Event-Specific Intensities For Recurrent Events

Why Condition on Event History?

For recurrent outcomes, the chance of another event can depend on what has already happened. A person's time since an earlier event, number of previous recurrences, and observed covariates may all be relevant to the timing of the next event. An event-specific intensity expresses the instantaneous recurrence rate for a person with a particular observed history. It is therefore a conditional quantity, not simply the average number of events in a population divided by the population's follow-up time.

The distinction matters when histories differ across people or change over time. A marginal event rate averages over whichever histories are represented in the population or risk set at a given time. That average can change because the composition of the risk set changes, even if the event-specific intensities for given histories do not.

Definition: For individual $i$, let $N_i(t)$ be the number of recurrent events observed by time $t$, and let $\mathcal H_i(t-)$ denote the individual's observed history just before $t$. This history can include prior event times and recorded covariates available by $t$. The event-specific intensity $\lambda_i(t\mid\mathcal H_i(t-))$ is the instantaneous rate of an event at $t$, conditional on that history and on being at risk. In a short interval of length $dt$, $$ P\{N_i(t+dt)-N_i(t)=1\mid\mathcal H_i(t-)\} = Y_i(t)\lambda_i(t\mid\mathcal H_i(t-))\,dt+o(dt), $$ where $Y_i(t)$ is 1 when the individual is at risk for another event at $t$, and 0 otherwise. The intensity has units of events per unit time; it is not itself a probability.

The notation $t-$ means immediately before $t$. Conditioning on the history just before the interval is important: the information used to describe the current rate must not include the event whose occurrence is being described. If $Y_i(t)=0$, the individual is not currently able to contribute an event to this recurrent-event process, so the event increment has zero intensity under this definition.

Event Order and the At-Risk Condition

The intensity can be indexed by recurrence order. Write $T_{ij}$ for the time of individual $i$'s $j$th recurrence, and set $T_{i0}=0$. For the $j$th recurrence, the individual enters the relevant risk set only after the $(j-1)$st recurrence and leaves it when the $j$th recurrence occurs. Let $C_i$ denote the end of observation, and let a terminal event, if relevant, also remove the individual from the recurrence risk set.

Definition: The $j$th-event counting process is $N_{ij}(t)=I(T_{ij}\leq t)$. A corresponding at-risk indicator is $$ Y_{ij}(t)=I(T_{i,j-1}<t\leq T_{ij},\ C_i\geq t,\ \text{no terminal event before }t). $$ The event-order-specific intensity $\lambda_{ij}(t\mid\mathcal H_i(t-))$ is defined by $$ P\{N_{ij}(t+dt)-N_{ij}(t)=1\mid\mathcal H_i(t-)\} = Y_{ij}(t)\lambda_{ij}(t\mid\mathcal H_i(t-))\,dt+o(dt). $$

The risk indicator makes the conditioning explicit. A person who has not yet had a first recurrence contributes to the first-event risk set, but not to the second-event risk set. After the first recurrence, that person can contribute to the second-event risk set. The precise time scale used to describe $t$ can vary; the definition above does not choose between time measured from study entry and time measured from a previous event.

An event-order-specific intensity need not be the same across event orders. Nor must it depend on event order alone: two people at risk for their second recurrence can have different intensities because their earlier histories or covariates differ. In continuous time, this formulation generally assumes that events do not occur simultaneously for the same individual, so an increment over a sufficiently short interval is either zero or one event with probability up to a term of smaller order than $dt$.

Worked Example: Interpreting a History-Conditional Intensity

Suppose a hypothetical model describes the next infection recurrence among patients who are currently under observation and have had two previous recurrences. At a particular time, for patients with the specified observed history, the model gives an intensity of $0.12$ recurrences per month. Interpret this value and approximate the chance of recurrence in the next $0.1$ month.

For a short interval $dt$, the conditional event probability is approximately the intensity multiplied by the interval length: $$ P(\text{recurrence in next }dt\mid\mathcal H(t-)) \approx \lambda(t\mid\mathcal H(t-))dt. $$

Substituting the hypothetical intensity and $dt=0.1$ month gives $$ 0.12\ \text{month}^{-1}\times 0.1\ \text{month}=0.012. $$ Thus the short-interval probability is approximately $0.012$, or $1.2\%$, for a patient with that history who is at risk. The intensity itself remains a rate, $0.12$ per month; it is not a $12\%$ monthly probability. The approximation is for a short interval and does not assert that the probability over a full month is exactly $0.12$.

