Why Joint Uncertainty Needs Its Own Calculation
A state may have several possible outgoing transitions. For example, a patient in a stable state might move to either of two different states. Each outgoing transition has a cumulative-intensity curve, and both estimates use the same people at risk in the origin state. That shared denominator can make it tempting to assume the curves must be correlated. But a common risk set is not, by itself, enough to determine their covariance.
The distinction matters when reporting or comparing multiple estimated curves. Marginal variance calculations describe uncertainty in each curve separately; a covariance calculation describes how their estimation errors vary jointly. Here the target is the covariance between Nelson–Aalen estimates of cumulative transition intensities for two distinct transitions with the same origin state.
The corresponding true cumulative transition intensity is \(A_{hj}(t)=\int_0^t \lambda_{hj}(u)\,du\), where \(\lambda_{hj}(u)\) is the transition intensity from \(h\) to \(j\) among individuals currently in \(h\). Thus, the cumulative-intensity curve accumulates instantaneous rates over time; it is not itself a transition probability.
Assumptions and the Covariance Result
The result below uses the usual counting-process conditions for Nelson–Aalen estimation. The state at each time is correctly observed, the risk set \(Y_h(u)\) is determined by information available just before \(u\), and censoring is non-informative for the transition process conditional on the observed history. Each outgoing transition has its specified intensity while an individual is in state \(h\). For the simple cross-covariance result, event times follow a continuous-time model, so one individual cannot make two distinct outgoing transitions at the same instant. The estimates are considered only while the relevant risk sets are nonzero.
For a transition \(h\to j\), define its compensated counting process by subtracting its accumulated intensity from its observed count. Under the stated intensity model, this is a martingale. The Nelson–Aalen estimation error is exactly an integral of that martingale with the inverse risk set as its weight:
For two different destinations \(j\ne k\), the transition counting processes for the same individual cannot jump together: leaving \(h\) for \(j\) precludes leaving \(h\) for \(k\) at that same instant. Under the continuous-time assumptions, the martingales for these distinct transitions therefore have zero predictable cross-variation. Integrating them with predictable weights such as \(1/Y_h(u)\) preserves zero cross-covariance. This gives the central result:
The diagonal entries of the covariance matrix are not zero. The usual Nelson–Aalen variance estimate for one transition is \(\widehat{\operatorname{Var}}\{\hat A_{hj}(t)\}=\sum_{u\le t}dN_{hj}(u)/Y_h(u)^2\), where the sum is over observed event times and \(dN_{hj}(u)\) is the number of \(h\to j\) events at \(u\). The joint covariance matrix for two distinct outgoing transitions therefore has the form
The same zero cross-covariance applies at different time horizons \(t\) and \(s\) for the two distinct transitions, provided the assumptions continue to hold. Each marginal variance depends on the risk set at the events for that transition: an event when few individuals remain at risk contributes more than an event when the risk set is large.
Worked Covariance Calculations
Worked Example: Two Outgoing Transitions With a Shared Risk Set
Suppose a hypothetical cohort has one origin state \(h\) and two possible destinations, \(j\) and \(k\). There are no tied event times. The observed event times, risk-set sizes immediately before each event, and event types are:
| Time | \(Y_h(u)\) | Observed transition |
|---|---|---|
| 1 | 12 | \(h\to j\) |
| 2 | 10 | \(h\to k\) |
| 3 | 9 | \(h\to j\) |
| 4 | 8 | \(h\to k\) |
At time 4, the cumulative-intensity estimates are \(\hat A_{hj}(4)=1/12+1/9=0.1944\) and \(\hat A_{hk}(4)=1/10+1/8=0.2250\). The marginal variance estimates are
The risk set is shared, but each event is for only one of the two transitions, and the continuous-time assumptions apply. The cross-covariance estimate is therefore \(0\), not a positive or negative quantity obtained merely from the common denominators. The estimated covariance matrix is \(\begin{pmatrix}0.01929&0\\0&0.02563\end{pmatrix}\).
Worked Example: Uncertainty in a Difference of Cumulative Intensities
Use the estimates in the preceding hypothetical event table to examine the difference \(\hat A_{hj}(4)-\hat A_{hk}(4)\). Its estimated value is \(0.1944-0.2250=-0.03056\). For any two random quantities \(U\) and \(V\), \(\operatorname{Var}(U-V)=\operatorname{Var}(U)+\operatorname{Var}(V)-2\operatorname{Cov}(U,V)\). Here the cross-covariance estimate is zero, so
The estimated standard error of the difference is \(\sqrt{0.04492}=0.2120\). The difference is in cumulative-intensity units, not a difference in probabilities. This calculation illustrates why the covariance entry matters: if a nonzero cross-covariance were justified, it would change the uncertainty of the difference through the term \(-2\operatorname{Cov}(U,V)\).
