Why Recurrent Events Need Their Own Rate
A first-event analysis asks when a subject experiences an initial event. But many clinical outcomes can occur repeatedly: a patient may have several disease episodes or hospital admissions during follow-up. After the first occurrence, the subject may remain under observation and at risk of another event. A model for the first event alone does not describe the rate of these later occurrences.
In joint modeling, the event rate may depend on a biomarker trajectory that is measured with error and represented by an underlying latent trajectory. The key quantity is then a conditional recurrent-event intensity: the instantaneous expected rate of event accumulation, given the latent trajectory, covariates, and event history up to just before that time.
The history in this definition matters: the intensity is conditional on what has happened before \(t\), not merely on baseline covariates. The risk indicator makes clear when the rate is zero because a subject is not under observation or is no longer at risk. For an ordinary recurrent-event analysis, a subject can remain at risk after an event and contribute further increments to \(N_i(t)\). This differs from a first-event analysis, in which the event-free risk period ends at the first event.
A Trajectory-Linked Intensity Model
A common model expresses the conditional intensity as a baseline rate multiplied by an exponential function of covariates and a pre-specified summary of the latent trajectory. Write \(\psi_i(t)\) for that trajectory summary. Its precise definition depends on the scientific question and must be specified as part of the model; the general formulation below does not require a particular choice.
The baseline intensity has units of events per unit time. The exponential terms are dimensionless, so they multiply the baseline rate without changing its units. Holding time, covariates, and other modeled conditions fixed, a one-unit increase in \(\psi_i(t)\) multiplies the conditional intensity by \(\exp(\alpha)\). This is a conditional rate comparison, not a causal effect and not automatically a comparison of marginal rates across subjects.
For the model to be meaningful, the intensity must be nonnegative and determined by information available just before \(t\), together with the latent trajectory on which we condition. Event times and risk periods must be defined consistently, and the trajectory summary must be well-defined over the relevant risk time. In the examples below, event times are treated as observed exactly, and follow-up is assumed to end for reasons adequately handled by the stated observation process. Those assumptions support interpretation of the calculations; they do not make a latent trajectory directly observed.
Worked Example: Calculating a Conditional Recurrent-Event Rate
Suppose a hypothetical subject is under observation, has covariate \(X_i=1\), and has trajectory summary \(\psi_i(t)=2\) at a particular time. Let the baseline intensity at that time be \(0.08\) events per person-year, \(\beta=\log(1.5)\), and \(\alpha=\log(1.25)\). Assume the displayed model applies and there is no additional history term.
Substituting into the intensity model gives $$ \lambda_i(t)=0.08\exp\{\log(1.5)(1)+\log(1.25)(2)\} =0.08(1.5)(1.25)^2. $$
Since \((1.25)^2=1.5625\), the rate is \(0.08(1.5)(1.5625)=0.1875\) events per person-year. This is the model’s instantaneous conditional event rate at that time, given the stated trajectory summary and covariate. It is not a 18.75% probability of an event in a year.
If the rate remained constant for the next half-year, the integrated conditional intensity over that interval would be \(0.1875(0.5)=0.09375\) expected events. Under a constant-rate Poisson process over that interval, the probability of at least one event would be \(1-\exp(-0.09375)\approx0.0895\), or about 8.95%. The probability calculation uses the additional constant-rate Poisson-process assumption; it is not obtained by treating the intensity itself as a probability.
How Intensity Differs From a First-Event Cause-Specific Hazard
For competing first events, the cause-specific hazard for cause \(k\) is defined among subjects who have experienced no event of any cause by time \(t\). If \(T\) is the time to the first event and \(J\) is its cause, the cause-specific hazard is the instantaneous rate of a first event of cause \(k\), conditional on \(T\ge t\). Its risk set ends for a subject as soon as any first event occurs.
