Selecting Promising-Zone Conditional-Power Cutoffs

Why Translate Conditional-Power Cutoffs?

A promising-zone rule is often stated using conditional power: results below a lower cutoff are not promising enough to trigger expansion, while results between lower and upper cutoffs may trigger it. But a design team also needs to know what interim results correspond to those probability cutoffs. Expressing the rule as thresholds on the interim Z statistic makes the decision rule easier to implement, audit, and communicate.

The translation depends on more than the two cutoffs. It also depends on the assumed effect used to calculate conditional power, the interim and planned final information, and the final rejection boundary. The cutoffs therefore do not define universal Z-statistic thresholds. This tutorial derives the conversion in a one-sided normal model and shows how to interpret the resulting region. It does not specify the size of any sample-size increase.

Model and Cutoff Rule

Assume a prespecified interim analysis at information \(I_1\), with planned final information \(I=I_1+I_2\), where \(I_2>0\) is the planned information remaining after the interim. Use the canonical normal model: the interim score is \(S_1=\sqrt{I_1}Z_1\), and the future score increment is independent of the interim data and has distribution \(S_2\sim N(\theta I_2,I_2)\) under the effect \(\theta\) used for the conditional-power calculation. Thus \(Z_1=S_1/\sqrt{I_1}\). The effect \(\theta\) is measured per square root of information; \(I_1\) and \(I_2\) are on the matching information scale.

For this derivation, the final one-sided test rejects \(H_0\) when the final statistic \(Z\) is at least a prespecified boundary \(c\). For example, a fixed one-sided normal test at significance level \(\alpha\) uses \(c=z_{1-\alpha}\), the \(1-\alpha\) standard-normal quantile. The normal model, information values, effect used for prediction, and rejection boundary are treated as specified for the calculation. In an application, the independent-increments approximation and variance assumptions must be appropriate for the design.

Definition: Conditional power at an interim value \(Z_1=z\), under assumed effect \(\theta\), is the probability—calculated under the specified model—that the final statistic will cross the rejection boundary, conditional on the interim result. A promising-zone rule specifies lower and upper cutoffs \(p_L\) and \(p_U\), with \(0<p_L<p_U<1\), for this conditional power.

A common operational convention is: if conditional power is below \(p_L\), do not expand; if it is at least \(p_L\) but below \(p_U\), consider the prespecified expansion; and if it is at least \(p_U\), do not expand. The rule must state what happens at equality. This convention treats a high conditional power as not needing expansion and a low conditional power as insufficiently promising. A low value does not, by itself, require stopping for futility.

Deriving the Interim-Statistic Thresholds

At final analysis the statistic is the accumulated score divided by the square root of total information. Conditional on \(Z_1=z\), the interim score is fixed at \(\sqrt{I_1}z\), while the future score has the distribution stated above. Therefore:

Z=\frac{S_1+S_2}{\sqrt{I_1+I_2}} \quad\Longrightarrow\quad Z\mid Z_1=z\sim N\left( \frac{\sqrt{I_1}z+\theta I_2}{\sqrt{I_1+I_2}}, \frac{I_2}{I_1+I_2} \right).

The conditional power is the probability that this normally distributed final statistic is at least \(c\). Standardizing by its conditional standard deviation gives:

\operatorname{CP}(z;\theta) = P(Z\ge c\mid Z_1=z,\theta) = \Phi\left( \frac{\sqrt{I_1}z+\theta I_2-c\sqrt{I_1+I_2}}{\sqrt{I_2}} \right),

where \(\Phi\) is the standard normal cumulative distribution function. To find the interim statistic giving a specified conditional power \(p\), set this expression equal to \(p\), apply \(\Phi^{-1}\), and solve for \(z\):

z(p)= \frac{c\sqrt{I_1+I_2}-\theta I_2+\sqrt{I_2}\,\Phi^{-1}(p)} {\sqrt{I_1}}.

Since \(\Phi^{-1}(p)\) increases as \(p\) increases, \(p_L<p_U\) implies \(z(p_L)<z(p_U)\). The conditional-power interval from \(p_L\) to \(p_U\) is therefore an interval on the interim Z scale, not a disconnected set of interim results.

