From Conditional Power to a Reproducible Decision
An interim conditional-power calculation is informative only if it leads to an unambiguous action. A rule must state which conditional-power values leave the planned sample size unchanged, which values permit an increase, what target the increase is meant to reach, and what to do if the target is unreachable. The rule must also identify the data and assumptions used in the calculation. Otherwise, different people could see the same interim results and make different decisions.
Here, a deterministic conditional-power re-estimation rule means a prespecified mapping from interim information to a final planned sample size (and, if specified, an associated action). Deterministic describes the mapping: once its inputs are fixed, the rule returns one decision. It does not mean that trial outcomes or conditional power are nonrandom.
Model and Conditional-Power Calculation
For a concrete rule, suppose independent observations follow \(X_i\sim N(\mu,\sigma^2)\), with known \(\sigma\). The trial tests \(H_0:\mu=0\) against \(H_a:\mu>0\) at a fixed one-sided significance level \(\alpha\), using one final test. At an interim analysis, \(n_1\) observations have been recorded and their mean is \(\bar{x}_1\). The proposed final total sample size is \(N>n_1\). The re-estimation calculation assumes a prespecified effect \(\delta^*>0\) for the future observations. This effect is an assumption used to calculate conditional power, not a claim that the true effect is known.
The assumptions matter: observations are independent and normally distributed with the stated known variance; the future observations are independent of the interim observations; and the final test, critical value, and assumed effect are specified. With one final look, the critical value is \(c=z_{1-\alpha}\), the \(1-\alpha\) quantile of the standard normal distribution. This calculation is not, by itself, a replacement for any error-control method required by a design with multiple looks.
Let \(m=N-n_1\) denote the number of future observations. Given the interim data and the assumed effect, their sum has mean \(m\delta^*\) and variance \(m\sigma^2\). The final test statistic is \(Z_N=\sum_{i=1}^N X_i/(\sigma\sqrt{N})\), and rejection occurs when \(Z_N>c\). Thus its conditional mean and standard deviation are
Standardizing the event \(Z_N>c\) gives the conditional-power formula:
where \(\Phi\) is the standard normal cumulative distribution function. The numerator compares the conditional expected total response with the final rejection boundary; the denominator is the standard deviation of the remaining random contribution on the same scale. This derivation also shows why the interim data, assumed effect, variance, test, and candidate \(N\) must all be specified to reproduce the calculation.
Specify the Rule Before Applying It
A promising-zone rule can be written using a lower cutoff \(L\), an upper cutoff \(U\), and a target conditional power \(T\), chosen so that \(0<L<U<T<1\). Let \(N_0\) be the planned total sample size and \(N_{\max}\) the prespecified maximum. For clarity, define the following policy: below \(L\), do not increase the sample size; at or above \(U\), also retain \(N_0\); between the cutoffs, select the smallest eligible final size that reaches \(T\), or \(N_{\max}\) if no eligible size reaches it. The lower zone is not an automatic instruction to stop: any futility action must be separately specified.
- If \(C<L\), set \(N_{\text{new}}=N_0\).
- If \(L\leq C<U\), choose the smallest eligible \(N\in[N_0,N_{\max}]\) for which \(CP(N)\geq T\). If no eligible \(N\) meets the target, set \(N_{\text{new}}=N_{\max}\).
- If \(C\geq U\), set \(N_{\text{new}}=N_0\).
This is one policy, not a universal definition of a promising zone. A different policy could prescribe a futility action in the lower zone or a different response above the upper cutoff. Reproducibility requires writing down the chosen policy rather than relying on the label “promising zone.”
Worked Examples
Worked Example: Select the Smallest Size That Meets the Target
Suppose, hypothetically, that \(n_1=40\), \(\bar{x}_1=0.12\) units, \(\sigma=1\) unit, \(\delta^*=0.20\) units, \(\alpha=0.025\), and \(N_0=100\). Use \(L=0.25\), \(U=0.60\), \(T=0.80\), and \(N_{\max}=240\). Assume eligible final sizes are integer totals. Since \(c=z_{0.975}\approx1.96\), the interim conditional power at the planned size is
Because \(0.25\leq0.3590<0.60\), the rule calls for an increase aimed at conditional power \(0.80\). For \(N=215\),
This is just below the target. For \(N=216\),
To verify that 216 is the smallest qualifying integer size, write \(t=N-40\) and \(a=40(0.12)=4.8\). The argument of \(\Phi\) is \(q(N)=a/\sqrt{t}+0.20\sqrt{t}-1.96\sqrt{1+40/t}\). For \(t\geq60\), its derivative is
because \(0.20t\geq12>4.8=a\), and the second term is positive. Thus \(q(N)\), and hence \(CP(N)\), increases for eligible \(N\geq100\). Since \(CP(215)<0.80\) and \(CP(216)>0.80\), the rule selects \(N_{\text{new}}=216\) observations in total.