A recurrent-event intensity instead concerns increments in a count process. A subject who has already had an event can remain at risk and contribute subsequent events. The two quantities may have similar-looking rate formulas, but they condition on different histories and describe different processes. In particular, the first-event hazard does not describe a person’s rate of second and later events.
| Feature | Recurrent-event intensity | First-event cause-specific hazard |
|---|---|---|
| What is counted? | All recurrent-event increments in \(N_i(t)\) | Only a first event of the specified cause |
| Who remains at risk after an event? | Potentially, if still under observation and at risk | No; the first event ends the event-free risk period |
| Conditioning history | Latent trajectory, covariates, and history just before \(t\) | Being free of all first events through time \(t\) |
| Units and interpretation | Conditional event rate per unit time | Conditional first-event rate per unit time |
Worked Example: A Recurrence After the First Event
Consider a hypothetical patient followed for 12 months, with recorded recurrent events at months 2 and 7. Assume the patient remains under observation and at risk after each event. Over the interval from month 7 to month 10, suppose the conditional recurrent-event intensity is constant at \(0.10\) events per month, given the trajectory and covariates.
The integrated intensity for those three months is $$ \int_7^{10}0.10\,du=0.10(3)=0.30 $$ expected events, conditional on the model information. This does not mean the patient has a 30% probability of an event. If, in addition, a constant-rate Poisson-process assumption is appropriate, the probability of at least one event in that interval is \(1-\exp(-0.30)\approx0.2592\).
At month 8, this patient is no longer in the event-free risk set for a first-event cause-specific hazard: the first event occurred at month 2. The recurrent-event intensity can nevertheless be defined at month 8, because the patient remains under observation and at risk for another occurrence. Thus, the numerical rate \(0.10\) events per month describes recurrence only; it cannot be read as the patient’s first-event cause-specific hazard at month 8.
Why Conditioning on the Latent Trajectory Matters
The model above is conditional on \(m_i\), which is latent rather than directly observed. A conditional rate for a subject with a given trajectory is not generally equal to the rate obtained by substituting an average trajectory into the model. Because the model is nonlinear in \(\psi_i(t)\), averaging over trajectory uncertainty and applying the model are not interchangeable operations.
Worked Example: Averaging Over Two Possible Trajectories
Suppose a hypothetical population has two equally likely latent trajectory groups at a fixed time. Their trajectory summaries are \(\psi=0\) and \(\psi=2\). There are no covariates, the baseline intensity is \(0.05\) events per year, and \(\alpha=\log(2)\). The model gives conditional rates $$ \lambda(\psi=0)=0.05\exp\{\log(2)(0)\}=0.05 $$ and $$ \lambda(\psi=2)=0.05\exp\{\log(2)(2)\}=0.20 $$ events per year.
The average of these two conditional rates is \(\{0.05+0.20\}/2=0.125\) events per year. The average trajectory summary is \((0+2)/2=1\), but substituting it into the model gives \(0.05\exp\{\log(2)(1)\}=0.10\) events per year. The plug-in rate of \(0.10\) is not the average conditional rate of \(0.125\).
Over one year, if each group follows its stated constant-rate Poisson process, the population probability of at least one event is $$ \frac{1}{2}(1-\exp(-0.05))+\frac{1}{2}(1-\exp(-0.20)) \approx 0.1150. $$
Using the average rate \(0.125\) as though everyone followed one Poisson process would instead give \(1-\exp(-0.125)\approx0.1175\). The difference occurs because a mixture of processes with different fixed rates need not itself be a single constant-rate Poisson process. The conditional model and the population-level quantity answer different questions.
Justification: From Short-Term Rates to Expected Counts
An intensity is useful because it accumulates over time. Let \(\lambda_i(u\mid m_i,X_i,\mathcal{H}_i(u-))\) be the conditional intensity while subject \(i\) is at risk. Over a short interval, its defining approximation is $$ E\{dN_i(u)\mid m_i,X_i,\mathcal{H}_i(u-)\} = Y_i(u)\lambda_i(u\mid m_i,X_i,\mathcal{H}_i(u-))\,du+o(du), $$ where \(dN_i(u)\) is the event-count increment. Summing these expected increments across a partition of a follow-up interval and taking the limit as the partition becomes finer yields the integrated intensity.
