Defining the Promising Zone

Why Define a Promising Zone?

At an interim analysis, an unblinded sample-size re-estimation rule may allow a prespecified increase only when the interim result is neither so weak that continuation is considered futile nor so strong that the planned design already has sufficient evidence. The promising zone formalizes this middle region using conditional power. It is a rule for identifying interim results at which an increase may be considered; defining the region does not, by itself, specify the size of the increase or guarantee that the trial will reject its null hypothesis.

The definition depends on choices made before examining the interim result: the conditional-power model, the effect used in that model, a lower cutoff for futility, and an upper cutoff for sufficient evidence. Changing any of these choices can move the zone. The introductory material on conditional power establishes the idea of calculating the probability of final rejection conditional on interim data; here the focus is how prespecified thresholds partition the possible interim results.

Definition Under a Normal Model

Consider a one-sided design in which rejection at the final analysis occurs when the final standardized statistic $Z_{\mathrm{final}}$ is at least $c$. At the interim, let $Z_{\mathrm{int}}=z$, and let $r$ be the fraction of total planned information that remains, so the fraction already accrued is $1-r$, with $0<r<1$. Let $\theta$ be the assumed standardized effect at the planned final information. In the canonical normal model, conditional on the interim statistic and this assumed effect, the final statistic has distribution

Z_{\mathrm{final}}\mid Z_{\mathrm{int}}=z,\theta \sim N\!\left(\sqrt{1-r}\,z+\theta r,\ r\right).

This model assumes valid information measurements, a normal statistic with independent future increments conditional on the interim data, and the specified effect and final boundary. For example, the model is appropriate when the relevant variance is known or treated as fixed for this calculation and the planned analysis uses the stated standardized boundary. If the actual design has different final boundaries or a different conditional distribution, its conditional-power calculation must reflect those features instead.

Definition: Under the stated model, the conditional power at interim value $z$ is the probability of crossing the final boundary, conditional on the observed interim statistic and the assumed effect: $$ CP(z;\theta,r,c) = P(Z_{\mathrm{final}}\ge c\mid Z_{\mathrm{int}}=z,\theta) = \Phi\!\left( \frac{\sqrt{1-r}\,z+\theta r-c}{\sqrt r} \right). $$ For prespecified cutoffs $L$ and $U$, where $0<L<U<1$, the futility region consists of interim results with $CP<L$; the promising zone consists of results with $L\le CP<U$; and the sufficient-evidence region consists of results with $CP\ge U$. A protocol may specify different conventions at equality, but it must do so in advance.

The labels describe a decision rule, not certainty about the eventual outcome. “Futility” means that conditional power is below a chosen threshold under the stated model; it does not prove that the treatment has no effect. “Sufficient evidence” means the conditional power meets an upper benchmark; it does not mean final rejection is assured. In the promising zone, the prespecified rule may permit consideration of a sample-size increase. A rule might instead prescribe a particular response or no change, but those operational details are separate from the definition of the zone.

How Conditional-Power Cutoffs Become Interim Boundaries

Because the normal cumulative distribution function $\Phi$ is strictly increasing, conditional power increases with $z$ in this one-sided model. For a chosen conditional-power value $q$, define $z_q$ as the interim statistic at which conditional power equals $q$. Inverting the conditional-power equation gives the boundary:

q = \Phi\!\left( \frac{\sqrt{1-r}\,z_q+\theta r-c}{\sqrt r} \right) \quad\Longrightarrow\quad z_q = \frac{c+\sqrt r\,\Phi^{-1}(q)-\theta r}{\sqrt{1-r}}.

The direction of the inequalities follows from monotonicity: values below $z_L$ are below the lower conditional-power cutoff, values from $z_L$ up to but not including $z_U$ are promising, and values at or above $z_U$ meet the upper cutoff. Thus the promising zone can be written $z_L\le Z_{\mathrm{int}}<z_U$, using the stated equality convention.

Worked Example: Calculating Conditional Power at an Interim Result

Suppose the remaining information fraction is $r=0.40$, the assumed standardized effect is $\theta=2.50$, the final boundary is $c=1.96$, and the interim statistic is $z=1.00$. These are hypothetical design values. Assume the canonical normal model and final rejection rule stated above apply.

