Why Interim Results Need More Than a “Good” or “Bad” Label
An interim result can be too weak to justify continuing, strong enough that the planned study is already adequate, or intermediate: promising, but not yet enough to meet the design’s target for final power. These categories are useful only when tied to a prespecified rule. Otherwise, labels such as “promising” can become subjective, and choices made after seeing unblinded results can undermine the planned analysis.
This tutorial treats conditional power as the measure used to classify interim evidence. The introductory material on target conditional power and target final power motivates the target; here the emphasis is on how cutoffs create decision regions and how each region maps to an action. Conditional power depends on the assumed effect, the information already observed, and the planned information remaining. It is therefore a design quantity under stated assumptions, not a direct probability that the study will succeed.
A Model for Classifying Interim Evidence
For a concrete rule, suppose the endpoint supports a one-sided test of a positive treatment effect, and the final rejection criterion is \(Z_{\mathrm{final}}>c\). Under a normal approximation with known variance and independent increments of information, let \(I\) be the planned total information, \(I_1\) the information at the interim analysis, and \(t=I_1/I\) the information fraction. Let \(z\) be the observed interim Z-statistic. Assume a planning effect whose standardized drift per square root of information is \(\theta\). Under that assumed effect, the conditional distribution at the planned final information is
Thus, the conditional power under this model is
where \(\Phi\) is the standard normal cumulative distribution function. This expression applies when \(0<t<1\), the normal approximation and information model are suitable, and the stated effect and final boundary are the ones used for the calculation. In a trial using another endpoint model or a different analysis, the conditional-power calculation must match that design.
The classification rule also needs explicit cutoffs. Let \(C_F\) be the futility cutoff and \(C_T\) the target conditional power, with \(0<C_F<C_T<1\). One common rule is:
| Conditional power | Region | Example prespecified action |
|---|---|---|
| \(CP<C_F\) | Futility | Stop for futility; do not claim efficacy |
| \(C_F\leq CP<C_T\) | Promising | Increase sample size according to the specified rule, subject to its maximum |
| \(CP\geq C_T\) | Sufficiently favorable | Keep the planned sample size; do not increase |
The inequalities are part of the rule. In particular, a result exactly at \(C_F\) belongs to the promising region, while one exactly at \(C_T\) belongs to the sufficiently favorable region. A protocol can choose different conventions, but it must state them unambiguously.
Why Conditional Power Moves with the Interim Result
The normal independent-increments model gives a short justification for the formula and explains the classification. Write the interim statistic as \(Z_{I_1}=[W(I_1)+\theta I_1]/\sqrt{I_1}\), where \(W\) has independent normal increments with variance equal to information. Conditional on \(Z_{I_1}=z\), the increment from \(I_1\) to \(I\) is independent of the observed interim data. Splitting the final statistic into the observed part and this future increment gives
The last term has conditional mean zero and variance \((I-I_1)/I=1-t\). This yields the stated conditional normal distribution. Because the mean increases with \(z\), conditional power also increases with \(z\) when the other design quantities are fixed. Consequently, the conditional-power cutoffs divide the interim Z-statistic scale into ordered regions in this model. The numerical cutoff values still depend on the assumed effect, information fraction, final boundary, and target values.
Worked Example: Classifying Three Interim Results
Suppose the final one-sided boundary is \(c=1.96\), the information fraction is \(t=0.50\), and the planning effect gives \(\theta\sqrt I=3.00\) at the planned final information. Let the futility cutoff be \(C_F=0.20\) and the target be \(C_T=0.80\). Classify hypothetical interim Z-statistics \(z=-0.50,\ 0.50,\) and \(1.80\).
For these values, the conditional mean is \(\sqrt{0.50}z+3.00(1-0.50)\), and the conditional standard deviation is \(\sqrt{0.50}=0.7071\). For \(z=-0.50\), the mean is \(0.7071(-0.50)+1.50=1.146\). Hence
Since \(0.125<0.20\), this interim result is in the futility region; the stated rule stops for futility.
For \(z=0.50\), the mean is \(0.7071(0.50)+1.50=1.854\), so
Since \(0.20\leq 0.440<0.80\), this result is promising. The rule assigns the prespecified sample-size action for that region.
For \(z=1.80\), the mean is \(0.7071(1.80)+1.50=2.773\), giving
Since \(0.875\geq0.80\), this result is sufficiently favorable under the rule, so the planned sample size is retained. These labels describe the stated conditional-power rule; they do not by themselves establish that the final treatment effect is statistically significant.
Turning the Regions into Sample-Size Actions
Classification and action should be written as a single operational rule. The rule needs to specify the effect assumption used for conditional power, both cutoffs, the action at each region, and any lower or upper sample-size constraints. In a promising region, “increase the sample size” is incomplete unless the rule states how much to increase it and what happens if the calculated value exceeds the permitted maximum. The rule should also say whether the decision is made at one interim look or at specified multiple looks.
Use the observed interim data and the prespecified model, effect assumption, final criterion, and planned information.
Compare the calculated value with \(C_F\) and \(C_T\), applying the stated boundary conventions.
Stop for futility, use the specified increase in the promising region, or retain the planned sample size when the result is sufficiently favorable.
Enforce sample-size constraints and use an inferential procedure whose error control is valid for the adaptation actually performed.
