Why Censoring Changes the Calculation
A transition can be counted only while a subject is observed and occupies its origin state. If follow-up ends before a transition, that subject contributes information to the risk set up to the censoring time, but not afterward. This is the key accounting rule behind a Nelson–Aalen estimate under right censoring.
For a specified transition \(j\to k\), the estimator adds an increment at each observed \(j\to k\) event. The size of the increment depends on how many subjects are still observable and in state \(j\) immediately before that event. A subject censored earlier is no longer in that denominator; a subject who has made a competing transition out of \(j\) is no longer there either.
The left limit \(X_i(t-)\) means the state just before time \(t\). Thus, a subject who is in state \(j\) immediately before a transition at \(t\) belongs to the risk set for that event. We assume there are no tied transition and censoring times in the examples; the convention for ties requires separate treatment. A transition to a state other than \(k\) is not counted in \(N^{jk}\), but it removes the subject from the risk set for transitions out of \(j\).
Assumptions and Interpretation
The estimate is used when subjects are followed from a common time origin, or when their observed state histories otherwise provide the appropriate risk sets. The transition intensity \(\alpha_{jk}(t)\) is the instantaneous rate of transition \(j\to k\) at time \(t\), conditional on being in state \(j\) just before \(t\) and on the history specified by the model. For the simple pooled estimator to estimate a single transition intensity, subjects at risk must have a common intensity at a given time, or the model must otherwise specify how their intensities relate. If relevant covariates or histories cause different intensities, an unadjusted pooled estimate need not represent each subgroup's intensity.
Right censoring must also be non-informative for the target intensity: conditional on the modeled information, censoring must not selectively remove subjects according to their future transition experience. Censoring may depend on measured history if that history is properly accounted for, but dependence on unobserved future transition risk can bias the risk-set estimate. Finally, there must be subjects at risk over the times at which the cumulative intensity is to be estimated. Where nobody is at risk, the data provide no increment information for that transition.
These conditions explain why censoring is not itself a transition event. A censored subject contributes to the denominator until censoring, then is removed. The estimate is a cumulative intensity, not the probability of making the transition by time \(t\). In particular, when several transitions can leave state \(j\), \(\widehat A_{jk}(t)\) alone does not account for the competing exits needed to determine the probability of occupying a later state.
Worked Calculations
Worked Example: Censoring Before a Target Event
Suppose six hypothetical subjects all begin in state \(j\). Their first observed exit from \(j\), or their censoring time, is:
| Subject | Observed time | Outcome |
|---|---|---|
| A | 2 | Transition \(j\to k\) |
| B | 3 | Censored |
| C | 4 | Transition \(j\to l\) |
| D | 5 | Transition \(j\to k\) |
| E | 6 | Censored |
| F | 7 | Transition \(j\to k\) |
At time 2, all six subjects are observed in \(j\) immediately before the event, so the first \(j\to k\) increment is \(1/6\). Subject B is then censored at time 3 and leaves the later risk sets. At time 4, subject C makes a competing transition to \(l\), removing C from the risk set for later exits from \(j\). Therefore, at time 5, three subjects are at risk and D's increment is \(1/3\). E is censored at time 6, leaving F alone at risk at time 7.
The estimate through time 7 is $$ \widehat A_{jk}(7)=\frac{1}{6}+\frac{1}{3}+\frac{1}{1} =0.1667+0.3333+1=1.500. $$ The units are cumulative intensity on the study's time scale. The censoring at times 3 and 6 contributes no numerator increment, while the competing transition at time 4 reduces the later risk set.
Worked Example: Subjects Entering the Origin State Later
Consider four hypothetical subjects whose fully observed state histories show when they enter \(j\). Their follow-up continues until the listed exit or censoring:
| Subject | Enters \(j\) | Exit or censoring time | Outcome |
|---|---|---|---|
| A | 1 | 4 | \(j\to k\) |
| B | 2 | 5 | Censored |
| C | 1 | 3 | \(j\to l\) |
| D | 2.5 | 6 | \(j\to k\) |
The risk set for \(j\to k\) is not the number originally enrolled. Each subject joins it upon entering \(j\), and leaves it upon censoring or any transition out of \(j\). At time 4, A, B, and D are in \(j\), so the increment is \(1/3\). C's transition to \(l\) at time 3 has already removed C. At time 5, B is censored; at time 6, D is the only subject still at risk, giving an increment of \(1/1\). Thus $$ \widehat A_{jk}(6)=\frac{1}{3}+1=1.333. $$ This calculation assumes the state histories are observed as described and that censoring is non-informative for the target intensity. A subject's time in another state before entering \(j\) does not put that subject in the risk set for \(j\to k\).
