Worked Promising-Zone Rule Calculation

From an Interim Result to a Design Action

A promising-zone rule translates an interim result into a specified action: stop for futility, continue at the planned sample size, or increase the sample size. The calculation is not just a power calculation at a new size. It uses the observed interim statistic to assess the chance of crossing the final efficacy boundary, conditional on the data already observed and on a stated assumption about the future treatment effect.

The introductory material on conditional power and promising zones establishes the role of these thresholds. Here the emphasis is on carrying one complete rule through its numerical branches, including integer-size search and a maximum-size constraint. The calculations below are hypothetical and use a one-sided normal test.

Definition: At an interim look, conditional power is the probability, under a specified model for future outcomes and conditional on the interim data, that the final test will reject \(H_0\) in the prespecified direction. In a planning-effect calculation, the future treatment effect is set to the effect assumed in the design, not estimated anew from the interim result.

Model, Notation, and the Rule

Suppose a normally distributed endpoint is analyzed with known variance and a fixed, upper one-sided final critical value \(c=1.960\), corresponding to a one-sided significance level of approximately 0.025. Assume independent increments of information: the future observations are independent of the interim observations, and the standardized interim and future contributions combine according to their information fractions. Use a normal approximation for the final statistic. These assumptions make the conditional distribution below applicable; in a real design they must be checked against the endpoint, analysis, and allocation plan.

Let \(n_1\) be the information accrued at the interim look, \(N\) a candidate final information total, and \(z_1\) the observed interim Z-statistic. Assume the planning effect gives a final standardized mean of 4.000 at the originally planned total \(N_0=200\). Under that planning effect, the conditional final statistic has standard deviation \(\sqrt{1-n_1/N}\) and mean

\mu_c(N;z_1)=\sqrt{\frac{n_1}{N}}z_1+ 4\sqrt{\frac{N}{200}}\left(1-\frac{n_1}{N}\right).

Consequently, the planning-effect conditional power at candidate total \(N>n_1\) is

CP(N;z_1)= \Phi\!\left( \frac{\mu_c(N;z_1)-1.960}{\sqrt{1-n_1/N}} \right).

For the worked rule, take \(n_1=40\), \(N_0=200\), a futility threshold of 0.200, a target conditional power of 0.800, and a maximum permitted total of \(N_{\max}=400\). The action is based on conditional power evaluated at the originally planned total:

If \(CP(N_0;z_1)<0.200\), stop for futility.
If \(CP(N_0;z_1)\geq0.800\), continue at \(N_0\), with no increase.
If \(0.200\leq CP(N_0;z_1)<0.800\), enter the promising zone. Increase to the smallest allowable \(N\) for which \(CP(N;z_1)\geq0.800\), subject to \(N\leq N_{\max}\). If no allowable size reaches the target by \(N_{\max}\), use the cap and report that the target was not reached.

The thresholds and maximum size are design choices, not universal constants. The numerical examples use integer information units as candidate totals. If the study requires a particular allocation or block size, candidate totals must instead be restricted to the permitted sizes and rounded accordingly.

A Complete Rule Calculation

Worked Example: Futility Stop

Suppose the hypothetical interim result is \(z_1=-5.000\). First evaluate conditional power at the planned total \(N_0=200\). Here \(n_1/N_0=40/200=0.200\), so

\begin{aligned} \mu_c(200;-5) &=\sqrt{0.200}(-5)+4\sqrt{\frac{200}{200}}(0.800)\\ &=-2.2361+3.2000=0.9639,\\ CP(200;-5) &=\Phi\!\left(\frac{0.9639-1.960}{\sqrt{0.800}}\right)\\ &=\Phi(-1.1135)\approx0.133. \end{aligned}

The conditional power is below 0.200, so this outcome meets the specified futility-stop condition. The rule stops the trial at the interim; it does not search for a larger sample size. This is a rule-based stopping decision, not a conclusion that the null hypothesis is true.

Worked Example: Continue at the Planned Size

Now suppose the hypothetical interim result is \(z_1=0.500\). At \(N_0=200\), the conditional mean is

\begin{aligned} \mu_c(200;0.5) &=\sqrt{0.200}(0.500)+4(1)(0.800)\\ &=0.2236+3.2000=3.4236,\\ CP(200;0.5) &=\Phi\!\left(\frac{3.4236-1.960}{\sqrt{0.800}}\right)\\ &=\Phi(1.6359)\approx0.949. \end{aligned}

Since 0.949 exceeds the target 0.800, the rule continues with the planned total \(N=200\); it does not increase the sample size. This example also illustrates why the classification is made at the planned total, not by inspecting conditional power at an arbitrarily selected smaller candidate size.

Worked Example: Promising-Zone Increase With an Integer Search

Suppose instead that the hypothetical interim statistic is \(z_1=-3.000\). At \(N_0=200\),

\begin{aligned} \mu_c(200;-3) &=\sqrt{0.200}(-3)+4(1)(0.800)\\ &=-1.3416+3.2000=1.8584,\\ CP(200;-3) &=\Phi\!\left(\frac{1.8584-1.960}{\sqrt{0.800}}\right)\\ &=\Phi(-0.1136)\approx0.455. \end{aligned}

This is between 0.200 and 0.800, so the result is in the promising zone. Search candidate totals for the first one that reaches 0.800. For clarity, the argument of \(\Phi\) can be written using future information \(m=N-40\):

A(N)= z_1\sqrt{\frac{40}{N-40}} +4\sqrt{\frac{N-40}{200}} -1.960\sqrt{\frac{N}{N-40}}, \qquad CP(N;z_1)=\Phi(A(N)).

