Defining Terminal Success for Binary Response

Why Define Terminal Success First?

In a single-arm trial, interim monitoring asks how likely it is that the trial will eventually meet its prespecified success criterion. That question is only well-defined after the criterion is stated precisely. If the final decision is based on a posterior probability exceeding a threshold, the terminal event is not “the posterior probability at the interim look is high.” It is the event that the final data will produce a posterior probability meeting the prespecified rule.

This distinction determines what quantity is monitored. The final criterion specifies which possible final response totals count as success. At an interim look, the predictive probability of terminal success is the probability, given the current data and model, that the eventual total will fall in that set. Deriving the predictive distribution is a separate calculation; here the aim is to define the set exactly.

Model and Posterior Criterion

Suppose each participant has a binary response, defined using the same clinical endpoint and assessment window throughout the trial. Let \(p\) be the response probability. Assume responses are conditionally independent Bernoulli observations with common probability \(p\), and use a Beta prior \(p\sim\operatorname{Beta}(a,b)\), where \(a>0\) and \(b>0\). These assumptions give a posterior distribution after any observed number of responses.

If \(r\) responders are observed among \(n\) participants, the likelihood is proportional to \(p^r(1-p)^{n-r}\), so the posterior distribution is \(\operatorname{Beta}(a+r,b+n-r)\). A common final rule compares the posterior probability that \(p\) exceeds a clinically chosen benchmark \(p_0\) with a prespecified threshold \(\gamma\), where \(0<p_0<1\) and \(0<\gamma<1\).

Here \(I_{p_0}(u,v)\) is the Beta cumulative distribution function evaluated at \(p_0\) for a \(\operatorname{Beta}(u,v)\) random variable. Under a strict rule, terminal success means \(q_N(R_N)>\gamma\). Under an inclusive rule, it means \(q_N(R_N)\geq\gamma\). Those rules can differ when a possible final count gives posterior probability exactly \(\gamma\), so the inequality must be specified rather than left implicit.

For fixed \(N\), the posterior probability \(q_N(r)\) increases as \(r\) increases. One justification is that moving from \(r\) to \(r+1\) changes the posterior density by a factor proportional to \(p/(1-p)\), an increasing function of \(p\). This shifts posterior mass toward larger values of \(p\), and therefore cannot decrease the probability that \(p>p_0\). Consequently, the qualifying final counts form an upper tail of the possible counts.

A Useful Identity for a Uniform Prior

When the prior is uniform, \(a=b=1\), the posterior probability can be evaluated through a binomial tail. For \(r\) responses among \(N\) participants, the posterior is \(\operatorname{Beta}(r+1,N-r+1)\). The identity below lets us verify a cutoff using a finite sum rather than a Beta integral.

To see why, the Beta density with integer parameters gives \(I_{p_0}(r+1,N-r+1)=\sum_{j=r+1}^{N+1}\binom{N+1}{j}p_0^j(1-p_0)^{N+1-j}\). This is \(P(Z\geq r+1)\). Taking its complement yields \(P(Z\leq r)\), as stated. The identity is for evaluating the posterior probability under the uniform prior; it is not itself the predictive distribution of future trial responses.

Translate the Final Event to an Interim State

At an interim analysis, let \(m\) participants have observed binary outcomes and let \(x\) of them be responders. If the final sample size remains \(N\), then \(N-m\) outcomes remain. Write \(Y\) for the number of responders among those remaining participants. The final total is \(x+Y\), so a terminal event \(R_N\geq c\) becomes a condition on \(Y\).

Because \(Y\) is an integer between 0 and \(N-m\), this translation also identifies the boundary cases. If \(c-x\leq 0\), the observed responses already meet the terminal count criterion, so every possible completion is a terminal success. If \(c-x>N-m\), even a response from every remaining participant cannot meet the criterion, so terminal success is impossible. Otherwise, success requires at least \(c-x\) responses among the remaining participants.

Assumptions, Checks, and Alternatives

The translation depends on the final rule and data structure being held fixed. In particular, it assumes a prespecified final sample size, a binary response defined consistently for each participant, a known benchmark \(p_0\), and a specified prior and posterior criterion. The binomial model further assumes conditionally independent responses with a common response probability. If outcomes are clustered, response probabilities vary across participants, or the endpoint definition changes during the trial, the stated posterior and resulting cutoff may not represent the intended design.

Before using a cutoff, check the posterior probability at the cutoff and at the immediately lower count. For a strict rule, verify \(q_N(c)>\gamma\) and, when \(c>0\), \(q_N(c-1)\leq\gamma\). For an inclusive rule, verify \(q_N(c)\geq\gamma\) and \(q_N(c-1)<\gamma\). These adjacent-count checks catch off-by-one errors and confirm that the inequality matches the protocol. Also inspect whether \(c\) is feasible: \(c>N\) means no possible final count succeeds, while \(c\leq0\) means all possible counts satisfy a count-form event.

A frequent error is to monitor the interim posterior probability \(P(p>p_0\mid\text{data observed so far})\) as though it were the probability of eventual success. These are different quantities. The former concerns the unknown response rate given current data; the latter concerns whether future data will produce a final result meeting the terminal rule. The correction is to first define the terminal event from the final rule, then calculate its predictive probability under the model.

Another alternative is a direct frequentist count rule, such as declaring success if \(R_N\geq c\), with \(c\) selected to control operating characteristics under specified response rates. That too produces a terminal count event, but its justification differs from a posterior-probability criterion. A rule based on the final posterior mean exceeding a target is also not interchangeable with a rule based on \(P(p>p_0\mid\text{data})\); the same posterior mean can accompany different posterior probabilities, depending on the posterior distribution’s spread and the benchmark’s location.

The posterior cutoff is sensitive to the prior, benchmark, posterior-probability threshold, and planned sample size. These inputs should be clinically and statistically justified before monitoring begins. The event’s definition alone does not show that the design has adequate power, acceptable error rates, or a clinically meaningful endpoint. Nor does crossing the event establish a causal treatment effect: in a single-arm design, the inference is about the response probability under the model and its comparison with the chosen benchmark.