Bootstrap Confidence Intervals For Transition Probabilities

Why Bootstrap the Full Transition-Probability Estimate?

An Aalen–Johansen transition probability is built from a sequence of estimated transition increments. Those increments are estimated from the same subjects, so their uncertainty is dependent. A bootstrap confidence interval should reflect that dependence and the nonlinear matrix multiplication that combines the increments. The direct approach is to resample subjects and repeat the entire estimation procedure—not to resample already-computed transition probabilities or treat separate increments as independent.

This tutorial focuses on a pointwise interval: an interval for one specified transition probability, from one specified origin state and at one specified time pair. Its coverage statement concerns that one target. It does not automatically give simultaneous coverage for a curve, several transitions, or a collection of time points.

Definition: Let $\widehat{\mathbf A}(u)$ denote the estimated matrix of transition-increment contributions at event time $u$. The Aalen–Johansen estimate from time $s$ to time $t$, with $s \le t$, is the ordered product $$ \widehat{\mathbf P}(s,t)=\prod_{s<u\le t}\left(\mathbf I+\widehat{\mathbf A}(u)\right). $$ The entry $\widehat P_{ij}(s,t)$ estimates the probability of being in state $j$ at time $t$, conditional on being in state $i$ at time $s$, under the model and sampling assumptions for the estimator.

The product is taken in chronological order, with earlier increments on the left in the row-vector convention used here. At each event time, the diagonal contribution for a state accounts for departures from that state, while off-diagonal contributions account for observed transitions into other states. The estimate is calculated from the risk sets and transition counts at every event time between $s$ and $t$.

For uncertainty estimation, the independent sampling units must be resampled. In the usual setting these are subjects, each represented by their complete observed history: entry or starting state, transition times and destinations, and censoring information. Resampling individual event records instead would break the within-subject structure and would not reproduce the sampling variation of the estimator.

Subject-Level Bootstrap Procedure

Assume subjects are independent and sampled from the target population. The Aalen–Johansen estimator must also be appropriate for the observation scheme; in particular, its usual consistency relies on suitable independent-censoring conditions. Resampling cannot repair informative censoring, an incorrect state definition, or a target population that the sampled subjects do not represent.

For an estimate conditional on one origin state, resample subjects from the relevant cohort and retain each selected subject’s complete record. If the target instead averages over baseline states using an estimated initial-state distribution, recompute that distribution in each replicate. If the target specifies a fixed initial-state distribution, keep that distribution fixed. The resampling scheme should match the estimand.

Fit the original estimate.
Calculate the Aalen–Johansen product integral and record the target entry $\widehat P_{ij}(s,t)$.

Resample subjects.
Draw $n$ subjects with replacement from the $n$ observed subjects. A selected subject contributes their entire observed trajectory; repeated selections create multiplicities in the bootstrap sample.

Rebuild the estimator.
Recalculate risk sets, transition counts, and every estimated increment in that resample. Multiply the resulting matrices in time order to obtain $\widehat P^{*}_{ij}(s,t)$.

Repeat and take quantiles.
Repeat the resampling $B$ times. Use the empirical quantiles of the $B$ bootstrap estimates as the limits of a percentile interval.

For a nominal $100(1-\alpha)\%$ percentile interval, let $q^*_{\alpha/2}$ and $q^*_{1-\alpha/2}$ be the corresponding empirical quantiles of the bootstrap estimates. Then

\left[q^*_{\alpha/2},\,q^*_{1-\alpha/2}\right].

For example, a 95% interval uses the 2.5th and 97.5th percentiles. These are quantiles of the bootstrap distribution of the estimated probability; they are not a statement that there is a 95% probability the fixed population probability lies in the computed interval. Under appropriate regularity conditions, the method aims for repeated-sampling coverage near its nominal level.

Why the Whole Product Integral Must Be Recomputed

The bootstrap works by approximating the sampling distribution of the estimator through repeated samples from the empirical distribution of subjects. Each resample changes the subjects represented in the risk sets. That can change several transition increments at once, and the resulting changes combine through matrix multiplication. Recomputing only the final estimate’s standard error, or perturbing each increment separately without preserving their joint variation, does not in general reproduce this distribution.

