Piecewise-Exponential PK/PD Hazard Models

Why Use a Piecewise-Exponential Hazard?

A PK/PD time-to-event analysis may need a baseline hazard that changes over time without imposing a single smooth shape on the entire follow-up period. A piecewise-exponential model addresses this by dividing time into intervals and allowing a separate baseline hazard in each one. Within an interval, the hazard is constant for a given exposure value; across intervals, it can change abruptly. This makes the model flexible enough to represent broad changes in baseline risk while retaining direct formulas for survival and event probabilities.

The model is useful when the scientific question concerns how exposure relates to event hazard and when a small set of time intervals is a reasonable representation of baseline risk. Its simplicity does not remove the need to choose interval boundaries carefully or to distinguish a hazard from an event probability.

Model Definition and Assumptions

Definition: Let the follow-up time be divided by cut points $0=\tau_0<\tau_1<\cdots<\tau_J$. For individual $i$, let $C_{ij}$ be the exposure value assigned to interval $j$, and let $\lambda_j>0$ be that interval’s baseline hazard. A piecewise-exponential proportional-hazards model is $$ h_i(t)=\lambda_j\exp(\beta C_{ij}),\qquad \tau_{j-1}\le t<\tau_j. $$ Here $h_i(t)$ is the instantaneous event rate at time $t$, $\lambda_j$ has units of inverse time, and $\beta$ has units inverse to those of exposure. The exposure hazard ratio for a one-unit increase in $C$, within an interval, is $\exp(\beta)$.

In this formulation, each interval has its own baseline hazard, while the exposure coefficient is shared across intervals. Thus, the exposure effect is proportional across intervals: the hazard ratio for the same exposure contrast does not depend on the interval. The exposure value $C_{ij}$ is treated as constant within interval $j$. It may be a fixed subject characteristic or a prespecified interval-level exposure summary; the model does not itself specify how a changing concentration should be summarized.

The principal assumptions are that the event hazard is constant within each interval for a given exposure, exposure is represented appropriately by $C_{ij}$, and the exposure effect has the stated log-linear form with a common $\beta$. Standard time-to-event likelihood analysis also requires an appropriate treatment of censoring; in particular, the usual right-censored likelihood assumes censoring is non-informative conditional on modeled information. Event times and interval membership must be recorded on a time scale consistent with the chosen cut points.

A useful consequence is that the model’s hazard is not the same quantity as the probability of an event during an interval. For a constant hazard $h$ over duration $\Delta$, the event probability depends on both $h$ and $\Delta$. Longer intervals have greater event probability even when the hazard rate is unchanged.

Deriving Probabilities Across Intervals

Within interval $j$, write $h_{ij}=\lambda_j\exp(\beta C_{ij})$, and let $\Delta_j=\tau_j-\tau_{j-1}$. Conditional on being event-free at the start of that interval, the survival probability over an elapsed time $u$ satisfies $S(u)=\exp(-h_{ij}u)$. This follows from the constant-hazard relationship $dS(u)/du=-h_{ij}S(u)$, with $S(0)=1$. Therefore, the conditional probability of an event before the interval ends is $1-\exp(-h_{ij}\Delta_j)$.

P(\text{event in interval }j\mid \text{event-free at }\tau_{j-1}) =1-\exp(-h_{ij}\Delta_j).

To obtain an unconditional probability, the individual must first survive all earlier intervals. Define the cumulative hazard through the end of interval $k$ as $H_{ik}=\sum_{j=1}^{k}h_{ij}\Delta_j$. The probability of remaining event-free through that point is $\exp(-H_{ik})$. Consequently, the probability of an event in interval $k$, measured from the beginning of follow-up, is the probability of surviving earlier intervals multiplied by the conditional event probability in interval $k$.

P(\text{event in interval }k) =\exp\left(-\sum_{j=1}^{k-1}h_{ij}\Delta_j\right) \left[1-\exp(-h_{ik}\Delta_k)\right].

The probability of an event by the end of interval $k$ is $1-\exp(-H_{ik})$. It also equals the sum of the unconditional event probabilities over intervals 1 through $k$: the interval-specific events are mutually exclusive, and the probability of surviving all $k$ intervals is the remaining probability.

Key distinction: $h_{ij}$ is a rate per unit time, whereas $1-\exp(-h_{ij}\Delta_j)$ is a dimensionless conditional probability. Do not report the hazard as though it were an interval event probability.

