Stochastic Curtailment for Discrete Outcomes

Why Discrete Outcomes Need Explicit Enumeration

For a binary endpoint, each participant contributes a success or failure. At an interim analysis, the number of successes is therefore an integer, and the number of possible future success counts is finite. This discreteness matters: the conditional probability of meeting a final rejection rule can jump substantially when one interim success is added. A continuous normal approximation may smooth over precisely the jumps that determine a stochastic-curtailment decision.

The introductory material on stochastic curtailment established the principle of using the chance of eventual rejection, given the data observed so far, to consider stopping early for futility. Here the calculation is made directly for binomial outcomes. The final rejection rule is kept fixed; the task is to enumerate which future counts would satisfy it.

Definition: Suppose $n_1$ participants have been observed at an interim analysis, with $x$ successes, and the planned final sample size is $N$. Let $m=N-n_1$ be the number of future participants, and let $Y$ be their number of successes. For a specified success probability $p$ and fixed final rejection set $\mathcal{R}$, the conditional probability of eventual rejection is $$ Q_p(x)=P_p(x+Y\in\mathcal{R}\mid X_1=x). $$ Under independent Bernoulli outcomes with a common success probability $p$, $Y\mid X_1=x\sim\operatorname{Binomial}(m,p)$. Thus $$ Q_p(x)=\sum_{y=0}^{m} I(x+y\in\mathcal{R}){m\choose y}p^y(1-p)^{m-y}, $$ where $I(\cdot)$ is 1 when its condition is true and 0 otherwise.

The model assumes a fixed final sample size, independent future outcomes, and the same success probability for each future participant. The fixed rejection set specifies exactly which final success totals lead to rejection. The probability $p$ must also be specified: for example, it may be the null value $p_0$, or a prespecified alternative value. The calculation is conditional on the interim count $x$, but it is not a probability that $H_0$ is true.

Upper-Tail Rules: Find the Required Future Count

For a one-sided upper-tail rule, suppose the final test rejects when the total number of successes is at least $c$. Then $\mathcal{R}=\{c,c+1,\ldots,N\}$. Given $x$ interim successes, eventual rejection requires at least $c-x$ future successes. The summation can therefore be shortened to the feasible future counts:

Q_p(x)= \begin{cases} 1, & c-x\leq 0,\\ \displaystyle\sum_{y=c-x}^{m}{m\choose y}p^y(1-p)^{m-y}, & 1\leq c-x\leq m,\\ 0, & c-x>m. \end{cases}

The first case means the interim results already meet the final success-count criterion, so no future outcome can undo it. The last case means even success by every remaining participant would not reach the criterion. In the middle case, enumerate all future counts from the required minimum through $m$.

Worked Example: Calculating an Upper-Tail Probability

Suppose 5 participants have been observed, with 2 successes. Four more will be observed, and the fixed final rule rejects if there are at least 5 successes among all 9 participants. Assume independent Bernoulli outcomes with common success probability $p=0.60$. Find the conditional probability of eventual rejection.

The interim count is $x=2$, the number remaining is $m=4$, and the total must reach $c=5$. Thus $Y$ must be at least $5-2=3$. The model gives $Y\mid X_1=2\sim\operatorname{Binomial}(4,0.60)$, so

Q_{0.60}(2)=P(Y\geq 3) ={4\choose 3}(0.60)^3(0.40)+(0.60)^4 =4(0.216)(0.40)+0.1296 =0.4752.

Under the specified $p=0.60$ model and the observed interim count, the chance of satisfying this final rule is 0.4752. This is a conditional probability under a stated model, not a guarantee of success and not evidence that the assumed response probability is correct.

Using the Probability for a Curtailment Decision

A futility policy needs a prespecified reference probability and a threshold. For example, a rule might stop for futility when $Q_{p_{\mathrm{ref}}}(x)\leq\gamma$, where $p_{\mathrm{ref}}$ is the response probability chosen for the calculation and $\gamma$ is the chosen probability threshold. A null-based policy uses $p_{\mathrm{ref}}=p_0$; another policy could specify a clinically relevant alternative value. The policy should state which probability is used rather than silently substituting an estimate from the interim data.

With an upper-tail final rule, $Q_p(x)$ cannot decrease as $x$ increases: if one interim success is added, the required number of future successes is no greater. However, because $x$ is integer-valued, the probability changes in steps, not smoothly. Consequently, several adjacent interim counts may share the same decision, and a small change in $\gamma$ may move the stopping boundary by a whole success.

Worked Example: Finding a Discrete Futility Boundary

Consider a hypothetical single-group trial with $n_1=8$ participants observed and $m=10$ still planned. The fixed final rule rejects if the total number of successes is at least 12. Use $p_{\mathrm{ref}}=p_0=0.50$, and stop for futility if the conditional probability of meeting that rule is at most $\gamma=0.05$. Compare interim counts $x=3$ and $x=4$.

The future count has distribution $Y\sim\operatorname{Binomial}(10,0.50)$. At $x=3$, at least $12-3=9$ future successes are required:

Q_{0.50}(3) =P(Y\geq9) =\frac{{10\choose9}+{10\choose10}}{2^{10}} =\frac{10+1}{1024} =0.01074.

Since $0.01074\leq0.05$, the rule stops for futility at $x=3$. At $x=4$, at least 8 future successes are required:

Q_{0.50}(4) =P(Y\geq8) =\frac{{10\choose8}+{10\choose9}+{10\choose10}}{2^{10}} =\frac{45+10+1}{1024} =0.05469.

Since $0.05469>0.05$, the rule continues at $x=4$. The discrete boundary between these two counts is therefore clear for this policy. The final success-count rule itself has not changed; the futility rule only determines whether the trial stops without rejecting before reaching the planned final sample size.

