Transition Counts and Risk-Set Indicators

Why Transition-Specific Counts Need Transition-Specific Risk Sets

In a multi-state model, an observed event is not fully described by its time alone: the individual moves from an origin state to a destination state. A transition count records how often a particular move occurs. Its corresponding risk set records who could make that move, based on the state occupied immediately before each time and whether the person is under observation. These two processes are basic bookkeeping objects for transition-specific analyses.

The key distinction is between being observed and being at risk for a specified transition. An individual can be observed while occupying a state that is not the origin of the transition under consideration. That person contributes to observation, but not to that transition’s risk set. Conversely, a move into a state can change the individual’s risk status for several possible outgoing transitions at once.

Definition: Let $X_i(t)$ be the state occupied by individual $i$ at time $t$, with a right-continuous path, and let $X_i(t-)$ denote the state immediately before $t$. Let $\mathcal{E}$ be the set of allowed directed transitions $h\to k$, where $h\ne k$. If individual $i$ is observed on $(L_i,C_i]$, define the transition counting process $$ N_i^{hk}(t)=\sum_{0<s\le t} I\{L_i<s\le C_i,\ X_i(s-)=h,\ X_i(s)=k\},\qquad h\to k\in\mathcal{E}, $$ and the transition-specific at-risk indicator $$ Y_i^{hk}(t)=I\{L_i<t\le C_i\}\,I\{X_i(t-)=h\}. $$ Here $I\{\cdot\}$ equals 1 when its condition holds and 0 otherwise. $N_i^{hk}(t)$ counts observed $h\to k$ transitions through time $t$; $Y_i^{hk}(t)$ is 1 when individual $i$ is observed and occupies the origin state $h$ immediately before $t$.

The observation-window convention in this definition treats $L_i$ as the entry time and $C_i$ as the last observed time, with observations through $C_i$ included. A transition known to occur at $C_i$ is counted; a transition after $C_i$ is not. If event and censoring times coincide but the event status is ambiguous, the data protocol must specify whether the event was observed. This convention also makes $Y_i^{hk}(t)$ reflect the state just before an event at $t$: a person can be at risk for $h\to k$ immediately before making that transition.

Reading the Processes From a State Path

For a fixed transition, $N_i^{hk}(t)$ starts at zero and is nondecreasing. It increases by one at each observed $h\to k$ transition and does not change at other transitions. In a model allowing recurrent moves, it can increase more than once for the same person. The indicator $Y_i^{hk}(t)$ is either zero or one at each time. It is not a count of transitions; it records whether the person is currently eligible to make the specified move.

When the state space has several possible outgoing transitions from $h$, a person in $h$ is at risk for each of those allowed transitions. For example, if both $h\to k$ and $h\to r$ are allowed, then $Y_i^{hk}(t)=Y_i^{hr}(t)=1$ while the person is observed in $h$. These indicators do not assert that both moves occur. They describe the alternatives available from the current origin state.

Worked Example: Counting Moves Along One State Path

Suppose one hypothetical individual is observed from time 0 through time 10, with path $A\to B$ at time 2, $B\to C$ at time 5, and $C\to B$ at time 8. The allowed transitions include these three moves. Find the counts through time 6 and identify the transition-specific risk status at times 3, 5, and 6.

By time 6, the observed $A\to B$ move at time 2 and $B\to C$ move at time 5 have occurred, while the $C\to B$ move at time 8 has not. Thus $$ N^{AB}(6)=1,\qquad N^{BC}(6)=1,\qquad N^{CB}(6)=0. $$

At time 3 the person is in $B$, so $Y^{BC}(3)=1$, whereas $Y^{AB}(3)=Y^{CB}(3)=0$. At time 5, the state immediately before the transition is still $B$, so $Y^{BC}(5)=1$; the transition count $N^{BC}$ jumps at 5, and after that time the person occupies $C$. At time 6, $Y^{CB}(6)=1$, and the other two indicators are zero. This illustrates why risk status at an event time is determined from the pre-transition state.

