Writing an Executable Re-Estimation Rule

From a Design Goal to a Decision Rule

A promising-zone design is not executable merely because it names a lower and upper conditional-power cutoff. At the interim analysis, the team must know exactly which data to use, what assumptions enter the calculation, which sample sizes are eligible, how a target is searched, and what to do if the target cannot be reached. If those choices are left to judgment at the interim, the same observed data could lead to different actions.

The earlier material on choosing promising-zone cutoffs from design goals addresses how cutoffs can be selected. Here the task is to express such choices as a deterministic algorithm. The numbers below describe a hypothetical, one-sided, two-arm trial. They illustrate an operational rule, not a claim that the final test retains its nominal type I error after sample size is changed.

Definition: An executable re-estimation rule is a prespecified mapping from permitted interim inputs to a specified action. It defines the input data and analysis, model and conditional-power assumptions, decision cutoffs, allowable final sample sizes, rounding or search convention, and actions for every possible result, including boundary cases and failure to attain a target.

Model and Conditional-Power Calculation

Suppose the outcome is continuous, treatment groups are independently randomized in equal numbers, and the outcome variance \(\sigma^2\) is known and common to both groups. Let \(n\) denote the number of participants per arm at a look. For a prespecified positive treatment effect \(\Delta\), define the standardized drift per square root of per-arm sample size as \(\delta=\Delta/(\sigma\sqrt{2})\). The standardized treatment-control statistic at sample size \(n\), denoted \(Z_n\), then has mean \(\delta\sqrt{n}\) under that effect and variance 1.

Assume the planned interim occurs at \(n_1\) per arm and the final analysis at \(N>n_1\) per arm. The canonical normal model assumes independent information increments. In particular, conditional on the interim statistic \(Z_{n_1}=z_1\), the final statistic has variance \(1-n_1/N\). Under the fixed planning drift \(\delta\), its conditional mean is

m_N(z_1;\delta) = \sqrt{\frac{n_1}{N}}\,z_1 + \delta\frac{N-n_1}{\sqrt{N}}.

For a one-sided upper efficacy boundary \(c\), conditional power at candidate final sample size \(N\) is the probability, under the stated effect assumption and conditional on the interim statistic, that the final statistic exceeds \(c\):

CP(N\mid z_1;\delta) = P(Z_N>c\mid Z_{n_1}=z_1;\delta) = \Phi\!\left( \frac{m_N(z_1;\delta)-c}{\sqrt{1-n_1/N}} \right).

This is conditional power under a fixed assumed effect, not a probability that the treatment effect is true. The calculation requires a prespecified analysis direction and boundary, a defined interim look, and a compatible model for the statistic and its information. If variance is estimated rather than known, or the actual analysis uses a different statistic, the conditional-power calculation must be specified for that method rather than silently reusing this formula.

A Complete Hypothetical Rule

Consider the following fully specified design. Sample sizes are per arm, with equal allocation; thus a final value \(N=160\) means 320 participants in total. The interim is fixed at \(n_1=80\) per arm. The positive direction is prespecified, the final upper efficacy boundary is \(c=1.96\), and conditional power uses \(\delta=0.20\). Let the baseline final size be \(N_0=160\), the maximum be \(N_{\max}=240\), the promising-zone cutoffs be 0.30 and 0.70, and the target conditional power for an increase be 0.80. The hypothetical rule treats the lower futility stop as binding.

Validate the interim input.
Use the prespecified analysis of the first 80 participants per arm. Compute and record the signed interim statistic \(z_1\), the analysis-data version, and any required data-quality checks. Do not substitute an unplanned look or a different effect estimate.

Calculate baseline conditional power.
Compute \(CP(160\mid z_1;0.20)\) using \(c=1.96\) and the stated formula.

Apply the zone cutoffs.
If conditional power is below 0.30, stop for futility. If it is at least 0.70, retain \(N=160\) per arm. If it is at least 0.30 but below 0.70, enter the promising zone and proceed to the sample-size search. The inequalities assign values exactly on either cutoff to one action.

