Conditional Power Curves Across Interim Results

What a Conditional-Power Curve Shows

A conditional-power curve displays how the probability of crossing a planned final efficacy boundary changes over possible interim Z statistics. It answers a conditional question: if the interim statistic were \(z\), how likely would final efficacy be under a specified model for the remaining data? Drawing the curve makes the dependence on the interim result visible, rather than reducing it to the value at one observed result.

The curve depends on more than the interim Z statistic. It also depends on the final information level (or, in a simple fixed-variance design, the final sample size), the information available at interim, the assumed effect, and the final efficacy boundary. To compare curves meaningfully, state which of these quantities change and which are held fixed.

Definition: A conditional-power curve is the function that assigns to each possible interim statistic \(z\) the probability, under a stated model and assumed effect, that the final test statistic crosses its specified efficacy boundary, conditional on the interim statistic being \(z\).

The calculations below use a one-sided upper efficacy boundary. They describe the probability of final boundary crossing if the trial continues to the fixed final analysis. If a design has an interim stopping rule, only interim results for which the trial continues are operationally relevant; the mathematical curve can still be defined outside that range.

Normal Canonical Model and the Curve Formula

Assume information increases from \(I_1\) at interim to \(I_2\) at the final analysis, with \(0<I_1<I_2\). Write \(t=I_1/I_2\) for the information fraction. The normal canonical model assumes that the standardized statistic has independent normal information increments, with a common effect per square root of information. Let \(\eta\) denote the assumed mean of the final Z statistic under that effect. Thus, under the assumed effect, the final statistic has mean \(\eta\) and variance 1 before conditioning on interim data.

Let \(Z_1\) be the interim statistic and \(Z_2\) the final statistic. A fixed final upper efficacy boundary is denoted by \(c\); final efficacy occurs when \(Z_2\ge c\). The independent-increment model gives the following conditional distribution:

Z_2\mid Z_1=z \sim N\!\left(\sqrt{t}\,z+(1-t)\eta,\;1-t\right).

The second argument of \(N\) is the variance. The conditional mean combines the information already observed, represented by \(\sqrt{t}\,z\), with the assumed effect’s contribution from the remaining information, represented by \((1-t)\eta\). The remaining variance is \(1-t\).

If \(\Phi\) is the standard normal cumulative distribution function, the conditional power at interim value \(z\) is therefore

CP(z;\eta,t,c) =P(Z_2\ge c\mid Z_1=z) =1-\Phi\!\left( \frac{c-\sqrt{t}\,z-(1-t)\eta}{\sqrt{1-t}} \right) =\Phi\!\left( \frac{\sqrt{t}\,z+(1-t)\eta-c}{\sqrt{1-t}} \right).

This expression requires a specified assumed effect, a fixed final information level and boundary, and the normal independent-increment model. In a setting where information is proportional to sample size, \(t=n_1/N\), where \(n_1\) is the interim sample size and \(N\) is the fixed final sample size. If those quantities are not proportional to information, use the information fraction rather than the sample-size fraction.

Why the Curve Has Its Shape

For fixed \(t\), \(\eta\), and \(c\), conditional power increases as the interim Z statistic increases. To see this, set \(A(z)=(\sqrt{t}\,z+(1-t)\eta-c)/\sqrt{1-t}\). Then \(CP(z)=\Phi(A(z))\), so

\frac{d\,CP(z)}{dz} =\phi(A(z))\sqrt{\frac{t}{1-t}}>0,

where \(\phi\) is the standard normal density. Thus, the curve is increasing. For a one-sided boundary, it approaches 0 as \(z\) tends to negative infinity and 1 as \(z\) tends to positive infinity. These properties are useful diagnostics: a decreasing curve under this model usually indicates a sign or standardization error.

The formula also gives the interim Z value associated with any target conditional power \(q\), where \(0<q<1\):

z_q= \frac{c+\sqrt{1-t}\,\Phi^{-1}(q)-(1-t)\eta}{\sqrt{t}}.

This is a point on the curve, not by itself a decision rule. For example, it identifies where a plotted curve reaches 80%; it does not determine how a trial should change its sample size.