Worked Example: Applying a Capped Increase
Suppose a hypothetical two-arm normal-endpoint trial plans 240 participants in total, with 120 providing interim data. Under the planning effect and final test, the conditional-power calculation at this information fraction gives \(CP=0.44\). The prespecified cutoffs are \(C_F=0.20\) and \(C_T=0.80\). The rule says to increase total enrollment by 20% in the promising region, subject to a maximum of 300 participants.
The interim result is promising because \(0.20\leq0.44<0.80\). The proposed total is
The maximum is not binding, since \(288\leq300\). The action is therefore to continue toward 288 participants in total, with the allocation and operational rounding handled as prespecified. This calculation applies the given rule; it does not claim that 288 participants will necessarily achieve the target conditional power. The conditional power used to classify the interim result was calculated for the stated original plan, whereas the rule’s increase determines the subsequent design.
Worked Example: A Result Near the Target Cutoff
Use the normal model in the first example, with \(t=0.50\), \(c=1.96\), and \(\theta\sqrt I=3.00\). Consider the hypothetical interim result \(z=1.50\). The conditional mean is
Therefore,
With a target of \(0.80\), this value is just above the target, so the stated rule retains the planned sample size. If rounding had produced \(0.80\) exactly, the boundary convention in the table would still place it in the sufficiently favorable region. In practice, retain enough numerical precision to classify near-boundary values reliably, and report the rule and the unrounded calculation when the decision is close.
Assumptions, Diagnostics, and Alternatives
The classification is meaningful only if its inputs are credible. The normal independent-increments calculation assumes an appropriate normal approximation, correctly specified information, and a final test boundary matching the planned analysis. It also requires a clear effect assumption: conditional power calculated under the planning effect can differ substantially from conditional power calculated under the observed interim effect. The assumed effect is a design choice, not a fact revealed by the interim data.
Useful checks include verifying the information fraction and final boundary, checking that the planned variance and endpoint model remain appropriate, confirming that the sample-size cap is applied as written, and assessing whether small changes in the interim estimate would change the region. A classification that changes with a tiny data perturbation is a warning that the decision is near a cutoff; it is not a reason to alter the cutoff after observing the data. With multiple interim looks, the rule must specify which look governs each action and how repeated opportunities to stop or increase are handled.
A futility action is often nonbinding: stopping for futility does not reject \(H_0\), and continuing despite crossing a nonbinding futility boundary need not invalidate type I error control solely because the trial continued. That does not make every adaptive analysis valid. In particular, if sample size is changed using unblinded treatment-effect information, applying the original fixed-sample final Z boundary without checking the design’s error control can give an incorrect type I error rate. The final inferential method and its calibration must account for the adaptation.
The main alternative is to use an effect-estimate or Z-statistic threshold directly rather than conditional-power thresholds. Under a fixed model and fixed design assumptions, conditional power is generally monotone in the interim Z-statistic, so thresholds can often be translated between these scales. They are not interchangeable without specifying the model and effect assumption: a conditional-power rule makes that assumption visible, while a direct threshold rule may hide it. Another alternative is to make no unblinded sample-size change and continue to the original planned analysis. It avoids this particular adaptation, though it does not solve problems such as an inadequate planning effect or an unsuitable variance assumption.
The method’s limitations are consequential. Conditional power is sensitive to its assumed effect, and interim estimates are noisy. A promising classification does not guarantee success, while a futility classification does not prove the treatment is ineffective. The region labels organize decisions; they are not posterior probabilities about whether \(H_0\) is true. Prespecified actions, valid final inference, and transparent reporting are essential companions to the classification.
Common Mistakes
- Calling every result below the target futile. The correction is to use two cutoffs when the design distinguishes futility from a promising region; intermediate conditional power can trigger an increase.
- Treating conditional power as the probability that the treatment works. It is a probability of satisfying the final criterion conditional on interim data under a stated model and effect assumption.
- Increasing sample size without stating the action precisely. Specify the increase rule, the maximum, and how constraints and rounding are handled.
- Assuming the original final test remains valid automatically. An unblinded data-dependent sample-size change may affect type I error; use an inferential procedure valid for the adaptation.
- Changing a cutoff after seeing which region the data enter. Prespecify thresholds and boundary conventions to avoid post hoc decision-making.
Check Your Understanding
Answer each question using the stated classification rule. The solutions follow the questions.
- If \(C_F=0.15\), \(C_T=0.75\), and \(CP=0.15\), which region applies under the inequalities used here?
- For \(C_F=0.20\) and \(C_T=0.80\), classify \(CP=0.79\) and \(CP=0.80\).
- Why must a conditional-power calculation state the effect assumption used?
- A promising-region rule proposes 330 participants but has a maximum of 300. What must be checked before applying the action?
- Does a sufficiently favorable conditional-power classification establish that the final result is statistically significant? Explain.
Solutions. (1) Since the promising interval includes its lower endpoint, \(CP=0.15\) is promising, not futile. (2) \(0.79\) is promising; \(0.80\) is sufficiently favorable and receives the no-increase action. (3) Conditional power depends on the assumed effect, so different assumptions can produce different values and classifications from the same interim data. (4) Apply the prespecified cap, giving no more than 300 participants, and follow the design’s stated constraints and inferential plan. (5) No. It indicates that the conditional-power target was met under the stated assumptions; statistical significance depends on the observed final data and the valid final test.