Why Represent Illness and Death as States?
A study of disease progression may record more than whether a person has died. A person might remain healthy, develop an illness and later die, or die without a recorded illness. Those histories have different clinical meanings, even if a summary records only the final endpoint. The three-state illness-death structure represents this distinction by describing each person as moving among defined health states over time.
The structure is a model of possible histories, not a claim that every disease follows the same course. Researchers must define “healthy” and “ill” for the clinical question and data available. For example, “ill” might mean a first diagnosed event, a confirmed disease stage, or a specific complication. A change in the operational definition can change which transitions are observed.
Let \(X(t)\) denote a person’s state at time \(t\). A person starting in state \(0\) can remain healthy, move to state \(1\), or move directly to state \(2\). If the person enters state \(1\), the person can remain ill or move to state \(2\). The structure excludes \(1\to0\), and it excludes any transition out of state \(2\). These restrictions describe the model’s permitted paths; they do not prove that recovery is biologically impossible. If recovery is relevant to the question, the state structure must be changed rather than treating it as an allowed transition here.
The Three Transition Types
Each arrow represents a distinct change of state. A transition is not simply a label for a person’s final outcome: it identifies the state occupied immediately before and after a change. The distinction between the two routes to death is central. A direct death is a \(0\to2\) transition; a death after illness is a \(1\to2\) transition.
| Transition | Interpretation | What must be true immediately before it? |
|---|---|---|
| \(0\to1\) | Healthy to ill | The person is in state \(0\) |
| \(0\to2\) | Healthy directly to dead | The person is in state \(0\) |
| \(1\to2\) | Ill to dead | The person has entered state \(1\) and is still there |
The word “directly” means that no state-1 illness transition was observed between healthy and dead in the specified model and data. It does not necessarily mean that no illness occurred in reality: an illness could have been unmeasured, undiagnosed, or excluded by the operational definition. The state definitions and observation schedule therefore affect how histories are classified.
Worked Example: Classifying Individual Histories
Consider three hypothetical participants followed from a healthy starting state. Participant A is diagnosed with the illness at month 5 and dies at month 14. Participant B remains healthy and dies at month 9. Participant C is diagnosed at month 7 and is alive and ill at the end of follow-up at month 12. Write each observed history as a state path.
Participant A: The observed sequence is \(0\to1\to2\), with transitions at months 5 and 14. The first transition is healthy to ill; the second is ill to dead.
Participant B: The sequence is \(0\to2\), with a direct healthy-to-dead transition at month 9. There is no recorded \(0\to1\) transition before death.
Participant C: The observed sequence is \(0\to1\), with entry into illness at month 7. At month 12 the participant remains in state 1; there is no observed transition to state 2 during the follow-up.
All three records are compatible with the structure. A hypothetical record showing \(0\to1\to0\) would not be compatible: it contains recovery, a transition the three-state structure excludes.
Allowed Histories and a Structural Consequence
For a person who begins healthy, the allowed paths follow directly from the arrows. Before any transition, the person is in state \(0\). From state \(0\), the person may enter illness or die. If the person enters illness, the person may remain ill or die, but cannot return to healthy under this structure. Once dead, no further transition occurs.
This gives a simple exhaustive description of routes to death. Any observed death belongs to one of two routes: direct death from state \(0\), or death after first entering state \(1\). To justify that these are exhaustive, begin at state \(0\). If the first transition is \(0\to2\), the route is direct and ends in the absorbing state. Otherwise, a first transition to state \(1\) is required before any \(1\to2\) transition can occur. Since there is no other allowed entry to state \(2\), there is no third route in this structure.
The argument relies on the model’s starting state and allowed transitions. If people may enter the study already ill, their initial state is 1, so their observed route can begin \(1\to2\); it would be inaccurate to describe that record as a healthy-to-ill-to-dead history. Entry rules and initial states must therefore be stated before interpreting paths.
Worked Example: Checking Whether a Proposed Path Is Allowed
A hypothetical record begins with a participant in state 0. The recorded states at successive event times are \(0,1,2\), with event times 4 and 10 months. Check the transitions and describe the record.
The first change is \(0\to1\), which is allowed and represents illness at month 4. The second change is \(1\to2\), also allowed, representing death at month 10 after illness. Thus the full path \(0\to1\to2\) is valid.
Now suppose a data-processing file instead lists \(0,1,0,2\), at months 4, 8, and 10. The first change is allowed, but the second is \(1\to0\), which is not. The path is therefore invalid under the specified structure. If the month-8 observation reflects genuine recovery, this is not a reason to silently relabel the record; it is evidence that a no-recovery structure does not represent the observed course and may need revision.
The check is structural rather than statistical: each adjacent pair of states must match one of the permitted arrows. The number of months between transitions does not make a forbidden transition allowable.