Worked Example: The Lower and Upper Zones Give Different Reasons for No Increase
Keep the hypothetical design and cutoffs above, but suppose first that the interim mean is \(\bar{x}_1=-0.10\) units. The planned-size conditional-power argument is
This is below \(L=0.25\), so the stated rule retains \(N_0=100\). It does not automatically stop the trial; stopping would require a separately prespecified action.
Now instead suppose \(\bar{x}_1=0.35\) units. Then
This value is at least \(U=0.60\), so the rule again retains \(N_0=100\), now because the interim result lies above the promising zone. Notice that the conditional power is slightly below \(T=0.80\). The rule nevertheless does not increase the sample size there: that is the policy specified by the upper cutoff.
Worked Example: The Maximum Size Does Not Reach the Target
Return to \(\bar{x}_1=0.12\) units and the promising-zone calculation, but suppose the prespecified cap is \(N_{\max}=160\). The interim conditional power remains about \(0.3590\), so the rule calls for an increase. At the cap,
The cap's conditional power is below \(T=0.80\). No eligible size up to 160 can meet the target in this example: the increasing-\(N\) result established above applies for \(N\geq100\). The rule therefore selects \(N_{\text{new}}=160\), not a size that it claims will achieve the target. The report should say that the maximum was selected and the target was not reached.
Checks, Comparisons, and Limitations
A rule is easier to audit when its decision can be traced from inputs to output. Before implementation, check that the conditional-power calculation uses the intended interim observations, assumed effect, variance, test direction, critical value, and planned size. Then check that the zone boundaries include their equality cases, the eligible sizes are defined, and the cap case has an explicit action. In the examples above, equality at \(L\) enters the increase zone, while equality at \(U\) enters the no-increase zone.
A further check is whether conditional power changes as expected over the eligible sizes. It need not be safe to assume monotonicity for every model and input. If it is not established, the rule should search the full eligible set for the smallest size meeting the target rather than rely on a numerical search that assumes a single crossing. Also check for implementation errors such as mixing a total sample size with a per-group size, using a two-sided critical value in a one-sided calculation, or changing the effect assumption after seeing the interim result.
| Approach | Decision principle | Important distinction |
|---|---|---|
| Fixed sample size | Continue with the originally planned \(N_0\) regardless of the interim conditional power. | Simple and does not use this conditional-power re-estimation mapping. |
| Threshold-based deterministic rule | Use prespecified zones; in the increase zone, seek the target subject to eligible sizes and the cap. | Actions are explicit, including cases in which the target cannot be reached. |
| Unrestricted target mapping | Seek the target from the interim result without the same zone-based no-increase policy. | Can prescribe increases in circumstances where the threshold rule deliberately does not. |
Conditional power is conditional on observed interim data and on the assumed future effect. It is not the probability that \(H_0\) is true, nor does reaching a conditional-power target establish that the trial has a particular final power under the unknown true effect. The rule is also not automatically a guarantee of type I error control. Sample-size adaptation can affect the operating characteristics of the final procedure; the full design and test must use an appropriate error-control justification. Finally, an unblinded decision can create operational and interpretive risks even when the mapping is deterministic. A written, prespecified algorithm helps make the decision auditable, but does not remove those risks.
Check Your Understanding
Answer each question, then compare your reasoning with the solution given.
- If \(CP(N_0)=0.20\), \(L=0.25\), and the rule says to retain \(N_0\) below \(L\), what sample size is selected? Solution: \(N_{\text{new}}=N_0\). The specified lower-zone action is no increase; it is not automatically a futility stop.
- If \(CP(N_0)=U=0.60\) under the rule above, which zone applies? Solution: The upper zone applies because the definition assigns \(C\geq U\) to that zone. The rule retains \(N_0\).
- In the first worked example, why is 215 not selected even though its conditional power is close to 0.80? Solution: The target is a threshold, not an approximate goal: \(CP(215)\approx0.7984<0.80\). The next integer size, 216, gives approximately \(0.8006\) and is the smallest qualifying size.
- If the cap has conditional power below the target, may the report state that the target was attained? Solution: No. It should state that the rule selected the cap and that the target was not attained within the eligible range.
- Does a deterministic conditional-power rule, by itself, prove that the final test controls type I error at \(\alpha\)? Solution: No. Determinism makes the decision reproducible; error control must be justified for the complete adaptive design and final testing procedure.