Worked Example: Tracing a Curve at Three Interim Results

Suppose the information fraction is \(t=0.50\), the assumed final noncentrality is \(\eta=2.80\), and the one-sided final efficacy boundary is \(c=1.96\). Calculate conditional power at interim Z statistics \(z=0\), \(1\), and \(2\).

Here \(\sqrt{1-t}=\sqrt{0.5}\), and the conditional-power argument simplifies to

\frac{\sqrt{0.5}\,z+0.5(2.80)-1.96}{\sqrt{0.5}} =z-0.79196.

At \(z=0\), \(CP(0)=\Phi(-0.79196)\approx0.2143\). At \(z=1\), \(CP(1)=\Phi(0.20804)\approx0.5824\). At \(z=2\), \(CP(2)=\Phi(1.20804)\approx0.8864\).

Under these assumptions, the probability of crossing the final boundary rises from about 21.4% to 88.6% across the three interim results. These are conditional probabilities under the assumed effect and model, not probabilities that the assumed effect is true.

Comparing Fixed Final Sample Sizes

A fixed final sample size determines the final information level and therefore affects both \(t\) and \(\eta\). To compare sample sizes, hold the interim information, assumed effect scale, and efficacy boundary fixed, then recompute the curve for each final size. The resulting curves answer different conditional questions: the same interim Z value is being evaluated against different amounts of planned remaining information.

Worked Example: Comparing Two Fixed Final Information Levels

Consider a hypothetical normal model in which information is measured in participant-equivalent units, the interim information is \(n_1=100\), and the assumed effect contributes \(0.15\) units of noncentrality per square root of information. Use a one-sided boundary \(c=2.00\), and compare fixed final information levels \(N=300\) and \(N=400\) at interim \(z=1\). Assume independent normal increments and information proportional to these units.

For \(N=300\), \(t=100/300=1/3\), and \(\eta=0.15\sqrt{300}=2.59808\). The standardized threshold in the conditional-power formula is

\frac{2-\sqrt{1/3}(1)- (2/3)(0.15\sqrt{300})}{\sqrt{2/3}} = \frac{2-\sqrt{1/3}-0.15\sqrt{300}(2/3)}{\sqrt{2/3}} \approx -0.378937.

Therefore \(CP=1-\Phi(-0.378937)=\Phi(0.378937)\approx0.6476\).

For \(N=400\), \(t=100/400=0.25\), and \(\eta=0.15\sqrt{400}=3.00\). The conditional normal mean at \(z=1\) is \(\sqrt{0.25}(1)+0.75(3.00)=2.75\), and its standard deviation is \(\sqrt{0.75}=0.8660\). Hence

CP =1-\Phi\!\left(\frac{2.00-2.75}{0.8660}\right) =1-\Phi(-0.8660) =\Phi(0.8660) \approx0.8068.

The larger fixed final information level gives higher conditional power in this comparison, about 80.7% rather than 64.8%. The comparison relies on the stated common effect scale and information model; it is not a universal claim that changing final sample size always moves every conditional-power curve in the same way.

Worked Example: Locating an 80% Curve Level

Suppose \(t=0.40\), \(\eta=2.50\), and \(c=1.96\). Find the interim Z statistic at which the one-sided conditional-power curve reaches 80%.

Use \(\Phi^{-1}(0.80)\approx0.84162\). Substituting into the level formula gives

z_{0.80} = \frac{1.96+\sqrt{0.60}(0.84162)-(0.60)(2.50)}{\sqrt{0.40}} \approx \frac{1.11192}{0.63246} \approx1.758.

Thus, under these fixed assumptions, conditional power is 80% at an interim Z statistic of approximately 1.758. Since the curve is increasing, values above 1.758 have conditional power greater than 80%, and values below it have conditional power less than 80%. This is a descriptive reading of the curve, not a sample-size adaptation rule.

Alternative Boundaries and Comparisons

The curve is sensitive to the efficacy boundary. Raising \(c\) makes the final event harder to attain and lowers conditional power at every fixed \(z\); lowering \(c\) raises it. Increasing the assumed effect \(\eta\) also raises the curve at every fixed interim result. These changes follow directly from the argument of \(\Phi\), and they should be distinguished from changes in the observed interim statistic.

