Estimating a Piecewise-Exponential Joint Model

Why Partition Time for a Joint Model?

A piecewise-exponential survival model represents the baseline hazard by a separate constant rate in each prespecified time interval. This gives a flexible alternative to choosing a single parametric baseline shape, while keeping the survival likelihood straightforward to construct. In a joint model, the survival component is conditional on a subject’s random effects: the same latent subject characteristics that help describe the longitudinal measurements may also enter the event hazard.

The central bookkeeping task is to determine how much time each subject contributes to each interval and which interval contains an observed event. Those quantities determine the cumulative-hazard term and the event-hazard term in the likelihood. They do not, by themselves, make the entire hazard constant within an interval: a time-varying longitudinal association can still make the total hazard vary there.

Definition: Choose cutpoints \(0=\tau_0<\tau_1<\cdots<\tau_K\), with the last interval extending as far as needed. The piecewise-constant baseline hazard is \(\lambda_0(t)=\lambda_k\) for \(\tau_{k-1}\le t<\tau_k\), where each \(\lambda_k>0\) has units of inverse time. For subject \(i\), let \(T_i\) be observed follow-up time and \(\delta_i=1\) indicate an observed event (\(\delta_i=0\) indicates right censoring). Define the interval exposure \(Y_{ik}=\max\{0,\min(T_i,\tau_k)-\tau_{k-1}\}\), with the upper endpoint interpreted as infinity for the final interval. Let \(d_{ik}=1\) if subject \(i\) has an event in interval \(k\), and \(0\) otherwise.

With this convention, a subject contributes exposure only up to \(T_i\). If the subject has an event, exactly one \(d_{ik}\) equals 1; if the subject is censored, all \(d_{ik}=0\). Events exactly at an internal cutpoint can be assigned to the interval beginning at that cutpoint. For a continuous event-time distribution, the probability of an event at any particular cutpoint is zero, but a consistent convention is still needed in data processing.

Conditional Survival Likelihood

Let \(b_i\) denote subject \(i\)’s random effects, and let \(\eta_i(t;b_i)\) be the dimensionless survival linear predictor, which may depend on fixed covariates and on the longitudinal process evaluated using \(b_i\). Write the hazard conditional on those random effects as

h_i(t\mid b_i)=\lambda_0(t)\exp\{\eta_i(t;b_i)\}.

The cumulative hazard through time \(T_i\) is \(H_i(T_i\mid b_i)=\int_0^{T_i}h_i(u\mid b_i)\,du\). Under conditionally independent, noninformative censoring, the event-time contribution, omitting the censoring distribution, is the hazard at the observed time raised to the event indicator, multiplied by survival through follow-up:

L_{S,i}(b_i) = \{h_i(T_i\mid b_i)\}^{\delta_i} \exp\{-H_i(T_i\mid b_i)\}.

The hazard factor appears only for an observed event. For a censored subject, the likelihood contribution is the probability of surviving to the censoring time, conditional on the model variables. This is a density contribution when \(\delta_i=1\), so its units are inverse time; a right-censoring contribution is dimensionless.

Substituting the piecewise baseline gives the general expression

L_{S,i}(b_i) = \left[ \lambda_{k(i)}\exp\{\eta_i(T_i;b_i)\} \right]^{\delta_i} \exp\left\{ -\sum_{k=1}^{K}\lambda_k \int_{\tau_{k-1}}^{\tau_{k-1}+Y_{ik}} \exp\{\eta_i(u;b_i)\}\,du \right\},

where \(k(i)\) is the interval containing the event time when \(\delta_i=1\). The integral upper endpoint is the end of the subject’s exposure in that interval. Intervals with zero exposure contribute zero to the cumulative hazard. If \(\eta_i(t;b_i)\) is constant within each exposed interval, write its value as \(\eta_{ik}(b_i)\); the cumulative-hazard term then reduces to \(\sum_k\lambda_kY_{ik}\exp\{\eta_{ik}(b_i)\}\). If the predictor varies over time, replacing the integral by \(Y_{ik}\exp\{\eta_i(T_i;b_i)\}\) is generally incorrect.

Worked Example: Assigning Exposure and the Event Interval

Suppose the cutpoints are 0, 2, and 5 years, followed by an interval from 5 years onward. A hypothetical subject has an event at \(T_i=3.4\) years. The exposures are

\(Y_{i1}=\min(3.4,2)-0=2\) years, \(Y_{i2}=\min(3.4,5)-2=1.4\) years, and \(Y_{i3}=0\). The event falls in the interval from 2 to 5 years, so \((d_{i1},d_{i2},d_{i3})=(0,1,0)\).

