Conditional Intensities From Event-Time Hazards

From an Event-Time Hazard to a Counting-Process Rate

An event-time hazard describes the instantaneous event rate among subjects who have not yet had the event. A counting process records event occurrences over time, but it also needs to account for who is currently under observation and eligible to experience the event. The connection is direct: at times when a subject is at risk, the counting-process rate is the conditional event-time hazard; when the subject is not at risk, that rate is zero.

The risk indicator is essential. A hazard for an event time is not automatically the intensity of the observed counting process over the whole timeline. Entry, failure, and censoring determine when the event process is observable and at risk. In particular, with delayed entry, the risk indicator and a local probability statement need care at the exact entry time.

Write \(N(t)\) for the number of events observed by time \(t\), and \(Y(t)\) for the predictable indicator that the subject is at risk immediately before \(t\). Thus \(Y(t)=1\) when the subject has entered observation, has not yet had the event, and has not been censored just before \(t\); otherwise \(Y(t)=0\). Under the usual continuous-time assumptions, the counting-process intensity is

This equality is an almost-everywhere-in-time statement: changes to the value at isolated time points do not change the rate integrated over time. At an ordinary time strictly inside an at-risk interval, it gives the familiar local approximation

This display should not be read as a pointwise assertion at every boundary. At a delayed-entry time, the risk indicator may equal zero at the boundary while an event can occur immediately after entry. Likewise, an interval beginning before entry can cross the entry time and contain an observed event. The correct local approximation applies at regular times away from such boundaries, or in the almost-everywhere sense used to define the intensity.

Assumptions Behind the Link

The formula is simple, but it rests on a coherent definition of the risk set and of the conditional hazard. The hazard must describe the event rate conditional on the relevant history among subjects who are actually at risk. If covariates change over time, their values used in the hazard must be available just before the event time; in counting-process terminology, they must be predictable.

• Continuous event timing: the event-time distribution has no point mass at the times under consideration. A hazard-rate approximation in \(h\) does not represent a positive instantaneous probability of an event at a fixed time.
• Locally integrable hazard: the conditional hazard is finite and integrable over bounded intervals, so the event probability over a short interval is of order \(h\).
• Correct risk indicator: \(Y(t)\) records observation and eligibility immediately before \(t\). It must turn off after an event or censoring and must not turn on before entry.
• Compatible conditioning: the event-time hazard conditions on the information used to describe the subject’s current event risk. If a latent event-time hazard is being inferred from censored data, censoring must be noninformative conditional on the modeled history, or its dependence must be handled explicitly.
• Simple event process: a subject cannot experience two jumps of the same event-counting process at the same instant. This is standard for a single-event time with continuous distribution.

These conditions concern the rate representation, not a particular estimation method. For example, independence between censoring and the event time is not needed merely to define the intensity of the observed process conditional on a sufficiently rich observed history. It is needed, or must be replaced by an appropriate model, when the observed risk sets are used to learn about a latent event-time distribution.

Why the Risk Indicator Multiplies the Hazard

At a regular time \(t\), condition on the history just before \(t\). If \(Y(t)=0\), the subject is not eligible for an observed event at that time, so the observed counting process has no event probability to first order in \(h\). If \(Y(t)=1\), the conditional event probability is the event-time hazard times the interval length, up to a smaller-order term. Combining the two cases gives

where \(\Delta N(t,h)=N(t+h)-N(t)\). Since \(Y(t)\) takes only the values zero and one, the two cases can be written as \(Y(t)\lambda(t\mid\mathcal F_{t-})h+o(h)\). This argument is valid at regular times, but the boundary qualification matters: a risk indicator can change within \([t,t+h)\).

For example, if entry occurs at \(L\) and the predictable convention is \(Y(t)=I(L<t\leq T\wedge C)\), then \(Y(L)=0\). The value at the single time \(L\) does not affect any time integral. But it does affect a pointwise expression that uses \(Y(L)\) to approximate the event probability over an interval starting at \(L\). This is why the intensity identity is understood almost everywhere in time, rather than as a pointwise local-probability formula at every boundary.

Delayed Entry: The Boundary Is Not an Ordinary Time

Delayed entry provides a useful stress test of the formula. Suppose the subject enters at \(L=5\) days and then has a constant post-entry event hazard of \(0.08\) per day. Before day 5 there is no observed at-risk time. If an interval crosses day 5, its event probability depends on the part of the interval after entry, not on the full interval length.

The same principle applies at exit boundaries, including event and censoring times. The risk indicator is defined just before time \(t\), so it can be one at the event time itself, allowing the event jump to be counted. Immediately afterward, the subject is no longer at risk for that first event. Boundary conventions are important for recording jumps correctly, while isolated values do not change an integral with respect to ordinary time.

Competing Event Types

If an event can occur for one of several mutually exclusive causes, define a counting process \(N_k(t)\) for cause \(k\), and let \(\lambda_k(t\mid\mathcal F_{t-})\) be its cause-specific event-time hazard among subjects who remain event-free and at risk. With a common risk indicator \(Y(t)\), the cause-specific intensity is

The total event rate while at risk is the sum of the cause-specific rates. This follows because, over a sufficiently short interval, the probability of two distinct causes both occurring is of smaller order than \(h\) under the continuous-time assumptions.

Diagnostics, Alternatives, and Limitations

A useful diagnostic is to check whether the proposed hazard and risk indicator describe the same population and time scale. If the hazard is defined among event-free subjects but \(Y(t)\) also includes subjects who have already experienced the event, their product assigns a positive rate where no first event can occur. If delayed entry is ignored, the model can assign event risk before observation begins. If a time-varying covariate is measured only after the event, it cannot be used as a predictable covariate in the pre-event hazard.

Other warning signs include a claimed positive point mass at a fixed event time despite using a continuous hazard, simultaneous jumps that violate the assumed simple-process structure, or intervals so long that the first-order approximation \(\lambda h\) is treated as an exact probability. For a constant hazard, the exact probability is \(1-\exp(-\lambda h)\), not \(\lambda h\); the approximation is appropriate when \(h\) is small.

The formula is a representation, not a guarantee that the hazard is correctly specified or identifiable from the data. The observed process may not reveal the latent event risk after censoring. Informative censoring, unmeasured time-varying predictors, or misspecified cause-specific hazards can undermine substantive conclusions even when the risk-indicator multiplication is algebraically correct. Conversely, an isolated entry-boundary discrepancy in a pointwise expansion is not evidence that the almost-everywhere intensity representation fails.

Common Mistake

Incorrect reasoning: “At the entry time \(L\), the predictable risk indicator is zero, so the event probability over \([L,L+h)\) must be zero to first order.” This incorrectly applies a regular-time local approximation at a boundary. The value \(Y(L)=0\) describes risk immediately before \(L\); it does not say the subject remains outside the risk set throughout the interval after \(L\).

Correction: State the hazard-to-intensity relation as an almost-everywhere-in-time identity, or restrict a pointwise local expansion to times away from entry and exit boundaries. For an interval crossing entry, calculate risk over the portion after entry. The boundary value itself does not affect the time-integrated rate.