outcome without evidence. (3) No. A small sample may simply not contain an allowed transition. (4) No. If relapse can follow remission, remission is not absorbing and the relevant outgoing arrow must be represented. (5) No. Under the stated definition, death is absorbing; an apparent transition to living status calls for checking the state definitions or records.