Discrete Predictive Probabilities and Boundary Ties

Why Discreteness Matters

In single-arm monitoring with a binary response, the number of responses still to come is an integer. A predictive probability is therefore a sum of probabilities over particular future counts, not a continuously varying quantity. As the interim data change, the predictive probability can jump from one value to another without ever taking values in between. This matters when an efficacy or futility rule compares that probability with a prespecified threshold: the comparison at equality must be defined in advance, and the terminal response requirement must be translated into the correct inclusive count boundary.

The discussion assumes one planned interim look, a fixed total sample size, observed binary outcomes for all participants enrolled by that look, and no pending outcomes. Conditional on a constant response probability \(p\), responses are independent Bernoulli observations. The prior is \(p\sim\mathrm{Beta}(a,b)\), with \(a,b>0\). These assumptions lead to a beta-binomial distribution for the number of future responses. The logic about integer thresholds applies more generally to any discrete predictive distribution.

Definition: Let \(n\) be the number of participants with observed outcomes at the interim, \(x\) the number of responses among them, \(N\) the fixed total sample size, and \(m=N-n\) the number of future outcomes. Let \(Y\) be the number of responses among those \(m\) participants. If terminal success requires at least \(r\) responses in all \(N\) participants, the predictive probability of terminal success is \(P(x+Y\ge r\mid x,n)\), equivalently \(P(Y\ge r-x\mid x,n)\). Since \(Y\) is integer-valued and \(0\le Y\le m\), only counts from \(\max(0,r-x)\) through \(m\) can contribute.

Predictive Probabilities Are Discrete

After observing \(x\) responses and \(n-x\) nonresponses, beta-binomial conjugacy gives \(p\mid x,n\sim\mathrm{Beta}(a+x,b+n-x)\). Write \(\alpha=a+x\) and \(\beta=b+n-x\). For any possible future count \(y\), the predictive probability averages the binomial probability conditional on \(p\) over this posterior distribution:

P(Y=y\mid x,n) =\int_0^1 {m\choose y}p^y(1-p)^{m-y} \frac{p^{\alpha-1}(1-p)^{\beta-1}}{B(\alpha,\beta)}\,dp ={m\choose y}\frac{B(\alpha+y,\beta+m-y)}{B(\alpha,\beta)}.

Here \(B(u,v)\) is the beta function. The integral follows by collecting the powers of \(p\) and \(1-p\) in the integrand. The resulting integral is \(B(\alpha+y,\beta+m-y)\), which gives the stated ratio. Equivalently, using the rising factorial \((c)_k=c(c+1)\cdots(c+k-1)\), with \((c)_0=1\), the probability is

P(Y=y\mid x,n) ={m\choose y}\frac{(\alpha)_y(\beta)_{m-y}}{(\alpha+\beta)_m}, \qquad y=0,1,\ldots,m.

The predictive probability of terminal success is the sum of these point probabilities over the eligible integer counts. Because \(Y\) has only \(m+1\) possible values, the sum can change only as the posterior parameters, interim data, or terminal boundary change. It is not generally correct to interpolate between attainable values as if the predictive probability were continuous.

Worked Example: Translating a Terminal Requirement Into a Count Boundary

Suppose the prior is \(p\sim\mathrm{Beta}(1,1)\). At an interim after \(n=4\) participants, \(x=2\) have responded. The planned total is \(N=8\), and terminal success is at least \(r=5\) responses. Find the predictive probability of terminal success.

There are \(m=N-n=8-4=4\) future outcomes. The posterior is \(\mathrm{Beta}(1+2,1+4-2)=\mathrm{Beta}(3,3)\). The terminal requirement \(x+Y\ge5\) becomes \(2+Y\ge5\), so success requires \(Y\ge3\), not \(Y>3\). Thus the eligible future counts are \(y=3\) and \(y=4\).

