Expected Sample Size From Stopping Probabilities

Why Expected Sample Size Depends on Stopping Probabilities

A group sequential design specifies planned sample sizes at successive looks, but a trial that stops early does not always reach its maximum planned sample size. The expected sample size summarizes this feature by averaging the sample size at which the trial actually stops over its possible stopping looks. It is not the maximum sample size, nor is it the sample size at a “typical” look chosen without regard to how likely that look is.

The calculation is straightforward once the stopping probabilities are defined correctly. At each look, the relevant probability is the probability that the trial first stops at that look. These probabilities weight the corresponding planned sample sizes. The same expectation can also be calculated from the probabilities of reaching each look, which makes explicit how much sample size is added as the trial proceeds.

In the efficacy-boundary setting, a trial stops at an earlier look if its test statistic crosses a stopping boundary. At the final look it stops whether or not the boundary is crossed. Thus the final first-stopping probability includes both trials that cross at the final look and trials that reach that look without crossing an earlier boundary. This distinction matters when using stagewise rejection probabilities: an earlier stagewise rejection probability is a first-stopping probability for an efficacy-only design, but the final first-stopping probability generally also includes final-look nonrejections.

The probabilities depend on the parameter setting. Write \(E_0(N)\) for the expected sample size under \(H_0\), and \(E_1(N)\) for the expected sample size under a specified alternative. The planned sample sizes are the same in both calculations, but the probabilities of stopping at each look can differ substantially.

Two Equivalent Formulas

Because \(N=n_k\) whenever \(T=k\), its expectation is the weighted average of the possible sample sizes, using the probabilities of stopping at those looks:

The probabilities must describe mutually exclusive first stops and collectively cover all trial outcomes. Consequently, \(\sum_{k=1}^{K}\pi_{k,\theta}=1\). This gives an immediate check on any set of inputs.

There is a second formula that emphasizes sample-size increments. Reaching look \(k\) means that the trial has not stopped at any earlier look, so \(P_\theta(T\geq k)=1-\sum_{j=1}^{k-1}\pi_{j,\theta}\). The sample size starts at \(n_1\), and each subsequent look adds \(n_k-n_{k-1}\) participants if it is reached.

Why the Increment Formula Is Correct

The equivalence follows by writing the sample size at the random stopping look as the initial sample size plus the increments for every later look that is reached. Let \(I(T\geq k)\) equal 1 if the trial reaches look \(k\), and 0 otherwise. If \(T=1\), no later increment is added; if \(T=r\), exactly the increments through look \(r\) are added. Therefore, for every possible value of \(T\),

Taking expectations and using \(E_\theta[I(T\geq k)]=P_\theta(T\geq k)\) gives the increment formula. Equivalently, substituting \(P_\theta(T\geq k)=\sum_{j=k}^{K}\pi_{j,\theta}\) into that formula and collecting the coefficient of each \(\pi_{j,\theta}\) yields \(\sum_{j=1}^{K}n_j\pi_{j,\theta}\). This argument requires no equal spacing between looks: each increment can have a different size.

Worked Calculations

Using Stopping Probabilities Correctly

For an efficacy-only design, stagewise rejection probabilities under a parameter setting can supply the first-stopping probabilities at the earlier looks: rejection at look \(k\) means the trial did not reject at an earlier look and rejects now. At the final look, however, use the probability of reaching that look as the final stopping probability, not just the probability of final-look rejection. The trial stops at the maximum sample size even when it does not reject. If probabilities are supplied as conditional probabilities among trials that reach a look, they must first be converted to unconditional first-stopping probabilities before using the weighted-look formula.

The model used to calculate the probabilities must match the intended design and parameter setting. In the group sequential calculation, the stopping probabilities are determined by the scheduled looks, the stopping boundaries, and the distribution of the test statistics across looks under \(H_0\) or the specified alternative. A probability set that omits outcomes, counts some outcomes at more than one look, or uses the wrong parameter setting will produce a misleading expectation even if the arithmetic is flawless.

• Check the total: first-stopping probabilities should be nonnegative and sum to 1, subject only to small numerical rounding error.
• Check the final-look entry: it includes all probability mass that reaches the final look, including nonrejection there.
• Check the bounds: when \(n_1\leq N\leq n_K\), the computed expectation must satisfy \(n_1\leq E_\theta(N)\leq n_K\).
• Check the reach probabilities: \(P_\theta(T\geq k)\) cannot increase as \(k\) increases, because reaching a later look requires reaching each earlier look.

The weighted-look formula is usually quickest when a complete set of first-stopping probabilities is available. The increment formula is often more revealing when the question is how much of the maximum sample size is avoided by stopping before each added cohort. They are algebraically equivalent, so comparing both is a useful arithmetic check. A direct simulation can also estimate expected sample size, but it introduces Monte Carlo error and is not a substitute for correctly calculated stopping probabilities.

Common Pitfalls and Limitations

• Using an unweighted average of the planned sample sizes. Averaging the \(n_k\)'s equally assumes every look is equally likely to be the stopping look. The correction is to weight each \(n_k\) by its first-stopping probability.
• Using conditional probabilities as if they were unconditional. A conditional probability of stopping at look \(k\), given that the trial reached it, is not \(\pi_{k,\theta}\). Convert it using the probability of reaching look \(k\) before applying the weighted sum.
• Leaving out final-look nonrejections. In an efficacy-only design, the probability of rejection at the final look is not generally the full probability of stopping there. Include every trial that reaches the planned maximum.
• Mixing null and alternative probabilities. Use one parameter setting consistently for each expectation. \(E_0(N)\) and \(E_1(N)\) can differ even though the sample-size schedule is identical.

This calculation concerns the sample size associated with the specified stopping rule and planned looks. It does not by itself give expected calendar duration, account for enrollment that must exceed the analyzable sample because of loss to follow-up, or determine operating characteristics such as power. Such quantities require additional models or calculations. In particular, expected sample size is not a replacement for checking that the design meets its error-rate and power goals.