How Interim Timing Affects Predictive Probability

Why Interim Timing Matters

In a single-arm phase 2 trial, an interim decision depends not only on how many responses have been observed, but also on how many patients have already been assessed. If the trial is still early, many future outcomes can change whether the planned terminal criterion is reached. At a late look, fewer outcomes remain, so the observed record may constrain the final result more strongly. Yet a late look is not automatically more favorable: observed nonresponses also provide information, and a patient who has not yet been observed is not a response.

This tutorial studies that interaction using the beta-binomial model for a binary response. The focus is the predictive probability (PP) of satisfying a prespecified terminal success rule, conditional on the interim data. It is a probability about the future terminal result under the model, not the posterior probability that the response rate exceeds a particular value.

Model and Interim Quantities

Assume that each patient has a binary response outcome, responses are independent conditional on a common response probability \(p\), and \(p\) has a beta prior, \(p\sim\mathrm{Beta}(a,b)\), with \(a,b>0\). Outcomes are treated as fully observed at the interim look. The planned sample size and the terminal rule are fixed, and there are no changes to the endpoint definition or response assessment process. These assumptions give a simple reference model; departures such as heterogeneous response probabilities or pending outcomes require different modelling.

After observing \(x\) responses among \(m\) patients, conjugacy gives \(p\mid x,m\sim\mathrm{Beta}(a+x,b+m-x)\). Conditional on \(p\), the future response count satisfies \(Y\mid p,x,m\sim\mathrm{Binomial}(r,p)\). Integrating over the posterior uncertainty in \(p\) gives the beta-binomial predictive distribution.

Here \(B(u,v)\) is the beta function. The predictive probability of terminal success is the upper tail beginning at the number of additional responses needed:

If \(d>r\), terminal success is impossible and the sum is zero. If \(d=0\), the terminal response-count criterion has already been met and the probability is one, provided the success rule depends only on reaching at least \(K\) responses. These edge cases are useful checks on an implementation.

Why the Predictive Distribution Has This Form

For any particular future count \(y\), first condition on \(p\). The probability of exactly \(y\) responses among \(r\) patients is \(\binom{r}{y}p^y(1-p)^{r-y}\). Average this probability over the posterior distribution of \(p\), whose density is proportional to \(p^{a+x-1}(1-p)^{b+m-x-1}\). The integral is a beta-function ratio:

The integration matters: treating \(p\) as known at its posterior mean would instead give a binomial predictive distribution and omit posterior uncertainty. In the beta-binomial model, future outcomes are positively associated marginally because they share the uncertain \(p\). The formula therefore combines uncertainty about the response rate with the randomness of future patient outcomes.

Worked Examples: Matched Early and Late Looks

Reading Timing Effects Correctly

The interim state affects PP through several connected quantities: the posterior distribution of \(p\), the number of patients remaining \(r\), and the gap \(d\) between the current response count and the terminal target. These do not act in isolation. More remaining patients create more opportunities to reach \(K\), but they also mean that more outcomes are uncertain. More observed patients sharpen the posterior in light of both responses and nonresponses. And the required future count changes whenever \(x\) changes.

Consequently, there is no general rule that PP must rise or fall as an interim look moves later. For instance, if the response count stays fixed while additional nonresponses accrue, PP can fall sharply, as in the second example. If responses accrue and the remaining required count falls, PP may rise, as in the first and third examples. A comparison should state the complete interim state—at least \(x\), \(m\), \(r\), and \(K\)—rather than reporting an observed response rate alone.

A useful distinction is between predictive probability and two related quantities. The posterior probability \(P(p>p_0\mid x,m)\) concerns whether the unknown response rate exceeds a specified benchmark \(p_0\). Predictive probability concerns whether future outcomes lead to a specified terminal result. Fixed-parameter conditional power instead calculates future success probability at a chosen value of \(p\), rather than averaging over its posterior. Each answers a different question; they should not be substituted for one another without an explicit design rationale.

Diagnostics, Limitations, and a Common Pitfall

Before interpreting an interim PP, verify the arithmetic and the assumptions. The remaining sample size must equal \(N-m\), the needed count must equal \(\max(0,K-x)\), and the predictive probabilities over \(y=0,\ldots,r\) must sum to one. The PP must lie between zero and one; it is zero if the target is unreachable and one if the response-count target has already been reached. When feasible, calculate the same tail by summing its terms and by subtracting the lower tail; agreement is a useful computational check.

A model-based PP can be poorly calibrated if response outcomes are not adequately represented by a common-probability model—for example, if patient subgroups have materially different response rates, eligibility or assessment changes over time, or the prior is not appropriate for the clinical context. It also depends on the endpoint and terminal threshold being specified correctly. PP describes the probability of the defined terminal rule under the assumed model; it is not by itself a clinical benefit assessment or a guarantee about the realized outcome. A stopping boundary for acting on PP is a separate design choice.

• Mistake: comparing observed response rates alone. Two looks with the same rate can have different \(x\), \(m\), \(r\), and \(d\), and therefore different predictive tails. Correction: report the response count, assessed sample size, remaining sample size, terminal target, and PP together.
• Mistake: treating the posterior mean as a known response probability. Plugging that mean into a binomial tail omits uncertainty in \(p\). Correction: integrate over the posterior, using the beta-binomial distribution under the stated model.
• Mistake: interpreting PP as the probability that the treatment works. PP is conditional on the observed data, model, prior, planned sample size, and terminal rule. Correction: describe it as the model-based probability of meeting the specified terminal criterion.