Expected Sample Size Under Stochastic Curtailment

Why Expected Sample Size Depends on the Interim Data

Under stochastic curtailment, a trial may stop early for futility when the conditional probability of eventual success is sufficiently small. The decision depends on the interim data, which vary from trial to trial. Consequently, the sample size is random: some hypothetical trials stop at an interim analysis, while others continue to the planned maximum. Expected sample size averages those possible stopping times over the distribution of the interim data.

The expectation is not a single property of the stopping rule alone. It also depends on the data-generating parameter under which the expectation is calculated. A rule may stop frequently when the true effect is small and rarely when it is large, even though the conditional-success calculation used to make each decision is unchanged.

Definition: Suppose a design has planned sample size \(N\), and let \(N_{\mathrm{stop}}\) be the sample size at which a trial stops under its prespecified stochastic-curtailment rule. The expected sample size under a specified true parameter value \(\theta\) is \(E_{\theta}(N_{\mathrm{stop}})\), the average stopping sample size over repeated trials generated under \(\theta\). It is not necessarily the sample size expected under the effect used to calculate conditional success.

One Interim Analysis: The Basic Calculation

Consider a design with one interim analysis after \(n_1\) participants and a planned final analysis at \(N\). Let \(A\) be the event that the conditional probability of success at the interim analysis is at or below a prespecified curtailment threshold \(\gamma\). The design stops at \(n_1\) on \(A\); otherwise, it continues to \(N\). Thus \(N_{\mathrm{stop}}\) takes just two values.

N_{\mathrm{stop}} = \begin{cases} n_1, & A,\\ N, & A^c. \end{cases} \qquad E_{\theta}(N_{\mathrm{stop}}) = n_1P_{\theta}(A)+NP_{\theta}(A^c) = n_1+(N-n_1)P_{\theta}(A^c).

Equivalently, \(E_{\theta}(N_{\mathrm{stop}})=N-(N-n_1)P_{\theta}(A)\). The expected saving from the maximum is therefore \((N-n_1)P_{\theta}(A)\): the number of participants avoided if the trial stops, multiplied by the probability of stopping under \(\theta\). The key task is to find \(P_{\theta}(A)\) from the distribution of the interim data.

Deriving the Stopping Probability in a Normal Model

For a transparent derivation, use a one-sided normal group-sequential model with independent increments and known variance. Let \(I\) be the planned final information, let \(t=I_1/I\) be the fraction of information available at the interim analysis, and let \(Z_1\) be the standardized interim statistic. Under a true effect \(\theta\), assume \(Z_1\sim N(\theta\sqrt{tI},1)\). The final success criterion is \(Z_{\mathrm{final}}\ge c\), where \(c\) is a fixed standardized boundary. These assumptions specify the model for the example; applications must use the information and statistic appropriate to their design.

For the conditional-success calculation, use a specified design effect \(\theta_d\). By independent increments, conditional on observing \(Z_1=z\), the final statistic has mean \(\sqrt{t}z+\theta_d\sqrt{(1-t)I}\) and variance \(1-t\). Thus the conditional probability of success is the upper-tail probability that this normal final statistic exceeds \(c\).

CP(z) = \Phi\!\left( \frac{\sqrt{t}\,z+\theta_d\sqrt{(1-t)I}-c} {\sqrt{1-t}} \right).

Here \(\Phi\) is the standard normal cumulative distribution function. Stop when \(CP(z)\le\gamma\), with \(0<\gamma<1\). Since \(\Phi\) is increasing, this condition is equivalent to \(z\le b\), where \(b\) is the interim-statistic cutoff obtained by solving the inequality for \(z\).

b = \frac{c+\sqrt{1-t}\,\Phi^{-1}(\gamma) -\theta_d\sqrt{(1-t)I}} {\sqrt{t}}.

Under the true effect \(\theta\), the interim statistic is normal with mean \(\theta\sqrt{tI}\) and variance 1. Therefore the stopping probability is \(P_{\theta}(Z_1\le b)=\Phi(b-\theta\sqrt{tI})\). Substitution into the two-outcome expectation gives the expected sample size. Notice the two effects in the calculation: \(\theta_d\) determines the curtailment boundary, while \(\theta\) determines how often the interim data fall below that boundary.