The conditional-power formula is $$ CP(1.00;2.50,0.40,1.96) = \Phi\!\left( \frac{\sqrt{0.60}(1.00)+2.50(0.40)-1.96}{\sqrt{0.40}} \right). $$

Substituting and calculating, $$ \frac{0.7745967+1.0000-1.9600}{0.6324555} = \frac{-0.1854033}{0.6324555} \approx -0.2931, \qquad CP\approx\Phi(-0.2931)=0.3847. $$

If the prespecified illustrative cutoffs are $L=0.20$ and $U=0.80$, this interim result is in the promising zone because $0.20\le0.3847<0.80$. This classification does not mean there is a 38.47% probability that the null hypothesis is true. It means that, under the specified effect and model, the conditional probability of crossing the final boundary is about 38.47%.

Worked Example: Finding the Two Interim-Statistic Boundaries

Continue with $r=0.40$, $\theta=2.50$, and $c=1.96$. For illustration, retain the conditional-power cutoffs $L=0.20$ and $U=0.80$. The thresholds are hypothetical and are used here to show how a promising zone is constructed, not to recommend particular cutoffs.

For the lower boundary, use $\Phi^{-1}(0.20)\approx-0.8416212$: $$ z_L = \frac{1.96+\sqrt{0.40}\,\Phi^{-1}(0.20)-2.50(0.40)} {\sqrt{0.60}} = \frac{1.96+\sqrt{0.40}(-0.8416212)-1.0000} {\sqrt{0.60}} \approx 0.5522. $$

For the upper boundary, use $\Phi^{-1}(0.80)\approx0.8416212$: $$ z_U = \frac{1.96+\sqrt{0.40}\,\Phi^{-1}(0.80)-2.50(0.40)} {\sqrt{0.60}} = \frac{1.96+\sqrt{0.40}(0.8416212)-1.0000} {\sqrt{0.60}} \approx 1.927. $$

The arithmetic uses the unrounded quantiles and square roots before rounding the results. The resulting regions are $Z_{\mathrm{int}}<0.5522$ for conditional power below 0.20, $0.5522\le Z_{\mathrm{int}}<1.927$ for the promising zone, and $Z_{\mathrm{int}}\ge1.927$ for conditional power at least 0.80. In practice, the protocol should implement the rule using prespecified numerical precision and a stated equality convention.

Why the Assumptions and Effect Choice Matter

The inverse formula is a direct consequence of the assumed conditional distribution, not a universal conversion from an interim $Z$-statistic to a zone. The standardized effect $\theta$ is part of the conditional-power calculation. It may be a design effect specified in advance or an effect estimate used according to a prespecified procedure. In either case, the choice must be clear: the same interim statistic can be classified differently under different assumed effects.

Worked Example: The Same Interim Result Under a Different Assumed Effect

Suppose $r=0.40$, $c=1.96$, and $z=1.00$ remain unchanged, but the assumed standardized effect is now $\theta=1.50$. The normal-model assumptions are the same as in the first example. With the illustrative futility cutoff $L=0.20$, calculate: $$ CP = \Phi\!\left( \frac{\sqrt{0.60}(1.00)+1.50(0.40)-1.96}{\sqrt{0.40}} \right) = \Phi\!\left( \frac{0.7745967+0.6000-1.9600}{0.6324555} \right). $$

The standardized value is $$ \frac{-0.5854033}{0.6324555}\approx-0.9256, \qquad CP\approx\Phi(-0.9256)=0.1773. $$

This is below $L=0.20$, so the result falls in the futility region under this assumed effect, even though the same interim statistic gave conditional power 0.3847 under $\theta=2.50$. The example shows why the effect assumption is not a minor computational detail: it helps define the decision regions and must be justified and prespecified.