General Rejection Sets and Two-Sided Rules

Not every final rule is an upper-tail threshold. A two-sided binomial test may reject for totals in a lower tail or an upper tail, and a fixed test may have a rejection set chosen by a particular testing procedure. In these cases, do not assume that all success totals above one cutoff reject. Instead, list which future counts produce a final total in $\mathcal{R}$, then sum their binomial probabilities.

Worked Example: Enumerating a Two-Sided Rejection Set

Suppose the final sample has $N=8$ participants. For illustration, take the fixed rejection set $\mathcal{R}=\{0,1,7,8\}$. Four participants have been observed, with $x=1$ success, so $m=4$ remain. Assume independent Bernoulli outcomes with $p=0.70$. Find the probability of eventual rejection.

A final total in $\mathcal{R}$ requires $1+Y\in\{0,1,7,8\}$. Since $Y$ must be between 0 and 4, the only feasible rejection total is 1, which requires $Y=0$. Totals 0, 7, and 8 cannot occur from this interim count and these four remaining participants. Therefore

Q_{0.70}(1) =P(Y=0) ={4\choose0}(0.70)^0(0.30)^4 =0.008100.

The probability is 0.008100 under the specified model. This example uses an illustrative fixed rejection set; it does not claim that this set is a particular nominal-level test. The key calculation is to retain only feasible future counts that lead to rejection.

Why the Enumeration Works—and What Can Go Wrong

The calculation follows from the independence of future outcomes from the observed interim outcomes under the stated Bernoulli model. Conditional on $X_1=x$, each of the $m$ remaining outcomes still has success probability $p$, so their sum has a binomial distribution. The final rejection event is exactly the set of future counts for which the total $x+Y$ belongs to $\mathcal{R}$. Summing the probabilities of those mutually exclusive future counts gives the conditional probability of rejection.

P_p(x+Y\in\mathcal{R}\mid X_1=x) =\sum_{y:\,x+y\in\mathcal{R}}P_p(Y=y\mid X_1=x).

If the only early stop is for futility and stopping means the trial will not reject, this rule cannot add rejection outcomes to the fixed final test: rejection still requires a final outcome in $\mathcal{R}$. Thus it does not increase the probability of rejection under $H_0$ relative to applying that same final rule to everyone. This observation does not justify adding an early efficacy rule, changing the final rule, or using a data-dependent rejection set; those changes require their own calibration.

Before relying on the binomial enumeration, check that the final sample size and rejection rule really are fixed, and that the future outcomes can reasonably be treated as independent with a common success probability. Calendar-time trends, clustering, unequal response probabilities, or informative missing outcomes can invalidate the simple binomial model. For a two-arm comparison, the treatment and control counts generally require a joint model; treating their difference as one binomial count is not justified. In such settings, derive the appropriate conditional distribution rather than applying this formula unchanged.

Exact enumeration is usually straightforward when $m$ is moderate and is preferable to a normal approximation when a decision depends on a small tail probability or a boundary count. A normal approximation can give a quick rough check, but it may be inaccurate in a tail, especially with small samples or probabilities near 0 or 1. Simulation can be useful for more complicated outcome models, but it introduces Monte Carlo error; direct enumeration of a binomial distribution does not. Regardless of the computational method, the assumed $p$, the rejection rule, and the futility threshold must be explicit.

Common Pitfalls

Using the wrong future-count range. For an upper-tail rule, calculate $c-x$, then restrict to feasible counts from 0 to $m$. If $c-x>m$, the probability is zero; if $c-x\leq0$, it is one.
Counting non-rejecting outcomes. Under a two-sided rule, include only future counts that put the final total in the rejection set. A large future count is not automatically a rejection if the rule is not upper-tailed.
Treating the probability as certainty or as evidence about $H_0$. $Q_p(x)$ is calculated assuming a particular value of $p$. It is not the probability that the trial will succeed with certainty, nor the probability that a hypothesis is true.
Ignoring discreteness. A probability threshold need not correspond to a smooth boundary. Compute the values at the neighboring integer counts rather than interpolating between them.

Key takeaway: For discrete outcomes, stochastic curtailment is an exact finite sum: identify the future counts that would satisfy the fixed final rejection rule, calculate their binomial probabilities under the specified response rate, and compare the sum with the prespecified futility threshold.

Check Your Understanding

Try each question, then check the solutions.

There are 6 participants remaining, and the final rule requires at least 9 total successes. The interim count is $x=2$. What minimum future count is required?
With $m=3$, $p=0.40$, and $x=1$, a final upper-tail rule requires at least 3 total successes. Calculate $Q_p(x)$.
Why is it incorrect to calculate a two-sided rejection probability by summing every future count above a single cutoff when the rejection set includes both tails?
If an upper-tail rule has $Q_p(x)=0.03$ and the futility threshold is $\gamma=0.05$, what does the specified rule do? What does the value 0.03 mean?

Solutions. (1) The required future count is $9-2=7$, which exceeds the 6 remaining participants; eventual rejection is impossible and the conditional probability is zero. (2) At least $3-1=2$ of the 3 future participants must succeed, so $Q_{0.40}(1)={3\choose2}(0.40)^2(0.60)+(0.40)^3=0.288+0.064=0.352$. (3) A two-sided rejection set also includes lower-tail totals, and only future counts leading to either rejection tail should be included. (4) The rule stops for futility because $0.03\leq0.05$. The value 0.03 is the conditional probability of eventual rejection under the specified $p$, interim count, and final rule; it is not the probability that $H_0$ is true.