Piecewise-Exponential Transition Regression

Why Use a Piecewise-Exponential Transition Model?

A transition hazard can change over time even when covariate effects are of interest. A parametric proportional-hazards model represents that change through a chosen baseline-hazard form; a piecewise-exponential model instead divides time into intervals and allows a separate constant baseline rate in each. This can represent a changing baseline without requiring a smooth curve, while retaining multiplicative covariate effects within each interval.

The model applies to a specified transition $r\to s$. A person contributes time at risk for this transition only while occupying state $r$, and may contribute multiple event times if the path permits repeated $r\to s$ transitions. The intervals are defined on the time scale chosen for the analysis, such as time since study entry or time since entering a state. That choice must be meaningful for the transition and consistent with the risk-time records.

Definition: Let $0=\tau_0<\tau_1<\cdots<\tau_K$ be prespecified cutpoints, defining intervals $I_k=[\tau_{k-1},\tau_k)$. For a subject at risk in state $r$, the piecewise-exponential intensity for transition $r\to s$ is $$ \lambda_{rs,i}(t\mid \mathcal H_i(t))= Y_i^r(t)\lambda_{0k}\exp\{\boldsymbol\beta_k^\mathsf{T}\mathbf z_i(t)\}, \qquad t\in I_k. $$ Here $Y_i^r(t)$ indicates whether subject $i$ is in state $r$ and under observation just before $t$, $\lambda_{0k}>0$ is the baseline rate in interval $k$, $\mathbf z_i(t)$ is a covariate vector known just before $t$, and $\boldsymbol\beta_k$ is the covariate-effect vector. The baseline is constant within each interval, not necessarily across intervals.

The model can impose a common covariate effect, $\boldsymbol\beta_k=\boldsymbol\beta$ for every $k$, while allowing the baseline rates to differ. Alternatively, it can estimate a separate $\boldsymbol\beta_k$ in every interval. In either case, the hazard ratio for a one-unit covariate contrast within interval $k$ is determined by the corresponding coefficient; it does not describe an absolute difference in transition probability.

Likelihood With Repeated Transitions

Let $N_i^{rs}(t)$ count the number of $r\to s$ transitions made by subject $i$ by time $t$. For a given transition, its counting-process likelihood contribution has an event part and an exposure part. Each event contributes the intensity evaluated just before that event; all time at risk contributes the integrated intensity, including time across which the covariates or interval index change.

L_{rs}= \prod_i \left[ \prod_{j:\,t_{ij}^{rs}\text{ observed}} \lambda_{0,k(t_{ij}^{rs})} \exp\left\{\boldsymbol\beta_{k(t_{ij}^{rs})}^{\mathsf T} \mathbf z_i(t_{ij}^{rs})\right\} \right] \exp\left[ -\sum_i\int Y_i^r(t)\lambda_{0,k(t)} \exp\{\boldsymbol\beta_{k(t)}^\mathsf T\mathbf z_i(t)\}\,dt \right].

The product is over every observed event time $t_{ij}^{rs}$, not merely over whether the subject had any event in the interval. Thus, if a path leaves state $r$, later returns to $r$, and makes the same transition again, both events contribute their own hazard factors. The exposure integral still includes only the time the subject is at risk in state $r$, with the appropriate covariate values and interval-specific parameters.

This likelihood is justified under a correctly specified conditional transition intensity, predictable covariate values, and an observation/censoring process that is non-informative for the transition process conditional on the modeled history. The risk indicator must reflect the actual multi-state path. In a full multi-state likelihood, each possible outgoing transition has its own event contribution, while remaining in the origin state contributes exposure to the relevant outgoing intensities. A transition that removes a subject from state $r$ ends that subject's risk time for this particular transition until and unless the path returns to $r$.

Worked Example: Counting Two Events in One Interval

Consider a hypothetical subject whose path permits two $r\to s$ events within one interval. The subject is at risk for 10 time units with covariate $z=0$, then 5 time units with $z=1$. The two events occur at covariate values 0 and 1, respectively. Suppose the interval's baseline rate is $\lambda_0=0.04$ per time unit and $\beta=\log(1.5)$.

The event factors are $0.04\exp(0\cdot\beta)=0.04$ and $0.04\exp(\beta)=0.06$. The integrated intensity is $0.04(10)+0.04(1.5)(5)=0.70$. Hence this subject's transition contribution is

(0.04)(0.06)\exp(-0.70) =0.0024(0.4966) \approx 0.001192.

