Likelihood Factorization Across Competing Event Types

Why Factor a Likelihood by Event Type?

When a subject can experience one of several competing terminal events, each event type has its own intensity while the subject remains at risk. The observed path records both how long the subject remained event-free and which type of event occurred, if any. A likelihood built from type-specific intensities makes both contributions explicit.

The resulting expression is a product over event types. That is an algebraic factorization, not an assertion that the event types are independent. All types share the same risk process: an event of any type ends follow-up for a terminal competing-risks outcome. The type-specific intensities may also depend on common covariates, shared parameters, or the observed history.

Assume event types are mutually exclusive for a subject: at most one type can occur at any instant, and the first event of any type ends the at-risk period. Conditional on the observed history, the probability of one type-\(k\) event in a short interval of width \(dt\) is \(\lambda_k(t\mid\mathcal{F}_{t-};\theta)dt+o(dt)\); the probability of no event of any type is \(1-\sum_k\lambda_k(t\mid\mathcal{F}_{t-};\theta)dt+o(dt)\). These statements require finite integrated intensities over the observation period and intensities predictable from the history just before each time.

The Joint Path Likelihood

For an observed path through time \(\tau\), let \(dN_k(t)\) denote the increment in the type-\(k\) count at time \(t\). Multiplying the conditional contributions from event times and event-free time gives the path likelihood, up to a factor that does not depend on \(\theta\):

Equivalently, the exponential term is a product of type-specific terms, so the likelihood can be written as

The product over event times supplies a factor for every observed event, while the integrated intensities account for the absence of events over the full at-risk time. With \(\lambda_k=Y\alpha_k\), where \(\alpha_k\) is the event-type rate while at risk, the integral automatically stops contributing after the subject leaves the risk set. For terminal competing risks, an event of any type makes \(Y=0\), so all types share the same stopping time.

Why the Formula Holds

A short-interval argument shows how the event and no-event contributions combine. Suppose there is no event in an interval \([t,t+dt)\). Its conditional likelihood contribution is \(1-\sum_k\lambda_k(t)dt+o(dt)\). If instead a type-\(j\) event occurs, its contribution is \(\lambda_j(t)dt+o(dt)\). Across intervals with no event, multiplying the no-event contributions and taking the limit gives the exponential of minus the integrated total intensity. At each observed event time, the appropriate type-specific intensity supplies the event factor.

At a type-\(j\) event time \(t\), the event contribution is \(\lambda_j(t)dt\); in a continuous-time path density, the interval-width terms are part of the reference measure, leaving the factor \(\lambda_j(t)\). Multiplying these event factors by the event-free contribution yields the displayed likelihood. This argument presumes a simple process with no simultaneous jumps of distinct types. If simultaneous events can occur, their joint mechanism needs its own intensity or event contribution; the formula above does not describe it.

Worked Event-Time Likelihoods

Assumptions, Censoring, and Diagnostics

The likelihood is appropriate when the recorded event types are exhaustive for the outcomes being modeled, event times and types are observed accurately, and the stated intensities describe the conditional event rates given the available history. The at-risk process must be defined consistently and be predictable. For right-censored paths, follow-up contributes event-free exposure through the censoring time. Censoring can be omitted from the event likelihood when its mechanism is independent of future event occurrence conditional on the modeled history, or when a separate censoring model is handled appropriately. Informative censoring requires additional modeling or a valid adjustment; simply stopping the integral at censoring does not remove the bias.

Useful checks include verifying that each observed event has exactly one type, that every event occurs while the subject is at risk, and that the recorded risk time ends at the first competing event for terminal outcomes. Compare observed event counts with the fitted type-specific intensity contributions, and inspect whether fitted rates vary systematically with time or recorded covariates in ways the model does not capture. Zero fitted intensity at an observed event makes the likelihood zero, signaling a model-data incompatibility. Unmodeled dependence through shared latent factors can also make a conditional-intensity specification inadequate; dependence should be represented through the history or an appropriate joint model rather than dismissed because the expression factors.

A common mistake is to interpret \(\prod_k L_k(\theta)\) as evidence that the competing event processes are independent, or to give each type a separate risk period that continues after another type has occurred. The correction is to use one shared at-risk history, stop risk time at the first terminal event, and interpret the product as the factorization of a single joint path likelihood. Independence assumptions, if needed for a different model or sampling design, must be justified separately.