For a two-sided final efficacy rule \(|Z_2|\ge c\), the conditional probability includes both tails:

P(|Z_2|\ge c\mid Z_1=z) = 1-\Phi\!\left(\frac{c-m(z)}{\sqrt{1-t}}\right) +\Phi\!\left(\frac{-c-m(z)}{\sqrt{1-t}}\right), \qquad m(z)=\sqrt{t}\,z+(1-t)\eta.

This is not generally the same as the one-sided formula. Use the event that matches the planned efficacy criterion.

Conditional power also differs from two common summaries. Unconditional power averages over possible interim results under a specified effect; a curve gives the probability conditional on each particular interim result. Plug-in conditional power estimates or substitutes an effect from interim data, so its assumed effect can itself vary with the observed data. A curve computed for a fixed assumed effect instead holds that effect constant while varying \(z\). Keeping these questions separate prevents a plotted fixed-effect curve from being interpreted as a plug-in analysis.

Assumptions, Diagnostics, and Limitations

A curve is only as appropriate as its inputs and model. Before interpreting one, check that the final information and efficacy boundary are fixed for the calculation; the assumed effect is stated; and the normal independent-increment approximation is suitable for the statistic and trial design. In a simple normal model, known variance and the planned information scale support this calculation. With estimated nuisance parameters, non-normal outcomes, or a different information structure, the canonical approximation may need justification or a model-specific calculation.

Useful diagnostics include checking that the curve is increasing for the one-sided upper-boundary case, that its values stay between 0 and 1, and that a higher boundary lowers it. Check the information fraction carefully: it must be interim information divided by final information, not the remaining fraction. Also check whether \(\eta\) corresponds to the same fixed final information used to calculate \(t\). Mixing values from different final sample sizes creates a curve with no coherent design interpretation.

A conditional-power curve is model-based, not a guarantee of final success. Its numerical value can be highly sensitive to the assumed effect, especially when little information remains. It is also not, by itself, a complete assessment of a design’s error rates or of the consequences of adapting the design after observing interim results. Those questions require analysis of the full design and its decision rules.

Common Mistake

Wrong reasoning: “The curve gives the probability that the treatment effect is real, so a value above 80% confirms efficacy.”
Correction: Conditional power is the probability of a specified future boundary-crossing event, conditional on the interim statistic, under a specified assumed effect and model. It is not the probability that \(H_0\) is false, nor does it establish that the assumed effect is true.
Wrong reasoning: “A larger final sample size changes only the amount of remaining variance.”
Correction: Changing final information changes the information fraction and the assumed final noncentrality, so both parts of the curve formula must be updated consistently.

Key takeaway: For a fixed final information level, assumed effect, and efficacy boundary, the normal-model conditional-power curve is obtained by evaluating the final boundary-crossing probability across possible interim Z statistics. State those assumptions alongside the curve.

Check Your Understanding

Answer each question using the fixed-effect, one-sided normal canonical model unless the question says otherwise.

For \(t=0.25\), \(\eta=2.00\), \(c=1.96\), and \(z=1\), calculate conditional power.
With \(t\), \(\eta\), and \(z\) fixed, what happens to one-sided conditional power when the final boundary \(c\) increases? Explain from the formula.
For fixed \(t\), \(\eta\), and \(c\), is the one-sided conditional-power curve increasing or decreasing in \(z\)? Give the mathematical reason.
In one sentence, distinguish a fixed-effect conditional-power curve from unconditional power.

Solutions. (1) The conditional mean is \(\sqrt{0.25}(1)+(0.75)(2.00)=2.00\), and the standard deviation is \(\sqrt{0.75}=0.8660\). Thus \(CP=1-\Phi((1.96-2.00)/0.8660)=1-\Phi(-0.04619)\approx0.5184\). (2) It decreases, because increasing \(c\) decreases \((\sqrt{t}z+(1-t)\eta-c)/\sqrt{1-t}\), and \(\Phi\) is increasing. (3) It is increasing: its derivative is \(\phi(A(z))\sqrt{t/(1-t)}>0\). (4) A fixed-effect conditional-power curve gives boundary-crossing probabilities conditional on each interim result under one held-constant assumed effect, whereas unconditional power averages over possible interim results under a specified effect.