For this posterior, the beta-binomial denominator in the rising-factorial formula is \((6)_4=6\cdot7\cdot8\cdot9=3024\). The required probabilities are

P(Y=3\mid x,n) ={4\choose3}\frac{(3)_3(3)_1}{(6)_4} =\frac{4\cdot60\cdot3}{3024} =\frac{720}{3024}, \qquad P(Y=4\mid x,n) =\frac{(3)_4}{(6)_4} =\frac{360}{3024}.

Therefore \(P(x+Y\ge5\mid x,n)=(720+360)/3024=1080/3024=5/14\approx0.3571\). The interpretation is that, under this model and given the interim data, the probability of reaching at least five total responses by the end is about 35.7%.

Prespecifying and Applying Boundary Comparisons

A predictive-probability rule compares the probability of eventual terminal success with fixed design thresholds. Let \(\gamma_F\) be the futility threshold and \(\gamma_E\) the efficacy threshold, with \(0\le\gamma_F<\gamma_E\le1\). A common deterministic convention is to stop for futility when the predictive probability is at or below \(\gamma_F\), stop for efficacy when it is at or above \(\gamma_E\), and continue otherwise. Equality belongs to a stopping region under this convention. Other conventions are possible, but the comparisons must be specified before the interim data are examined.

\begin{aligned} \text{Stop for futility} &\quad\text{if}\quad P(x+Y\ge r\mid x,n)\le\gamma_F,\\ \text{Stop for efficacy} &\quad\text{if}\quad P(x+Y\ge r\mid x,n)\ge\gamma_E,\\ \text{Continue} &\quad\text{if}\quad \gamma_F<P(x+Y\ge r\mid x,n)<\gamma_E. \end{aligned}

The strict gap between the two thresholds prevents a predictive probability from satisfying both stopping rules at once. If the protocol instead uses strict comparisons, such as “greater than” for efficacy, that choice changes what happens at equality and must be stated explicitly. Equality should be evaluated using the unrounded probability, not a display rounded to a few decimal places.

Worked Example: An Exact Efficacy Boundary Tie

Use the interim data and model from the previous example: \(n=4\), \(x=2\), \(N=8\), prior \(\mathrm{Beta}(1,1)\), and terminal success defined as at least five responses. Suppose the design specifies efficacy stopping when the predictive probability is at least \(\gamma_E=5/14\), and futility stopping when it is at most \(\gamma_F=0.20\). Determine the decision.

The predictive probability was calculated as \(5/14\). Comparing without rounding, \(P(x+Y\ge5\mid x,n)=5/14=\gamma_E\), so the inclusive efficacy rule is met exactly. The interim decision is to stop for efficacy. The futility condition is not met because \(5/14\approx0.3571>0.20\).

If the protocol had instead specified stopping for efficacy only when the probability is strictly greater than \(5/14\), this exact tie would not trigger efficacy stopping. It would fall outside the efficacy region, and the decision would depend on the remaining prespecified rule. The data and probability do not decide whether equality counts as crossing; the written rule does.

Worked Example: An Exact Futility Boundary Tie

Retain the interim data and posterior \(\mathrm{Beta}(3,3)\), but suppose terminal success is defined as at least four responses in \(N=8\). At the interim, \(x=2\) responses have been observed, so success requires \(Y\ge2\) among the four remaining participants. Suppose the rule is futility stopping at or below \(\gamma_F=9/14\) and efficacy stopping at or above \(\gamma_E=0.80\).

From the beta-binomial probabilities for \(Y=2,3,4\), the point probabilities are \(864/3024\), \(720/3024\), and \(360/3024\), respectively. Hence

P(x+Y\ge4\mid x,n) =\frac{864+720+360}{3024} =\frac{1944}{3024} =\frac{9}{14} \approx0.6429.

This equals the futility threshold, so the inclusive rule stops for futility. It is below the efficacy threshold. The conclusion concerns the predictive chance of meeting the specified terminal response count, not whether the underlying response probability \(p\) is itself below some threshold.

Checking the Count Boundary and the Decision

The count calculation and the threshold comparison are separate steps. First translate terminal success into the required future count. Then sum the predictive mass over eligible counts. Only after that should the result be compared with the prespecified stopping thresholds. This separation is useful for detecting off-by-one errors and boundary ties.