Key result: Under the stated model, define \(z_L=z(p_L)\) and \(z_U=z(p_U)\). Then \(\operatorname{CP}(z;\theta)\) is below \(p_L\) when \(z<z_L\), is between the cutoffs when \(z_L\le z<z_U\), and is at least \(p_U\) when \(z\ge z_U\). The direction follows because conditional power increases with \(z\).

Worked Examples

Worked Example: Converting Two Cutoffs to Z Thresholds

Suppose a hypothetical one-sided design has \(I_1=50\), \(I_2=50\), final boundary \(c=1.96\), and assumed effect \(\theta=0.20\). The design team selects \(p_L=0.20\) and \(p_U=0.80\). Find the two interim-statistic thresholds.

Here \(I_1+I_2=100\), \(\sqrt{I_1}=\sqrt{I_2}=7.071\), and \(\theta I_2=10\). The standard-normal quantiles are approximately \(\Phi^{-1}(0.20)=-0.8416\) and \(\Phi^{-1}(0.80)=0.8416\). Substituting into the threshold formula:

z_L= \frac{1.96\sqrt{100}-10+\sqrt{50}(-0.8416)} {\sqrt{50}} = \frac{19.60-10-5.951}{7.071} \approx 0.516.

z_U= \frac{1.96\sqrt{100}-10+\sqrt{50}(0.8416)} {\sqrt{50}} = \frac{19.60-10+5.951}{7.071} \approx 2.199.

Thus the conditional power is between 0.20 and 0.80 for interim Z values from about 0.516 up to, but not including, 2.199. Under the stated operational convention, that is the region in which the prespecified rule considers expansion. These are thresholds under this particular effect, information schedule, and boundary—not generic cutoffs for every design.

Worked Example: How the Choice of Cutoffs Changes the Zone

Keep the hypothetical design in the previous example, but consider cutoffs \(p_L=0.10\) and \(p_U=0.75\). Calculate their Z thresholds and compare the resulting region with the one for cutoffs 0.20 and 0.80.

The relevant quantiles are \(\Phi^{-1}(0.10)=-1.2816\) and \(\Phi^{-1}(0.75)=0.6745\). Using the same information, boundary, and assumed effect:

z_L= \frac{19.60-10+7.071(-1.2816)}{7.071} = \frac{9.60-9.062}{7.071} \approx 0.076.

z_U= \frac{19.60-10+7.071(0.6745)}{7.071} = \frac{9.60+4.769}{7.071} \approx 2.032.

This rule's conditional-power zone is approximately \(0.076\le Z_1<2.032\), wider toward weaker interim results than the earlier zone, but with a lower upper boundary. That difference follows from both cutoffs changing: lowering the lower cutoff admits results with lower conditional power, while lowering the upper cutoff classifies more high-statistic results as already above the expansion region. Cutoff selection is a design choice; the arithmetic translates that choice but does not establish that it is appropriate.

Worked Example: Classifying Interim Results

For the design with \(p_L=0.20\), \(p_U=0.80\), and thresholds \(z_L\approx0.516\), \(z_U\approx2.199\), classify hypothetical interim results \(Z_1=0.40\), \(1.20\), and \(2.30\). Verify the middle result directly using conditional power.

Since \(0.40<0.516\), its conditional power is below 0.20, so the rule does not call for expansion. Since \(0.516\le1.20<2.199\), the middle result lies in the expansion region. Direct substitution gives:

\operatorname{CP}(1.20;0.20) = \Phi\left( \frac{7.071(1.20)+10-1.96(10)}{7.071} \right) = \Phi\left(\frac{-1.115}{7.071}\right) \approx \Phi(-0.1577) \approx 0.437.

The value 0.437 is between 0.20 and 0.80, confirming the classification. Finally, \(2.30>2.199\), so its conditional power is at least 0.80 and the rule does not call for expansion. In this case, a high interim statistic is considered sufficiently promising without an increase.

What Determines the Threshold Locations?

