Mapping Conditional Power to a Revised Sample Size

Why Map Conditional Power to Sample Size?

At an interim analysis, a conditional-power calculation answers a planning question: if the trial continues, how likely is it to cross the final efficacy boundary, given the interim result and a specified assumption about the treatment effect? When that probability is below a desired target, one can ask how much additional information would be needed to reach the target. This tutorial derives that mapping in the canonical normal model.

The answer depends on three inputs that must be kept distinct: the observed interim statistic, the effect assumed for future planning, and the final efficacy boundary. The effect assumption is not automatically the interim effect estimate. Changing the assumed effect changes the required information, sometimes substantially.

Definition: Let \(I_1\) be the information available at the interim analysis and \(I\) the proposed total information at the final analysis, with \(I>I_1\). The information ratio is \(r=I/I_1\). Given an observed interim statistic \(Z_1=x\), an assumed effect \(\delta\), and a final upper efficacy boundary \(c\), the conditional power is the probability of crossing \(c\) at the final analysis, conditional on \(Z_1=x\), under the assumed effect.

Model and Conditional-Power Formula

Assume a one-sided upper-boundary test with a known-variance normal model and information measured on a scale for which the effect per square root of information is \(\delta\). A standard canonical representation is

Z(I)=\frac{W(I)+\delta I}{\sqrt{I}},

where \(W(I)\) is standard Brownian motion indexed by information. Thus, increments in \(W\) over disjoint information intervals are independent and normally distributed, with mean zero and variance equal to the interval length. The assumed effect \(\delta\) is fixed for the calculation. The planned final analysis uses the same endpoint, variance model, allocation, and one-sided boundary \(c\).

Write the observed interim statistic as \(Z(I_1)=x\), and define \(\eta=\delta\sqrt{I_1}\), the assumed drift expressed on the interim information scale. The future information increment is \(I-I_1\). Conditional on the interim observation, the future Brownian increment remains independent and has variance \(I-I_1\). Therefore, the final statistic has conditional mean and variance

E_\delta\{Z(I)\mid Z(I_1)=x\} =\frac{x+\eta(r-1)}{\sqrt{r}}, \qquad \operatorname{Var}_\delta\{Z(I)\mid Z(I_1)=x\} =1-\frac{1}{r}.

These expressions apply for \(r>1\), so that there is positive future information. Let \(\Phi\) denote the standard normal cumulative distribution function. The conditional power is the probability that this conditional normal statistic exceeds \(c\):

CP(r;x,\eta,c) = \Phi\left( \frac{x+\eta(r-1)-c\sqrt{r}}{\sqrt{r-1}} \right).

Deriving the Sample-Size Equation

Suppose the target conditional power is \(\gamma\), where \(0<\gamma<1\). Define \(z_\gamma=\Phi^{-1}(\gamma)\). Since \(\Phi\) is increasing, setting the conditional power equal to \(\gamma\) is equivalent to setting its normal argument equal to \(z_\gamma\). Hence the required information ratio satisfies

x+\eta(r-1)-c\sqrt{r} = z_\gamma\sqrt{r-1}.

This is generally a nonlinear equation; it does not usually reduce to a simple proportional scaling of the current sample size. A convenient numerical parameterization is \(y=\sqrt{r-1}\), so \(r=1+y^2\) and \(y>0\). The equation becomes

f(y) = x+\eta y^2-c\sqrt{1+y^2}-z_\gamma y =0.

A root-finding method can solve this one-dimensional equation. A bracketed method, such as bisection, is useful when values of \(f\) with opposite signs have been found. After solving, substitute the resulting ratio into the conditional-power formula as a check; a root rounded too coarsely can otherwise make the reported power appear not to match the target.

Key result: Under the stated normal model, the target conditional-power ratio \(r\) solves \(x+\eta(r-1)-c\sqrt{r}=z_\gamma\sqrt{r-1}\). If information is proportional to sample size under an unchanged design, the corresponding total sample size is the interim sample size multiplied by \(r\).

Worked Example: Solving for the Information Ratio

Suppose the observed interim statistic is \(x=1.50\), the assumed interim-scale drift is \(\eta=2.00\), the final upper boundary is \(c=2.00\), and the target conditional power is \(\gamma=0.800\). Find the information ratio. All values are hypothetical.

The standard normal quantile for the target is \(z_{0.800}\approx0.8416\). With \(y=\sqrt{r-1}\), solve

f(y)=1.50+2.00y^2-2.00\sqrt{1+y^2}-0.8416y=0.

For example, \(f(1.08)\approx-0.0199\), while \(f(1.10)\approx0.0210\), so a root lies between 1.08 and 1.10. Numerical root-finding gives \(y\approx1.0898\), and therefore

r=1+y^2\approx1+(1.0898)^2\approx2.188.

To check, use \(r-1\approx1.1877\) and \(\sqrt{r}\approx1.4791\). The conditional mean is approximately \((1.50+2.00(1.1877))/1.4791=2.620\), and the conditional standard deviation is approximately \(\sqrt{1-1/2.188}=0.7368\). Thus the standardized distance above the boundary is \((2.620-2.00)/0.7368\approx0.8416\), giving conditional power \(\Phi(0.8416)\approx0.800\), as required.

