Zero-Risk-Set Boundaries In Intensity Estimation

Why an Empty Risk Set Matters

A transition-intensity estimate is built from observed transitions relative to the time individuals spend at risk in the transition's origin state. This construction works only when someone is actually at risk. At late follow-up, or during a gap in observed occupancy, an origin state may have no one in its risk set. The arithmetic then has a boundary: there is no denominator from which to calculate an increment.

This is more than a computational nuisance. An empty risk set means the data provide no direct exposure information about that transition at that time. It does not show that the transition is impossible, that its intensity is zero, or that the intensity is known to remain at its last estimated value. The earlier material on cumulative transition hazards and instantaneous intensities distinguishes an accumulated estimate from a rate at a particular time; the risk-set boundary determines where observed data can support such estimates.

Definition: For a transition from state $i$ to state $j$, let $Y_i(t)$ be the number of individuals observed and at risk in origin state $i$ just before time $t$, and let $N_{ij}(t)$ count observed $i\to j$ transitions by time $t$. At an observed transition time $t_k$, write $\Delta N_{ij}(t_k)$ for the number of such transitions at that time. The Nelson–Aalen increment is $$ \Delta \widehat A_{ij}(t_k)=\frac{\Delta N_{ij}(t_k)}{Y_i(t_k)}, $$ provided $Y_i(t_k)>0$. Its cumulative estimate is the sum of defined increments through the time of interest. Follow-up times with $Y_i(t)>0$ have observed risk-set support for transitions out of $i$; times with $Y_i(t)=0$ do not.

The risk set is transition-specific through its origin state, not its destination. Anyone at risk in state $i$ contributes to the denominator for each outgoing transition, including transitions to competing destinations. An individual leaves that origin-state risk set when they are no longer observed at risk in $i$, for example because they transition out of $i$ or are censored. The risk-set convention at tied times must be applied consistently; in the examples below, each stated risk-set count is the number at risk just before the listed time.

Why the Increment Is Undefined

If $Y_i(t_k)=0$, then no individual is at risk in state $i$ just before $t_k$. Under coherent event recording, an observed $i\to j$ transition at that time cannot occur: a recorded transition must have an individual at risk in its origin state. Thus $\Delta N_{ij}(t_k)=0$ as well. Substitution into the ratio would produce $0/0$, which is undefined, not zero.

Y_i(t_k)=0 \quad\Longrightarrow\quad \frac{\Delta N_{ij}(t_k)}{Y_i(t_k)} \text{ is undefined, not }0.

This can also be understood from the cumulative-hazard estimator's exposure logic. Where $Y_i(t)>0$, the observed transition count can be scaled by the number at risk. Where $Y_i(t)=0$, there is no exposure in state $i$ during which an $i\to j$ transition could be observed. The data therefore supply no empirical increment for that time. In counting-process notation, one may define the integrand $1/Y_i(t)$ to be zero when $Y_i(t)=0$, because the observed transition count also has no jump there. That convention makes the accumulated estimator computable; it does not turn an unsupported intensity into a zero intensity.

Worked Examples at the Boundary

Worked Example: An Empty Risk Set After Observed Events

Suppose a hypothetical study follows individuals in state $i$. At time 2, four are at risk and one makes an observed $i\to j$ transition. At time 5, two are at risk and one makes an $i\to j$ transition. By time 6, the remaining individual has left the observed risk set; at time 7, nobody is at risk in $i$. Calculate the observed increments and interpret the boundary.

At time 2, the risk set is positive, so the increment is defined: $$ \Delta \widehat A_{ij}(2)=\frac{1}{4}=0.25. $$

At time 5, it is also defined: $$ \Delta \widehat A_{ij}(5)=\frac{1}{2}=0.50. $$ The cumulative estimate through time 5 is $0.25+0.50=0.75$.

At time 7, $Y_i(7)=0$. There is no observed transition from $i$ then, and $0/0$ is not an increment. The cumulative estimator can be carried forward numerically as 0.75 under the convention of adding no observed jumps after time 5. But this plateau records the absence of further observed increments; it does not establish that the $i\to j$ intensity is zero after time 6. The observed data have no risk-set support there.

Worked Example: A Temporary Gap in Support

Consider a second hypothetical setting in which three individuals are at risk in state $i$ at time 2, and one makes an $i\to j$ transition. No one is observed at risk in $i$ from time 4 through time 8. At time 9, an individual transitions into $i$ from another state. At time 10, that individual is at risk and makes an $i\to j$ transition. Find the increments and explain what happens during the gap.

At time 2: $$ \Delta \widehat A_{ij}(2)=\frac{1}{3}\approx 0.3333. $$ There is no observed risk-set exposure during the stated gap, so no increment can be calculated there. After the new arrival to $i$, the risk set is positive at time 10. The increment is: $$ \Delta \widehat A_{ij}(10)=\frac{1}{1}=1. $$ The cumulative estimate after the event at time 10 is $1/3+1=4/3$, or approximately 1.333.

This example shows why “the risk set has become empty” need not describe a permanent boundary. A later transition into state $i$ can create new support. The estimates are supported at the times when individuals are at risk, but the gap itself supplies no information about the intensity. In particular, the return of support at time 10 does not retrospectively provide exposure during times 4 through 8.

Worked Example: Zero Events Versus Zero Exposure

At time 4 in a hypothetical multi-state study, five individuals are at risk in state $i$. Two transition to a competing destination $k$, and none transition to $j$. Later, at time 8, no one is at risk in state $i$. Compare the $i\to j$ increments at those times.