Worked Probability Calculations

Worked Example: Conditional and Unconditional Risk Across Two Intervals

Suppose follow-up is divided into intervals of 1 week and 2 weeks. A hypothetical model has $\lambda_1=0.05000$ per week and $\lambda_2=0.08334$ per week, with $\beta=\log(0.895)=-0.11093$ per mg/L. The exposure is $1$ mg/L in both intervals. Assume the stated exposure values are constant within their intervals and use the model’s common exposure coefficient.

The exposure multiplier in either interval is $\exp(\beta C)=\exp(\log(0.895)\times1)=0.895$. Thus the interval hazards are $h_1=(0.05000)(0.895)=0.04475$ per week and $h_2=(0.08334)(0.895)=0.07459$ per week. These are constant within their respective intervals, as required by the model.

The conditional probability of an event in interval 1 is $1-\exp[-(0.04475)(1)]=1-\exp(-0.04475)\approx0.04376$, or about 4.376%. The probability of remaining event-free through week 1 is $\exp(-0.04475)\approx0.9562$.

Conditional on surviving to week 1, the event probability in interval 2 is $1-\exp[-(0.07459)(2)]=1-\exp(-0.14918)\approx0.1386$, or about 13.86%. This is not the unconditional probability of an event in interval 2. The unconditional probability is $\exp(-0.04475)[1-\exp(-0.14918)]\approx(0.9562)(0.1386)\approx0.1325$, or about 13.25%.

The probability of an event by the end of interval 2 is $1-\exp[-0.04475-(0.07459)(2)]=1-\exp(-0.19393)\approx0.1763$, or about 17.63%. This agrees, up to rounding, with the sum of the unconditional interval risks: $0.04376+0.1325\approx0.1763$. The calculation assumes no additional hazard change within either interval.

Worked Example: Exposure-Dependent Hazards in Each Interval

Consider two intervals with durations of 2 days and 3 days, baseline hazards $\lambda_1=0.06000$ and $\lambda_2=0.03000$ per day, and a common exposure coefficient $\beta=0.4000$ per mg/L. For a hypothetical individual, exposure is $0.5$ mg/L in interval 1 and $1.5$ mg/L in interval 2. Assume each assigned exposure is constant within its interval and that the log-linear exposure effect applies in both.

In interval 1, $h_1=0.06000\exp[(0.4000)(0.5)]=0.06000\exp(0.2)\approx0.07328$ per day. The cumulative hazard for that interval is $(0.07328)(2)\approx0.1466$, so the conditional event probability is $1-\exp(-0.1466)\approx0.1363$. Survival through interval 1 is therefore approximately $0.8637$.

In interval 2, $h_2=0.03000\exp[(0.4000)(1.5)]=0.03000\exp(0.6)\approx0.05466$ per day. Its cumulative hazard is $(0.05466)(3)\approx0.1640$. Conditional on surviving the first interval, the interval 2 event probability is $1-\exp(-0.1640)\approx0.1512$. The unconditional probability of an event in interval 2 is approximately $(0.8637)(0.1512)=0.1306$.

The probability of an event by the end of day 5 is the sum of the mutually exclusive unconditional interval risks, approximately $0.1363+0.1306=0.2669$, or about 26.69%. Equivalently, using the total cumulative hazard, it is $1-\exp[-(0.1466+0.1640)]\approx0.2670$; the small difference is due to rounding intermediate values. The coefficient implies a hazard ratio of $\exp(0.4000)\approx1.492$ for a 1 mg/L exposure increase within either interval, not a 49.2 percentage-point increase in event probability.

Justification and Interpretation

The interval formulas follow directly by chaining survival probabilities. If the hazard is constant at $h_{i1}$ over duration $\Delta_1$, survival is $\exp(-h_{i1}\Delta_1)$. Conditional survival over the next interval is $\exp(-h_{i2}\Delta_2)$. Multiplying these conditional probabilities gives $\exp(-h_{i1}\Delta_1)\exp(-h_{i2}\Delta_2)=\exp[-(h_{i1}\Delta_1+h_{i2}\Delta_2)]$. Repeating this argument for $k$ intervals gives $\exp(-H_{ik})$; subtracting from 1 gives the event probability by the end of interval $k$. The product rule also explains why an event probability in a later interval must include survival up to that interval.

