A Worked Transition-Intensity Calculation

From Risk-Set Counts to Cumulative Curves

A transition-intensity calculation is built from two quantities at each event time: how many individuals were still in the relevant state just before that time, and how many made the transition of interest at that time. Keeping those quantities separate for each transition is essential when a state has competing exits. The same individual can be at risk for either exit, but once an exit occurs the individual is no longer at risk for another transition out of the original state.

This tutorial works through a small cohort with two possible transitions from state 0. It shows how censoring changes later risk sets, how tied events contribute, and how the Nelson–Aalen increments accumulate into two step curves. The curves summarize cumulative transition hazards; they are not, by themselves, probabilities of occupying another state.

Definition: For a transition from state \(j\) to state \(k\), let \(Y_j(t)\) be the number of observed individuals in state \(j\) and under observation immediately before time \(t\). Let \(dN_{jk}(t)\) be the number of observed \(j\to k\) transitions at time \(t\). The Nelson–Aalen increment for this transition at \(t\) is \(dN_{jk}(t)/Y_j(t)\), provided \(Y_j(t)>0\). Its cumulative transition-hazard estimate through time \(t\) is the sum of these increments over event times up to \(t\).

Hypothetical Cohort and Risk Sets

Suppose 12 hypothetical individuals begin in state 0 at time 0. From state 0, they can move to state 1 or state 2. Follow-up records transitions or right-censoring; there are no other ways to leave state 0. The following event history is hypothetical and is given only to demonstrate the calculation. At time 4, one individual makes each of the two transitions, so the outgoing transitions are tied.

Time	Observed event	\(Y_0(t)\), just before time \(t\)	\(dN_{01}(t)\)	\(dN_{02}(t)\)
1	One \(0\to1\) transition	12	1	0
2	One individual censored	11	0	0
3	One \(0\to2\) transition	10	0	1
4	One \(0\to1\) and one \(0\to2\) transition	9	1	1
5	One individual censored	7	0	0
6	One \(0\to2\) transition	6	0	1

The risk-set column is determined by who remains in state 0 and under observation just before each time. There are 12 individuals at risk just before time 1. The transition at time 1 removes one, leaving 11. Censoring at time 2 removes another, leaving 10 before time 3. The time-3 transition leaves 9 before time 4; both tied transitions at time 4 are counted from this same pre-event risk set of 9. After those two departures, 7 remain before time 5. Censoring at time 5 leaves 6 before time 6.

A censoring time contributes no transition count to either \(dN_{01}\) or \(dN_{02}\), but it reduces the later risk set. An observed transition to state 1 is also an exit from state 0: it reduces the state-0 risk set for both outgoing transitions, not only for \(0\to1\).

Calculating Both Nelson–Aalen Curves

For each event time, divide the count for a particular transition by the state-0 risk set just before that time. The two transitions have separate numerators but share \(Y_0(t)\), because both are possible exits for individuals in state 0.

Worked Example: The Two Cumulative Estimates

Calculate the \(0\to1\) and \(0\to2\) Nelson–Aalen increments at every event time, and find each cumulative estimate through time 6.

For \(0\to1\), the increment at time 1 is \(1/12\). At time 4 it is \(1/9\), and the other event times have no \(0\to1\) events. Thus the cumulative estimate is

\widehat A_{01}(t)= \begin{cases} 0, & t<1,\\ 1/12\approx0.08333, & 1\le t<4,\\ 1/12+1/9=7/36\approx0.1944, & t\ge4. \end{cases}

For \(0\to2\), the increments occur at times 3, 4, and 6. They are \(1/10\), \(1/9\), and \(1/6\), respectively. Therefore,

\widehat A_{02}(t)= \begin{cases} 0, & t<3,\\ 1/10=0.1000, & 3\le t<4,\\ 1/10+1/9=19/90\approx0.2111, & 4\le t<6,\\ 1/10+1/9+1/6=17/45\approx0.3778, & t\ge6. \end{cases}

These are cumulative transition-hazard estimates, so their numerical values are dimensionless. The event times and the horizontal axis use the study's time units. Each curve is right-continuous under the displayed convention: its value includes the increment at the time shown. For example, at time 4 the estimates are \(7/36\approx0.1944\) for \(0\to1\) and \(19/90\approx0.2111\) for \(0\to2\).

Key calculation rule: At a tied event time, use the risk set immediately before the time for every transition count at that time. Here both increments at time 4 use denominator 9: \(1/9\) for \(0\to1\) and \(1/9\) for \(0\to2\). The risk set is not reduced between tied events.

