Beta-Binomial Predictive Success for Binary Outcomes

Why Use a Beta-Binomial Calculation?

At an interim analysis of a trial with a binary endpoint, the observed responses do not determine whether the trial will ultimately meet its go criterion. The remaining participants may have different numbers of responses, and each possible final count produces a different posterior distribution. Beta-binomial predictive success calculates the probability, given the interim data and model, that the final posterior decision will be “go.”

The calculation has two linked parts: first, determine which future response counts would satisfy the final posterior rule; second, calculate the predictive probability of those counts and add them. This tutorial specializes predictive averaging to a binary response rate with a beta posterior. It is not the current probability that the response rate exceeds a benchmark: it is the probability, conditional on the current data and model, that future data will make the final rule pass.

Model and Final Go Rule

Let \(p\) be the probability of a binary response for a participant in the population of interest. Suppose the prior distribution is \(p\sim\operatorname{Beta}(a_0,b_0)\), where \(a_0,b_0>0\). Conditional on \(p\), responses are independent Bernoulli outcomes with the same response probability \(p\). At the interim analysis, \(x\) responses have been observed among \(n\) participants. Conjugacy gives the interim posterior \(p\mid\text{interim data}\sim\operatorname{Beta}(a,b)\), where \(a=a_0+x\) and \(b=b_0+n-x\).

Suppose \(m\) responses remain to be observed. Let \(Y\) denote the number of responses among those remaining participants. Conditional on \(p\), \(Y\) has a binomial distribution. The final analysis updates the interim posterior using the future count \(Y=y\). If the final go rule compares the response rate with a benchmark \(p_B\), it may be written as: go if the posterior probability that \(p>p_B\) is at least \(\gamma\), where \(0<\gamma<1\) is fixed in advance.

This definition assumes that the endpoint is recorded as a binary response, the remaining participants share the response probability \(p\) with those already observed, and responses are conditionally independent given \(p\). It also assumes the beta prior and the go rule are appropriate for the design question. The binomial model does not automatically account for clustering, changing response probabilities over time, informative missingness, or different response probabilities across treatment groups.

Deriving the Beta-Binomial Predictive Distribution

The predictive probability of a future count is obtained by averaging its binomial probability over the interim posterior for \(p\). The beta density is \(p^{a-1}(1-p)^{b-1}/B(a,b)\), where \(B(a,b)\) is the beta function. Therefore:

The last equality follows because the integrand contains \(p^{a+y-1}(1-p)^{b+m-y-1}\), whose integral from 0 to 1 is \(B(a+y,b+m-y)\). This is the beta-binomial predictive distribution. Its probabilities sum to one because they were obtained by integrating a binomial distribution that sums to one for every \(p\):

For each possible \(y\), the final posterior tail probability is the beta-distribution tail \(P(p>p_B\mid Y=y,\text{interim data})\). Calculate that tail, compare it with \(\gamma\), and include the predictive probability for \(y\) in the sum only if it passes. This enumeration is exact up to numerical evaluation of beta functions and beta tail probabilities.

Worked Calculations

Interpretation, Checks, and Failure Modes

A predictive success probability of 0.3000 means that, under the specified model and conditional on the interim data, the probability that future responses will lead the final posterior rule to say go is 30%. It does not mean that the current posterior probability of benefit is 30%, nor that the probability the response rate exceeds the benchmark is 30%. Those are different probabilities about different events.

A reliable calculation keeps three objects separate: the final posterior tail for each hypothetical future count, the predictive probability of that count, and the final sum over counts that pass. Common errors include reversing a beta tail, using the interim posterior tail as if it were predictive success, and including counts that have substantial predictive probability but fail the posterior threshold. In the first worked example, \(P(p>0.5)\) for \(\operatorname{Beta}(4,2)\) is 0.8125, not 0.6875; and the count \(y=1\) has predictive mass 0.4000 but does not meet the 0.65 final threshold. Therefore predictive success there is 0.3000, not 0.7000.

Useful diagnostics include verifying that the predictive masses are nonnegative and sum to one, confirming that the final posterior parameters add the future successes to \(a\) and future failures to \(b\), and checking that the qualifying set behaves sensibly as the go threshold changes. Raising \(\gamma\) cannot add a qualifying count. As another check, if no future count passes, predictive success is zero; if every possible count passes, it is one.

Exact enumeration is particularly convenient because the future count has only \(m+1\) possible values. A plug-in calculation that fixes \(p\) at a point estimate omits uncertainty about \(p\) and generally gives a different predictive distribution. Simulation can approximate the same posterior predictive quantity in more complex settings, but for this beta-binomial case enumeration avoids simulation error. Neither exact calculation nor simulation protects against a misspecified model: dependence among participants, substantial heterogeneity in response probabilities, or changes in response rates over time may make the common-\(p\) binomial assumption inappropriate. The result is also conditional on the chosen prior, benchmark, and posterior threshold; it does not by itself establish clinical value or determine whether the decision rule is well calibrated for a broader operating objective.