Why Resample Subjects?
A transition-specific Nelson–Aalen curve is estimated from event counts and risk sets that evolve over time. Its uncertainty therefore reflects more than variation in the number of transitions: the same subject can contribute to several transition risk sets, and their timing and censoring are linked by that subject’s path. To represent this dependence, a nonparametric bootstrap resamples subjects, carrying each selected subject’s full observed history into the replicate.
This tutorial concerns uncertainty for cumulative transition-hazard curves, not transition probabilities. For a transition from state \(j\) to state \(k\), write \(Y_j(u)\) for the number at risk in state \(j\) immediately before time \(u\), and \(N_{jk}(t)\) for the number of observed \(j\to k\) transitions by time \(t\). The Nelson–Aalen estimate adds the observed transition count divided by the risk set at each transition time.
If a subject is selected more than once, each copy counts as a separate bootstrap subject. The copies share the same observed history, so repeated event times are possible and contribute as ties. Use the same transition-specific event and risk-set definitions, and the same tie convention, as in the original analysis.
Assumptions and the Bootstrap Procedure
The usual subject bootstrap treats the observed subjects as independent draws from the study’s target population, or from the relevant sampling units under the design. The observed follow-up and censoring process must also support estimation of the transition hazards of interest; in standard right-censored settings this includes an appropriate non-informative censoring assumption, possibly conditional on measured covariates. Resampling does not make informative censoring harmless or supply information beyond observed follow-up.
For a simple independent-subject sample, draw \(n\) indices \(I_{b1},\ldots,I_{bn}\) independently and uniformly from \(1,\ldots,n\) for bootstrap replicate \(b\). The replicate curve for transition \(j\to k\) is
where the sum is over times at which that replicate has an observed \(j\to k\) transition, \(dN^{*(b)}_{jk}(u)\) is the number of such transitions at \(u\), and \(Y^{*(b)}_j(u)\) is the replicate risk set just before \(u\). A time in the original data or on a shared display grid with zero replicate risk has no replicate transition there, so it contributes no increment. The replicate curve remains the cumulative sum over its own observed transition times; no arbitrary increment is assigned.
Draw \(n\) subject labels with replacement. If the sampling design has strata, resample within strata; if the independent sampling units are clusters, resample clusters rather than individual records.
Carry every sampled subject’s path, transition times, and censoring or exit information together. Do not resample transition-specific records independently.
Rebuild the transition-specific risk sets and event counts in the replicate, then calculate the Nelson–Aalen curve at the times of its observed target transitions.
Repeat many times. At a chosen time \(t\), use the distribution of \(\widehat A^{*(b)}_{jk}(t)\) to estimate uncertainty for the original \(\widehat A_{jk}(t)\).
Worked Examples
Worked Example: Recalculating One Replicate
Consider four hypothetical subjects, all initially in state 0 and followed from time 0. The target transition is \(0\to1\). Subjects A and B make this transition at times 2 and 3, respectively; subjects C and D have no target transition and are censored at times 3.5 and 5. There is no delayed entry.
| Subject | Observed history |
|---|---|
| A | In state 0 until \(0\to1\) at time 2 |
| B | In state 0 until \(0\to1\) at time 3 |
| C | In state 0 until censored at time 3.5 |
| D | In state 0 until censored at time 5 |
In the original data, all four are at risk just before time 2, so the increment is \(1/4=0.25\). Just before time 3, B, C, and D remain at risk, giving an increment of \(1/3\approx0.3333\). Thus the estimated curve at time 3 is
Now suppose a bootstrap draw selects A, A, C, and D. At time 2, four copies are at risk and two have the target transition, so the increment is \(2/4=0.5\). There is no subject with a transition at time 3 in this replicate. Its curve at time 3 is therefore \(0.5\), with no increment added at time 3. This illustrates why bootstrap event counts and risk sets must both be rebuilt from the sampled histories.
Worked Example: Forming a Pointwise Percentile Interval
For a separate hypothetical study, suppose the original estimate of a particular transition’s cumulative hazard at time 4 is \(0.42\). A small set of 20 subject-level bootstrap replicates gives the following sorted curve estimates at time 4:
\(0.22,\ 0.25,\ 0.28,\ 0.30,\ 0.32,\ 0.34,\ 0.36,\ 0.38,\ 0.40,\ 0.42,\ 0.44,\ 0.46,\ 0.48,\ 0.50,\ 0.52,\ 0.54,\ 0.56,\ 0.60,\ 0.64,\ 0.70.\)
To illustrate the percentile calculation, use linear interpolation with zero-based position \(h=(B-1)p\), where \(B=20\) and \(p\) is the desired percentile. For the 2.5th percentile, \(h=19(0.025)=0.475\), between the first two ordered values. The interpolated endpoint is
For the 97.5th percentile, \(h=19(0.975)=18.525\), between the nineteenth and twentieth values. The other endpoint is
The resulting illustrative 95% pointwise percentile interval is approximately \((0.234,\ 0.672)\) at time 4. It describes uncertainty in the cumulative transition-hazard estimate at that fixed time, not a 95% probability that the fixed parameter lies in this particular interval. Twenty replicates are far too few for reliable tail quantiles in an actual analysis; substantially more replicates are ordinarily needed, and Monte Carlo stability should be checked.