From Conditional Intensities to Marginal Rates

A marginal event rate averages over the histories represented in a population or in a current risk set. To see how that differs from the event-specific intensity, consider the expected number of events in a short interval. Conditional on the history, the expected increment is approximately $Y_i(t)\lambda_i(t\mid\mathcal H_i(t-))dt$. Averaging over histories gives the population-level expected increment.

Derivation: By the definition of intensity and the law of total expectation, $$ E\{N_i(t+dt)-N_i(t)\} = E\!\left[E\{N_i(t+dt)-N_i(t)\mid\mathcal H_i(t-)\}\right] = E\{Y_i(t)\lambda_i(t\mid\mathcal H_i(t-))\}\,dt+o(dt). $$ If the expectation of $N_i(t)$ is differentiable, division by $dt$ and passage to the limit give $$ \frac{d}{dt}E\{N_i(t)\} = E\{Y_i(t)\lambda_i(t\mid\mathcal H_i(t-))\}. $$

The derivative in this result is the expected event count gained per unit time per member of the original population. It averages the at-risk conditional intensities while assigning zero contribution to people not at risk. If instead the average is taken only among those currently at risk, the corresponding rate is $$ \frac{E\{Y_i(t)\lambda_i(t\mid\mathcal H_i(t-))\}}{E\{Y_i(t)\}}, $$ provided $E\{Y_i(t)\}>0$. This is an average over the current risk set, not necessarily over the baseline population.

These averages depend on the distribution of histories at the time of averaging. Thus, a marginal rate can change as people experience recurrences, leave follow-up, or have terminal events, even if the conditional intensity for a fixed history remains constant. Conversely, a stable marginal rate does not establish that everyone has the same conditional intensity.

Worked Example: Risk-Set Composition Changes the Average

Consider a hypothetical risk set with 100 people. Based on their observed histories, 90 have a conditional intensity of $0.02$ events per person-year and 10 have an intensity of $0.10$ events per person-year. Compute the average conditional intensity in this current risk set.

The total expected event contribution per year is the sum of the numbers at risk in each history group multiplied by their intensities: $$ 90(0.02)+10(0.10)=1.8+1.0=2.8 \quad\text{events per year}. $$ Dividing by 100 people at risk gives $$ \frac{2.8}{100}=0.028 \quad\text{events per person-year}. $$

The value $0.028$ is the risk-set average. It is not the conditional intensity for either history group: those are $0.02$ and $0.10$ events per person-year. If the risk set later contained 50 people from each group, while the group-specific intensities stayed the same, its average would instead be $$ \frac{50(0.02)+50(0.10)}{100} = \frac{1+5}{100} = 0.06 \quad\text{events per person-year}. $$ The difference between $0.028$ and $0.06$ comes solely from the changed mix of histories in the current risk set.

Estimating an Event-Order-Specific Rate

Observed data can provide a simple estimate when the target group has a common, constant intensity over the risk periods being summarized. In that restricted setting, the event count divided by total at-risk time estimates the intensity. This calculation is not a general solution for history-dependent intensities: if rates vary with time or history, pooling the risk periods can conceal those differences.

Worked Example: A Constant Second-Recurrence Intensity

In a hypothetical study, participants who have had one recurrence contribute a total of 40 person-months while at risk for a second recurrence. Three second recurrences are observed during those risk periods. Assume, for this illustration, that the second-recurrence intensity is constant and common across these risk periods, and that censoring does not depend on the unobserved timing of the recurrence after conditioning on the recorded history. Calculate the likelihood-based estimate.

Let $\lambda$ be the common intensity in recurrences per person-month. Under the constant-intensity assumption, the event-time likelihood is proportional to $\lambda^D\exp(-\lambda T)$, where $D=3$ is the number of observed events and $T=40$ person-months is total time at risk. The log likelihood, omitting terms not involving $\lambda$, is $$ \ell(\lambda)=3\log(\lambda)-40\lambda. $$ Differentiating and setting the score to zero gives $$ \frac{d\ell}{d\lambda}=\frac{3}{\lambda}-40=0, \qquad \widehat{\lambda}=\frac{3}{40}=0.075 \quad\text{recurrences per person-month}. $$

The estimate describes the assumed common second-recurrence intensity for these risk periods. It does not show that every person's intensity is $0.075$, and it should not be used as a history-adjusted estimate if intensity varies systematically across histories. The constant-intensity and censoring assumptions are what make this pooled calculation interpretable for the stated target.