A simple two-state calculation illustrates the procedure. Suppose state 1 can transition to state 2, and state 2 is absorbing. At an event time with $Y$ subjects at risk and $d$ observed transitions, the estimated increment for the first transition is $d/Y$. The transition matrix at that time is

\begin{pmatrix} 1-d/Y & d/Y\\ 0 & 1 \end{pmatrix}.

With several event times, these matrices are multiplied in chronological order. In every bootstrap sample, both $Y$ and $d$ must be recalculated from the resampled subject histories. This is the core justification for the method: it applies the same estimator to each resampled data set, allowing the empirical distribution to reflect variation in the entire calculation.

Worked Example: One Bootstrap Replicate in a Two-State Model

Consider a hypothetical cohort of four independent subjects, all initially in state 1. Subjects A, B, and D have transition times 1, 2, and 2, respectively. Subject C is censored at time 2. We estimate the probability of being in state 2 by time 2. For this illustration, an event at time 2 is counted among the events before censoring at that same recorded time.

In the original sample, at time 1 there are $Y=4$ at risk and $d=1$ transition. At time 2, the three subjects still at risk just before that time are B, C, and D, with $d=2$ transitions. The estimated probability of remaining in state 1 is therefore

\widehat P_{11}(0,2) =\left(1-\frac{1}{4}\right)\left(1-\frac{2}{3}\right) =\frac{3}{4}\cdot\frac{1}{3} =0.25.

Because state 2 is absorbing, $\widehat P_{12}(0,2)=1-\widehat P_{11}(0,2)=0.75$.

Now take one hypothetical bootstrap sample with multiplicities A: 2, B: 1, C: 1, and D: 0. This represents drawing four subjects with replacement. At time 1, $Y=4$ and $d=2$, so the probability of remaining in state 1 after that event time is $1-2/4=0.5$. At time 2, the resampled risk set contains one copy of B and one of C, so $Y=2$, $d=1$, and the remaining probability is $1-1/2=0.5$. Thus

\widehat P^{*}_{11}(0,2)=0.5(0.5)=0.25, \qquad \widehat P^{*}_{12}(0,2)=1-0.25=0.75.

This particular replicate happens to equal the original estimate. Other resamples have different multiplicities and produce other estimates. The interval comes from the distribution across many such replicates, not from this single calculation.

Worked Example: A Percentile Interval at One Year

Suppose a hypothetical study follows 80 independent subjects who begin in state 1. The target is the probability of being in state 2 at one year, conditional on starting in state 1. The original Aalen–Johansen estimate is 0.420. A subject-level bootstrap with $B=2{,}000$ replicates gives an empirical 2.5th percentile of 0.290 and a 97.5th percentile of 0.560.

The 95% percentile confidence interval is

[\,q^*_{0.025},q^*_{0.975}\,]=[0.290,0.560].

In context, the estimated one-year probability is 42.0%, with a pointwise 95% bootstrap percentile interval from 29.0% to 56.0%. The interval describes uncertainty in this one transition probability. It does not provide 95% simultaneous coverage for probabilities at every time point, nor does it imply that the event is caused by any particular exposure.

The stated bootstrap summaries are hypothetical output from the stated procedure. The count of 80 subjects and the estimate alone do not determine the interval: its limits depend on the full event and censoring histories and on the resulting resampled estimates.

Worked Example: Recomputing a Multi-State Product

In a hypothetical three-state system, state 3 is absorbing. In one bootstrap replicate, suppose the estimated event-time transition matrices are

\mathbf A_1= \begin{pmatrix} 0.8&0.2&0\\ 0&1&0\\ 0&0&1 \end{pmatrix}, \qquad \mathbf A_2= \begin{pmatrix} 1&0&0\\ 0&0.75&0.25\\ 0&0&1 \end{pmatrix}, \qquad \mathbf A_3= \begin{pmatrix} 1&0&0\\ 0&1&0\\ 0&0.1&0.9 \end{pmatrix}.

