Why Specify Allowed Transitions?
A multi-state model needs more than a list of states. It also needs a clear account of which state changes can occur directly. That account is part of the scientific specification: it determines which histories the model regards as possible and which transition parameters can exist. If a move is ruled out by the event definitions or study process, representing it as possible can add meaningless parameters. Conversely, incorrectly ruling out a possible move can force the model to describe the process inaccurately.
The key distinction is between a direct transition and being in one state at one time and another state later. A subject may move from state \(i\) to state \(j\) through an intermediate state even if a direct \(i\to j\) move is forbidden. Structural zeros concern the specified one-step moves or direct jumps, not every possible destination over an interval of time.
The state labels should describe the process at the time resolution relevant to the analysis. Each subject is assigned one state at a time, and a transition means an actual change from the current state to a different state. A transition rule must be justified by the scientific meaning of the states, a protocol, or a known feature of how events occur. It is not justified merely because a small data set happens not to contain a particular move.
Structural Zeros in Different Model Representations
In a discrete-time model, a transition matrix gives the conditional probabilities of the state at the next specified time point given the current state. A structural zero for a direct one-step move is a zero in the corresponding matrix entry. Each row must still sum to one: probability assigned to forbidden moves is assigned instead to permitted moves or to remaining in the same state, as appropriate to the model.
In a continuous-time model, direct jumps can be represented by transition intensities. For \(i\ne j\), \(q_{ij}(t)\) is the instantaneous rate of a jump from \(i\) to \(j\) at time \(t\), conditional on being in \(i\) just before \(t\), under the specified intensity model. Its units are inverse time. A forbidden direct move has \(q_{ij}(t)=0\) throughout the relevant time range. Diagonal entries are not direct state changes; in a generator matrix they are set to make each row sum to zero.
The representation matters. In a discrete-time model, a transition matrix describes changes between its specified successive observation times. In a continuous-time jump model, an intensity concerns an instantaneous direct jump. Neither should be confused with the probability of ever reaching a state, or with the probability of being in that state at a later time.
Worked Example: Listing the State Space Before Counting Moves
Worked Example: Listing the State Space Before Counting Moves
Suppose a hypothetical follow-up system records four states: \(N\), no documented complication; \(U\), active complication; \(R\), complication resolved; and \(D\), death. The system defines entry into \(R\) to require that the immediately preceding observed state was \(U\); no intervening state is permitted. Death is recorded as the final state, so no subsequent state change is recorded after \(D\). Specify the consequence for \(N\to R\), and summarize 12 hypothetical observed transitions: 5 \(N\to U\), 2 \(N\to D\), 3 \(U\to R\), and 2 \(U\to D\).
State space. The four states are \(S=\{N,U,R,D\}\). Including \(D\) is essential because the records include moves into death; it is not an unlisted outcome.
Restriction. By the stated event rule, an entry into \(R\) must be immediately preceded by \(U\). Thus a direct \(N\to R\) move is forbidden, and the structural-zero pair is \((N,R)\). This conclusion follows from the immediate-predecessor rule. The weaker statement that a subject must have been in \(U\) at some point would not suffice: a history such as \(N\to U\to N\to R\) would then be possible.
Observed transition proportions. There are \(5+2=7\) observed moves out of \(N\), of which \(5\) go to \(U\) and \(2\) to \(D\). Their observed proportions among moves out of \(N\) are \(5/7=0.7143\) and \(2/7=0.2857\). There are \(3+2=5\) moves out of \(U\), so the observed proportions are \(3/5=0.6000\) to \(R\) and \(2/5=0.4000\) to \(D\). In total, \(7+5=12\) transitions are reported.
These sample proportions describe the 12 recorded transitions; they do not establish the structural restriction. The rule about immediate entry into \(R\) does that. Also, the absence of observed moves such as \(U\to N\) does not make those moves structurally impossible unless an independent rule excludes them.
Checking Whether a Zero Is Structural
A structural zero is a statement about the model’s support: the model assigns no chance to that direct move. A sample zero is only a statement about what happened to be observed. These have different implications. If a possible transition is rare, it may simply be absent in a modest sample; treating it as impossible because its count is zero can bias the model and misrepresent uncertainty.