Search within the cap.
For the promising zone, evaluate candidate integer per-arm sizes \(N=160,161,\ldots,240\). Select the smallest candidate for which \(CP(N\mid z_1;0.20)\geq0.80\). If no candidate meets the target, set \(N=240\) and record that the cap binds and the target was not attained.

Implement and document the action.
For a selected final size \(N\), enroll \(N-80\) additional participants per arm. Record the calculated powers, candidate search, selected total and additional sample sizes, and the resulting action. Use the prespecified final analysis plan.

The search convention makes the rule determinate even if conditional power is not monotone over every candidate value: the algorithm checks the eligible integers in order and selects the first qualifying value. If a design requires only even per-arm sample sizes, that restriction must instead be stated in advance and the candidate grid changed accordingly.

Worked Examples

Worked Example: An Increase Reaches the Target

Suppose the hypothetical interim statistic is \(z_1=1.40\). At the baseline \(N=160\), the conditional mean is

m_{160} = \sqrt{\frac{80}{160}}(1.40) + 0.20\frac{160-80}{\sqrt{160}} = 0.989949+1.264911 = 2.254861.

The conditional standard deviation is \(\sqrt{1-80/160}=0.707107\). Therefore,

CP(160\mid1.40;0.20) = \Phi\!\left(\frac{2.254861-1.96}{0.707107}\right) = \Phi(0.41700) \approx0.6617.

This falls in the promising zone, since \(0.30\leq0.6617<0.70\). The algorithm searches the candidate integers. For example, at \(N=204\), the conditional mean is approximately 2.6131, the standard deviation is 0.7796, and conditional power is approximately 0.7989. At \(N=205\), the mean is approximately 2.6206, the standard deviation is 0.7809, and conditional power is approximately 0.8012. The prespecified integer search finds 205 as the first qualifying value. The rule therefore selects 205 per arm, or 410 participants total, and requires 125 additional participants per arm after the interim.

Worked Example: A Binding Futility Stop

Suppose instead that the observed interim statistic is \(z_1=-0.50\). At \(N=160\),

m_{160} = \sqrt{\frac{80}{160}}(-0.50) + 0.20\frac{80}{\sqrt{160}} = -0.353553+1.264911 = 0.911358.

Thus,

CP(160\mid-0.50;0.20) = \Phi\!\left(\frac{0.911358-1.96}{0.707107}\right) = \Phi(-1.483) \approx0.0690.

Because this is below 0.30, the action is to stop for futility. Under the stated hypothetical rule this stop is binding: the trial does not enroll further participants under this design. It is not a conclusion that the null hypothesis is true or that the treatment has no effect.

Worked Example: The Maximum Sample Size Binds

Let \(z_1=0.80\). At the baseline size \(N=160\), the conditional mean and power are

m_{160} = \sqrt{\frac{80}{160}}(0.80) + 0.20\frac{80}{\sqrt{160}} = 0.565685+1.264911 = 1.830596,

CP(160\mid0.80;0.20) = \Phi\!\left(\frac{1.830596-1.96}{0.707107}\right) = \Phi(-0.1830) \approx0.4274.

This is in the promising zone. At the cap \(N=240\), the conditional mean is

m_{240} = \sqrt{\frac{80}{240}}(0.80) + 0.20\frac{160}{\sqrt{240}} = 0.461880+2.065591 = 2.527471.

The conditional standard deviation is \(\sqrt{1-80/240}=0.816497\), giving

CP(240\mid0.80;0.20) = \Phi\!\left(\frac{2.527471-1.96}{0.816497}\right) = \Phi(0.6950) \approx0.7565.

Even at the largest permitted size, conditional power is below the 0.80 target. The rule therefore sets \(N=240\) per arm and records that the cap binds and the target is unattained. It must not report that the target was achieved merely because the maximum was selected.