For a cohort of $n$ individuals, the aggregate transition count and risk-set size are the sums of the individual processes:

N^{hk}(t)=\sum_{i=1}^{n}N_i^{hk}(t),\qquad Y^{hk}(t)=\sum_{i=1}^{n}Y_i^{hk}(t).

The aggregate count is the number of observed $h\to k$ moves through $t$, and the aggregate risk-set size is the number observed in state $h$ immediately before $t$. Both depend on the transition specified. It is generally incorrect to use the total number observed, or the number occupying any state, as $Y^{hk}(t)$.

Worked Example: Updating an Aggregate Risk Set

Consider four hypothetical individuals, all entering in state $A$ at time 0. Individual 1 moves $A\to B$ at time 2 and $B\to D$ at time 6, and is observed through time 8. Individual 2 moves $A\to B$ at time 4 and is censored at time 7. Individual 3 moves $A\to C$ at time 3 and $C\to D$ at time 5. Individual 4 remains in $A$ and is censored at time 5. Find the $A\to B$ count through time 6 and its risk-set sizes immediately before times 2, 3, 4, and 5.

Initially all four individuals are observed in $A$, so $Y^{AB}(2)=4$. The move by individual 1 at time 2 removes that person from the $A\to B$ risk set. Immediately before time 3, individuals 2, 3, and 4 remain observed in $A$, giving $Y^{AB}(3)=3$. Individual 3 moves to $C$ at time 3, so immediately before time 4, only individuals 2 and 4 are in $A$: $Y^{AB}(4)=2$. Individual 2 moves to $B$ at time 4; immediately before time 5, individual 4 is the only person still in $A$, so $Y^{AB}(5)=1$. Censoring individual 4 at time 5 ends that person’s observed risk time.

The observed $A\to B$ moves occurred at times 2 and 4. Therefore $$ N^{AB}(6)=1+1=2. $$ The move $A\to C$ at time 3 reduces the $A\to B$ risk set even though it is not an $A\to B$ event: the person leaves its origin state. Likewise, the $B\to D$ move at time 6 does not add to $N^{AB}$, because its origin is $B$, not $A$.

Delayed Entry, Re-entry, and State Balance

Delayed entry is handled by the lower endpoint $L_i$ of the observation window. A person is not included in any transition risk set before entry, even if their state before entry is known from other information. After entering, the person can contribute to the risk sets for transitions out of their current state. If a path returns to a state, the person can re-enter that state’s outgoing risk sets. This is essential in models that allow recurrent transitions or cycles.

Worked Example: Delayed Entry and a Return to the Origin State

A hypothetical individual enters observation at time 2 in state $U$, moves $U\to V$ at time 3, returns $V\to U$ at time 6, and is censored at time 9. Determine the times when the individual is at risk for $U\to V$, and calculate $N^{UV}(8)$.

Before time 2 the person is not observed, so $Y^{UV}(t)=0$ for $t\le 2$ under the stated open-entry convention. From just after entry up to the transition at time 3, the person is observed in $U$, so $Y^{UV}(t)=1$ for $2<t\le 3$, with the value at 3 determined by the pre-transition state. After moving to $V$, the person is no longer at risk for $U\to V$. The return to $U$ at time 6 makes the person at risk for $U\to V$ again for $6<t\le 9$. There is only one observed $U\to V$ move by time 8, hence $N^{UV}(8)=1$. The later occupancy of $U$ does not add a count unless another $U\to V$ move occurs.

The processes also obey a useful state-balance identity. Suppose an individual’s path is completely observed from time 0 through $t$, every change of state is recorded as one allowed transition, and there are no simultaneous transitions. For any state $h$, the change in its occupancy indicator equals the number of entries into $h$ minus the number of exits from $h$:

I\{X_i(t)=h\}-I\{X_i(0)=h\} = \sum_{g:g\to h}N_i^{gh}(t) - \sum_{k:h\to k}N_i^{hk}(t).