A useful check is to ask what fact would make the move impossible for every subject under the stated definitions. A protocol might require a recorded intermediate state before a treatment can begin. A physical or clinical definition might make one direct change incompatible with the meaning of the states. If the justification depends on what happened earlier but the current state labels do not retain that history, the exclusion needs further scrutiny. The state description or the model may need to reflect the relevant information rather than imposing a zero unsupported by the recorded states.
Define each state in terms of what is true at a given time, and check that the states cover the modeled situations without overlap.
Specify what counts as the next state change or next discrete-time state. Do not silently treat an eventual destination as a direct move.
Give the event rule, protocol, or state-definition reason that makes the direct move impossible. A zero observed count alone is not a justification.
Confirm that permitted sequences remain scientifically coherent and that intermediate routes are not accidentally removed along with a forbidden direct move.
Why a Zero Intensity Implies No Such Direct Jumps
The continuous-time interpretation can be justified using the counting process for a particular direct transition. Let \(N_{ij}(t)\) count direct jumps from \(i\) to \(j\) by time \(t\), and let \(Y_i(t)\) indicate whether the subject is in state \(i\) and at risk for that jump just before \(t\). Under the usual intensity formulation, the expected number of \(i\to j\) jumps over follow-up through \(\tau\) is the expected integrated intensity:
This expression assumes the intensity is correctly specified for the process and that the relevant counting-process expectation is well defined. If \(q_{ij}(t)=0\) for all relevant \(t\), the integrand is zero, so \(E\{N_{ij}(\tau)\}=0\). Since \(N_{ij}(\tau)\) is a nonnegative integer, it must then equal zero with probability one: if it were positive with any positive probability, its expectation would be positive. Thus the structural zero excludes direct \(i\to j\) jumps during that follow-up period under the model.
This argument does not exclude a route through other states. The count \(N_{ij}\) concerns only direct jumps; it does not count a sequence such as \(i\to k\to j\). Nor does it say that every observed zero in a finite sample implies a zero intensity. The direction of reasoning is from the specified zero intensity to the model-implied absence of that jump, not from an observed absence to a proven structural restriction.
Worked Example: A Forbidden Direct Move With an Allowed Route
Worked Example: A Forbidden Direct Move With an Allowed Route
Consider a hypothetical discrete-time process with states \(A,B,C\). The study rules state that a subject in \(A\) cannot move directly to \(C\), but can move from \(A\) to \(B\), and a subject in \(B\) can move to \(C\). A specified one-step transition model assigns probabilities \(0.70\) to \(A\to A\), \(0.30\) to \(A\to B\), \(0.60\) to \(B\to B\), and \(0.40\) to \(B\to C\). The remaining probability in row \(C\) is assigned to \(C\to C\).
The one-step probability of \(A\to C\) is the structural zero \(0\). The one-step row for \(A\) is valid because \(0.70+0.30+0=1.00\); the row for \(B\) is valid because \(0.60+0.40=1.00\).
A two-step route from \(A\) to \(C\) is possible: first move \(A\to B\), then \(B\to C\). Under the stated transition model, the probability of that particular two-step sequence is \[ 0.30\times 0.40=0.12. \] So the direct \(A\to C\) entry is zero, while a later arrival in \(C\) through \(B\) has positive probability. The calculation uses the specified conditional transition probabilities for the two successive steps; it does not reinterpret the structural zero as a prohibition on ever reaching \(C\).
Worked Example: Testing a Proposed Structural Zero Against the Event Rule
Worked Example: Testing a Proposed Structural Zero Against the Event Rule
In a hypothetical monitoring system, let \(S\) mean “screen negative,” \(P\) mean “confirmed positive,” and \(T\) mean “treatment started.” The protocol states that treatment can begin only when the immediately preceding recorded state is \(P\). A reviewer proposes setting the direct transition \(S\to T\) to zero.
The proposal is justified if the states are recorded at the event times needed to apply the protocol and the protocol’s “immediately preceding” condition is part of the process definition. Under those conditions, every treatment-start event has \(P\) immediately before it, so a direct \(S\to T\) transition is excluded.