Both events appear in the event product, and both covariate-specific exposure periods appear in the integral. Replacing the two event counts by a single indicator would incorrectly omit one factor of $0.04\exp(\beta)$.

Estimation and a Short Derivation

For clarity, first suppose covariates are constant within each subject's risk-time segments. Write $D_k$ for the number of $r\to s$ events in interval $k$, and let

A_k(\boldsymbol\beta_k)= \sum_i\int_{I_k}Y_i^r(t) \exp\{\boldsymbol\beta_k^\mathsf T\mathbf z_i(t)\}\,dt.

The interval's log-likelihood terms involving $\lambda_{0k}$ are $D_k\log(\lambda_{0k})-\lambda_{0k}A_k$, plus event terms involving covariates. Differentiating with respect to $\lambda_{0k}$ gives $D_k/\lambda_{0k}-A_k$. Setting this derivative equal to zero yields the profiled baseline estimate, conditional on the covariate coefficients:

\widehat{\lambda}_{0k}(\boldsymbol\beta_k) =\frac{D_k}{A_k(\boldsymbol\beta_k)},\qquad D_k>0.

The covariate coefficients are estimated by maximizing the resulting likelihood across intervals. For a common coefficient, the same $\boldsymbol\beta$ is used in every $A_k$ and event factor; for interval-varying coefficients, each interval has its own coefficient vector. The expression for $\widehat{\lambda}_{0k}$ remains valid in the latter case, using $\boldsymbol\beta_k$. When covariates vary continuously over time, the integral generally requires numerical evaluation.

If $D_k=0$, the likelihood is maximized at the boundary $\lambda_{0k}=0$; a finite log-rate estimate does not exist for that interval. If the denominator is zero because there is no risk time, the interval provides no information about its rate. An interval with events but little risk time can produce a very large rate estimate and substantial uncertainty.

Worked Example: Fitting Shared Covariate Effects

Suppose a hypothetical analysis has one binary covariate $x$, with $x=0$ as the reference, and two time intervals. In each cell, $D$ is the number of transition events and $E$ is the total time at risk. Assume $x$ is constant on these risk-time segments. The model has separate baseline rates $\lambda_{01},\lambda_{02}$ but one shared coefficient $\beta$.

Interval	$x$	Events $D$	Risk time $E$
1	0	4	400
1	1	4	200
2	0	6	300
2	1	6	150

For a proposed $\beta=\log(2)$, the weighted exposure denominators are $400+200(2)=800$ in interval 1 and $300+150(2)=600$ in interval 2. The total event counts are 8 and 12, so the profiled baseline estimates are

\widehat{\lambda}_{01}=\frac{8}{800}=0.0100,\qquad \widehat{\lambda}_{02}=\frac{12}{600}=0.0200

events per unit time. To verify the shared coefficient, the covariate-event score, after profiling the baselines, is the observed number of events among $x=1$ minus its model-expected number. In interval 1 the fitted expected fraction of events at $x=1$ is $2(200)/800=0.5$, matching $4/8$. In interval 2 it is $2(150)/600=0.5$, matching $6/12$. The score is therefore $(4-8(0.5))+(6-12(0.5))=0$, so $\widehat\beta=\log(2)=0.6931$ is the maximum-likelihood estimate for this data.

The estimated hazard ratio for $x=1$ versus $x=0$ is $\exp(\widehat\beta)=2.00$ in both intervals. The baseline rate doubles from interval 1 to interval 2, while the modeled covariate hazard ratio stays fixed. These are rates for the specified transition among those currently at risk, not probabilities of making the transition over an interval.

Allowing Covariate Effects to Vary by Interval

A common coefficient is a substantive restriction: the covariate's multiplicative effect is assumed constant over the chosen time intervals. If that restriction is implausible, estimate $\boldsymbol\beta_k$ separately or use a structured time-varying coefficient. Separate estimates require more events and can be unstable, especially when some intervals have few events or sparse covariate groups.

Worked Example: Comparing Interval-Specific Effects

Consider hypothetical risk-time data for the same binary covariate, now fitted with a separate coefficient in each interval.

Interval	$x$	Events $D$	Risk time $E$
1	0	4	400
1	1	4	200
2	0	6	300
2	1	3	300

Within interval 1, the estimated rates by covariate group are $4/400=0.0100$ and $4/200=0.0200$ per time unit. Their ratio is 2.00, giving $\widehat\beta_1=\log(2)=0.6931$. In interval 2, the rates are $6/300=0.0200$ and $3/300=0.0100$, so their ratio is 0.500 and $\widehat\beta_2=\log(0.500)=-0.6931$.

