Event Probability Over a Follow-Up Interval

From Conditional Survival to an Interval Probability

A hazard describes the instantaneous event rate among subjects who have remained event-free up to a given time. It is not itself the probability of an event over a follow-up interval. To answer an interval question, first obtain survival probabilities at the interval’s endpoints, conditional on the subject’s random effects, and then take their difference.

This distinction matters in a shared random-effects model because the event hazard, and therefore survival, depends on the subject’s random effects. Holding those effects fixed gives a subject-specific probability. It does not give a population-average probability over the distribution of random effects.

These are two different probability statements. The first is conditional on the random effects alone; it includes both subjects who fail before \(a\) (which are not counted in the interval) and subjects who survive to \(a\). The second additionally conditions on being event-free at \(a\). Make that distinction explicit whenever reporting an interval probability.

Why the Difference of Survival Probabilities Works

Suppose the conditional event-time distribution is continuous, as it is under a finite, integrable hazard model without point masses. The event \(\{a<T_i\leq b\}\) is the part of \(\{T_i>a\}\) that is removed by time \(b\). Consequently,

The same result follows by integrating the conditional event-time density. For a continuous hazard, that density is \(f_i(t\mid b_i)=\lambda_i(t\mid b_i)S_i(t\mid b_i)\). Since \(dS_i(t\mid b_i)/dt=-f_i(t\mid b_i)\), integration gives

For the probability conditional on surviving to \(a\), divide the first interval probability by \(P(T_i>a\mid b_i)=S_i(a\mid b_i)\). This gives \(P(a<T_i\leq b\mid T_i>a,b_i)=1-S_i(b\mid b_i)/S_i(a\mid b_i)\). This conditional probability is often useful for a follow-up question that starts at a landmark time, but it is not interchangeable with the probability from time zero.

The calculation assumes the conditional hazard specifies the event process over the entire interval, and that its cumulative hazard is well-defined. Random effects are held fixed throughout. If the observed data are censored, censoring affects what is observed about \(T_i\), not these probability identities; estimating the model from data requires additional assumptions about the observation and censoring processes.

Using Cumulative Hazard When the Hazard Changes

A hazard need not be constant over time. In that case, the same survival-difference calculation applies, but the cumulative hazard must account for each part of the interval. Integrating the hazard is essential; multiplying the hazard at one time by the interval length is only valid when the hazard is constant over that interval.

Substituting a Subject’s Random Effects

In a shared random-effects model, the hazard may depend on a latent longitudinal trajectory, which in turn depends on \(b_i\). Once the conditional hazard is specified, the interval calculation itself is unchanged: evaluate the cumulative hazard at both endpoints with the same \(b_i\), convert to survival, and subtract. The next example isolates this substitution step; it does not require treating noisy observed marker values as the hazard covariate.

Checks, Alternatives, and Limitations

A reliable calculation has several quick checks. First, cumulative hazard must be nondecreasing, so \(H_i(b\mid b_i)\geq H_i(a\mid b_i)\); hence survival must satisfy \(S_i(b\mid b_i)\leq S_i(a\mid b_i)\). The difference is therefore nonnegative and no larger than \(S_i(a\mid b_i)\). Second, all hazards and cumulative hazards must use compatible time units: if time is measured in years, a hazard stated per month must be converted or integrated on a consistent scale. Third, the conditional-survival ratio must use the same \(b_i\) at both endpoints.

The survival difference is usually the most direct approach when conditional survival is already available. Integrating \(\lambda_i(t\mid b_i)S_i(t\mid b_i)\) over the interval is an equivalent route and can be useful for numerical integration or checking a derivation. For a discrete-time event model with possible probability mass at particular times, endpoint conventions must be handled according to that model; the continuous-hazard density argument above does not apply unchanged. Under the continuous model used here, an event exactly at a fixed endpoint has probability zero.

A common mistake is to report \(1-S_i(b\mid b_i)\) as the probability of an event in \((a,b]\). That quantity is the probability of an event by \(b\), including events before \(a\). The correction is to subtract survival at the interval’s start: \(S_i(a\mid b_i)-S_i(b\mid b_i)\). A related mistake is to divide by \(S_i(a\mid b_i)\) without saying that the probability is now conditional on survival to \(a\).

The calculation is only as credible as the conditional hazard used to obtain \(H_i\). A misspecified time pattern, an invalid extrapolation of the hazard, or a model that does not represent relevant heterogeneity can produce misleading interval probabilities even when the subtraction is performed correctly. Likewise, an estimated conditional probability has uncertainty from parameter estimation and, when random effects are estimated, from uncertainty about those effects. A plug-in value alone does not express that uncertainty.