At time 4, the denominator is positive and the count for the transition of interest is zero: $$ \Delta \widehat A_{ij}(4)=\frac{0}{5}=0. $$ This is a defined increment: there was risk-set exposure, but no observed $i\to j$ transition at that event time. The two transitions to $k$ are part of the observed history and remove individuals from state $i$, but they do not count in the numerator for $i\to j$.

At time 8, $Y_i(8)=0$, so there is no denominator. Even though no $i\to j$ event is recorded, the increment is undefined, not $0/0=0$. The time-4 result describes an observed zero increment with five people at risk; the time-8 boundary describes no observed exposure from which to estimate an increment.

Supported Follow-Up and a Short Justification

For a transition $i\to j$, the relevant empirical support is determined by the origin-state risk process $Y_i(t)$. This support may begin or end at different times for different states, and it need not be a single uninterrupted interval if individuals can enter $i$ after a gap. A destination-specific event count does not by itself determine support: the denominator is the risk set in $i$.

The reason an empty risk set cannot yield a data-based increment follows directly from the ratio. Suppose the observed increment is formed by comparing the event count with the number at risk. If $Y_i(t_k)>0$, this comparison is defined and can distinguish no observed event from one or more observed events. If $Y_i(t_k)=0$, then there is no observed person-time at risk at that time, and the ratio has no defined value. Assigning zero would conflate “no event among people at risk” with “no people at risk.”

Identify the transition and time.
Specify the origin state $i$, destination $j$, and the event time or follow-up interval being assessed.

Check the origin-state risk set.
Use $Y_i(t)$ under the study's time and tie conventions. Do not substitute the number of events or the number ever observed in state $i$.

Calculate only when supported.
If $Y_i(t)>0$, calculate $\Delta N_{ij}(t)/Y_i(t)$. If $Y_i(t)=0$, record the increment as undefined or unsupported.

Report the support boundary.
State the times or intervals with observed risk-set support. If the risk set later becomes positive again, identify the gap rather than treating support as permanently ended.

A practical diagnostic is to inspect the origin-state risk-set counts alongside event counts, especially near the end of follow-up and around times when people enter or leave a state. A claimed transition event with a zero origin-state risk set signals inconsistent event coding, an incorrect risk-set calculation, or a tie-handling convention that needs review. A calculated nonzero increment with a zero denominator is an arithmetic or implementation error.

How to Report Boundaries Without Overstating the Data

A clear analysis distinguishes three situations. First, with positive risk and no event of type $i\to j$, the observed increment is zero. Second, with positive risk and one or more such events, the increment is a defined positive ratio. Third, with no one at risk, the increment is undefined and the transition intensity is unsupported by observed exposure at that time. These are different data situations and should not be described with the same phrase “no events.”

When reporting a cumulative estimate, it may be convenient to display its step function as flat after its last observed jump. That numerical convention does not justify interpretation of the flat section as a measured absence of transitions. State the last supported follow-up time, any unsupported gaps, and whether risk-set support resumes later. If a scientific question requires an intensity beyond the supported follow-up, answer it only through an explicitly stated model or other source of assumptions; do not silently extend the empirical estimate.

This boundary check complements, rather than replaces, the usual checks of state definitions, censoring, event-time recording, and risk-set construction. It also differs from choosing a smoother representation: smoothing cannot create observed exposure where the origin-state risk set is empty. The later topic of smoothed transition-intensity estimates addresses how to represent estimates over supported follow-up, not how to manufacture support.

Common Mistake and Alternatives

Common mistake: “There were no transitions after time 6, so the increment after time 6 is zero and the transition intensity is zero.” This reasoning ignores the denominator. If nobody is at risk after time 6, the increment is undefined, and the data do not identify the intensity there. A zero increment is justified only when the risk set is positive and the relevant event count is zero.

Approach	What it does	Key limitation
Restrict empirical interpretation to supported follow-up	Reports increments only where the origin-state risk set is positive, and identifies unsupported gaps or boundaries.	Does not answer questions about times with no observed exposure.
Carry the cumulative curve forward as a flat step	Preserves the accumulated value when no further observed increments are added.	A flat display after support ends is not evidence of zero intensity or constancy.
Use an explicit model to extend beyond observed support	Can produce predictions for times with no risk-set observations by imposing assumptions.	The result depends on model assumptions and is not a direct empirical estimate at unsupported times.

The most defensible default is to show where the data support estimation and avoid interpreting estimates outside that region. Model-based extrapolation may be useful for a particular scientific question, but it must be labeled as extrapolation and justified separately from the observed Nelson–Aalen increments.

Check Your Understanding

Answer each question, then compare your reasoning with the solutions.

If $Y_i(t)=6$ and $\Delta N_{ij}(t)=0$, is the increment defined, and what is its value?
If $Y_i(t)=0$ and $\Delta N_{ij}(t)=0$, why is the increment not zero?
Can a risk set for state $i$ be empty for a period and later become positive? Give the mechanism.
At a time with five people at risk in $i$, two transition to $k$, and none transition to $j$. What is the observed increment for $i\to j$?

Solutions. (1) Yes. The denominator is positive, so the increment is $0/6=0$, a defined zero increment. (2) The ratio is $0/0$, which is undefined; no one was exposed in the origin state, so the data provide no increment. (3) Yes. One or more individuals may later enter $i$, for example by transitioning from another state, restoring a positive risk set. (4) The risk set is the origin-state count, including people who may make competing outgoing transitions. The $i\to j$ increment is $0/5=0$; the two transitions to $k$ are not in its numerator.