Assumptions, Diagnostics, and Common Pitfalls

An intensity is defined relative to a specified event process, history, time scale, and risk condition. Before interpreting one, check that the event being counted is clear, that the history contains only information available before the time in question, and that people who cannot experience the event at that time are excluded from the risk set. The intensity is a conditional description; it is not automatically a causal effect of any covariate in the history.

For estimation from observed follow-up, censoring also matters. The conditional intensity among people still observed is a description of the observed process. To use it to represent the corresponding process beyond censoring, censoring must be independent of future recurrence timing conditional on the recorded history, or an appropriate correction for informative censoring must be used. A terminal event that prevents further recurrence is a different event from loss to follow-up: it removes a person from the recurrence risk set and can change the population being described.

Useful diagnostics include tabulating risk time and event counts by recurrence order and by important recorded-history groups, checking for sparse risk sets, and examining whether event rates vary substantially over time or history strata. If the common-intensity assumption is doubtful, a single pooled events-per-time calculation may be misleading. Sparse or empty event-order risk sets also make estimates unstable or undefined. A marginal-rate summary may still describe the observed population average, but it answers a different question.

Common mistake: A reader sees an intensity of $0.10$ per month and calls it a $10\%$ probability of recurrence in the next month. This confuses a rate with a finite-interval probability. For a sufficiently short interval of $0.1$ month, the first-order approximation is $0.10\times0.1=0.01$, or about $1\%$. The exact one-month probability requires additional assumptions about how the intensity behaves throughout that month. A second mistake is to interpret a marginal rate as the conditional intensity for a typical individual. The risk-set composition calculation shows why no such individual interpretation follows without further assumptions.

Quantity	What is averaged or conditioned on?	Typical interpretation
Event-specific intensity	Conditional on an individual's observed history and being at risk	Instantaneous recurrence rate for that history, in inverse-time units
Current risk-set average rate	Average of at-risk intensities over histories represented in the current risk set	Average event rate per unit of current at-risk person-time
Population expected-count rate	Average over the original population, with zero contribution when not at risk	Expected events gained per unit time per original population member

The event-specific intensity is useful when the scientific question concerns how the instantaneous recurrence rate depends on previously observed event history. It can distinguish, for example, whether the rate after a second recurrence differs from the rate after a first. The marginal rate instead summarizes an average over the histories that happen to be represented in a population or risk set. Neither is universally preferable: the choice depends on the estimand. Event-specific intensities also do not, by themselves, describe the overall expected event-count trajectory or establish causal effects; those require their own target definitions and assumptions.

Check Your Understanding

Answer each question, then compare your reasoning with the solutions.

A conditional intensity is $0.04$ events per week. What is the first-order approximation to the event probability over the next $0.25$ week?
Why can the current risk-set average rate change even when each history group's conditional intensity stays fixed?
Who is at risk for a third recurrence: a person who has had no recurrences, one recurrence, or two recurrences?
In a constant-intensity illustration, there are 6 events over 120 person-months at risk. What is the events-per-time estimate, and what assumption gives it a conditional-intensity interpretation?
Explain why an intensity of $0.10$ per month is not automatically a $10\%$ probability of recurrence in a month.

Solutions. (1) Multiply rate by interval: $0.04\ \text{week}^{-1}\times0.25\ \text{week}=0.01$, approximately a $1\%$ probability. (2) The average weights group-specific intensities by the histories represented in the risk set, so changing group proportions changes the average. (3) A person with two previous recurrences is at risk for a third, provided they remain under observation and have not had a terminal event that precludes recurrence. (4) The estimate is $6/120=0.05$ events per person-month; interpreting it as a common conditional intensity requires, among other conditions, a constant common intensity over the included risk periods and appropriate handling of censoring. (5) Intensity is a rate, not an interval probability; the probability depends on how the intensity behaves across the interval, while rate times interval is only a short-interval approximation.