Delayed Outcomes In Joint Dose-Finding

Why Pending Outcomes Matter

In dose-finding trials, participants may be assigned a dose before their efficacy and toxicity outcomes can be fully assessed. At an interim analysis, one participant may have completed both assessment windows, another may have completed only the toxicity window, and a recently enrolled participant may have had neither outcome yet. Treating every pending outcome as if it were absent discards information; treating it as a completed event-free assessment overstates the evidence for safety or lack of efficacy.

The relevant information is the time each participant has been observed without an event, as well as any event time already observed. A time-to-event likelihood uses both. It allows a dose assignment decision to use partial follow-up while distinguishing a participant who has had no event after one week from one who remained event-free through a full assessment window.

Likelihood Contributions With Unequal Follow-Up

Suppose participants assigned to dose \(d\) have an outcome time with density \(f_d(t)\) and survival function \(S_d(t)=P(T>t\mid d)\). If an event is observed at time \(y\), that observation contributes the event-time density \(f_d(y)\). If no event has been observed by time \(y\), the contribution is the probability of remaining event-free through \(y\), namely \(S_d(y)\). Thus, for one outcome, the likelihood contribution is

This expression also covers completed windows: if a participant is event-free at the end of the window, \(Y_{ik}=W_k\) and the contribution is \(S_d(W_k)\). A participant whose follow-up ends earlier contributes survival only through that shorter time. If efficacy and toxicity times are conditionally independent given dose and model parameters, their contributions multiply. If the model allows dependence, the appropriate joint time-to-event likelihood must be used instead; multiplying marginal likelihoods would then generally be incorrect.

For a transparent working model, suppose each outcome has a dose-specific exponential time distribution. Write \(\lambda_{dk}\) for its event rate per unit time. The model assumes a constant event hazard over the relevant window, so \(f_d(t)=\lambda_{dk}\exp(-\lambda_{dk}t)\) and \(S_d(t)=\exp(-\lambda_{dk}t)\). Substitution gives

For multiple participants at dose \(d\), define \(D_{dk}=\sum_i\Delta_{ik}\), the number of observed events, and \(A_{dk}=\sum_iY_{ik}\), the total observed time at risk, capped at the outcome-specific window. The likelihood becomes

The unequal follow-up times enter through \(A_{dk}\): someone observed event-free for one week contributes one week, not a full assessment window. Efficacy and toxicity have separate \(Y_{ik}\), \(W_k\), event counts, and exposure totals when their assessment schedules differ.

Updating Dose Estimates With Partial Follow-Up

A dose-finding procedure can use these likelihood contributions to update its dose-specific event-time estimates at each interim analysis. One convenient illustration is a Bayesian exponential model. For a particular dose and outcome, place a Gamma prior on the event rate, using a shape \(a\) and rate \(b\): \(\lambda\sim\mathrm{Gamma}(a,b)\). Here \(b\) is expressed in the same time units as the follow-up. Combining this prior with the likelihood above gives a Gamma posterior. The derivation follows directly by multiplying the prior density by the likelihood:

This is the kernel of \(\mathrm{Gamma}(a+D,b+A)\). Observed events increase the shape, while all observed follow-up, including time from participants whose outcomes are pending, increases the rate parameter. Under this model the posterior mean event rate is \((a+D)/(b+A)\). The formula is useful only under the exponential model and the stated prior; another time-to-event model would require its own likelihood and update.

From Updated Event Times to an Interim Dose Assignment

An event rate is not itself a dose decision. A design must connect the updated model to the trial's prespecified clinical priorities, such as the importance of efficacy relative to toxicity and the time windows in which those outcomes matter. Under an exponential model, a rate \(\lambda\) implies a probability \(1-\exp(-\lambda W)\) of observing the event by window \(W\). For efficacy, a higher event probability may be desirable; for toxicity, it may be harmful. Different windows matter: comparing rates alone can conceal the probability of events during the actual clinical assessment periods.

The following calculation illustrates the translation. It uses fixed rates to make the arithmetic transparent; in a Bayesian design, uncertainty in those rates should also be incorporated into the decision quantity.

Assumptions, Diagnostics, and Alternatives

The likelihood approach depends on a clear event definition, reliable event times, and prespecified assessment windows. Censoring must be noninformative for the event-time model: conditional on variables included in the model, the reason follow-up ends must not convey additional information about the unobserved event time. Administrative censoring at an interim cutoff is often compatible with this assumption when enrollment timing is unrelated to participants' outcomes. Loss to follow-up related to prognosis may not be. The exponential version adds a constant hazard assumption within each modeled dose and outcome. It should not be adopted merely for computational convenience if risk is expected to change substantially over time.

At interim looks, useful checks include tabulating event counts and accumulated follow-up separately by dose and outcome; checking that each participant's time is capped at the correct window; comparing enrollment and dropout patterns across doses; and examining whether observed event timing is consistent with the chosen time-to-event model. A dose with little follow-up can have a highly uncertain event rate even if no events have yet occurred. That uncertainty should be visible in the decision calculation, not hidden by a point estimate. If event hazards vary over time, alternative survival models can represent that pattern. If censoring is plausibly informative, sensitivity analyses or an explicit model for the dropout process may be needed.

Complete-case analysis waits for full assessment and avoids using partial follow-up, but can delay decisions and exclude much of the available information. A simple binary analysis that labels pending participants as event-free is faster but biased toward underestimating event risk when follow-up is incomplete. Some time-to-event dose-finding procedures use a weighting approximation for partially observed participants; a full survival likelihood instead uses the observed event time or survival time under its specified model. The likelihood approach is more explicit about exposure duration and windows, but its validity depends more directly on correct event-time and censoring assumptions.

A frequent mistake is to say, “No toxicity has been reported, so this participant is a toxicity-free observation.” If the participant has been followed for only one of a four-week toxicity window, the correct contribution is survival through one week, not survival through four weeks. Another mistake is to multiply efficacy and toxicity marginal contributions when the fitted model allows dependence. The correction is to use the joint likelihood implied by that model. These distinctions matter because early assignment decisions can otherwise look more reassuring than the available follow-up warrants.