Because \(\Phi\) is increasing, the target is met exactly when \(A(N)\geq\Phi^{-1}(0.800)\approx0.8416\). Checking the adjacent integer candidates around the crossing gives

\begin{aligned} A(264) &=-3\sqrt{\frac{40}{224}}+ 4\sqrt{\frac{224}{200}}- 1.960\sqrt{\frac{264}{224}}\\ &\approx0.8377, &CP(264;-3)&\approx0.799,\\ A(265) &=-3\sqrt{\frac{40}{225}}+ 4\sqrt{\frac{225}{200}}- 1.960\sqrt{\frac{265}{225}}\\ &\approx0.8506, &CP(265;-3)&\approx0.803. \end{aligned}

The integer search over allowable totals finds 265 as the first total meeting the target; 264 falls short, and the preceding candidate totals also fall short. Thus the revised total is \(N=265\), an increase of \(265-200=65\) information units over the planned total. It is below \(N_{\max}=400\), so the cap does not bind. The result is conditional on the planning effect and model; it does not mean that final power is guaranteed to be 0.800.

Checking Candidate Calculations and the Cap

A candidate-size calculation is useful only when its role in the rule is clear. For example, using \(z_1=0.500\) and \(N=80\) gives \(n_1/N=40/80=0.500\), so

\begin{aligned} \mu_c(80;0.5) &=\sqrt{0.5}(0.5)+4\sqrt{\frac{80}{200}}(0.5)\\ &=0.3536+1.2649=1.6185,\\ CP(80;0.5) &=\Phi\!\left(\frac{1.6185-1.960}{\sqrt{0.5}}\right) \approx\Phi(-0.4830)\approx0.315. \end{aligned}

The planning-effect contribution is \(4\sqrt{80/200}(0.5)=1.2649\), not 0.8944. At \(N=80\), conditional power is about 0.315 and is therefore in the promising range under the 0.200 futility threshold. But in the \(z_1=0.500\) example, the rule first evaluates the originally planned total, where conditional power is about 0.949, and therefore makes no increase. The \(N=80\) value is a candidate calculation, not the action called for by that example’s rule.

The distinction matters when checking the target near a boundary. For the same \(z_1=0.500\), at \(N=140\), substitution gives a conditional mean of approximately 2.6577 and a standard deviation of \(\sqrt{100/140}\approx0.8452\):

CP(140;0.5) =\Phi\!\left(\frac{2.6577-1.960}{0.8452}\right) =\Phi(0.8256)\approx0.795.

At \(N=141\), the corresponding values are approximately 2.6721 and 0.8464, giving

CP(141;0.5) =\Phi\!\left(\frac{2.6721-1.960}{0.8464}\right) =\Phi(0.8414)\approx0.7999.

Thus 141 remains strictly below 0.800; a calculation rounded to “about 0.800” is not enough to establish that the target is met. At \(N=142\), the conditional mean is approximately 2.6864 and the standard deviation is approximately 0.8475, so

CP(142;0.5) =\Phi\!\left(\frac{2.6864-1.960}{0.8475}\right) \approx\Phi(0.8571)\approx0.804.

In any real implementation, evaluate the prescribed candidate sizes, including constraints from allocation and rounding, and apply the cap explicitly. If the target remains unmet at \(N_{\max}\), report that fact rather than describing the capped design as having attained the target.

Why the Calculation Matters—and Where It Can Fail

Conditional power is a decision input, not a guarantee of the eventual outcome. Its value depends on the assumed future effect, the variance model, the interim statistic, and the final rejection boundary. If the planning effect is optimistic or the variance assumption is inappropriate, the resulting size recommendation may be misleading. Unblinded adaptation also has inferential implications: changing sample size in response to interim treatment-effect information can affect the operating characteristics of the final procedure. This arithmetic alone does not establish that the overall design controls its intended error rate; the design’s testing and analysis plan must address that issue.

A common mistake is to calculate conditional power at the proposed increased size and use that value to decide whether the original result was in the promising zone. The correction is to apply the rule in order: classify using conditional power at \(N_0\), then, only in the promising zone, search for a revised size. Another mistake is to treat a value rounded to 0.800 as certainly meeting the target. Retain enough precision to distinguish values just below and above the threshold, and verify the final decision using unrounded calculations.

A fixed-size design avoids interim-driven size changes but does not respond to interim evidence. A promising-zone rule allows a prespecified increase in a middle range of conditional power, while a futility-only rule can stop for low conditional power without adding participants. These approaches answer different design objectives; none removes the need to state the effect and variance assumptions. The rule is also limited by the maximum size: a cap can prevent the conditional-power target from being reached.

Key takeaway: Calculate conditional power at the originally planned total to classify the interim result. Only after identifying the promising zone should you search permitted candidate totals for the target, check the cap, and report whether the target was actually attained.

Check Your Understanding

Use the worked rule and assumptions above. Answers are given after the questions so that each decision can be checked.

For \(z_1=-5.000\), why does the rule stop rather than search for a larger total?
For \(z_1=0.500\), does the candidate result at \(N=80\) determine the final action? Explain.
For the promising-zone example with \(z_1=-3.000\), why is \(N=264\) not selected while \(N=265\) is?
If no allowable candidate total at or below \(N_{\max}\) reaches conditional power 0.800, what should be reported?

Solutions. (1) At the planned total, conditional power is approximately 0.133, below the 0.200 futility threshold, so the prescribed action is stopping. (2) No. Classification uses \(CP(200;0.5)\approx0.949\), which is above the target; the rule continues at \(N=200\). The \(N=80\) calculation is only a candidate value. (3) The integer search gives \(CP(264;-3)\approx0.799\), below the target, and \(CP(265;-3)\approx0.803\), above it; 265 is the first allowable total meeting the target. (4) Report the capped total and explicitly state that the conditional-power target was not reached.