More precisely, under the usual integrability conditions for the intensity, the difference between the count and its accumulated conditional rate has conditional mean-zero increments. Consequently, conditional on the latent trajectory and covariates, $$ E\{N_i(\tau)-N_i(0)\mid m_i,X_i\} = E\left\{ \int_0^\tau Y_i(u)\lambda_i(u\mid m_i,X_i,\mathcal{H}_i(u-))\,du \;\middle|\;m_i,X_i \right\}. $$
If the conditional intensity is deterministic over the interval once \(m_i\) and \(X_i\) are given, the expectation on the right reduces to the integral of that rate while at risk. If it also depends on event history, the rate can change as events occur, and the conditional expectation must account for that history. This is why one cannot generally calculate a probability of one or more events from an integrated intensity alone without additional process assumptions.
Define when the subject is observed and at risk, including whether observation continues after a recurrence.
State the baseline intensity, covariates, latent trajectory summary, and any history dependence included in the model.
Integrate the conditional intensity to obtain an expected count under the model; invoke additional assumptions before turning it into an event probability.
Interpretation, Diagnostics, and Limitations
The coefficient \(\alpha\) describes a conditional association between the specified trajectory summary and the recurrent-event intensity. For example, \(\exp(\alpha)=1.25\) means that, at the same time and covariate values and under the specified conditioning, a one-unit higher \(\psi_i(t)\) corresponds to an intensity 1.25 times as large. It does not establish that changing the biomarker would cause the recurrence rate to change. Unmeasured factors, model misspecification, or dependence between the latent trajectory and event process may contribute to the association.
Diagnostics should ask whether the fitted model describes both the event timing and the accumulation of counts. Compare observed and model-based cumulative event counts across time and across groups defined by predicted rates. Check whether residual event increments, often formed by subtracting the fitted integrated intensity from the observed count, show systematic departures from their expected pattern. Large discrepancies can signal a misspecified baseline intensity, an inadequate trajectory summary, omitted dependence on event history, or poor handling of observation and censoring.
A further limitation is that the conditional intensity is not directly observed when the biomarker trajectory is latent. Its interpretation depends on the longitudinal model and on how uncertainty about that trajectory is handled. Replacing the latent trajectory by noisy observed measurements can distort the association. Likewise, a conditional rate comparison should not be presented as a marginal population comparison without accounting for variation in trajectories and covariates.
Common Pitfalls
- Calling a rate a probability. An intensity of \(0.2\) events per year is a conditional rate, not a 20% probability. A probability requires a time interval and assumptions about the event process.
- Stopping the recurrent process at the first event by default. That changes the estimand to a first-event quantity. State whether subjects remain at risk after recurrence.
- Using the first-event risk set for later recurrences. A cause-specific hazard conditions on no previous first event; a recurrent-event intensity can apply after prior events.
- Plugging in an average trajectory without justification. With an exponential link, the rate at the average trajectory need not equal the average rate across trajectories.
- Interpreting a conditional association causally. A trajectory-linked rate model describes a conditional statistical association unless additional causal assumptions and a suitable design justify more.
Check Your Understanding
Try each question, then compare your reasoning with the solutions.
- A subject has already experienced a first event but remains under observation. Can a recurrent-event intensity still be defined? How does this differ from the first-event cause-specific hazard?
- In the model, \(\alpha=\log(1.4)\). What is the conditional intensity ratio for a one-unit increase in \(\psi_i(t)\), holding other modeled terms fixed?
- Why is an intensity of \(0.30\) events per month not automatically a 30% probability of an event in the next month?
- Two trajectory groups have rates \(0.04\) and \(0.16\) events per year and are equally common. What is the average conditional rate, and why can it differ from the model rate at the average trajectory summary?
Solutions. (1) Yes, if the subject remains observed and at risk for recurrence; the count process can continue to accumulate events. The first-event cause-specific hazard no longer applies to that subject after the first event, because its risk set requires no prior event. (2) The ratio is \(\exp(\alpha)=1.4\), so the conditional intensity is 1.4 times as large for the one-unit increase. (3) An intensity is a rate with units of events per month, not a probability. Converting a rate into an event probability requires a specified interval and assumptions about how the rate behaves and how events are generated. (4) The average rate is \((0.04+0.16)/2=0.10\) events per year. Under a nonlinear link, averaging the conditional rates need not equal evaluating the model at the average trajectory summary; those are distinct calculations.