Why the Derivation and the Audit Trail Matter

The conditional-power formula follows from the independent-increments representation. Write \(t=n_1/N\), and express the final statistic as

Z_N=\sqrt{t}\,Z_{n_1}+\sqrt{1-t}\,Z_{\mathrm{inc}},

where the standardized increment \(Z_{\mathrm{inc}}\) is independent of the interim statistic and, under drift \(\delta\), has distribution \(N(\delta\sqrt{N-n_1},1)\). Conditioning on \(Z_{n_1}=z_1\) fixes the first term. The conditional mean is therefore

\sqrt{t}\,z_1+\sqrt{1-t}\,\delta\sqrt{N-n_1} = \sqrt{\frac{n_1}{N}}z_1+\delta\frac{N-n_1}{\sqrt{N}},

and the conditional variance is \((1-t)\times1=1-n_1/N\). Standardizing the event \(Z_N>c\) gives the stated conditional-power expression. This justification also shows why the interim fraction, assumed effect, and final boundary cannot be omitted from an implementation specification.

An audit record should allow an independent reviewer to reproduce both the input and the decision. At minimum, retain the interim data cutoff and analysis version, \(z_1\), the assumed \(\delta\), \(c\), baseline and maximum sizes, zone cutoffs, target, candidate grid, unrounded calculation values, selected action, and additional enrollment required. Specify how missing or invalid interim data are handled before the study; an analyst should not improvise a replacement input after seeing the result.

Diagnostics, Alternatives, and Limits

A rule is going wrong operationally when its recorded inputs cannot be reproduced, different implementations assign a cutoff value to different branches, the candidate search uses a different sample-size unit than the cap, or the reported target is inconsistent with the computed conditional power. Check calculations at boundary values and test the implementation on values just below, exactly at, and just above each cutoff. Independently recompute several cases, including one where the cap binds. The earlier material on interim-effect instability, variance estimation, and sensitivity to interim timing is relevant when judging whether the specified input and model are credible; an executable rule does not remove those sources of uncertainty.

Conditional-power rules differ from direct effect-threshold rules: the former use the interim statistic together with an assumed future effect and final boundary, while the latter classify interim results by an effect estimate or another threshold. A fixed-size design avoids data-dependent sample-size selection but cannot respond to the observed interim result. A maximum-size rule with a cap limits burden, but when the cap binds it may fail to attain its target, as in the third example. These approaches answer different design questions; the choice should be explicit rather than hidden in operational discretion.

Finally, this algorithm specifies what to do, not whether the resulting inferential procedure controls type I error. Data-dependent sample-size changes can affect the validity of a final test. The re-estimation rule and the final testing method must therefore be considered together, and the inferential calibration must be justified separately. Conditional power is a planning quantity under its stated assumptions; it is not a guarantee of success, a posterior probability, or a substitute for checking operating characteristics.

Common pitfall: “The conditional power is close to the target, so we can round the sample size until the target looks met.” That is not an executable rule: it leaves the candidate set, rounding direction, and decision to discretion. Specify the integer grid and selection method beforehand, calculate using unrounded values, and report the achieved power at the selected size. If no candidate meets the target, apply the stated cap action and say that the target was not attained.

Check Your Understanding

Answer each question, then compare your reasoning with the solutions.

In the stated rule, what action applies if baseline conditional power is exactly 0.70?
Why is the assumed planning drift needed in addition to the observed interim statistic?
If the promising-zone search finds no candidate with conditional power at least 0.80, what must the record say?
At \(n_1=80\), \(N=160\), \(z_1=1.40\), and \(\delta=0.20\), calculate the conditional standard deviation and state the approximate conditional power.
Does selecting a sample size by this rule alone establish that the final test has type I error 0.025? Explain.

Solutions. (1) The rule retains \(N=160\), because the no-increase branch is defined by conditional power at least 0.70. (2) The interim statistic supplies the observed first-stage evidence, while \(\delta\) specifies the assumed drift for the unobserved future information; conditional power depends on both. (3) It must select the cap \(N=240\) and document that the target was not attained. (4) The standard deviation is \(\sqrt{1-80/160}=\sqrt{0.5}=0.707107\); the conditional power is approximately 0.6617. (5) No. The rule defines sample-size actions but does not by itself justify the type I error of the final test after a data-dependent change.