From Information to Sample Size

The information ratio becomes a sample-size ratio only when the information per participant remains fixed. This is a reasonable direct conversion for a normal model with unchanged allocation, outcome variance, and measurement plan. If these change, information must be recalculated from the design rather than assumed proportional to the participant count.

Worked Example: Converting the Ratio to Participants

Suppose the trial has accrued 60 participants per group at the interim analysis, and the planned total was 120 per group. Use the ratio \(r\approx2.188\) from the preceding example. Assume equal allocation and fixed information per participant, with no change to the endpoint or variance model. Find the revised total.

The interim sample size is 60 per group. Under the stated proportionality assumption, the revised total per group is

n_{\mathrm{rev,group}}=60(2.188)\approx131.3.

The corresponding total across both groups is approximately \(2(131.3)=262.6\) participants, compared with the original planned total of \(2(120)=240\). This calculation gives a continuous target. Actual enrollment uses whole participants and must respect the trial's allocation plan; the resulting implemented size may therefore differ slightly from 262.6. The conversion does not imply that 262.6 is itself an allowable enrollment count.

When a Finite Target Cannot Be Reached

The equation need not have a positive solution for every combination of interim result, assumed effect, boundary, and target. A particularly clear case occurs when the assumed effect is zero and the observed interim statistic is \(x=0\). For any \(r>1\), the conditional-power argument is

\frac{-c\sqrt{r}}{\sqrt{r-1}} = -c\sqrt{\frac{r}{r-1}}<-c \qquad (c>0).

Consequently, conditional power is always below \(\Phi(-c)\), approaching that value only as \(r\) grows without bound. If the target exceeds \(\Phi(-c)\), no finite information ratio can meet it in this case.

Worked Example: Recognizing an Unattainable Target

Suppose \(x=0\), the assumed effect is \(\eta=0\), and the final boundary is \(c=2.00\). Could any finite information increase attain a target conditional power of \(0.800\)?

Here, for every \(r>1\),

CP(r;0,0,2) = \Phi\left(-2\sqrt{\frac{r}{r-1}}\right) < \Phi(-2)\approx0.0228.

Since \(0.800\) is far above this upper limit, there is no finite solution. Increasing information does not create an assumed positive effect; the conditional probability approaches about 0.0228, not 0.800. The appropriate conclusion is that the target is unattainable under these inputs, not that a larger numerical sample size should be reported.

Diagnostics, Alternatives, and Limitations

A calculated ratio should be checked both numerically and substantively. Confirm that \(r>1\), that the root solves the stated equation to the working precision, and that direct substitution gives conditional power close to the target. Also check that the assumed effect, boundary, information scale, and interim statistic use compatible definitions. If a numerical solver returns no root, inspect the function over the feasible range and consider whether the target is unattainable or whether there are multiple roots; do not assume every target has exactly one solution.

The calculation differs from two common alternatives. An unconditional power calculation averages over possible interim results and asks about the probability of success from the start of the trial. Conditional power instead fixes the observed interim statistic and specifies an effect assumption for the future. A plug-in conditional-power calculation uses an effect estimated from interim data; a prespecified-effect calculation uses a chosen planning value. Those choices can yield different ratios even with the same \(x\), \(c\), and target.

The mapping is exact only under its model assumptions. It can be an approximation when variance is estimated, outcomes are non-normal, information per participant changes, or the test statistic does not follow the canonical normal model. In a group sequential design, the final boundary used here must be the applicable final efficacy boundary, not an arbitrary fixed-\(Z\) cutoff substituted without justification. Conditional-power-based sample-size adaptation also does not by itself establish that the complete adaptive design controls its intended error rate. The design's operating characteristics require evaluation under its full adaptation and testing procedures.

Common pitfall: It is incorrect to multiply the original planned sample size by the solved ratio when the ratio is defined relative to interim information. The ratio \(r=I/I_1\) compares proposed final information with information already accrued. Under proportionality, multiply the interim sample size by \(r\); separately check the resulting total against the original plan and the actual allocation.

Check Your Understanding

Answer each question, then compare with the brief solutions below.

In the canonical model, what does \(r\) represent, and why must \(r>1\) for the conditional-power formula used here?
For a target conditional power \(\gamma\), what value is assigned to the standardized distance from the conditional mean to the boundary?
If the interim sample size is 80 per group and the solved ratio is \(r=1.50\), what continuous revised total per group follows under fixed information per participant?
With \(x=0\), \(\eta=0\), and \(c=2\), can a target of 0.10 be attained at a finite \(r\)? Explain using the limiting conditional power.
Name one assumption under which multiplying the interim sample size by \(r\) is not justified without further calculation.

Solutions. (1) \(r=I/I_1\), the proposed final-to-interim information ratio; \(r>1\) ensures positive future information and a nonzero conditional variance. (2) It equals \(z_\gamma=\Phi^{-1}(\gamma)\). (3) \(80(1.50)=120\) participants per group. (4) No: conditional power remains below \(\Phi(-2)\approx0.0228\) for every finite \(r\), so it cannot reach 0.10. (5) For example, a change in allocation, variance, or information per participant invalidates the direct proportional conversion unless information is recalculated.