To justify the identity, consider the change at each transition time. A transition into $h$ changes $I\{X_i(\cdot)=h\}$ by $+1$, and contributes exactly one to an incoming count. A transition out of $h$ changes the indicator by $-1$, and contributes exactly one to an outgoing count. A transition between two states neither of which is $h$ changes neither side. Summing these increments over the observed path gives the identity. It provides a direct consistency check: if the counts imply a different final occupancy from the recorded path, at least one transition, state label, or time ordering has been recorded incorrectly.

Assumptions, Checks, and Common Errors

Constructing these processes is a data-representation task, not an estimation procedure. It requires a clearly specified state space and allowed-transition set, correctly ordered state paths, and observation windows that say when each individual is under observation. A transition must be assigned to its actual origin and destination states. For the definitions above, state paths are taken to be right-continuous, transitions occur at distinct times for an individual, and a transition known to occur at the censoring boundary is treated as observed. If simultaneous changes are possible, the data must specify their ordering or encode them using an appropriate joint transition convention before these simple single-jump counts are used.

Several checks are useful in practice:

At a recorded $h\to k$ event time $s$, verify $Y_i^{hk}(s)=1$ under the chosen event-observation convention and verify that $N_i^{hk}$ increases by one.
Confirm that a person leaves the risk sets for transitions out of $h$ after leaving $h$, and enters the outgoing risk sets for the new state while still observed.
Check that censoring stops future observed contributions but does not erase transitions already observed.
Where the full path is observed, compare transition counts with the state-balance identity.
Check that counts are nondecreasing integers and that each individual’s count jumps only at recorded transitions of the specified type.

A common mistake is to call everyone under follow-up “at risk for $h\to k$.” That reasoning confuses observation with transition eligibility. The correction is to use both parts of $Y_i^{hk}(t)$: the person must be observed at $t$, and $X_i(t-)$ must equal the origin state $h$. A person observed in state $r\ne h$ contributes zero to this transition’s risk set, even if they remain under follow-up.

Transition-specific counts and risk sets also differ from an overall event count and a state-occupancy total. An overall count can combine events with different origins or destinations; it cannot identify how many times a particular move occurred. A state-occupancy total counts people in a state, while a transition risk set additionally respects the observation window and is defined just before time $t$. The earlier material on marked and multivariate counting processes provides the broader view of keeping event types distinct. These definitions alone do not estimate a transition rate or establish a probabilistic model; they provide the correctly indexed data processes such analyses require.

Key takeaway: For each allowed move $h\to k$, count only observed jumps from $h$ to $k$, and include an individual in its risk set only while observed and occupying $h$ immediately before the time in question. The origin state, not merely follow-up status, determines transition-specific eligibility.

Check Your Understanding

Answer each question, then check the solutions below.

An observed individual moves $P\to Q$ at time 3 and $Q\to R$ at time 7. What are $N^{PQ}(6)$ and $Y^{QR}(7)$, assuming observation includes both event times?
Four people are observed in state $A$ immediately before time 5. At time 5, one moves $A\to B$, one moves $A\to C$, and the other two remain in $A$. What is $Y^{AB}(5)$ and how much does $N^{AB}$ increase at time 5?
A person enters observation at time 4 in state $M$, moves $M\to N$ at time 6, and is censored at time 9. What is $Y^{MN}(5)$, and is the person at risk for $M\to N$ at time 7?
Explain why an observed person in state $C$ is not included in $Y^{AB}(t)$, even if $A\to B$ is an allowed transition.

Solutions. (1) $N^{PQ}(6)=1$, because the $P\to Q$ move has occurred once by time 6. Immediately before time 7 the person is in $Q$, so $Y^{QR}(7)=1$. (2) Immediately before time 5, all four occupy $A$, so $Y^{AB}(5)=4$. The $A\to B$ move makes $N^{AB}$ increase by 1; the $A\to C$ move is not counted in $N^{AB}$. (3) At time 5 the person is observed in $M$, so $Y^{MN}(5)=1$. At time 7 the person has already moved to $N$, so they are not at risk for $M\to N$. (4) The indicator requires $X_i(t-)=A$, as well as being under observation. Occupancy of $C$ fails the origin-state condition, so the person contributes zero to this risk set.

Transition Counts and Risk-Set Indicators

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