For a censored subject followed to \(T_i=4\) years, the exposures would instead be 2, 2, and 0 years, with all event indicators equal to zero. Thus, the subject contributes survival through 4 years but no event-hazard factor.

Worked Example: Calculating a Conditional Event Contribution

Consider hypothetical intervals \([0,2)\) and \([2,4)\) years, extended as needed. For one subject with \(T_i=3\) years and an event, suppose the conditional hazard multipliers \(\exp\{\eta_i(t;b_i)\}\) are constant at 1.2 and 4 in the two intervals, and \(\lambda_1=0.20\) and \(\lambda_2=0.10\) per year. The cumulative hazard is

\(H_i(3\mid b_i)=(0.20)(1.2)(2)+(0.10)(4)(1)=0.48+0.40=0.88.\)

The event occurs in the second interval, so the hazard at the event time is \((0.10)(4)=0.40\) per year. Therefore,

\(L_{S,i}(b_i)=0.40\exp(-0.88)\approx0.40(0.4147829)=0.1659\) per year.

This is a conditional event-time density contribution, not a probability that the event occurs at exactly year 3. The random effects are held fixed in this calculation; the joint-model likelihood still has to account for their distribution.

From Conditional Contributions to the Joint Likelihood

Let \(y_i\) collect subject \(i\)’s longitudinal measurements, with conditional density \(f_Y(y_i\mid b_i)\), and let \(b_i\) have density \(\phi(b_i;D)\), where \(D\) is the random-effects covariance matrix. Conditional on \(b_i\), the model treats the longitudinal and event processes as independent, with their dependence represented through the shared random effects and any specified association structure. The subject’s observed-data likelihood contribution is therefore

L_i = \int f_Y(y_i\mid b_i)\, L_{S,i}(b_i)\, \phi(b_i;D)\,db_i, \qquad L=\prod_{i=1}^{n}L_i.

This integral is essential: the observed-data likelihood is not obtained by substituting an estimated random effect into \(L_{S,i}(b_i)\) and treating that value as known. In general, the integral has no closed form and must be evaluated numerically or handled through an estimation algorithm. The piecewise intervals make the baseline part explicit, but do not remove the integration over \(b_i\).

The event-time likelihood above assumes the censoring mechanism is noninformative conditional on the modeled variables and that event times are recorded on the time scale used to define the intervals. With delayed entry at \(A_i>0\), the survival contribution must be conditioned on surviving to entry: its survival factor becomes \(\exp\{-[H_i(T_i\mid b_i)-H_i(A_i\mid b_i)]\}\), with the event hazard still included when \(\delta_i=1\). Ignoring delayed entry would count survival time before the subject entered observation.

How the Baseline Rates Enter Estimation

A useful derivation shows why event counts and exposure times are central. First hold the other parameters and random effects fixed, and suppose \(\eta_i(t;b_i)=\eta_{ik}(b_i)\) within interval \(k\). Ignoring terms that do not depend on \(\lambda_k\), the conditional log likelihood for that interval is

\ell_k(\lambda_k) = D_k\log(\lambda_k)-\lambda_k A_k, \quad D_k=\sum_i d_{ik}, \quad A_k=\sum_iY_{ik}\exp\{\eta_{ik}(b_i)\}.

Here \(A_k\) is weighted exposure, with units of time, and \(D_k\) is the number of events. Differentiating and setting the score to zero gives \(D_k/\lambda_k-A_k=0\). If \(D_k>0\) and \(A_k>0\), the conditional maximizer is \(\widehat{\lambda}_k=D_k/A_k\), which has units of inverse time. The second derivative is \(-D_k/\lambda_k^2<0\), so this stationary point is a maximum.

This ratio is not automatically the maximum-likelihood estimate in the full joint model. The random effects are unobserved, their distribution must be integrated, and their posterior distribution depends on the parameters. In an EM formulation, the analogous rate update uses conditional expectations of the relevant event counts and weighted exposures given the observed data. In direct maximum likelihood, the integrated likelihood is optimized numerically. In either approach, intervals with no events require care: the simple conditional likelihood has no positive finite maximizer for that interval’s rate when \(D_k=0\); its supremum occurs as \(\lambda_k\) approaches zero. Constraints, a different time partition, or a regularizing modeling choice may be needed.

