Conditional-Power Rules Versus Effect-Threshold Rules

Why Compare These Rules?

An unblinded interim review may use the observed treatment effect to decide whether to increase the planned sample size. One approach asks whether the conditional power of the current design is high enough under a specified assumption about future data. Another compares the interim effect estimate directly with a cutoff. They can sound like different methods, but under a fixed model and matched assumptions, a conditional-power target is simply a particular effect threshold written on a probability scale.

The distinction matters when cutoffs are chosen separately, or when the assumed future treatment effect, variance, remaining sample size, or efficacy boundary differs between calculations. In those cases the rules can make different decisions for the same interim result. The earlier material on conditional power and promising-zone calculations provides the ingredients; here the focus is on translating one decision rule into the other and checking whether the comparison is fair.

Model and Matched-Rule Relationship

Consider a one-sided normal-increments model with known standard deviation $\sigma$. Let $n_1$ be the information accrued at the interim analysis, $n_2$ the planned additional information, and $N=n_1+n_2$ the total information if the study continues. Let $\hat{\delta}_1$ be the interim estimate of the treatment effect, measured in the original outcome units. For the derivation below, the interim score is $Y_1=n_1\hat{\delta}_1$, with variance $n_1\sigma^2$, and the future score $Y_2$, conditional on the interim data, is assumed independent and normally distributed with mean $n_2\delta_p$ and variance $n_2\sigma^2$. Here $\delta_p$ is the effect assumed for the future data in the conditional-power calculation.

The final one-sided statistic is $Z=(Y_1+Y_2)/(\sigma\sqrt{N})$, and success means $Z\ge z_\alpha$, where $z_\alpha$ is the upper-tail standard-normal critical value for significance level $\alpha$. Conditional power is the probability of that event given the interim data and the specified future-effect assumption. This calculation treats $\delta_p$ as fixed; it is not a probability that the treatment effect is true, nor a guarantee of the trial’s eventual outcome.

Definition: Under this model, conditional power at the interim result is $$ CP(\hat{\delta}_1;\delta_p) = \Phi\left( \frac{n_1\hat{\delta}_1+n_2\delta_p-z_\alpha\sigma\sqrt{N}} {\sigma\sqrt{n_2}} \right), $$ where $\Phi$ is the standard normal cumulative distribution function. A conditional-power rule calls for expansion when $CP\ge 1-\beta$, for a chosen target $1-\beta$.

Because $\Phi$ is strictly increasing, applying the target $CP\ge1-\beta$ is equivalent to requiring its argument to be at least $z_{1-\beta}$, the standard-normal quantile with cumulative probability $1-\beta$. Rearranging gives a direct threshold on the interim estimate:

\hat{\delta}_1 \ge c_{\mathrm{CP}} = \frac{z_\alpha\sigma\sqrt{N} +z_{1-\beta}\sigma\sqrt{n_2} -n_2\delta_p}{n_1}.

Thus, with the same $n_1,n_2,\sigma,\delta_p,\alpha$, and conditional-power target, a direct effect-threshold rule using $c_{\mathrm{CP}}$ makes exactly the same decision as the conditional-power rule. This equivalence is a consequence of the assumed model and fixed planning quantities, not a general identity between any two rules bearing these names.

Worked Example: Translating an 80% Conditional-Power Target

Suppose a hypothetical trial has $n_1=40$ information units at interim and plans $n_2=60$ more, so $N=100$. Let $\sigma=1$ outcome unit per information unit, use a one-sided $\alpha=0.05$ boundary $z_\alpha=1.645$, and assume $\delta_p=0.30$ outcome units for future data. The expansion target is $CP\ge0.80$, so $z_{1-\beta}=z_{0.80}=0.8416$.

The equivalent interim Z-statistic cutoff is found by substituting the planning values:

Z_{I,\mathrm{cut}} = \frac{1.645\sqrt{100}+0.8416\sqrt{60}-60(0.30)} {\sqrt{40}} = \frac{16.45+6.5190-18}{6.3246} \approx 0.7857.

