Binary-Endpoint Posterior Go Rules Against a Benchmark

From a Clinical Benchmark to a Go Rule

For a binary endpoint, each participant either meets the response definition or does not. A trial may ask whether the treatment's response rate is not merely above a historical or control benchmark, but above that benchmark by an amount considered clinically meaningful. A posterior go rule makes this question operational: calculate the posterior probability that the response rate exceeds the clinically meaningful threshold, then compare that probability with a prespecified cutoff.

This is a rule for a particular decision, not a claim that the treatment causes a response or that the benchmark is itself known without uncertainty. The benchmark is treated as fixed here. If it is estimated from data, its uncertainty would need to be represented rather than silently ignored.

The inequality is strict in the event of interest: the posterior probability concerns rates greater than \(p_*\), not rates equal to it. Under a continuous prior and posterior, a single point has probability zero, so replacing \(>\) with \(\geq\) in the event would not change the probability. The operational comparison with \(\gamma\), however, should be specified in advance, including whether equality meets the rule.

Model and Posterior Probability

Suppose \(n\) participants have evaluable binary outcomes, with \(x\) responses. The basic single-rate model assumes that the outcomes are independent Bernoulli observations with a common response probability \(p\). Consequently, \(X\mid p\sim\operatorname{Binomial}(n,p)\). A beta prior, \(p\sim\operatorname{Beta}(a,b)\), has density proportional to \(p^{a-1}(1-p)^{b-1}\) on \(0<p<1\), where \(a,b>0\). This model assumes that the endpoint and response assessment are defined consistently, and that the observed outcomes can reasonably be treated as independent with a shared response probability.

Combining the binomial likelihood with the beta prior gives the posterior \(p\mid x\sim\operatorname{Beta}(a+x,b+n-x)\). The probability needed by the rule is the posterior upper-tail probability at the clinical threshold:

where \(I_z(r,s)\) is the cumulative distribution function of a \(\operatorname{Beta}(r,s)\) random variable evaluated at \(z\). The go decision is \(q\geq\gamma\); otherwise the rule gives no-go. The decision is conditional on the data, model, prior, threshold, and cutoff. It is not a probability that the treatment is effective in a frequentist sense, nor does it make the clinical threshold or model assumptions certain.

A useful exact identity is available for a uniform prior, \(p\sim\operatorname{Beta}(1,1)\). The posterior is then \(\operatorname{Beta}(x+1,n-x+1)\), and its upper-tail probability can be written as a binomial cumulative probability:

To justify the identity, let \(F(p)=\sum_{k=0}^{x}{n+1\choose k}p^k(1-p)^{n+1-k}\). Differentiating and cancelling adjacent terms gives \(F'(p)=-p^x(1-p)^{n-x}/B(x+1,n-x+1)\), the negative beta posterior density. Since \(F(1)=0\), integration of that density from \(p_*\) to 1 yields \(F(p_*)\). Thus, the sum uses \(n+1\) trials and includes outcomes \(k=0,\ldots,x\). This identity is a calculation shortcut for the uniform-prior case, not a general replacement for evaluating the beta posterior tail under other priors.

Prespecification, Checks, and Alternatives

The rule is only interpretable if the ingredients are chosen and reported clearly. State the endpoint's response definition, the analysis population and handling of missing outcomes, the benchmark, the absolute margin, the prior, and the cutoff. The threshold must be a plausible probability: if \(p_B+\Delta>1\), the requested improvement cannot occur on the response-rate scale and the rule needs reconsideration. Also clarify whether the benchmark represents a relevant comparator population and whether differences in eligibility, follow-up, or response assessment make it misleading.

The common-binomial model can fail if response probabilities vary substantially across sites or participants, observations are dependent, or outcome ascertainment differs between participants. Loss to follow-up can also distort the analysis if missing responses are simply excluded without justification. Review response definitions, follow-up completeness, site patterns, and the plausibility of a common response rate. If the design or data call for a richer model, do not treat a simple beta-binomial calculation as automatically adequate. Prior sensitivity is also important, particularly with a small sample: compare the decision under reasonable prespecified priors and report when the go conclusion changes.

A posterior go rule differs from simply checking whether the observed fraction \(x/n\) exceeds \(p_*\). The observed fraction ignores uncertainty and can cross the threshold in a small trial without producing a high posterior probability. Conversely, a large sample can give strong posterior evidence of exceeding a threshold even when the observed fraction is only modestly above it. A posterior credible interval criterion is another possible decision convention, but it is not interchangeable with a tail-probability rule unless its endpoint and probability cutoff are aligned exactly. A classical hypothesis test answers a different question using its own error-rate calibration; the posterior cutoff \(\gamma\) is not automatically a frequentist significance level.

A go result is evidence relative to the chosen benchmark and threshold, not proof of benefit, safety, or generalizability. A no-go result likewise need not establish ineffectiveness; uncertainty may remain substantial. These rules compress a posterior distribution into an action, so the reported probability and assumptions should accompany the action rather than be hidden behind the word “go.”