Worked Example: Covariance at Different Time Horizons
Consider another hypothetical data set with no tied events. For transition \(h\to j\), one event occurs at time 1 with \(Y_h(1)=10\); for transition \(h\to k\), one event occurs at time 2 with \(Y_h(2)=8\). Compare \(\hat A_{hj}(1)\) with \(\hat A_{hk}(2)\). They equal \(1/10=0.1000\) and \(1/8=0.1250\), respectively. The first curve is observed at time 1 and the second at time 2, but both concern distinct transitions from the same origin state. Under the continuous-time assumptions,
The later horizon does not, by itself, create cross-covariance between the two transition-specific estimation errors. Each estimate still depends on its own transition events and their risk-set sizes. This conclusion concerns the cumulative-intensity estimates; it should not be transferred automatically to other quantities derived from them.
What the Shared Risk Set Does—and Does Not—Imply
The denominator \(Y_h(u)\) is shared because the same individuals in state \(h\) are at risk of each outgoing transition. This affects the precision of both curves: small risk sets give larger contributions to their marginal variance estimates. Also, when one transition occurs, that individual leaves the origin state and is no longer at risk for any outgoing transition from \(h\). These facts couple the data structure and the risk-set history.
They do not, under the model above, create a nonzero first-order covariance between the two Nelson–Aalen estimation errors. The covariance calculation is about the joint random errors after accounting for the transition intensities and the observed risk process, not about whether the estimates use the same data. In particular, do not infer the covariance sign from the fact that an event of one type rules out an event of another type for that individual at the same instant.
This is specific to distinct transition-specific cumulative intensities under the stated model. It does not establish that all estimates involving the same people are independent, nor does it give the covariance of transition probabilities or other functions of the estimated intensities. Those are different targets and require their own uncertainty calculation.
Diagnostics and Failure Modes
Before using the zero cross-covariance result, check the event-time and observation structure. The continuous-time argument relies on distinct transition martingales having no common jumps for an individual. A dataset with genuinely discrete observation intervals, or timestamps rounded so that distinct events appear tied, may not support that argument when analyzed as if all times were distinct. Multiple individuals can also have events recorded at the same rounded time; the aggregate table then does not preserve the individual-level timing information needed to assess the assumption.
Other warnings include an origin-state risk set that is incorrectly defined, informative loss to follow-up, or a model that omits relevant history needed to specify the transition intensity. These can undermine the intensity model or its martingale representation. A zero cross-covariance under a misspecified analysis is not evidence that the underlying estimation errors are in fact uncorrelated.
When event times are inherently discrete or ties arise from coarse recording, do not mechanically interpret a zero from the continuous-time calculation as the complete joint uncertainty. Use a covariance method appropriate to the observation scheme and the dependence structure, retaining the relevant individual-level information. The key diagnostic is not simply whether two event counts appear in the same recorded time row, but whether the data and model justify the continuous-time, no-common-jump setup.
Comparison and Limitations
| Approach | What it provides | Key caution |
|---|---|---|
| Separate Nelson–Aalen variance estimates | Marginal uncertainty for each cumulative-intensity curve. | Does not, by itself, state the covariance between curves. |
| Joint martingale covariance calculation | Marginal variances and cross-covariances under the specified transition-intensity model. | The zero cross-covariance conclusion depends on the continuous-time and no-common-jump assumptions. |
| Methods adapted to discrete times or tied observations | Joint uncertainty reflecting the actual observation and dependence structure. | Requires a model or covariance procedure appropriate to that structure; the continuous-time formula is not a substitute. |
The martingale calculation is a focused result for cumulative transition-intensity estimates. It does not account for uncertainty introduced by smoothing, shared estimated model parameters, or a later transformation of the curves. Such procedures can change the covariance structure and must be handled as part of the method used. Keep the estimand clear: covariance between Nelson–Aalen cumulative-intensity estimates is not interchangeable with covariance between downstream probability estimates.
Common Mistake
Check Your Understanding
Answer each question, then compare your reasoning with the solutions below.
- Two distinct transitions leave the same origin state, and the continuous-time assumptions hold. Does their shared risk set imply a nonzero Nelson–Aalen cross-covariance?
- For one transition, events occur with risk-set sizes 20 and 10. Calculate the Nelson–Aalen variance estimate from these events.
- Suppose two outgoing transition estimates have variance estimates 0.03 and 0.05, with cross-covariance 0 under the stated model. What is the estimated variance of their difference?
- Why should events recorded at the same rounded time prompt a check before applying the continuous-time zero cross-covariance result?
Solutions. (1) No. A shared risk set affects both marginal variance estimates, but under the stated assumptions distinct outgoing-transition martingales have zero cross-covariance. (2) The estimate is \(1/20^2+1/10^2=0.0025+0.0100=0.0125\). (3) The variance is \(0.03+0.05-2(0)=0.08\). (4) Rounded ties may obscure the event-time structure; if the data are genuinely discrete or common jumps are possible under the observation scheme, the continuous-time argument may not apply, so a suitable covariance method is needed.