Since $Z_I=\hat{\delta}_1\sqrt{40}$ in this model, the equivalent effect-estimate cutoff is $c_{\mathrm{CP}}=0.7857/\sqrt{40}\approx0.1242$ outcome units. The rule expands when the interim estimate is at least $0.1242$ units.

For a hypothetical interim estimate of $0.13$, the conditional power is

CP = \Phi\left( \frac{40(0.13)+60(0.30)-1.645\sqrt{100}}{\sqrt{60}} \right) = \Phi\left(\frac{6.75}{\sqrt{60}}\right) = \Phi(0.8714) \approx0.808.

The conditional-power rule expands because $0.808\ge0.80$; the equivalent effect-threshold rule also expands because $0.13\ge0.1242$. The estimate is in outcome units, while the conditional power is a model-based probability of crossing the specified final boundary.

When Assumptions Change, So Can the Decision

A fair comparison holds the assumptions fixed. In practice, a conditional-power calculation may be updated to use a different future effect, while a direct threshold remains at its originally specified value. That is not a contradiction: the two rules no longer encode the same decision criterion. The following example isolates this issue while keeping the interim outcome and other design quantities unchanged.

Worked Example: Replanning the Future Effect

Use the same hypothetical design as above and suppose the interim estimate is $0.13$. If the future-effect assumption is changed from $0.30$ to $0.20$ outcome units, the updated conditional power is

CP = \Phi\left( \frac{40(0.13)+60(0.20)-16.45}{\sqrt{60}} \right) = \Phi\left(\frac{0.75}{7.7460}\right) = \Phi(0.0968) \approx0.539.

This updated conditional-power rule does not meet its $0.80$ target. If it is compared with the old direct cutoff of $0.1242$, however, that direct rule still calls for expansion, since $0.13\ge0.1242$. The disagreement arises because the old cutoff encodes $\delta_p=0.30$, whereas the updated conditional power assumes $\delta_p=0.20$.

For an exact comparison under the new planning assumption, translate the $0.80$ target again:

c_{\mathrm{CP}} = \frac{1.645\sqrt{100}+0.8416\sqrt{60}-60(0.20)}{40} = \frac{10.9690}{40} \approx0.2742.

Under that matched cutoff, an interim estimate of $0.13$ does not trigger expansion, just as the updated conditional-power calculation does not. The direct threshold is now higher because a smaller assumed future effect makes the target harder to attain.

A Direct Threshold Need Not Be a Conditional-Power Target

A direct effect threshold may instead be selected as an operational policy—for example, because decision-makers want expansion only when the interim estimate exceeds a prespecified clinically meaningful value. Unless that threshold is calculated from the conditional-power formula, it need not match a conditional-power rule. It can still be a deliberate rule, but its rationale and consequences should be described on their own terms.

Worked Example: A Separately Chosen Effect Cutoff

Return to the hypothetical $n_1=40,n_2=60$ design with $\delta_p=0.30$, and suppose the direct rule requires an interim estimate of at least $0.15$ outcome units. Consider a hypothetical interim estimate of $0.14$. The direct rule does not expand, because $0.14<0.15$.

Under the stated conditional-power assumptions, however,

CP = \Phi\left( \frac{40(0.14)+18-16.45}{\sqrt{60}} \right) = \Phi\left(\frac{7.15}{7.7460}\right) = \Phi(0.9231) \approx0.822.

The conditional-power rule expands because $0.822\ge0.80$. The rules disagree because the direct cutoff $0.15$ is not the equivalent cutoff $0.1242$ derived for this conditional-power target and these assumptions. The example does not establish that one rule is universally preferable: it shows that the two rules answer different decision specifications.

Derivation, Diagnostics, and Practical Comparison

The key justification is the conditional distribution of the final statistic. Given $Y_1$, only $Y_2$ remains random. Under the stated model, $Y_2$ has mean $n_2\delta_p$ and variance $n_2\sigma^2$. Therefore the conditional mean and standard deviation of $Z$ are

E(Z\mid Y_1)=\frac{Y_1+n_2\delta_p}{\sigma\sqrt{N}}, \qquad SD(Z\mid Y_1)=\frac{\sqrt{n_2}\sigma}{\sigma\sqrt{N}} =\sqrt{\frac{n_2}{N}}.