A positive $\beta$ means higher exposure is associated with a higher hazard under the model; a negative $\beta$ means higher exposure is associated with a lower hazard. These are conditional hazard comparisons, not direct statements about causal effects or absolute risk differences. For two exposure values differing by $d$ units, the hazard ratio is $\exp(\beta d)$, holding the interval fixed. Absolute risk also depends on baseline hazards, interval durations, and survival in earlier intervals.

Choosing Intervals, Diagnostics, and Alternatives

Interval boundaries determine where the baseline hazard is allowed to change. They should be selected with the event-time scale, scientific context, and amount of information in mind. Too few wide intervals may leave substantial within-interval changes unrepresented; too many narrow intervals can produce unstable estimates, especially when few events occur in some intervals. Boundaries should be chosen without using outcome patterns in a way that makes the resulting analysis misleading.

Useful checks include tabulating events and person-time within each interval, examining whether fitted interval hazards are implausibly extreme, and comparing observed event patterns with model-based survival predictions. A fitted hazard near zero or very large in an interval with sparse information can signal weak estimation rather than a reliable biological feature. Examine whether residual or predictive patterns suggest systematic misfit over time or across exposure levels. Also verify that the exposure values and units used to construct $C_{ij}$ match the coefficient interpretation, and that each person contributes risk time only while event-free and under observation.

A piecewise-exponential model is not the only way to represent baseline hazard. A Weibull model imposes a smooth parametric hazard shape, which can be more concise but may not capture abrupt changes. A Cox proportional-hazards model leaves the baseline hazard unspecified, avoiding a parametric baseline shape but providing a different route to absolute-risk prediction. A discrete-time model can be appropriate when outcomes are recorded only in periods rather than at exact event times; its link and probability model must be interpreted on that scale. These approaches are alternatives, not interchangeable labels for the same assumptions.

The piecewise-exponential model has important limits. It treats hazards as constant between cut points, so results can depend on the chosen partition. It also imposes the stated exposure functional form and, with a shared $\beta$, the same exposure hazard ratio across intervals. If either restriction is implausible, the model may not represent the exposure-response pattern well. A good fit in observed intervals does not establish that the exposure effect is causal, and the model does not automatically address confounding, informative censoring, or unmeasured heterogeneity.

Common Mistakes

Using a hazard as an event probability. A hazard of $0.08$ per week is a rate, not an 8% weekly risk. For a constant hazard of $0.08$ per week over 1 week, the conditional probability is $1-\exp(-0.08)\approx0.0769$.
Forgetting prior survival. The conditional probability in interval 2 applies only to people event-free at its start. The unconditional interval 2 probability also multiplies by the probability of surviving interval 1.
Ignoring interval duration. The same hazard produces different event probabilities over different interval lengths because the cumulative hazard is $h\Delta$.
Interpreting $\exp(\beta)$ as an absolute-risk ratio. It is a hazard ratio for a one-unit exposure contrast under the model, not a direct multiplier of interval or cumulative event probability.
Assuming a more detailed partition is always better. Narrow intervals can leave too little event information to estimate their baseline hazards reliably.

Key takeaway: Specify the interval hazards first, convert each hazard to a conditional event probability using its interval duration, and then account for prior survival to obtain unconditional risks. Always interpret exposure effects on the hazard scale separately from absolute event probabilities.

Check Your Understanding

Work through the questions, then compare your results with the solutions.

A constant hazard is $0.10$ per day over a 2-day interval. What is the conditional probability of an event during the interval?
A person survives the first interval with probability $0.90$. Given survival to the start of the second interval, the event probability there is $0.20$. What is the unconditional probability of an event in interval 2?
In the model $h=\lambda\exp(\beta C)$, what is the hazard ratio for an exposure increase of 2 units if $\beta=-0.30$ per exposure unit?
State one assumption made by the piecewise-exponential model and one diagnostic that could reveal a problem with it.

Solutions. (1) $1-\exp[-(0.10)(2)]=1-\exp(-0.20)\approx0.1813$, or 18.13%. (2) The unconditional interval 2 probability is $(0.90)(0.20)=0.18$, or 18%. (3) The hazard ratio is $\exp[(-0.30)(2)]=\exp(-0.60)\approx0.5488$; the specified exposure increase is associated with a 45.12% lower hazard, holding the interval and other modeled conditions fixed. (4) One assumption is a constant hazard within each interval for a given exposure value. A diagnostic is to inspect observed event patterns and fitted predictions across time; systematic within-interval discrepancies suggest the constant-hazard representation may be inadequate.