Why a promising-zone increase changes power
An interim result can be neither strong enough to make success likely at the planned maximum nor so weak that increasing the sample size seems unhelpful. A design may therefore specify an interval of interim results—the promising zone—in which the maximum sample size is increased. The key quantity is not just the conditional power at one interim result. Final power averages conditional rejection probabilities over the distribution of all possible interim results.
This tutorial quantifies that gain in a one-sided normal model. It separates two questions that must not be confused: how much additional information raises conditional power, and how to make the adaptive final test retain its prespecified type I error. The introductory material on conditional power and promising-zone rules supplies the design motivation; here the emphasis is on the power calculation and its assumptions.
Under the original design, write the final statistic as \(Z_0=\sqrt{t}Z_1+\sqrt{1-t}Z_2\), where \(Z_2\) is the standardized score from the remaining information. Conditional on \(Z_1=z\), \(Z_2\) has variance 1 and mean \(\delta\sqrt{I_0-I_1}\) under the specified alternative. Under \(H_0\), its mean is zero. The original design rejects when \(Z_0>c\), where \(c=z_{1-\alpha}\) is the one-sided standard-normal critical value.
A promising zone is a prespecified interval \(L\leq Z_1\leq U\), with \(L<U\). In that interval, the design changes the total information to \(I'\), while outside it the original plan remains in force. The thresholds and the expanded information must be specified by the design; they are not selected after examining which choice gives the most favorable power.
Conditional power and preserving the error rate
For a given interim value \(z\), rejection under the original plan requires \(Z_2>(c-\sqrt{t}z)/\sqrt{1-t}\). Define this required standardized future-stage value as \(q(z)\). The conditional power at \(z\) is the probability that the future-stage statistic exceeds this value under the alternative.
If the total information is increased to \(I'\), the future-stage statistic is based on \(I'-I_1\) information. A simple comparison of its conditional power with the original conditional power is meaningful only if the final test remains valid. In particular, keeping the same critical value \(c\) for a newly standardized final statistic need not preserve the original type I error when the information choice depends on \(Z_1\).
One exact solution in this model is to preserve the original test's conditional error: for each observed \(z\), retain the original probability of rejection under \(H_0\), conditional on \(Z_1=z\). Under the original design that probability is \(1-\Phi(q(z))\). Thus, after an information increase, reject when the new standardized future-stage statistic exceeds the same \(q(z)\). Under \(H_0\), its conditional distribution is still standard normal, so its conditional rejection probability is unchanged. Consequently, its overall rejection probability under \(H_0\), averaged over \(Z_1\), remains \(\alpha\).
For \(\delta>0\) and \(I'>I_0\), \(\delta\sqrt{I'-I_1}>\delta\sqrt{I_0-I_1}\), so \(CP'(z)>CP_0(z)\) at every finite \(z\). This is a conditional statement under the model and the stated test rule. The overall power gain is the conditional gain averaged over the alternative distribution of the interim result, but only over the promising zone.
Here \(\phi\) is the standard-normal density, so the interim statistic under the specified alternative has distribution \(N(\delta\sqrt{I_1},1)\). The original fixed-design power is \(\Phi(\delta\sqrt{I_0}-c)\). Adding the integral to that baseline gives the adaptive design's power. The integration weights each local gain by how likely that interim result is under the alternative; an impressive conditional gain in a rarely reached part of the zone may contribute little overall.
Worked calculations
Worked Example: A Local Conditional-Power Gain
Suppose that, at a particular interim result \(z=0\), the conditional terminal statistic has standard deviation \(0.8\), and the one-sided rejection boundary is \(1.96\). Under the specified alternative, the mean contribution from the remaining data is \(0.6404\) under the original plan and \(0.96\) after expansion. Calculate the conditional powers and their difference.
For this illustration, conditional power is the probability that the conditional terminal statistic exceeds 1.96. Under a normal distribution with standard deviation \(0.8\), the original conditional power is \(\Phi((0+0.6404-1.96)/0.8)=\Phi(-1.6495)\approx0.0495\).
After expansion it is \(\Phi((0+0.96-1.96)/0.8)=\Phi(-1.25)\approx0.1056\). The local gain is approximately \(0.1056498-0.0495226=0.0561272\), or 5.61 percentage points. This is a conditional comparison at \(z=0\), not an overall power gain; an overall calculation also needs the interim-result distribution and the promising-zone limits.
Worked Example: Averaging Gains Across a Promising Zone
Assume the normal independent-increments model with \(I_1=36\), \(I_0=100\), \(I'=144\), \(\delta=0.20\), and \(c=1.96\). The promising zone is \(0\leq Z_1\leq1.5\). Approximate the overall power gain using three midpoint bins: \([0,0.5)\), \([0.5,1)\), and \([1,1.5]\).
The interim information fraction is \(t=36/100=0.36\), so \(\sqrt t=0.6\) and \(\sqrt{1-t}=0.8\). Therefore \(q(z)=(1.96-0.6z)/0.8=2.45-0.75z\). The original and expanded future-stage mean contributions are \(0.20\sqrt{64}=1.6\) and \(0.20\sqrt{108}\approx2.0785\). Thus \(CP_0(z)=\Phi(0.75z-0.85)\) and \(CP'(z)=\Phi(0.75z-0.3715)\).