A second arithmetic check illustrates a different information fraction. Suppose $r=0.50$, $z=1.80$, $c=1.96$, and $\theta=2.80$. Then: $$ CP = \Phi\!\left( \frac{\sqrt{0.50}(1.80)+2.80(0.50)-1.96}{\sqrt{0.50}} \right) = \Phi\!\left( \frac{1.2728+1.4000-1.9600}{0.7071} \right) \approx\Phi(1.0080)=0.8433. $$

With an illustrative sufficient-evidence cutoff of 0.80, this result is in the sufficient-evidence region. The numerator is $0.7128$, and $0.7128/0.7071$ is about $1.0080$; retaining unrounded square roots gives the same reported value. This check also shows why intermediate arithmetic should be verified before assigning a region near a cutoff.

Interpreting the Rule and Diagnosing Problems

The promising zone matters because it separates three different interim situations without treating conditional power as an observed final result. In comparison, a fixed-sample design has no interim-based zone or sample-size change. A futility-only rule can identify weak results but does not distinguish a middle range for possible expansion from a strong range where the planned sample size is retained. A promising-zone rule makes that distinction through two conditional-power thresholds. It is still only one component of an adaptive design.

Several checks help identify a faulty definition or implementation:

Check the information convention. Here $r$ is the fraction remaining, not the fraction already accrued. Using the wrong fraction changes both the conditional distribution and the cutoffs.
Check monotonicity. With $0<r<1$, conditional power increases with $z$, since the coefficient of $z$ inside $\Phi$ is $\sqrt{1-r}/\sqrt r>0$. A calculation that assigns lower conditional power to a larger interim statistic signals an error or a different model.
Check the effect and boundary. Confirm that the assumed effect, final critical value, information fraction, and one-sided direction match the prespecified design.
Check numerical cutoffs. Recalculate values near a boundary using unrounded inputs. Rounding an interim statistic or a threshold too early can change its assigned region.
Check that the model remains credible. Poorly estimated variance, non-independent increments, changed outcome definitions, or departures from the planned data-generating assumptions can make the nominal conditional power a poor guide.

A common mistake is to say, “The interim result is in the promising zone, so the trial has enough evidence to reject $H_0$.” That reasoning confuses a conditional probability benchmark with a realized final test result. The correction is that zone membership states only where the interim result lies relative to prespecified conditional-power cutoffs; final rejection depends on the eventual analysis and its valid inferential procedure.

Another important limitation is that defining a zone does not establish type I error control. An unblinded sample-size change can affect the distribution of the final test statistic under $H_0$. Whether the overall design controls its planned significance level depends on the complete adaptation and testing procedure, not merely on the labels “futility,” “promising,” and “sufficient evidence.” Conditional power is also model-dependent: a value calculated under one assumed effect is not a model-free probability of success.

Key takeaway: In the stated normal model, a promising zone is the interval of interim results whose conditional power lies between two prespecified cutoffs. Its boundaries follow by inverting the conditional-power equation. The zone identifies where a prespecified increase may be considered; it does not alone determine the increase, assure rejection, or guarantee error control.

Check Your Understanding

Answer each question using the definitions and assumptions in this tutorial.

If $L=0.15$, $U=0.75$, and an interim result has conditional power 0.10, which region contains it?
In the normal model, does conditional power increase or decrease as the one-sided interim statistic $z$ increases? Give the mathematical reason.
For $r=0.50$, $\theta=2.00$, $c=1.96$, and $z=0$, calculate conditional power to three significant figures.
Can the same interim statistic belong to different regions under two different assumed effects? Explain using the conditional-power formula.
Does being in the sufficient-evidence region guarantee that the trial will reject $H_0$? Why or why not?

Solutions. (1) It is in the futility region because $0.10<0.15$. (2) It increases: the argument of $\Phi$ has slope $\sqrt{1-r}/\sqrt r>0$ with respect to $z$, and $\Phi$ is increasing. (3) The standardized value is $(0+2.00(0.50)-1.96)/\sqrt{0.50}=-0.96/\sqrt{0.50}\approx-1.3576$, so conditional power is $\Phi(-1.3576)\approx0.0873$. (4) Yes. The assumed effect $\theta$ appears in the numerator as $\theta r$, so changing it can change conditional power and hence zone membership. (5) No. The region says conditional power meets a prespecified benchmark under the model; the final statistic may still fail to cross its rejection boundary.

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