Worked Example: Expected Sample Size Under the Design Effect

Suppose a hypothetical trial plans \(N=100\) participants and has an interim analysis at \(n_1=50\), so \(t=0.5\) and \(I=100\) in equal-information-per-participant units. Use a one-sided final boundary \(c=1.645\), a design effect \(\theta_d=0.20\), and a curtailment threshold \(\gamma=0.10\). Calculate expected sample size if the true effect is also \(\theta=0.20\).

The assumptions are the independent-increment normal model above, a fixed final boundary, and a stopping decision based only on the interim conditional-success probability. The standard normal quantile is \(\Phi^{-1}(0.10)=-1.2816\). The cutoff is

b = \frac{1.645+\sqrt{0.5}(-1.2816)-0.20\sqrt{50}} {\sqrt{0.5}} = \frac{1.645-0.9062-1.4142}{0.7071} \approx -0.9552.

Under the true effect \(0.20\), the interim mean is \(0.20\sqrt{50}=1.4142\). Hence \(P_{\theta}(A)=P(Z_1\le -0.9552)=\Phi(-0.9552-1.4142)=\Phi(-2.3694)\approx0.0089\). The expected sample size is

E_{\theta}(N_{\mathrm{stop}}) =50+50(1-0.0089) =99.555 \approx 99.56\text{ participants}.

Under this true effect, the interim data rarely trigger futility, so the expected sample size is close to the planned maximum of 100 participants. This is an average over repeated hypothetical trials, not a prediction that any individual trial will enroll 99.56 participants.

Worked Example: Changing the True Effect, Not the Rule

Keep the design and stopping rule from the previous example, but calculate expected sample size under the null effect, \(\theta=0\). The cutoff remains \(b\approx-0.9552\), because the conditional-success rule still uses \(\theta_d=0.20\). The interim statistic under the null has mean zero, so \(P_0(A)=\Phi(-0.9552)\approx0.1698\).

E_0(N_{\mathrm{stop}}) =50+50(1-0.1698) =91.51\text{ participants}.

The same design has different expected sample sizes under the two true effects: approximately 99.56 under \(\theta=0.20\) and 91.51 under \(\theta=0\). The lower expectation under the null arises because more interim statistics fall below the futility cutoff. It does not mean the trial is guaranteed to stop early under the null.

Discrete Interim Outcomes

For discrete data, the conditional-success probability and the interim-data distribution may both be discrete. The general expectation formula remains valid, but the stopping probability is found by summing probabilities for all interim outcomes that trigger curtailment. A normal approximation is not needed when exact probabilities are manageable. Also specify how equality at the threshold is handled; unlike a continuous statistic, a discrete conditional probability can equal the threshold with positive probability.

Worked Example: Exact Expected Sample Size for a Binomial Outcome

Consider a hypothetical trial with a binary outcome, an interim analysis after \(n_1=4\) participants, and a planned total of \(N=8\). Success is defined as at least 5 responses among all 8 participants. For the curtailment calculation, assume a response probability \(p_d=0.60\) for each of the 4 future participants and stop if the conditional probability of final success is at most \(\gamma=0.20\). Calculate expected sample size when the true response probability is \(p=0.40\).

Given \(x\) responses among the first 4 participants, the conditional success probability is the probability that at least \(5-x\) of the remaining 4 respond, calculated under \(p_d=0.60\). If \(x=0\), success is impossible. If \(x=1\), all 4 future participants must respond, so the conditional probability is \(0.60^4=0.1296\). If \(x=2\), at least 3 of 4 must respond, giving \(4(0.60^3)(0.40)+(0.60^4)=0.4752\). For \(x=3\), the probability of at least 2 future responses is \(1-(0.40^4)-4(0.60)(0.40^3)=0.8208\). For \(x=4\), only one further response is needed, giving \(1-(0.40^4)=0.9744\).

Thus the rule stops for \(x=0\) or \(x=1\), and continues for \(x=2,3,4\). Under the true response probability \(p=0.40\), the interim count \(X\sim\operatorname{Binomial}(4,0.40)\), so \(P_{0.40}(X\le1)=0.60^4+4(0.40)(0.60^3)=0.1296+0.3456=0.4752\). The expected sample size is

E_{0.40}(N_{\mathrm{stop}}) =4(0.4752)+8(1-0.4752) =1.9008+4.1984 =6.0992\text{ participants}.

