Worked Example: Time-to-Event Response Adaptation

Why follow-up matters at an interim

A time-to-event outcome is not summarized adequately by counting how many participants have had an event by the interim. Participants who have not yet had an event may have very different amounts of follow-up: one might have been observed for two months and another for ten. Both are event-free at the interim, but they provide different amounts of information about the event rate. A participant whose follow-up ends without an event is right-censored; that participant contributes observed event-free time, not an observed event.

This worked example constructs an allocation update at a prespecified interim using event counts and total follow-up. To keep the calculation transparent, it uses a constant-hazard exponential model and a Gamma prior. The chosen adaptation rule rewards arms with greater posterior predictive event-free survival at a specified horizon. That is appropriate when the event is undesirable. If the event is a desirable response, the score must instead be defined to reward the clinically desirable outcome, such as the probability of response by a horizon.

Model and allocation rule

For each treatment arm \(j\), suppose event times have an exponential distribution with constant hazard \(\lambda_j>0\). The hazard is the instantaneous event rate among participants who have not yet experienced the event; its units are events per unit of time. The model assumes independent participants within an arm, a common hazard for participants in that arm, and independent, non-informative censoring conditional on the model. In particular, censoring times must not selectively reveal information about a participant's event risk that is absent from the model. The exponential assumption also rules out a changing hazard over time.

Use a Gamma prior with shape \(a_j\) and rate \(b_j\), written \(\lambda_j\sim\operatorname{Gamma}(a_j,b_j)\). The rate parameterization has density proportional to \(\lambda_j^{a_j-1}e^{-b_j\lambda_j}\); \(b_j\) has units of time. With \(d_j\) observed events and \(T_j\) total observed follow-up time, the posterior is \(\operatorname{Gamma}(a_j+d_j,b_j+T_j)\). Censored observations enter \(T_j\) in exactly the same way as event observations, but only event observations enter \(d_j\).

Let \(U_j(\tau)\) be the posterior predictive probability that a new participant assigned to arm \(j\) remains event-free through horizon \(\tau\). Conditional on \(\lambda_j\), this probability is \(e^{-\lambda_j\tau}\). Averaging over the posterior gives a predictive score. In this example, allocation uses a prespecified exploration mixture: with probability \(1-\epsilon\), use scores proportionally, and with probability \(\epsilon\), allocate uniformly across the \(K\) available arms. This guards against assigning zero probability to an arm.

The score is a predictive survival probability, not the posterior probability that an arm is best. The allocation rule is a design choice, not a universal consequence of Bayesian updating. Its horizon, exploration fraction, model, and clinical interpretation should be chosen before using interim outcomes to change assignment.

Worked allocation update

Why the posterior update has this form

The likelihood contribution can be seen directly. An event observed at time \(t\) has exponential density \(\lambda e^{-\lambda t}\). A participant censored after \(c\) units contributes the probability of remaining event-free through \(c\), namely \(e^{-\lambda c}\). Multiplying the contributions over participants gives one factor of \(\lambda\) for each observed event and an exponential involving the sum of all observed times. Hence \(L(\lambda)\propto\lambda^d e^{-\lambda T}\). Multiplying this by the Gamma prior density yields a kernel proportional to \(\lambda^{a+d-1}e^{-(b+T)\lambda}\), the kernel of \(\operatorname{Gamma}(a+d,b+T)\).

To obtain the predictive formula, integrate the conditional survival probability against that posterior. A Gamma distribution with shape \(a+d\) and rate \(b+T\) has Laplace transform \(E(e^{-s\lambda})=((b+T)/(b+T+s))^{a+d}\) for \(s\geq0\). Setting \(s=\tau\) gives \(U(\tau)\). This derivation shows precisely where censoring enters: it increases \(T\), and therefore updates the posterior rate, while leaving the posterior shape unchanged unless an event is observed.

Worked Example: What a Censored Participant Contributes

Worked Example: Changing the Prediction Horizon

Checks, alternatives, and limitations

Before acting on an update, inspect the data and the model rather than treating the computed probabilities as self-validating. Confirm that the event definition and time origin are the same across arms, that time units are consistent, and that person-time is summed once per participant. Check that each event increments \(d_j\) exactly once and that each participant without an event contributes only the follow-up actually observed. A negative exposure, an impossible event time, or a sum that excludes censored participants indicates a data-processing error.

The exponential model is convenient, but its constant-hazard assumption may be implausible when event risk changes over time. Plotting event-time patterns or comparing observed rates across follow-up intervals can reveal warning signs, although sparse interim data may not diagnose the shape reliably. Informative censoring, major differences in case mix across arms, or time-varying hazards can also undermine the interpretation of the simple posterior scores. If these concerns are material, the model and allocation rule need a more suitable prespecified survival model; the calculation here should not be applied unchanged.

This method differs from several common alternatives. Equal randomization does not use interim outcomes and can be preferable when simplicity or a stable concurrent comparison is paramount. Allocating in proportion to raw event-free fractions ignores unequal follow-up and treats a short censoring time like a long one. Using the observed event rate \(d_j/T_j\) as a plug-in score does use exposure, but it does not average predictive survival over uncertainty in the hazard. The Gamma-exponential rule performs that averaging explicitly. None of these allocation choices alone establishes treatment benefit or replaces the trial's planned clinical and inferential analyses.

A common mistake is to count every participant without an event as an event-free success and ignore how long that participant was observed. The correction is to record that participant as censored at the observed follow-up time: add that time to \(T_j\), but do not add one to \(d_j\). Another mistake is to interpret \(\pi_A\approx0.5043\) as a 50.43% posterior probability that A is the better treatment. It is instead the probability of assignment to A under the specified randomized allocation rule.