Delayed Binary Endpoints With Fixed Follow-Up

Why Fixed Follow-Up Changes the Update

A binary endpoint may be defined as whether a patient has experienced a specified outcome by a fixed time after assignment: for example, response by day 28. The endpoint is binary, but it is not known at enrollment. Until the patient reaches the end of the window and the assessment is available, the response is pending, not a success or a failure. A response-adaptive design must therefore define both the endpoint and the information available when each new allocation is made.

The fixed window gives a clear endpoint definition, but it creates a predictable delay: patients enrolled recently have had less time to complete follow-up. An allocation rule based only on outcomes already observed can be implemented without guessing those patients' responses. Its cost is that the current estimate may use only older patients. The method is credible when its outcome and observation assumptions fit the trial and when that limitation is recognized.

Endpoint, Maturity, and Allocation Information

Suppose there are \(K\) treatment arms. Patient \(j\) is assigned to arm \(A_j\) at enrollment time \(e_j\). Let \(W\) be the prespecified follow-up window, and define \(Y_j=1\) if the patient meets the binary response definition by \(e_j+W\), and \(Y_j=0\) otherwise. The endpoint definition must specify what counts as response, the assessment time or window, and how an assessment not obtained at the required time is handled. The response is not defined by whether it has already been reported.

Let \(C_j\) denote the time the required endpoint information is available to the allocation system. At decision time \(t\), patient \(j\)'s response is mature and observed if \(e_j+W\leq t\), \(C_j\leq t\), and the endpoint is recorded. Patients whose windows have not ended are structurally pending. A patient whose window has ended but whose endpoint remains unrecorded is overdue or incomplete; that is a different status and should be monitored separately.

Let \(\mathcal{F}_{j-1}\) denote the information available immediately before assigning patient \(j\). It can include previous assignments, mature endpoint values, enrollment times, and the status of pending follow-up. A mature-only allocation rule chooses probabilities \(\pi_{j,k}=P(A_j=k\mid\mathcal{F}_{j-1})\), for \(k=1,\ldots,K\), using the mature responses rather than unobserved responses. It must be implementable: the rule cannot depend on \(Y_i\) if \(Y_i\) is still unknown at that decision. A common protection against excluding an arm is to require \(\pi_{j,k}\geq\epsilon/K\) for every arm, where \(0<\epsilon<1\).

For a basic response model, assume that, conditional on assignment to arm \(k\) and any prespecified covariates used in the model, responses have a stable probability \(p_k\) of being 1 over the trial. For the simple calculation below, suppose \(Y_j\mid A_j=k\sim\operatorname{Bernoulli}(p_k)\), with conditionally independent patient responses. This working model excludes unaccounted changes in response over calendar time and dependence between patients. Random assignment probabilities are chosen from the recorded history; they do not make the unknown response of a pending patient observable.

Estimating Response and Turning Estimates Into Probabilities

For arm \(k\), let \(n_k(t)\) be the number of mature, observed endpoints and \(s_k(t)\) the number of responses among them. When \(n_k(t)>0\), the empirical estimate is \(\hat p_k(t)=s_k(t)/n_k(t)\). If an arm has no mature observations, its empirical rate is undefined; a rule must specify what to do rather than silently assign it a rate of zero.

One possible allocation illustration uses a \(\operatorname{Beta}(1,1)\) prior for each arm's response probability. Its posterior mean based on the mature outcomes is \(q_k=(s_k+1)/(n_k+2)\). For a power \(\gamma>0\), define \(r_k=q_k^\gamma/\sum_{\ell=1}^{K}q_\ell^\gamma\). A probability-floor version of the rule is \(\pi_{j,k}=(1-\epsilon)r_k+\epsilon/K\). This is one transparent example of a rule, not a uniquely required choice. Its use presumes that the response estimates are relevant to the population currently enrolling, that the endpoint data are correctly recorded, and that the allocation system uses only data available at that time.

Why Pending Outcomes Must Not Be Coded as Failures

The timing distinction can be seen directly from a fixed-window example. Suppose an update occurs at the end of calendar day 60, the window is 28 days, and assessments are available as soon as the window ends. A patient enrolled on day 30 has completed the window by day 58 and may contribute an observed endpoint. A patient enrolled on day 40 cannot complete the window before day 68. On day 60 that patient's endpoint is unknown, even if the patient has not yet shown a response. Coding the latter patient as a failure changes the response definition from “response by day 28” to a mixture of that endpoint and “response observed so far.”

A short justification clarifies how pending patients enter a response-data calculation. Under the independent Bernoulli working model, if the observed pattern is fixed or ignorable with respect to the unobserved response, the response likelihood for mature observations on arm \(k\) is

A pending patient's response has not been observed. If pending status is determined by enrollment time and the fixed window, independently of that patient's eventual \(Y_j\), summing over the two possible responses contributes \(\sum_{y=0}^{1}P(Y_j=y\mid A_j=k)=p_k+(1-p_k)=1\). It contributes no observed response to the mature-outcome likelihood. This argument does not establish that every observed-only likelihood is valid: it depends on the stated independence and observation assumptions.

In particular, a Bernoulli response model alone does not justify ignoring outcome-dependent observation. If a response is more likely to be recorded when \(Y=1\), then the response distribution among recorded cases differs from the population response distribution. For example, let \(p=P(Y=1)=0.60\), with observation probabilities \(P(R=1\mid Y=1)=0.90\) and \(P(R=1\mid Y=0)=0.50\). Then

Thus the observed cases would have a response proportion of about 0.730 under this selection mechanism, not 0.600. If observation can depend on the unobserved outcome, a valid analysis needs a model for observation and the unobserved responses, or another justified method for handling the missing data. The simple mature-outcome calculation should not be presented as the joint observed-data likelihood in that setting.

Implementation Checks and Alternatives

A protocol should specify the response definition, the window \(W\), the time at which an endpoint becomes usable, and how records arriving after that time are handled. It should also state which patients enter each update, how zero mature counts are handled, what allocation probabilities are calculated, and how minimum probabilities are enforced. Operationally, the allocation system should retain enrollment dates and distinguish “window not complete” from “window complete but endpoint unavailable.”

Useful checks include verifying that no endpoint enters an update before its window ends, that each mature response matches the endpoint definition, and that the allocation probabilities are nonnegative and sum to 1. Review mature sample sizes by arm and the number and age of overdue records. If recent enrollment has a different case mix or response probability from older enrollment, an older mature sample may poorly represent current patients. Check for calendar-time changes, arm-specific follow-up completion, and patterns of missing assessments; do not assume these are harmless just because the endpoint window is fixed.

Mature-only adaptation is one option among several. Waiting for a cohort or a specified number of endpoints reduces the frequency of updates but can delay enrollment or produce periods with no new information. Predictive imputation can use pending patients in an allocation calculation, but it adds model assumptions and must distinguish a prediction from an observed response. Mature-only updating avoids imputing pending responses, at the cost of slower feedback. These alternatives do not change the endpoint's definition, and none makes an informative observation process ignorable.

The fixed-window setup also has limits. It does not by itself guarantee stable response probabilities, equal outcome ascertainment across arms, or absence of calendar trends. Nor does a response-adaptive allocation probability determine the final treatment-effect analysis: the trial's analysis must account for its design and the endpoint's observation process. The main practical safeguard is to keep the response definition, maturity rule, and allocation information set aligned.