The replicate’s product from the first to the last event time is $\mathbf A_1\mathbf A_2\mathbf A_3$. For a subject starting in state 1, the row calculation is

(1,0,0)\mathbf A_1=(0.8,0.2,0), \qquad (0.8,0.2,0)\mathbf A_2=(0.8,0.15,0.05),

and then

(0.8,0.15,0.05)\mathbf A_3 =(0.8,\,0.15+0.05(0.1),\,0.05(0.9)) =(0.8,0.155,0.045).

This replicate therefore contributes 0.155 to the bootstrap distribution for the state-1-to-state-2 probability at the selected horizon. In an actual analysis, the matrices in each replicate must be estimated from that replicate’s risk sets and transition counts; the numbers here illustrate the multiplication, not a shortcut for obtaining the bootstrap distribution.

Diagnostics, Alternatives, and Limitations

Before interpreting the interval, check that the resampling scheme matches the data collection and the estimand. If subjects are independent, resample subjects. If subjects are sampled in dependent clusters, resampling individuals independently understates dependence; the independent clusters generally need to be resampled instead. If the estimate is conditional on a particular starting state, verify that resampling preserves the intended starting-state cohort. When baseline state frequencies are part of the target, decide explicitly whether those frequencies are estimated anew or fixed by definition.

Inspect the bootstrap distribution, not only its two endpoints. Useful warning signs include many identical estimates, a large mass at zero or one, very wide or irregular tails, and replicates in which the target transition cannot be estimated because the relevant risk set is empty. These patterns can occur when transitions are rare, follow-up is limited, censoring is heavy, or the cohort is small. Increasing $B$ can reduce Monte Carlo noise in the quantiles, but it cannot create information missing from the observed subjects. Report $B$, the resampling unit, the quantile method, and any replicates that could not yield the target estimate.

The percentile interval has the advantage of using the estimated sampling distribution directly and often staying within the probability range. It may nevertheless have poor coverage in small samples or when the bootstrap distribution is highly skewed or concentrated at a boundary. A normal or delta-method interval uses an estimated standard error and a normal approximation instead; it can be computationally simpler, but its approximation can also be poor and its endpoints may fall outside the probability range. Neither approach removes the need to assess the data support and assumptions.

A further limitation is multiplicity across targets. If one computes separate pointwise intervals at many time points, each interval has its own nominal pointwise interpretation; the collection does not automatically provide a confidence band with the same coverage. The bootstrap procedure described here answers the specified single-target question. A simultaneous inferential target requires a method designed for joint coverage.

Common Pitfall

Common mistake: Resample the observed event times or transition records independently, calculate separate bootstrap intervals for each increment, and combine the endpoints as if the increments were independent. This breaks the link between a subject’s history, risk-set membership, censoring, and transitions. It also ignores how uncertainty propagates through the ordered matrix product. The correction is to resample complete independent subject histories and recompute the entire Aalen–Johansen product integral in every replicate.

Key takeaway: A subject-level bootstrap interval for $\widehat P_{ij}(s,t)$ is obtained by repeatedly resampling complete subject records, reconstructing all risk sets and transition increments, and recalculating the full ordered product. Percentile limits are pointwise and rely on both an appropriate estimator and a resampling design that matches the sampling units.

Check Your Understanding

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

For a pointwise 90% percentile interval, which bootstrap quantiles are used, and what does the interval cover?
Why should a subject’s complete history, rather than each transition record separately, be resampled?
A bootstrap distribution for a transition probability has many values equal to zero. Does increasing the number of replicates necessarily resolve the problem?
In a three-state model, why must the event-time matrices be multiplied in the same time order in every bootstrap replicate?

Solutions. (1) Use the 5th and 95th percentiles of the bootstrap estimates. The interval is a pointwise interval for the specified transition probability at the specified time pair, not a simultaneous band. (2) A subject can contribute to several risk sets and may have a sequence of transitions or censoring; resampling the full history preserves those dependencies. (3) No. More replicates can stabilize the numerical estimate of the quantiles, but a large mass at zero may reflect sparse data or a boundary feature of the estimator. (4) Matrix multiplication is generally noncommutative, so changing the order changes the product and the transition probabilities it represents.