Under the alternative, \(Z_1\sim N(1.2,1)\), since \(\delta\sqrt{I_1}=0.20\sqrt{36}=1.2\). The bin probabilities and midpoint gains are approximately:
| Bin | Alternative probability | Midpoint | Conditional-power gain | Weighted gain |
|---|---|---|---|---|
| [0, 0.5) | 0.1269 | 0.25 | 0.1732 | 0.02197 |
| [0.5, 1) | 0.1788 | 0.75 | 0.1889 | 0.03377 |
| [1, 1.5] | 0.1972 | 1.25 | 0.1794 | 0.03538 |
For example, in the first bin the midpoint gain is \(\Phi(-0.1840)-\Phi(-0.6625)\approx0.426991-0.253832=0.173159\approx0.1732\); multiplying by its probability gives \(0.1269(0.17316)\approx0.02197\). Summing the weighted gains gives approximately \(0.0911\). The original fixed-design power is \(\Phi(0.20\sqrt{100}-1.96)=\Phi(0.04)\approx0.5160\). The adaptive power is therefore approximately \(0.5160+0.0911=0.6071\). The integral has been approximated by a three-bin midpoint calculation, so these figures are approximate, not an exact evaluation.
Worked Example: Comparing Two Expansion Sizes
Keep the information, effect, critical value, and zone from the preceding example, but expand instead to \(I'=196\). Estimate the gain over the same three bins and compare it with the increase to \(I'=144\).
The future-stage mean contribution is now \(0.20\sqrt{196-36}=0.20\sqrt{160}\approx2.5298\). The threshold \(q(z)=2.45-0.75z\) is unchanged because the conditional-error rule retains the original null rejection probability at each interim value. Hence \(CP'(z)=\Phi(0.75z+0.0798)\).
At the three midpoints \(0.25,0.75,1.25\), the conditional-power gains are approximately \(0.3516,0.3528,0.3106\), respectively. Weighting by the same alternative bin probabilities gives \(0.1269(0.3516)+0.1788(0.3528)+0.1972(0.3106)\approx0.1690\). The approximate final power is \(0.5160+0.1690=0.6850\). In this example, increasing the maximum information from 144 to 196 raises the estimated gain over the original design from about 9.11 to 16.90 percentage points. The larger increase costs more information; power alone does not determine whether it is an attractive design choice.
Interpretation, diagnostics, and limitations
The calculations rely on a specific set of assumptions: a correctly specified normal model, known variance, independent increments, a single interim look at the planned information, and a fixed treatment effect for power evaluation. If the variance is estimated, the endpoint is binary, information is not proportional to sample size, or the interim timing changes, these equations may not apply without modification. The value of \(\delta\) is the effect used to evaluate operating characteristics; it is not necessarily the effect estimated at interim.
A useful diagnostic is to check the calculation in stages. Confirm that the alternative interim mean is \(\delta\sqrt{I_1}\), that the future information is \(I-I_1\), and that the same \(q(z)\) is used before and after expansion under conditional-error preservation. Check that the promised zone is integrated with the alternative density—not the null density—when calculating power under the specified alternative. Also verify that the estimated gain is between zero and the probability of reaching the zone, since the conditional-power difference cannot exceed one.
There are two important comparisons. A fixed design uses \(I_0\) for everyone and has power \(\Phi(\delta\sqrt{I_0}-c)\), with no adaptation. A promising-zone design uses more information only for interim results in the specified interval and retains the original plan elsewhere. A naïve approach that increases the information but uses the original fixed critical value without recalibrating the adaptive test can produce a different type I error; a resulting power increase is not a fair comparison if type I error has also increased. Conditional-error preservation makes the comparison interpretable at the original error rate in this model.
The midpoint integration above is a transparent approximation, not a design-analysis standard. For a final design, evaluate the integral with sufficiently accurate numerical quadrature, check the calculation independently, and assess power at the treatment effects relevant to the scientific question. Moreover, a gain under one assumed effect does not guarantee a similar gain at other effects. The method quantifies power for the specified rule and model; it does not establish that the effect assumption is realistic or that the added sample size is worthwhile.
Check Your Understanding
Answer each question, then compare with the solutions below.
- In the normal model, what is the distribution of \(Z_1\) under a specified alternative with effect \(\delta\)?
- Why does retaining \(q(z)\) preserve the conditional type I error after an information increase?
- Why is a conditional-power gain at one interim value not the same as the overall power gain?
- With a positive effect, what happens to conditional power when future information increases under the stated model and test rule?
Solutions. (1) \(Z_1\sim N(\delta\sqrt{I_1},1)\). (2) Under \(H_0\), the future-stage standardized statistic remains \(N(0,1)\); exceeding the same \(q(z)\) therefore has the same conditional rejection probability \(1-\Phi(q(z))\). (3) Overall gain averages local gains using the alternative distribution of \(Z_1\), and includes only the promising zone. (4) It increases, because the future-stage mean \(\delta\sqrt{I-I_1}\) increases while \(q(z)\) is fixed; thus the normal probability of exceeding the boundary increases.