Why represent transition hazards with Poisson regression?
A multi-state process generates event histories: individuals occupy states, remain at risk for some duration, and may then move to another state or be censored. A piecewise-exponential model describes each transition hazard as constant over specified time intervals. The same likelihood can be expressed using Poisson regression, provided each record has the correct exposure and the log of that exposure is used as an offset. This representation lets familiar regression software fit a transition model, but it is valid because of a likelihood identity—not because event indicators are ordinary independent count outcomes.
The key data-construction task is to divide each subject’s follow-up into intervals on which both the applicable hazard segment and the covariate values are constant. For each such period, record the time at risk for every available outgoing transition and mark which transition, if any, occurred at its end.
Intervals may be defined by cutpoints \(0=\tau_0<\tau_1<\cdots<\tau_M\). Follow-up must also be split at observed state changes, censoring times, and changes in time-varying covariates. On a resulting record, the linear predictor \(\eta_r=\alpha_{km}+x_r^\mathsf{T}\beta_k\) is fixed, and the hazard is \(\exp(\eta_r)\). The exposure \(e_r\) is measured in the time unit used to define the hazard, such as years.
From the transition likelihood to the Poisson likelihood
Suppose a subject occupies state \(h\) for a record of length \(e_r\), with constant outgoing transition intensities \(\lambda_{k}\) for \(k\) leaving \(h\). The probability of no transition during that exposure is \(\exp(-e_r\sum_k\lambda_k)\). If transition \(k^\ast\) occurs at the record’s end, its event-density contribution is \(\lambda_{k^\ast}\). Thus the record’s path-likelihood contribution is
where \(d_{rk}=1\) for the realized transition and 0 for other outgoing transitions; if the record ends without a transition, all those indicators are 0. The same expression applies across records by multiplying their contributions. For independent subjects, the full likelihood is the product of their path likelihoods. Censoring contributes no event factor when it is noninformative conditional on the modeled history.
For one transition-specific record, a Poisson model with mean \(\mu_r=e_r\lambda_r\) has likelihood contribution proportional to \(\exp(-e_r\lambda_r)(e_r\lambda_r)^{d_r}/d_r!\). Because exposure is observed, \(e_r^{d_r}/d_r!\) does not depend on the regression parameters. Removing that factor leaves the same parameter-dependent contribution as the transition likelihood. Taking logarithms gives the familiar offset form:
The second expression is the Poisson log-likelihood with \(\log(e_r)\) as an offset and \(\eta_r\) as the regression predictor. This establishes the equivalence: maximizing the Poisson likelihood with the offset gives the same coefficient estimates as maximizing the corresponding piecewise-exponential transition likelihood. The likelihoods can differ by factors involving observed exposures and factorials, but those factors do not affect the estimates.
Create a new record at every hazard cutpoint, covariate change, state transition, or censoring time.
For each period in state \(h\), create a transition-specific record for each modeled transition leaving \(h\), with exposure equal to the period’s duration.
Set \(d_r=1\) only for the transition that occurred; set other outgoing transition indicators to 0. Include the interval-specific baseline term and covariates that apply to that record.
Use a Poisson log-link model with \(\log(e_r)\) as a fixed offset, not as an estimated coefficient.
Worked applications
Worked Example: A Competing Transition in the Last Interval
Consider one hypothetical subject who begins in state A and remains there until a transition to state C at time 2.5 years. There are cutpoints at 0, 1, and 2 years. In the three intervals, the A-to-B rates are 0.10, 0.20, and 0.30 per year, while the A-to-C rates are 0.05, 0.10, and 0.40 per year. Assume these rates are constant within their stated intervals and that the event time and transition type are observed.
The subject contributes 1 year of A-state exposure in each of the first two intervals and 0.5 years in the third. For each interval, make one record for A-to-B and one for A-to-C. The event indicators are 0 for both transitions in the first two intervals; in the final interval they are 0 for A-to-B and 1 for A-to-C.
The cumulative hazard across the three periods is \(1(0.10+0.05)+1(0.20+0.10)+0.5(0.30+0.40)=0.80\). The path likelihood is therefore
The Poisson likelihood kernel across the six records is \(\exp(-0.80)(0.5\times0.40)=0.08987\), since only the final A-to-C row has an event and its exposure is 0.5 years. The kernels differ by the parameter-independent factor 0.5. Dividing by that factor recovers the transition likelihood. Both likelihoods therefore give the same estimates when the rates are unknown and represented by the corresponding regression predictors.
Worked Example: Estimating a Constant Hazard Ratio
Suppose hypothetical follow-up consists of two covariate groups at risk for the same transition, with a constant hazard within the observation period. Group 0 contributes 200 person-years and 4 transitions; group 1 contributes 150 person-years and 6 transitions. Let \(z=0\) for group 0 and \(z=1\) for group 1, and use \(\lambda(z)=\exp(\alpha+\beta z)\). Assume independent subjects, correct event classification, and noninformative censoring.
The group-specific Poisson means are \(200\exp(\alpha)\) and \(150\exp(\alpha+\beta)\), with \(\log(200)\) and \(\log(150)\) as their respective offsets. For this two-group model, the maximum-likelihood rate in each group is its event count divided by its exposure:
Consequently, \(\hat{\alpha}=\log(0.0200)=-3.912\) and \(\hat{\beta}=\log(0.0400/0.0200)=\log(2)=0.6931\). The fitted hazard ratio comparing group 1 with group 0 is \(\exp(\hat{\beta})=2\). Under this model, group 1 has twice the transition hazard of group 0 at any time in the modeled period. This is a model-based association, not by itself a causal effect.
