Why the Structure of Stage Progression Matters
A sequence of disease stages can look like a simple ladder: a patient moves from an earlier stage to the next later stage. But a diagram that permits only adjacent forward moves makes a stronger claim than one that also permits jumps over stages. A diagram that allows movement back to an earlier stage makes a different claim again. These structural choices determine which patient histories the model can represent.
Here, a stage is a state category defined by the clinical question—for example, disease severity at a particular time. The order of the labels expresses a scientific ordering, not a numerical measurement of distance between states. Stage III is later or more severe than Stage II in the specified ordering; that does not mean the difference between them equals the difference between Stages I and II.
A useful starting point is to define states that are mutually exclusive and collectively cover the disease conditions the model is intended to represent. Then state the stage ordering and decide which transitions are allowed. If death is included, it is usually represented as a separate absorbing state: once entered, there are no outgoing arrows. Whether death can occur directly from each disease stage is a clinical and modeling choice, rather than a consequence of the stage ordering.
Three Structures for Ordered Stages
For clarity, suppose the disease stages are \(1,2,\ldots,K\), ordered from earliest to latest. The following terms describe distinct structures. In this tutorial, adjacent-stage strictly progressive means that a patient may advance by one stage at a time, but cannot regress or skip a stage. This explicit definition avoids relying on “progressive” alone, which can be used more broadly.
These diagrams should be read as lists of allowed moves, not as predictions that every patient follows every arrow. For example, the arrow \(1\to3\) allows a direct transition from Stage 1 to Stage 3. It does not say how often that happens. Conversely, if \(1\to3\) is absent from a strictly adjacent structure, a recorded direct transition challenges that structure unless the apparent jump can be explained by the observation process or state definitions.
The mathematical distinction can be stated using the stage index. Under adjacent-stage strict progression, every disease-stage transition satisfies \(j-i=1\). Under monotone progression with skips, every such transition satisfies \(j-i\geq1\). A regression-permitting diagram includes at least one allowed transition with \(j-i<0\). These constraints concern individual transitions, not only the first and last stage in a patient's history.
Worked Example: Drawing an Adjacent-Stage Structure
Worked Example: Drawing an Adjacent-Stage Structure
In a hypothetical clinic, the proposed model has three ordered disease stages and death. The clinical team specifies that disease can advance only to the next stage, while death may occur from any disease stage. The diagram is intended to describe allowed pathways, not estimated transition rates.
Let the states be \(1\), \(2\), \(3\), and \(D\), where \(D\) denotes death. The specified disease-stage arrows are \(1\to2\) and \(2\to3\). The specified death arrows are \(1\to D\), \(2\to D\), and \(3\to D\). State \(D\) has no outgoing arrows.
A hypothetical set of 10 observed disease-stage transitions is: six \(1\to2\), three \(2\to3\), and one \(1\to3\). The first two transition types satisfy the adjacent-stage rule because \(2-1=1\) and \(3-2=1\). The last does not: \(3-1=2\), so it skips Stage 2. Thus, the histories contain \(6+3=9\) adjacent transitions and one transition inconsistent with the proposed disease-stage structure.
The arrow \(1\to D\) does not count as a skipped disease stage: death is a separate endpoint, not Stage 2 or Stage 3. Whether the observed \(1\to3\) should lead to changing the model depends on whether stages were reliably observed between assessments. If the patient could have passed through Stage 2 between visits, the observation schedule may have hidden an intermediate transition. If continuous or sufficiently frequent observation confirms a direct jump, the adjacent-only structure does not represent that history.
Checking a Structure Against Histories
A proposed diagram is a structural restriction on possible histories. To check one, list the successive observed states for each patient, turn each consecutive pair into a transition, and compare each transition with the allowed arrows. If a history contains \(1,2,3\), its transitions are \(1\to2\) and \(2\to3\); the diagram need not contain a direct arrow \(1\to3\), because the patient did not make that direct move. If a history contains \(1,3\), it does require an allowed \(1\to3\) move at the level of observation.
This check depends on what “observed transition” means. If assessments occur only at scheduled visits, a patient seen in Stage 1 at one visit and Stage 3 at the next may have passed through Stage 2 in between. The data then show a change between observed states, not necessarily a biologically instantaneous jump. The modeler must decide whether the diagram describes transitions in the underlying disease process or only changes between recorded assessments, and define states and observation times accordingly.
Worked Example: Allowing a Skipped Stage
In a second hypothetical dataset, three stages are assessed often enough that the investigators regard the recorded transition pairs as direct moves. Among 12 observed disease-stage transitions, there are five \(1\to2\), four \(2\to3\), two \(1\to3\), and one \(3\to2\).
For a monotone skip-permitting structure, all disease-stage transitions must move forward, so \(j-i\geq1\). The first three types satisfy this condition: \(2-1=1\), \(3-2=1\), and \(3-1=2\). Together they account for \(5+4+2=11\) forward transitions. The remaining transition has \(2-3=-1\), so \(3\to2\) is a regression and is not allowed by a monotone structure.