Standardizing the event $Z\ge z_\alpha$ using this conditional normal distribution gives the conditional-power formula above. Equivalently, writing the numerator as $Y_1+n_2\delta_p-z_\alpha\sigma\sqrt{N}$ and dividing by $\sigma\sqrt{n_2}$ yields the same expression. Setting this standardized quantity at least $z_{1-\beta}$ and substituting $Y_1=n_1\hat{\delta}_1$ gives the direct effect cutoff. This derivation depends on the normal-increments model, the assumed future drift, and the fixed final boundary.

Before comparing or implementing rules, verify that both use the same interim information, remaining information, variance scale, future-effect assumption, final boundary, and conditional-power target. Also verify that an effect cutoff uses the same effect units and direction. If the remaining sample size is changed, the cutoff must be recalculated for the new $n_2$; if a maximum sample size binds, the calculation must reflect the actual permitted continuation rather than an unattainable plan.

Useful diagnostics include checking that conditional power increases as the interim effect estimate increases, and that the derived effect cutoff reproduces the conditional-power target to rounding error. A non-monotone relationship in this fixed model, or disagreement between a computed conditional power and its translated cutoff, signals an implementation or unit error. The assumptions can also fail to describe the planned analysis: for example, a misspecified variance or future-effect assumption can make the reported conditional power poorly calibrated to actual prospects. Earlier diagnostic work on noisy interim effects and planning-effect misspecification is relevant when assessing those risks.

A conditional-power rule has the advantage of expressing continuation in terms of a modeled probability of meeting a specified final criterion. Its result is sensitive to the assumed future effect, however. A direct threshold is easy to communicate and may reflect a prespecified practical objective, but its probability implications are not apparent unless it is calibrated. When the threshold is obtained by algebraically translating a conditional-power target, the two formulations have no decision difference under the same assumptions. When cutoffs or assumptions differ, compare the rules across plausible interim estimates and planning scenarios rather than treating their labels as evidence of equivalence.

Neither calculation by itself resolves every design issue. It presumes a specified final test and a continuation plan, and it does not establish that the adaptation has acceptable overall operating characteristics. Those properties require evaluation of the complete design and analysis procedure. In particular, the conditional-power value should not be interpreted as the probability that the treatment effect is real, and crossing an expansion threshold should not be presented as proof that success will follow.

Common Mistake

Common mistake: “Both rules use the interim treatment effect, so they must make the same decision.” That is false when the effect threshold is chosen independently or when the conditional-power calculation uses different assumptions. The correction is to derive the threshold from the same $n_1,n_2,\sigma,\delta_p,\alpha$, and target power, then compare decisions using that matched cutoff. If a separately chosen threshold is used, describe it as a distinct decision policy and evaluate its consequences.

Check Your Understanding

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

Under the stated model, why does $CP\ge1-\beta$ translate into an interim effect-estimate threshold?
For $n_1=40,n_2=60,\sigma=1$, and a fixed planning effect, what must be held equal for a direct threshold and a conditional-power rule to be equivalent?
With the original example’s assumptions, does an interim estimate of $0.10$ meet the $0.80$ conditional-power target? Explain using the translated cutoff.
Why might a direct cutoff of $0.15$ disagree with the $0.80$ conditional-power rule in the worked example?

Solutions. (1) The standard normal cumulative distribution function is strictly increasing, so requiring its value to be at least $1-\beta$ is equivalent to requiring its argument to be at least $z_{1-\beta}$; rearrangement gives a cutoff on $\hat{\delta}_1$. (2) The interim and remaining information, variance, assumed future effect, final boundary, and target conditional power must match, and the effect estimate and cutoff must use the same units. (3) No. The equivalent cutoff is about $0.1242$ units, and $0.10<0.1242$; its conditional power is about $0.763$. (4) The $0.15$ cutoff was chosen separately, not derived from the conditional-power target; at an estimate of $0.14$, conditional power is about $0.822$, while the direct rule does not expand.