Worked Example: Splitting at a Covariate Change
A hypothetical subject remains at risk for one transition for 1 year. A binary external covariate is 0 for the first 0.4 years and 1 for the next 0.6 years, and the transition occurs at year 1. Suppose the fitted hazard is \(\lambda(t)=0.10\exp(\beta z(t))\) per year, with \(\beta=\log(2)\). The assumed rates are constant within each of these two records, and the event time and covariate change time are known.
Create a first record with \(e_1=0.4\), \(z_1=0\), \(d_1=0\), and a second with \(e_2=0.6\), \(z_2=1\), \(d_2=1\). The hazards are 0.10 and 0.20 per year, respectively. The path likelihood is
The Poisson kernel is \(\exp(-0.16)(0.6\times0.20)=0.1023\); its ratio to the path likelihood is 0.6, the event-row exposure. That factor is fixed by the observed record. If the covariate change were ignored and \(z=1\) were assigned to the entire year, the fitted hazard history would be wrong: the first 0.4 years would be incorrectly assigned hazard 0.20 rather than 0.10.
Assumptions and checks on the records
The representation requires the modeled intensity to be constant on each record, conditional on its covariates and interval. This is why cutpoints and observed covariate changes must be represented in the data. A transition ends exposure for all outgoing transitions from the state just left; it must not be treated as additional exposure in that state after the event. Independent subjects and an appropriate conditional censoring assumption support the usual likelihood interpretation. For histories with multiple transitions, construct records from the evolving state path: after a state change, only transitions leaving the new state are at risk.
Before fitting, check that each record has positive exposure, belongs to the state actually occupied, and uses the correct time interval and covariate values. For each event, verify that exactly one available transition has event indicator 1 and that its destination matches the observed path. Censored records have zero event indicators for the transitions at risk at censoring. Also check that the sum of record exposures for a transition agrees with the time subjects spent at risk for it. A cutpoint error, duplicated exposure, or event assigned to an unavailable transition changes the likelihood, not merely the appearance of the data.
The Poisson representation is useful for fitting and organizing piecewise-exponential models, but it does not make the underlying event process a sequence of arbitrary independent counts. The event indicators arise from transition histories, and their exposure is determined by those histories. Standard Poisson coefficient estimation uses the offset correctly; uncertainty calculations should respect the sampling structure, especially when subjects contribute multiple records. Model diagnostics should also examine whether the chosen piecewise baseline and covariate form adequately describe the transition hazards.
How this representation differs from alternatives
| Approach | What it uses | Key distinction |
|---|---|---|
| Poisson representation | Interval-level event indicators, exposures, and log-exposure offsets | Fits a piecewise-constant hazard model by maximizing an equivalent likelihood. |
| Direct piecewise-exponential likelihood | Transition times, risk histories, and integrated hazards | Expresses the same model directly through event and survival contributions, without requiring Poisson software. |
| Cox regression | Transition event times and risk sets | Uses a partial likelihood and does not require the same interval-specific parametric baseline-rate specification. |
The Poisson and direct likelihood approaches agree on coefficient estimates when they encode the same piecewise-exponential model and use the same records. They do not eliminate model choice: cutpoints determine where baseline rates may change, and a coarse partition can miss important time variation. Conversely, many intervals can leave little event information for estimating separate baseline rates. The Cox approach is a different modeling strategy, not a Poisson fit with a different offset.
Common mistakes and limitations
- Omitting the offset. Regressing event indicators on covariates without \(\log(e_r)\) does not fit the intended hazard model. The exposure is part of the expected event count, not an optional adjustment.
- Using the wrong exposure. The offset is the time at risk for that transition, not total study follow-up if the subject was in another state or not under observation.
- Failing to split records. A single row spanning a cutpoint or covariate change assigns one hazard to a period where the model specifies more than one.
- Marking every outgoing transition as an event. A realized move from A to C is one event for A-to-C, not also an event for A-to-B. The other transition contributes its risk exposure but no event.
- Interpreting a Poisson count literally. A row’s 0/1 event indicator is a likelihood device for a transition process. The equivalence does not license treating follow-up records as unrelated counts or ignoring how the state history generated them.
A further limitation is that piecewise-exponential rates change only at the specified cutpoints. The Poisson formulation is computationally convenient, but it does not repair an inadequate hazard model or incorrect transition history. Its central benefit is a transparent, exact connection between the transition likelihood and a familiar regression likelihood, conditional on correct interval-level construction.
Check Your Understanding
Answer each question, then compare your reasoning with the solutions.
- A transition record has exposure 2.5 months and event indicator 1. What is its Poisson offset if time is measured in months, and what does the event indicator represent?
- During a period in state A, a subject has two possible outgoing transitions and is censored without either occurring. How many outgoing-transition event indicators are 1?
- In a constant-rate group, 3 transitions occur over 120 person-years. What is the maximum-likelihood rate under the Poisson representation?
- A covariate changes halfway through a record, but the data remain in one row. What condition for the equivalence has been violated, and how should the record be corrected?
Solutions. (1) The offset is \(\log(2.5)=0.9163\); the indicator 1 means the modeled transition occurred in that record. (2) Neither indicator is 1; both are 0, while each available transition still receives the at-risk exposure. (3) The fitted rate is \(3/120=0.0250\) transitions per person-year. (4) The covariate, and therefore the modeled hazard, is not constant within the record. Split it at the change time, assign each row its own exposure and covariate value, and place an event indicator only on the row in which the event occurred.