The conditional probabilities determining which counts stop were calculated under \(p_d=0.60\), while the probability of encountering those counts was calculated under the true \(p=0.40\). Using the same probability for both tasks would answer a different question.

More Than One Interim Analysis

With several possible stopping times, expected sample size can be written in terms of the chance of reaching each analysis. Let the planned sample sizes be \(n_1<n_2<\cdots<n_K=N\), and suppose a trial that reaches analysis \(j\) has accrued \(n_j-n_{j-1}\) additional participants since the preceding analysis, with \(n_0=0\). Then

E_{\theta}(N_{\mathrm{stop}}) = \sum_{j=1}^{K}(n_j-n_{j-1}) P_{\theta}(\text{reach analysis }j).

To see why, write the total sample size used as the sum of the increments accrued at each stage. The increment \(n_j-n_{j-1}\) is accrued exactly when analysis \(j\) is reached. Taking expectations of this sum gives the formula, because the expected value of an increment multiplied by its reach indicator is that increment times the probability of reaching it. The probability of reaching an analysis must account for all earlier stopping decisions.

In particular, the probabilities of stopping at different looks are not generally obtained by adding marginal probabilities calculated separately at each look. The interim statistics are dependent, and a trial can stop only once. Use their joint distribution, or a simulation that applies the complete rule to each generated trial, to determine the probabilities of reaching each stage.

Interpretation, Diagnostics, and Limitations

Expected sample size is useful for describing the resource implications of a stopping rule across repeated trials. It complements the planned maximum: the maximum describes the largest sample allowed by the design, while the expectation describes the average sample under a specified data-generating scenario. Report the parameter scenario, the rule, and the unit of sample size together. For a design whose information per participant is unequal, calculate expected information or expected accrued sample size using the appropriate stage increments rather than silently treating information units as participant counts.

Check a calculation by confirming that \(n_1\le E_{\theta}(N_{\mathrm{stop}})\le N\) for the one-interim design, and that the stopping probability lies between 0 and 1. If the stopping probability is near zero, the expectation should be near \(N\); if it is near one, it should be near \(n_1\). For discrete outcomes, list every interim outcome, its conditional-success probability, and the decision it triggers before summing probabilities. These checks often reveal a reversed inequality, a mismatch between the design effect and the true effect, or an omitted outcome.

A fixed-sample design always uses \(N\), so its expected sample size is \(N\). A deterministic futility rule also yields an expected sample size, but it uses a prespecified interim-data boundary rather than defining the boundary through a conditional-success threshold. Stochastic curtailment makes that connection explicit: the threshold is applied to conditional success, and the distribution of interim data translates it into a probability of stopping. Expected sample size alone does not establish that a design has acceptable power, error rates, or estimation properties; those operating characteristics require separate evaluation under the full stopping rule.

Common pitfall: The conditional-success calculation uses the effect specified for that calculation, whereas expected sample size under \(\theta\) uses the distribution of interim data under \(\theta\). Substituting the design effect for the true effect in both places reports the expectation under that particular scenario only; it does not give a scenario-free expected sample size.

Check Your Understanding

Use the definitions and calculations above to answer each question. Solutions are included after the questions.

In a one-interim design, \(n_1=40\), \(N=100\), and the stopping probability under a stated true effect is 0.25. What is the expected sample size?
In the normal model, which parameter determines the interim cutoff, and which parameter determines the probability of falling below it?
Why must the expected sample size for a discrete interim outcome be calculated by summing over stopping outcomes?
For multiple analyses, why is it generally incorrect to add marginal stopping probabilities from each look?

Solutions. (1) \(E(N_{\mathrm{stop}})=40+60(1-0.25)=85\) participants. (2) The design effect \(\theta_d\), together with the threshold and other design quantities, determines the cutoff; the true effect \(\theta\) determines the interim-statistic distribution and thus the probability of crossing it. (3) A discrete statistic has a finite or countable set of outcomes, so the stopping event is the collection of outcomes whose conditional-success probability meets the rule; its probability is the sum of their probabilities. (4) The looks are dependent and only the first stopping decision applies; probabilities of reaching later analyses must incorporate earlier decisions and the joint interim-data distribution.