Now suppose instead that the data record only monthly status and a subject can test positive and start treatment between visits. A monthly record might change from \(S\) to \(T\) without showing the intermediate positive state. That observed pair does not necessarily represent a true direct jump: it may conceal \(S\to P\to T\) between assessments. The analyst must distinguish the actual state process from the coarsened observation schedule before imposing the zero.
For example, among 20 hypothetical recorded changes, suppose 8 are \(S\to P\), 5 are \(P\to T\), 4 are \(P\to S\), 2 are \(P\to D\), and 1 is \(T\to D\), where \(D\) denotes death. These counts sum to \(8+5+4+2+1=20\), and the observed count for \(S\to T\) is zero. That sample count alone does not prove a structural zero. The protocol and the observation-time definition determine whether the pair is impossible as a direct transition or only absent in these 20 recorded changes.
Common Mistakes, Alternatives, and Limitations
- Calling every sample zero structural. No observed \(i\to j\) transition can reflect limited follow-up or a small sample. Require a scientific or design-based reason for excluding the move.
- Confusing direct moves with eventual destinations. A zero for \(i\to j\) does not rule out reaching \(j\) through intermediate states. Check which transitions in a path are actually forbidden.
- Inferring impossibility from a vague history condition. “The subject must previously have been in \(U\)” allows intervening states unless the rule says otherwise. If the exclusion requires the immediately preceding state to be \(U\), state that explicitly.
- Leaving an outcome out of the state space. If death or another outcome is recorded as a destination, include it among the states before specifying the transition structure.
- Ignoring observation coarsening. A pair of states seen at consecutive visits may conceal unobserved intermediate events. The observed sequence need not be the sequence of actual direct jumps.
One alternative to a restricted transition structure is to allow every pair and estimate all corresponding transition parameters. That is appropriate when the moves are genuinely possible and data support estimation, but it can add parameters for impossible events and produce unstable estimates for rare ones. A different response to a restriction that depends on unrecorded prior history is to reconsider whether the state description contains the information needed to express the rule. The choice cannot be made from counts alone; it depends on the scientific process, the event definitions, and how states are observed.
Structural zeros also impose a strong limitation: if the restriction is wrong, the model cannot represent the excluded behavior, no matter how much data are collected. Before fitting a model, inspect the state definitions, protocol, observed histories, and time resolution together. If the data contain a purportedly forbidden direct transition, investigate coding, state assignment, and the rule itself rather than silently retaining an incompatible zero.
Check Your Understanding
Answer each question, then compare your reasoning with the solutions below.
- A discrete-time model has states \(L,M,H\). The one-step move \(L\to H\) is forbidden, but \(L\to M\) and \(M\to H\) are allowed. Is reaching \(H\) from \(L\) necessarily impossible?
- In 40 hypothetical transitions, no \(A\to B\) move is observed. What additional evidence is needed before \(A\to B\) can be treated as a structural zero?
- A rule says a patient can enter \(R\) only if they have previously been in \(U\). Does this alone rule out a direct \(N\to R\) move? Explain.
- If a direct-transition intensity \(q_{ij}(t)\) is zero throughout follow-up, what does the intensity argument imply about \(N_{ij}(\tau)\), and what does it not imply about indirect routes?
Solutions. (1) No. The two-step route \(L\to M\to H\) is allowed, so \(H\) may be reached even though the one-step \(L\to H\) move is forbidden. (2) A scientific, definitional, or protocol-based reason that makes the direct move impossible is needed; a finite-sample count of zero is not sufficient. (3) No. The history \(N\to U\to N\to R\) includes a prior visit to \(U\) but does not immediately precede entry into \(R\). To exclude \(N\to R\), the rule must require \(U\) immediately before \(R\), or the relevant history must be represented in the state definition. (4) Under the stated intensity assumptions, the expected direct-jump count is zero, and since the count is nonnegative it is zero with probability one. This excludes direct \(i\to j\) jumps; it does not exclude reaching \(j\) through one or more intermediate states.