The appropriate diagram therefore depends on the scientific interpretation of that one history. If Stage 3 to Stage 2 is a credible reduction in severity under the state definitions, the monotone structure is inadequate and a regression-permitting arrow \(3\to2\) is needed. If the change reflects measurement error or inconsistent staging, adding the arrow may encode a data-quality problem as if it were a genuine disease transition. The two \(1\to3\) transitions, by contrast, support allowing a skip in the diagram; they are not evidence of regression.
A Short Structural Justification
Why does an adjacent-stage model force every path from Stage 1 to Stage 3 to pass through Stage 2? Each allowed disease-stage move increases the index by exactly one. Starting from index 1, after one such move the index is 2; after a second it is 3. A direct move from 1 to 3 would increase the index by two in one transition, contradicting the rule \(j-i=1\).
More generally, for adjacent-stage progression, any path from stage \(i\) to a later stage \(j\) must visit each intervening stage in order. This follows by repeatedly applying the rule that the next stage index is the current index plus one. In a skip-permitting structure, only the requirement \(j>i\) remains for each forward move, so the intermediate states need not appear in the observed path.
Worked Example: Representing Regression
Worked Example: Representing Regression
In a third hypothetical example, a treatment-response study uses Stage 1 for lower disease burden, Stage 2 for higher burden, and Stage 3 for advanced disease. Investigators decide that improvement from Stage 2 to Stage 1 is clinically possible, but do not permit regression from Stage 3 in this particular model. They allow progression from Stage 1 to Stage 2 and from Stage 2 to Stage 3.
The disease-stage arrows are \(1\to2\), \(2\to3\), and \(2\to1\). The arrow \(2\to1\) represents regression in stage, while \(1\to2\) and \(2\to3\) represent progression. No arrow \(3\to2\) is included because this specified structure does not allow it.
Suppose 15 hypothetical observed disease-stage transitions consist of seven \(1\to2\), five \(2\to3\), and three \(2\to1\). The counts sum to \(7+5+3=15\). The three \(2\to1\) transitions demonstrate why a strictly progressive structure would fail to represent all these histories. The diagram above accommodates all 15 transition types because each has a corresponding arrow.
The absence of a recorded \(3\to2\) transition among these 15 observations does not by itself prove that such regression is impossible. The model disallows it because of the stated scientific specification, not because its observed count is zero. If later evidence or clinical knowledge makes that transition plausible, the structural specification should be reconsidered.
Diagnostics, Alternatives, and Limitations
A useful diagnostic is to tabulate consecutive observed state pairs and mark whether each has an arrow in the proposed diagram. An observed pair without an arrow signals a mismatch to investigate. Possible explanations include an overly restrictive structure, unrecorded intermediate stages, inconsistent state classification, or an error in the recorded history. The table cannot determine which explanation is correct by itself.
Keep two claims separate: “this transition was not observed” and “this transition is structurally impossible.” The first describes the sample; the second is a modeling restriction. Even a large dataset with no observed \(1\to3\) moves does not establish that the move can never occur. Conversely, a single well-documented move may expose an overly restrictive structural rule, though measurement and timing need review.
The main alternatives express different scientific commitments. Adjacent-stage progression enforces ordered movement one stage at a time. Monotone progression with skips permits direct forward transitions but excludes regression. A regression-permitting structure includes selected moves to earlier stages. Collapsing several stages into a broader state can simplify the diagram, but loses distinctions that may matter clinically. Choose among these by defining the target states and plausible pathways, not by making the diagram as simple as possible.
A diagram is also limited by the quality and resolution of the state measurements. Coarse assessment schedules can make sequential progression appear to be a direct skip. Noisy staging can create apparent regression. And an arrow establishes only that a move is permitted in the specified model; it does not quantify how likely or how rapid that move is. State diagrams do not, on their own, establish a Markov assumption or any particular transition model.
Check Your Understanding
Answer each question, then compare your reasoning with the solutions below.
- In an adjacent-stage model with stages 1, 2, and 3, is \(1\to3\) allowed? Give the structural reason.
- A monotone skip-permitting model records \(2\to4\). Is this compatible with its ordering rule? What about \(4\to2\)?
- Why is a patient observed in Stage 1 at one visit and Stage 3 at the next not necessarily proof of a biological direct jump?
- A transition has count zero in a dataset. What can and cannot be concluded from that count about whether the transition is structurally possible?
Solutions. (1) No. The adjacent-stage rule requires \(j-i=1\), but \(3-1=2\); a path from 1 to 3 must pass through 2. (2) \(2\to4\) is compatible because \(4-2=2\geq1\), so it is a forward skip. \(4\to2\) is incompatible because \(2-4=-2<0\), which is regression. (3) Stage 2 may have occurred between visits without being observed, so the recorded pair does not establish the underlying transition sequence. (4) A zero count says only that the transition was not observed in that dataset. It does not prove the transition impossible; a structural prohibition needs scientific justification.