Transition-Specific Quantities
When a model assigns a transition-specific instantaneous rate, the rate is indexed by the arrow, not just by the destination “death.” For a continuous-time process with single transitions, one common definition is
Here \(\mathcal H_{t-}\) denotes the observed history just before time \(t\), and \(h\) is a short time interval. The expression describes the instantaneous transition rate from state \(i\) to state \(j\), conditional on being in state \(i\) and the stated history. It applies when the limit exists and the continuous-time, single-transition formulation is appropriate. It does not assume that the current state alone summarizes the relevant past; that additional Markov assumption is a separate modeling choice.
In this structure, the possible transition-specific rates are \(\lambda_{01}\), \(\lambda_{02}\), and \(\lambda_{12}\). There is no \(\lambda_{10}\) because \(1\to0\) is disallowed, and no transition rate out of state 2 because death is absorbing. The two rates ending in death, \(\lambda_{02}\) and \(\lambda_{12}\), describe different transitions: the first applies while in state 0, and the second while in state 1. They should not be treated as one interchangeable “death rate.”
Worked Example: Summarizing Routes in a Hypothetical Cohort
Suppose 60 hypothetical participants all enter in state 0 and are completely followed to a fixed study endpoint. Eighteen enter state 1; of those 18, seven later die. Twelve others die directly from state 0. The remaining participants have no observed transition out of state 0 by the endpoint. Calculate the numbers and proportions in each route.
The illness-mediated death route contains the seven participants who first entered state 1 and later entered state 2. Its proportion of the original cohort is \(7/60=0.1167\), or about 11.7%.
The direct-death route contains 12 participants. Its proportion is \(12/60=0.2000\), or 20.0%.
The total number of deaths is \(7+12=19\), giving an overall death proportion of \(19/60=0.3167\), or about 31.7%. The arithmetic also checks the route accounting: \(7\) illness-mediated deaths plus \(12\) direct deaths equals \(19\) deaths. Of the 18 who entered illness, \(18-7=11\) did not enter state 2 during the observed follow-up. Of the 60 total, \(60-18-12=30\) had no observed transition out of state 0. The route-specific counts retain information that the combined death count alone does not.
These are descriptive proportions for the stated cohort and follow-up, not instantaneous rates or estimates of a causal effect. The calculation assumes the records fully classify these participants through the specified endpoint; censoring or unobserved transitions would require a different analysis.
Model Checks, Alternatives, and Limits
Before using the structure, check that it matches the scientific question and the observed data. In particular, verify the state definitions, the initial-state rule, and whether recovery is possible and recorded. Then inspect event histories for prohibited sequences, duplicated transitions, or an apparent transition recorded in the wrong order. A transition cannot originate in a state the person has not yet entered.
The three-state structure is useful when illness status and the route to death matter. A single time-to-death analysis can summarize when death occurs but, on its own, does not distinguish death from healthy state \(0\) from death after illness in state \(1\). Conversely, a binary indicator of whether illness ever occurred does not retain the timing and subsequent transition to death. The state structure keeps these routes explicit, but it does not by itself specify how transition probabilities or rates are estimated.
The structure has important limits. It treats illness as a single state, so it cannot distinguish severity levels or multiple forms of illness. Its exclusion of recovery may be inappropriate for a reversible condition. It also assumes that the recorded state labels adequately capture the clinical distinctions relevant to the analysis. If illness is missed between visits, a recorded direct death may not correspond to the true clinical path. More states or different allowed transitions may be needed when the scientific question requires them.
Check Your Understanding
Answer each question using the three-state structure. Solutions follow the questions.
- A person starts healthy, becomes ill at month 3, and is alive and ill at month 9. What is the observed path, and what transition has not occurred?
- Is \(0\to1\to0\to2\) allowed? Identify the transition that determines your answer.
- Among 40 hypothetical participants who all begin healthy, 8 die directly from state 0 and 6 die after entering state 1. What is the total number of deaths, and how many follow each route?
- Why should deaths from state 0 and deaths from state 1 not be represented as the same transition?
- A participant enters the study already ill and later dies. Why is it inaccurate to report the observed history as \(0\to1\to2\)?
Solutions. (1) The path is \(0\to1\); no \(1\to2\) transition has occurred by month 9. (2) No. The proposed path contains \(1\to0\), which is excluded. (3) There are \(8+6=14\) deaths: 8 direct \(0\to2\) transitions and 6 illness-mediated \(1\to2\) transitions. (4) Their origins differ: the person is in state 0 before a direct death and in state 1 before a death after illness. (5) The observed initial state is 1, not 0; the observed path is \(1\to2\), and the earlier healthy-to-ill transition was not observed in the study.