Why the Increment Has This Form

The Nelson–Aalen increment is a count per individual at risk. Its interpretation as an intensity increment relies on assumptions about how transitions are generated and observed. Suppose individuals are independent conditional on their observed histories, transitions are recorded at their correct times and types, and censoring is independent of future transitions conditional on the information used to define the risk set. For a pooled estimate to target one common transition intensity, individuals currently in state 0 must share that intensity for the transition being considered, conditional on the modeled history.

The common-intensity assumption can be relaxed for the expectation calculation, but then the target must be described carefully. If individual \(i\), while at risk in state 0, has transition intensity \(\alpha_{i,0k}(u)\), then over a short interval of length \(du\), the conditional expected number of \(0\to k\) transitions is approximately the sum of those individual intensities times \(du\). It is not generally the risk-set size times one individual's intensity.

E\{dN_{0k}(u)\mid\mathcal F_{u-}\} \approx \sum_{i\in R_0(u)}\alpha_{i,0k}(u)\,du = Y_0(u)\bar\alpha_{0k}(u)\,du, \qquad \bar\alpha_{0k}(u) = \frac{1}{Y_0(u)} \sum_{i\in R_0(u)}\alpha_{i,0k}(u),

Here \(R_0(u)\) is the set of individuals at risk in state 0 just before \(u\), and \(\mathcal F_{u-}\) denotes the observed history just before \(u\). The risk-set average \(\bar\alpha_{0k}(u)\) is defined only when \(Y_0(u)>0\). Under a common transition intensity \(\alpha_{0k}(u)\), it equals that common intensity. In either case, dividing the observed event count by the risk-set size gives a natural estimate of a short-interval intensity contribution.

At observed event times, the Nelson–Aalen estimator adds \(dN_{0k}(u)/Y_0(u)\). This produces a step-function estimate of cumulative transition hazard. The calculation is not an assertion that the observed ratio exactly equals an instantaneous intensity: the numerator is random, and the estimator accumulates observed increments over time.

Worked Example: Common and Heterogeneous Intensities

Suppose four hypothetical individuals are at risk for \(0\to2\) during a short interval, with individual transition intensities \(0.02\), \(0.04\), \(0.06\), and \(0.08\) per month. Assume these intensities are approximately constant over an interval of length \(du=0.01\) month, and that simultaneous transitions have negligible probability at this short scale.

The expected event count is approximately the sum of the four intensities times the interval length:

E\{dN_{02}\mid\mathcal F_{u-}\} \approx(0.02+0.04+0.06+0.08)(0.01) =0.20(0.01)=0.002.

The risk-set average intensity is \(0.20/4=0.05\) per month, so the equivalent expression is \(4(0.05)(0.01)=0.002\). If one transition is actually observed in a particular interval, the Nelson–Aalen increment is \(1/4=0.25\), not \(0.05\). The first quantity is a conditional expected count for a short interval; the second is the realized estimator increment at an observed event time. They answer different questions and have different time-scale interpretations.

If all four individuals instead shared a common intensity of \(0.05\) per month, the same expected count would follow from \(Y_0(u)\alpha_{02}(u)du=4(0.05)(0.01)=0.002\). This makes explicit the assumption needed when writing a pooled intensity as a single common value.

Interpreting and Checking the Curves

A cumulative transition-hazard curve increases by a discrete amount at each observed transition of its type and stays flat between such event times. Its increment depends on both the event count and the size of the risk set. One event among 12 at risk contributes less than one event among 6 at risk. The curve therefore is not simply the running total of events.

The estimate \(\widehat A_{01}(t)\) summarizes accumulated transition-specific hazard for leaving state 0 toward state 1; \(\widehat A_{02}(t)\) does the same for leaving state 0 toward state 2. These estimates are not transition probabilities. In particular, adding them does not yield the probability of having left state 0, and a cumulative hazard can exceed 1. Obtaining state-occupation or transition probabilities requires additional calculations.

A practical check is to recompute the risk set from the event history before each event time, then verify each increment separately. For this cohort, the sum of the two increments at time 4 is \(1/9+1/9=2/9\), matching two departures among the 9 individuals at risk. Through time 6, the sum of all transition-specific increments is

\widehat A_{01}(6)+\widehat A_{02}(6) = \left(\frac{1}{12}+\frac{1}{9}\right) + \left(\frac{1}{10}+\frac{1}{9}+\frac{1}{6}\right) \approx0.5722.

This sum is a useful arithmetic check of the separate increments, not a probability. It also confirms that the two tied transitions at time 4 each use the denominator 9.

Worked Example: Checking a Changed Event Type

Keep the same event times, censoring times, and risk-set sizes, but suppose the event at time 1 was \(0\to2\), not \(0\to1\). Recalculate the cumulative estimates through time 6.

The risk-set sizes do not change: an observed transition at time 1 still removes one individual from state 0. The \(0\to1\) transition now occurs only at time 4, so

\widehat A_{01}(6)=\frac{1}{9}\approx0.1111.