Write terminal success precisely.
For a requirement of at least \(r\) responses overall, write \(x+Y\ge r\). For a requirement of more than \(r\), write \(x+Y>r\); these are different rules when counts are integers.

Find the eligible future counts.
Rearrange the inequality to obtain the lower count boundary, then respect \(0\le Y\le m\). If the required count exceeds \(m\), terminal success is impossible; if it is at most zero, success is already guaranteed under that terminal definition.

Calculate and compare.
Sum the predictive probabilities over the eligible counts and compare the unrounded result with \(\gamma_F\) and \(\gamma_E\), applying the exact inclusive or strict comparisons specified by the design.

The sum must use mutually exclusive values of \(Y\), each counted once. A useful check is that the beta-binomial point probabilities for \(y=0,\ldots,m\) are nonnegative and sum to one. When a terminal boundary leaves no eligible counts, the predictive success probability is zero; when every possible count is eligible, it is one. These edge cases can reveal a mistaken inequality direction or a boundary outside the possible count range.

Common Mistakes, Diagnostics, and Alternatives

Off-by-one count boundary. Replacing “at least \(r\)” with “more than \(r\)” changes the requirement from \(x+Y\ge r\) to \(x+Y>r\). The correction is to write the terminal rule as an inequality before rearranging it.
Assuming a smooth probability scale. A small data change can move probability mass between integer outcomes and produce a jump. The correction is to calculate the discrete tail rather than infer values between attainable probabilities.
Rounding before deciding. A displayed value of 0.357 may hide a value just above or below a threshold, or an exact tie. Compare full-precision probabilities with the prespecified threshold, then round only for reporting.
Changing the tie rule after seeing the result. Choosing a strict or inclusive comparison to obtain a preferred decision undermines the prespecified rule. Document the boundary convention in the design and apply it consistently.
Substituting a posterior mean for a predictive probability. The posterior mean response probability is not the probability that the remaining count will reach a target. The predictive probability accounts for uncertainty in \(p\) and the integer distribution of future responses.

A continuous normal approximation may be convenient for some count distributions, but it can obscure the exact mass at a boundary and is especially unhelpful when only a few future participants remain. The beta-binomial sum directly respects the support \(0,\ldots,m\). A different choice is to define stopping directly through posterior evidence about \(p\), but that is a different decision target from the predictive probability of terminal success. Neither alternative removes the need to define equality behavior for a threshold rule. Predictive decisions also depend on the assumed prior, response model, terminal success definition, and planned sample size; a precise calculation does not eliminate sensitivity to those design choices.

Key takeaway: Integer-valued future counts create discrete predictive probabilities. State the terminal count inequality, sum the probability over exactly the eligible counts, and apply the prespecified inclusive or strict threshold rule to the unrounded result.

Check Your Understanding

Answer each question, then compare your reasoning with the solutions below.

At an interim, \(x=3\) responses have been observed and terminal success requires at least 7 responses overall. Write the condition on the future response count \(Y\).
A design stops for efficacy if predictive success probability is at least 0.40. The unrounded probability is exactly 0.40. Does the stated rule trigger efficacy stopping?
With \(n=4\), \(x=2\), \(N=8\), prior \(\mathrm{Beta}(1,1)\), and terminal success at five or more responses, what is the predictive probability of success?
Why can rounding to three decimal places give the wrong stopping decision near a threshold?

Solutions. (1) Success requires \(x+Y\ge7\), so \(3+Y\ge7\), or \(Y\ge4\). (2) Yes. “At least 0.40” is an inclusive comparison, so equality triggers efficacy stopping. (3) The posterior is \(\mathrm{Beta}(3,3)\), four outcomes remain, and success requires \(Y\ge3\). The probability is \(P(Y=3)+P(Y=4)=(720+360)/3024=5/14\approx0.3571\). (4) Rounding can make a value slightly above or below a threshold appear equal to it, or conceal an exact tie. Apply the specified rule to the unrounded value.