The formula makes the design dependencies explicit. Increasing \(p\) raises the threshold because stronger interim evidence is needed to achieve a higher conditional-power target. Increasing the assumed effect \(\theta\) lowers the threshold for a fixed \(p\): the model then expects more contribution from the remaining information. Changing the final boundary \(c\) or the information schedule also changes the thresholds. In particular, the same pair of conditional-power cutoffs can correspond to different interim Z regions in different designs.

These dependencies matter when selecting cutoffs. The lower cutoff determines how weak an interim result can be while still entering the zone where expansion is considered. The upper cutoff determines when the original plan is treated as sufficiently promising without expansion. Teams should specify the conditional-power assumption and the operational actions alongside the numeric cutoffs; reporting only the two probabilities leaves the interim decision rule incomplete.

Assumptions, Diagnostics, and Limitations

The conversion is appropriate only when the conditional distribution used in the calculation is credible. Check that the analysis is at the prespecified interim look, the information values correspond to the actual design, and the future increment is reasonably represented by an independent normal increment with the stated mean and variance. The assumed effect must be on the same scale as the information model. If nuisance parameters are estimated, or if the outcome, test statistic, or analysis plan does not support this approximation, the calculation may need a design-specific model rather than this formula.

A useful implementation check is to calculate conditional power at values just below, inside, and just above the proposed zone. It must increase as \(Z_1\) increases, and it must equal \(p_L\) and \(p_U\) at the calculated boundaries, apart from rounding. If a computed lower threshold exceeds the upper threshold despite \(p_L<p_U\), check the quantiles, signs, and units. If the threshold changes unexpectedly when a parameter changes, recompute from the displayed formula rather than relying on intuition alone.

A key limitation is that these thresholds describe a decision based on conditional power; they do not themselves determine a revised sample size. Nor do the cutoffs alone guarantee control of the type I error. An unplanned change to the sample size followed by an unadjusted final test can change the test's operating characteristics. Confirmatory use therefore requires the full adaptive design and its inferential procedure to be prespecified and appropriately calibrated.

Common Mistake

Pitfall: “The lower and upper conditional-power cutoffs are 0.20 and 0.80, so the interim Z thresholds are always fixed numbers.” This is incorrect: the conversion also uses the assumed effect, interim and remaining information, and final boundary. Compute both thresholds from the model and design actually specified; do not reuse values from a different information schedule or effect assumption.

Why the Translation Matters

Conditional-power cutoffs are intuitive probability statements, while interim-statistic thresholds are operational decision boundaries. Providing both makes it possible to see which observed results lead to which action and to verify that the rule is monotone and internally consistent. The threshold representation also makes sensitivity to assumptions visible. It should be read as a transparent description of the chosen promising-zone rule, not as evidence that the cutoffs are universally optimal or that expansion will necessarily improve the design.

Check Your Understanding

Use the stated normal-model threshold formula. Unless a question says otherwise, treat the rule as considering expansion for \(z_L\le Z_1<z_U\).

For \(I_1=I_2=25\), \(c=1.96\), \(\theta=0.20\), and \(p=0.50\), find \(z(p)\).
Why must \(z(0.80)\) exceed \(z(0.20)\) when the design parameters are fixed?
With the hypothetical design in Worked Example 1, classify \(Z_1=0.50\) using its lower threshold of approximately 0.516.
Does a conditional-power cutoff rule by itself guarantee that the final test has type I error \(\alpha\)? Explain briefly.

Solutions. (1) Since \(\Phi^{-1}(0.50)=0\), \(z(0.50)=[1.96\sqrt{50}-0.20(25)]/\sqrt{25}=(13.86-5)/5\approx1.772\). (2) The threshold increases with \(\Phi^{-1}(p)\), and \(\Phi^{-1}(0.80)>\Phi^{-1}(0.20)\); the other terms are unchanged. (3) Since \(0.50<0.516\), conditional power is below 0.20, so the rule does not call for expansion. (4) No. The cutoffs specify an adaptation decision, not the calibration of the final testing procedure; the full adaptive design must be assessed for type I error control.