The \(0\to2\) transitions now occur at times 1, 3, 4, and 6, with denominators 12, 10, 9, and 6. Thus

\widehat A_{02}(6) = \frac{1}{12}+\frac{1}{10}+\frac{1}{9}+\frac{1}{6} = \frac{83}{180} \approx0.4611.

The total remains approximately \(0.5722\), because the event times and denominators are unchanged and each observed departure from state 0 contributes one increment to exactly one outgoing transition. The allocation of the cumulative estimate between transition types changes with the event type.

Assumptions, Diagnostics, and Alternatives

The calculation is straightforward, but its interpretation depends on the data and observation process. Check that state membership and event type are well defined, that event times are recorded on a consistent scale, and that each individual contributes to a risk set only while in state 0 and under observation. A zero risk set makes an increment undefined; no transition-intensity increment can be estimated from an empty risk set. Sparse risk sets also make late increments highly sensitive to individual events.

Check the risk-set ledger. At each time, subtract earlier exits from state 0 and earlier censoring before recording the pre-event denominator. Do not subtract a tied event before calculating another increment at the same time.
Check transition coding. A transition to one competing destination removes an individual from risk for every other transition out of the original state.
Check censoring. If censoring depends on unobserved future transition behavior after conditioning on the available history, the observed risk sets may not represent the intended population risk sets.
Check heterogeneity. When individuals at risk have different intensities, the pooled estimate naturally relates to a risk-set average. Changes in the composition of that risk set can change this average over time.
Check the scale of interpretation. The curve is a cumulative hazard, not a probability or a smoothed instantaneous-rate curve.

A raw cumulative event count ignores the number at risk and is not a transition-intensity estimate. The Nelson–Aalen calculation adjusts each event count by its contemporaneous risk set. A smoothed transition-intensity estimate, as covered in the earlier material on smoothed transition-intensity estimates, instead aims to display a local rate over time and requires a smoothing choice. The stepwise Nelson–Aalen curve avoids that choice but can be visually irregular, especially with few individuals or sparse events. Neither curve alone gives the probability of being in a destination state.

The pooled calculation also has limits: it does not adjust for covariates or model individual-level intensity differences, and it depends on adequate observation of transitions and censoring. When these assumptions or the desired target are not credible, the curve should be described as a pooled, risk-set-based summary rather than as a common individual-level transition process.

Common Mistakes

Using the wrong denominator. At time 1, the \(0\to1\) increment is \(1/12\), not \(1/9\). Use the risk set immediately before the event time in question.
Changing the risk set between tied transitions. Both time-4 increments use 9 as the denominator. They are simultaneous observations from the same pre-event risk set.
Dropping censoring from the ledger. Censoring adds no event increment, but it does reduce later risk sets.
Treating an increment as a probability. An increment is an event count divided by the number at risk; cumulative increments estimate a cumulative hazard, not a state-occupation probability.
Assuming one common intensity without justification. With individual heterogeneity, the summed conditional expectation uses the sum of individual intensities, or equivalently the risk-set average times the risk-set size.

Key takeaway: For each outgoing transition, add its event count divided by the shared pre-event risk-set size at every observed event time. Build the risk set from all earlier departures and censoring, use the same denominator for tied events, and interpret the resulting step curve as a cumulative transition hazard.

Check Your Understanding

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

At a time with 8 individuals at risk in state 0, two make a \(0\to1\) transition and one makes a \(0\to2\) transition. What are the two Nelson–Aalen increments?
In the cohort above, why is the denominator at time 4 equal to 9 for both outgoing transitions?
If one individual is censored at time 2, what does that censoring contribute to the \(0\to1\) and \(0\to2\) increments at time 2, and how does it affect later calculations?
In the heterogeneous-intensity example, what is the risk-set average intensity, and what is the approximate expected count over \(0.01\) month?
State one reason that \(\widehat A_{01}(t)+\widehat A_{02}(t)\) should not be interpreted as the probability of leaving state 0.

Solutions: (1) The increments are \(2/8=0.25\) for \(0\to1\) and \(1/8=0.125\) for \(0\to2\); both use the pre-event risk set of 8. (2) Before time 4, 12 individuals have been reduced by the time-1 transition, time-2 censoring, and time-3 transition, leaving \(12-1-1-1=9\). Both time-4 events are tied and use that same pre-event count. (3) Censoring contributes zero to both event counts at time 2, but removes one individual from later risk sets. (4) The average is \((0.02+0.04+0.06+0.08)/4=0.05\) per month; the expected count is approximately \(4(0.05)(0.01)=0.002\). (5) The sum is a cumulative-hazard summary, not a probability; cumulative hazards are formed by adding risk-set-scaled event increments and are not state-occupation probabilities.