Thus the fitted hazard ratio for $x=1$ versus $x=0$ is 2.00 in interval 1 and 0.500 in interval 2. This hypothetical pattern suggests a change in the covariate association over time, but the point estimates alone do not establish that the effects differ: uncertainty and an appropriate inferential comparison are needed. That comparison must account for the model's parameterization and data support.

Assumptions, Diagnostics, and Alternatives

The model's main assumptions and practical checks are:

Piecewise-constant baseline: within each chosen interval, the baseline rate is constant. Plot event counts and risk time over fine time bands, inspect fitted transition rates, and consider whether important changes are being hidden within broad intervals.
Covariate form: the log hazard is linear in the specified covariates unless transformations or interactions are included. Use the earlier material on transition residual diagnostics and covariate forms to look for systematic lack of fit.
Effect restriction: a shared $\boldsymbol\beta$ assumes no covariate-effect change across intervals. Compare observed and fitted event patterns by interval and covariate; strong interval-specific discrepancies are a warning, though not by themselves a formal test.
Correct risk sets and timing: events must be assigned to the right interval, and exposure must stop and resume according to the subject's state path. Covariates used at an event must be available just before it.
Adequate information: intervals need sufficient risk time, events, and covariate overlap. Zero-event intervals, empty covariate groups, or near-separation can make coefficients weakly identified or estimates unstable.

The interval cutpoints should be chosen based on subject-matter timing and data support, not repeatedly adjusted until a preferred estimate appears. Sensitivity analyses using reasonable alternative cutpoints can reveal dependence on this choice. Because the fitted baseline may jump at cutpoints, the implied hazard can also jump there; this is a modeling device and should not be interpreted as evidence of an instantaneous biological change.

Compared with a parametric proportional-hazards model for transitions, the piecewise form is more flexible about baseline shape but replaces a smooth parametric baseline with cutpoint-dependent constants. Compared with a model that leaves the baseline hazard unspecified, it makes a more explicit baseline approximation and can give direct interval-rate estimates. Its flexibility is limited by the chosen partition, while a very fine partition can lead to noisy rates and many parameters. The additive-hazards approach uses a different, additive covariate structure; it is not a piecewise-exponential multiplicative model.

Common Mistakes and Limitations

Counting only whether an event occurred. An indicator of any event in an interval is not enough when repeated $r\to s$ events can occur. Include every event time and its covariate value in the event product.
Using all follow-up time as exposure. Include only time at risk for the specified transition, and resume exposure only if the subject returns to the origin state.
Calling a hazard ratio a probability ratio. $\exp(\beta)$ compares instantaneous transition rates at a given time and covariate contrast. It is not generally a ratio of transition probabilities or cumulative incidences.
Assuming separate baselines imply separate covariate effects. Interval-specific $\lambda_{0k}$ allow baseline rates to change; the covariate effects remain shared unless the $\boldsymbol\beta_k$ are also allowed to vary.

Piecewise-exponential regression does not eliminate modeling choices. It depends on time origin, interval boundaries, covariate specification, and valid transition risk sets. In sparse data, a complex interval-by-covariate model may fit the observed events but estimate effects too imprecisely for useful conclusions.

Key takeaway: Specify the time intervals, risk process, and covariate-effect structure before fitting. The likelihood must include every transition event and the integrated hazard over all time at risk; interval-specific baseline rates alone do not determine whether covariate effects are shared or time-varying.

Check Your Understanding

Answer each question, then compare with the solutions.

A subject makes the same transition twice in one interval. How many event-hazard factors belong in the likelihood contribution?
In one interval, there are 5 events and 250 units of risk time at $x=0$, and 3 events and 100 units at $x=1$. Under a model with a separate coefficient in that interval, calculate the group-specific fitted rates and the $x=1$ versus $x=0$ hazard ratio.
What does a shared covariate coefficient across intervals assume, and what is allowed to change across intervals?
Give two reasons an interval-specific coefficient estimate could be unstable or unavailable.

Solutions. (1) Two: each observed event contributes its own intensity factor, evaluated using the covariate value just before that event. (2) The rates are $5/250=0.0200$ and $3/100=0.0300$ per unit time. The fitted hazard ratio is $0.0300/0.0200=1.50$, so the corresponding coefficient is $\log(1.50)=0.4055$. (3) The multiplicative covariate effect is held constant across intervals, while each interval may have its own baseline rate. (4) Examples include few events, no events, little risk time, a covariate group with no risk time, or weak covariate overlap.