Worked Example: Why Complete Histories Matter
Suppose four hypothetical subjects begin in state A. Subject 1 moves \(A\to B\) at time 1 and \(B\to D\) at time 3. Subject 2 moves \(A\to B\) at time 2 and is censored at time 4. Subject 3 moves \(A\to C\) at time 1.5 and is followed to time 4. Subject 4 remains in A and is censored at time 4. Consider the transition-specific estimates at time 2 for \(A\to B\) and time 3 for \(B\to D\).
For \(A\to B\), the original increments are \(1/4\) at time 1 and \(1/3\) at time 2, so \(\widehat A_{AB}(2)=0.25+0.3333=0.5833\). For \(B\to D\), subjects 1 and 2 are both in B immediately before time 3, and subject 1 makes the transition. Thus \(\widehat A_{BD}(3)=1/2=0.5\).
Now draw the bootstrap sample consisting of subject labels 1, 1, 3, and 4. For \(A\to B\), two sampled copies transition at time 1, giving \(2/4=0.5\); there is no \(A\to B\) event at time 2, so the estimate at time 2 remains 0.5. For \(B\to D\), both copies of subject 1 are at risk just before time 3 and both make the transition. The increment is \(2/2=1\), so \(\widehat A^{*}_{BD}(3)=1\).
The same sampled labels determine both curves. Resampling the \(B\to D\) records separately could pair a \(B\to D\) event with no preceding entry into B, creating a history that the observed multi-state process could not have produced. Resampling subjects preserves the dependence and path feasibility needed to define the risk sets.
Why the Percentile Interval Works—and What It Does Not Say
The bootstrap uses the empirical distribution of complete subject histories as an approximation to the population distribution. Conditional on the observed data, repeated samples from that empirical distribution approximate how an estimator would vary across repeated samples from the target population. Applying the same estimator to each resample therefore approximates the sampling distribution of \(\widehat A_{jk}(t)\).
A percentile interval takes the lower and upper quantiles of that bootstrap distribution. For example, the 2.5th and 97.5th percentiles give a nominal 95% pointwise interval. This is a direct construction, not a guarantee of exact coverage: its performance depends on the sampling assumptions, sample size, and regularity of the estimator near the time considered.
The interval in the preceding example is pointwise: it concerns one specified transition at one specified time. Taking such intervals over many times does not create a 95% simultaneous band for the entire curve. A simultaneous band requires a method calibrated for the maximum deviation over time or another appropriate whole-curve criterion.
Diagnostics, Alternatives, and Limitations
Inspect the replicate curves as well as their quantiles. Useful checks include the proportion of replicates with no target events by the time of interest, the spread and skew of the replicate estimates, and whether interval endpoints change materially when the number of bootstrap replicates is increased. If the curves become erratic or highly variable late in follow-up, inspect the original risk sets and the number of subjects still under observation. A large number of resamples cannot compensate for a small risk set or scant tail data.
A time with zero replicate risk set at a shared time-grid point does not create an undefined Nelson–Aalen increment: there is no replicate transition there, and the curve does not jump. The separate inferential concern is whether the original data support a stable estimate at or beyond that part of follow-up. Do not interpret a flat tail caused by a lack of observed transitions as strong evidence of negligible hazard when few subjects remain at risk.
The analytical variance and covariance calculations for cumulative transition hazards provide a useful alternative or cross-check, especially when their approximations are adequate. The subject bootstrap has the advantage of carrying observed within-subject dependence through the calculation, but may be unstable with few independent subjects, rare transitions, or extreme censoring. Model-based methods are another alternative when a specified intensity model is appropriate; they rely on that model’s assumptions rather than only on resampling observed histories.
A common mistake is to resample event times, transition records, or transition-specific risk sets independently and then treat the resulting curves as if they represented subject-level sampling uncertainty. This breaks the link among a subject’s entry into a state, later transitions, and censoring. The correction is to resample the independent sampling units and carry each selected unit’s entire history into every transition-specific calculation.
Check Your Understanding
Answer each question, then compare your reasoning with the concise solutions below.
- In a subject-level bootstrap, what information must be copied when a subject is selected more than once?
- At a time on a shared display grid, a replicate has zero subjects at risk for a transition. What increment is added to its curve at that time, and why?
- A percentile interval is calculated for one transition at time 5. Does it automatically provide 95% coverage for the whole transition curve?
- Why can independently resampling \(A\to B\) and \(B\to D\) event records create an invalid multi-state replicate?
- Name one diagnostic that can reveal weak support for a late-follow-up interval.
Solutions. (1) Copy the complete observed history, including state path, event and censoring times, and entry or exit information. (2) Add no increment: zero risk means no replicate transition occurred there, so the curve does not jump. (3) No; it is pointwise at time 5, not automatically a simultaneous interval for all times. (4) Separate resampling can detach a later transition from the earlier path that placed the subject in its origin state. (5) For example, inspect the original late risk-set sizes or the proportion of bootstrap replicates with no target events by the time of interest.