Worked Example: A Conditional Baseline-Rate Estimate

Suppose a hypothetical analysis temporarily holds the predictors and random effects fixed. In the first interval, there are \(D_1=2\) events and \(A_1=10\) weighted person-years; in the second, there are \(D_2=1\) event and \(A_2=4\) weighted person-years. The conditional rate estimates are

\(\widehat{\lambda}_1=2/10=0.200\) per year and \(\widehat{\lambda}_2=1/4=0.250\) per year.

For example, if all interval predictors were zero, the weighted exposures would equal the ordinary total person-time in those intervals. If predictors are not zero, \(A_k\) instead sums each subject’s exposure multiplied by \(\exp\{\eta_{ik}(b_i)\}\). These calculated rates illustrate the conditional likelihood update; they are not, by themselves, estimates from the fully integrated joint likelihood.

Diagnostics, Alternatives, and Limitations

The partition determines how much flexibility the baseline hazard has. Cutpoints chosen too sparsely can fail to capture important changes in baseline risk; too many intervals can leave some intervals with little exposure or no events, producing unstable or boundary estimates. Inspect event counts and total (or predictor-weighted) exposure by interval, and compare fitted survival patterns with observed event-time patterns. Also verify that exposures sum to each subject’s observed follow-up time and that the event indicator is assigned to exactly one interval for an observed event.

A key modeling distinction is between a piecewise-constant baseline hazard and a piecewise-constant total hazard. If the longitudinal current-value predictor changes continuously over time, then \(\lambda_k\exp\{\eta_i(t;b_i)\}\) changes within interval \(k\). The interval exposure alone is no longer sufficient for the cumulative-hazard calculation; the integral of the time-varying multiplier must be evaluated. A numerical quadrature or a suitable analytic integral can be used, provided it accurately represents the specified predictor.

The main alternatives differ in how they represent baseline risk. A parametric baseline such as a Weibull hazard imposes a smooth functional form with relatively few parameters. A spline-based baseline offers smooth flexibility but requires spline basis and smoothing choices. A Cox baseline hazard leaves the baseline unspecified and uses partial-likelihood methods in its standard form; incorporating shared random effects and longitudinal information requires a joint likelihood treatment. The piecewise-exponential model is transparent and convenient for likelihood construction, but results can depend on interval placement and it does not guarantee a smooth hazard at cutpoints.

Common pitfall: It is tempting to multiply each interval’s baseline rate by the full interval width, even when a subject is censored or has an event before the interval ends. The correction is to use \(Y_{ik}\), the actual time contributed in that interval, not the interval’s full width. A second error is to use \(Y_{ik}\exp\{\eta_i(T_i;b_i)\}\) when the predictor varies inside the interval; the correct contribution is the integral of \(\exp\{\eta_i(u;b_i)\}\) over that subject’s exposed portion.

Key takeaway: Construct the survival contribution as an event hazard, when observed, times survival through actual follow-up. For a piecewise baseline, partition cumulative hazard by interval exposure; retain the time integral whenever the conditional predictor varies within an interval. Then integrate the conditional likelihood over the random effects to obtain the joint observed-data likelihood.

Check Your Understanding

Answer each question, then compare with the brief solutions below.

For cutpoints 0, 1, and 3 years, a subject is censored at 2.5 years. What are the interval exposures and event indicators?
If a subject has constant predictor multiplier 2 in an interval with baseline rate \(0.15\) per year and contributes 1.5 years there, what is that interval’s cumulative-hazard contribution?
Why is the factor \(\exp\{\eta_i(T_i;b_i)\}\) not generally sufficient to calculate cumulative hazard when the predictor varies with time?
Under what conditions does the conditional rate estimate reduce to events divided by ordinary person-time?

Solutions. (1) The exposures are 1 year in \([0,1)\), 1.5 years in \([1,3)\), and zero thereafter; both event indicators are zero because the subject is censored. (2) The contribution is \((0.15)(2)(1.5)=0.45\), dimensionless. (3) Cumulative hazard integrates the hazard multiplier over the entire exposed period, so the value at follow-up alone does not describe its values at earlier times. (4) The predictor multiplier must be 1 throughout the interval for each subject, so weighted exposure equals ordinary person-time; the calculation is still conditional on the other model quantities being held fixed.