Worked Example: Censoring Is Not a Target Event
Five hypothetical subjects start in \(j\). One is censored at time 1; one has \(j\to k\) at time 2; one has \(j\to l\) at time 3; one has \(j\to k\) at time 4; and the last is censored at time 5. For the target transition \(j\to k\), the risk set at time 2 has size 4, and the risk set at time 4 has size 2. Therefore, $$ \widehat A_{jk}(4)=\frac{1}{4}+\frac{1}{2}=0.750. $$ The censoring at time 1 is absent from both numerators. If it were incorrectly counted as a \(j\to k\) event, a spurious increment \(1/5\) would be added, producing \(0.950\). The \(j\to l\) event is also absent from the target numerator, although it removes that subject from the later risk set. This example shows why the outcome type and the risk-set update must be handled separately.
Why the Increment Formula Works
A short counting-process argument explains the estimator. Under the model, the conditional intensity of the observed \(j\to k\) counting process is \(Y^j(t)\alpha_{jk}(t)\): there is no opportunity for this transition unless someone is observed in \(j\). Hence $$ M^{jk}(t)=N^{jk}(t)-\int_0^tY^j(u)\alpha_{jk}(u)\,du $$ is a martingale under the stated intensity and censoring assumptions. In differential form, \(dN^{jk}(u)=dM^{jk}(u)+Y^j(u)\alpha_{jk}(u)\,du\). Using the piecewise-defined \(g^j(u)\) from the definition gives $$ \widehat A_{jk}(t)=\int_0^t g^j(u)\,dM^{jk}(u) +\int_0^t g^j(u)Y^j(u)\alpha_{jk}(u)\,du. $$ By construction, \(g^j(u)Y^j(u)=1\) when \(Y^j(u)>0\), and it is \(0\) when \(Y^j(u)=0\). The second term therefore accumulates the transition intensity only over times with a nonempty risk set. The first term is the martingale fluctuation around that accumulated intensity. This also shows why an empty risk set must not be handled by writing an undefined \(0/0\): the integrand is explicitly set to zero there, and no intensity is learned at such times.
Diagnostics, Alternatives, and Limitations
Before calculating, make a timeline for each subject and check that the risk-set membership agrees with the observed history. At each target event, verify three items: the subject was in \(j\) just before the event, the subject was not already censored, and every other currently observed subject in \(j\) is included in the denominator. A denominator that fails to decrease after a censoring time or a competing exit is a warning that the risk set has been misconstructed.
A second diagnostic is to examine whether censoring could be informative. If follow-up ends more often among subjects whose transition risk is high, even after conditioning on available information, the observed risk set may not represent the population still at risk. Similarly, strong variation in transition intensities by an omitted covariate makes a pooled estimate difficult to interpret as one common intensity. Report the population and conditioning information to which the estimate is intended to apply.
The Nelson–Aalen estimator is a nonparametric, event-time step estimate: it changes only at observed events of the target transition. A parametric transition-intensity model instead imposes a specified functional form and can provide estimates where event-time information is sparse, at the cost of model-form assumptions. The two methods answer an intensity question; neither makes the cumulative intensity itself a transition probability. Transition probabilities in a multi-state setting require accounting jointly for all outgoing transitions, rather than interpreting a single \(\widehat A_{jk}(t)\) as a probability.
Common mistakes include treating a censoring time as a target event, retaining a censored subject in later denominators, and using everyone enrolled as the denominator even after they have left state \(j\). Another subtle error is to write an expression such as \(\mathbf{1}\{Y^j(u)>0\}/Y^j(u)\) without defining its value when \(Y^j(u)=0\). The safe formulation is piecewise: use \(1/Y^j(u)\) for a positive risk set and zero otherwise. Finally, a numerically large cumulative intensity is not a probability exceeding one; it is a sum of rate-based increments, with a different interpretation and scale.
Check Your Understanding
Answer each question, then compare with the brief solutions below.
- Four subjects are observed in state \(j\) immediately before a \(j\to k\) event. What is the Nelson–Aalen increment?
- A subject moves from \(j\) to \(l\) before a later \(j\to k\) event. Does that subject contribute to the later denominator?
- Why does a subject censored at time 2 not appear in the risk set at time 4?
- What value is assigned to the estimator's integrand when \(Y^j(u)=0\), and why?
- In one sentence, explain why \(\widehat A_{jk}(t)\) is not generally the probability of a \(j\to k\) transition by time \(t\).
Solutions. (1) The increment is \(1/4=0.250\). (2) No; the subject has left the origin state \(j\), so it is no longer at risk for transitions out of \(j\). (3) Right censoring ends that subject's observed follow-up, so the subject is excluded from later risk sets. (4) The integrand is defined as zero when the risk set is empty; there is no event opportunity or information about the intensity then. (5) A cumulative intensity sums rate-based increments for one transition, whereas a transition probability must account for the possible competing transitions and state occupancy.