Worked Beta-Binomial Monitoring Calculations

From Interim Data to a Terminal Target

In a single-arm trial with a binary endpoint, interim monitoring often asks a concrete planning question: given the responses observed so far, how likely is the trial to finish with enough responses to meet its prespecified terminal success rule? The answer is a predictive probability. It averages over uncertainty about the response rate and the outcomes that have not yet been observed.

This tutorial works through that calculation for several hypothetical trials. The terminal success rule is held fixed within each example; the focus is translating the rule into a future-count tail, computing that tail, and interpreting it. A predictive probability is not, by itself, a stopping instruction. A protocol needs a separate decision criterion to turn a calculated probability into an action.

The calculation assumes that each participant has a binary response outcome, the response probability is constant across participants, and outcomes are conditionally independent given \(\theta\). The prior is Beta, the total sample size and terminal response requirement are fixed, and the \(n\) interim outcomes are observed without unresolved outcomes. These assumptions make the remaining outcomes exchangeable after integrating over uncertainty in \(\theta\). If response probabilities differ systematically by time, subgroup, or treatment exposure, this simple model may not be adequate.

The Calculation in a Usable Form

After observing \(x\) responses in \(n\) participants, the posterior distribution is \(\theta\mid x,n\sim\mathrm{Beta}(A,B)\), where \(A=a+x\) and \(B=b+n-x\). Conditional on \(\theta\), the future count has a binomial distribution with \(M\) trials. Integrating that binomial probability over the posterior gives the Beta-Binomial predictive mass:

Here \(\mathrm{B}(\cdot,\cdot)\) is the beta function. The predictive probability of terminal success is the sum of the masses for all future counts that meet the rule. Importantly, the count must be within the possible range from zero to \(M\).

When the required future count lies inside the possible range, it can be convenient to start with the probability of zero future responses and build the other masses by recurrence. If \(p_y=\Pr(Y=y\mid x,n)\), then

The recurrence follows by dividing the Beta-Binomial mass at \(y+1\) by the mass at \(y\), using \(\binom{M}{y+1}/\binom{M}{y}=(M-y)/(y+1)\) and the beta-function identity \(\mathrm{B}(u+1,v-1)/\mathrm{B}(u,v)=u/(v-1)\). It provides a compact arithmetic check: the masses should be nonnegative and sum to one, up to rounding.

Worked Calculations

Interpreting Results and Checking Calculations

A predictive probability answers a forward-looking question about the terminal count. It combines the uncertainty in the response rate after the interim data with the randomness of future binary outcomes. For a fixed \(N\) and \(r\), an interim result can therefore produce a probability near zero, near one, or somewhere between. The same observed response proportion can lead to different predictive probabilities when the remaining sample size or terminal target changes.

Several checks catch common calculation errors:

• Check the remaining sample size. \(M=N-n\) must be nonnegative. If \(n>N\), the assumed interim state and planned total are inconsistent.
• Check the posterior parameters. They are \(A=a+x\) and \(B=b+n-x\), not parameters formed by adding all \(n\) participants to the response component.
• Check the success tail and its support. The first contributing future count is \(\max(0,r-x)\), but if \(r-x\leq0\) the probability is 1; if \(r-x>M\) it is 0. A sum must never include impossible future counts.
• Check the mass calculation. The values \(p_0,\ldots,p_M\) should sum to 1, and the reported success probability must lie between 0 and 1.
• Keep the target fixed. Changing \(r\) changes the tail, even if the interim data and posterior are unchanged.

A frequent mistake is to treat the posterior mean response rate as if it were known and calculate a binomial tail using that single value. That plug-in calculation omits posterior uncertainty in \(\theta\). The Beta-Binomial calculation instead averages the future binomial probability over the entire posterior distribution. Another mistake is to describe the predictive probability as the probability that the treatment is effective or that the null hypothesis is false. It is neither: its event is specifically that the final response count reaches the stated threshold.

The main alternative calculation is direct enumeration of the Beta-Binomial masses, using the beta-function formula for every \(y\) from 0 to \(M\). The recurrence is a convenient computational shortcut; direct evaluation is a useful independent check, especially when \(M\) is modest. A simulation can also approximate the probability by repeatedly drawing \(\theta\) from the posterior and then drawing \(Y\mid\theta\), but simulation adds Monte Carlo error and is unnecessary for small counts when exact summation is feasible.

These calculations rely on a fixed sample size and terminal rule, complete binary outcomes at the interim analysis, and the model assumptions stated above. A misspecified prior or a changing response probability can affect the result. The probability also depends on the chosen terminal target: it does not independently establish clinical importance, safety, or a justified stopping decision. Monitoring rules require a prespecified interpretation of the predictive probability, not an informal reaction to a single number.