Unblinded Re-Estimation Using Conditional Power

Why Unblinded Conditional Power Is Used

At an interim analysis, unblinded arm-specific outcomes reveal both the estimated treatment difference and the information accumulated so far. A sample-size re-estimation rule can use those interim data to ask a forward-looking question: if the trial continues to a proposed final sample size, how likely is the fixed final test to reject its null hypothesis? That probability, conditional on the observed interim data and an assumption about future outcomes, is called conditional power.

This tutorial develops a plug-in rule: use the observed interim treatment difference as the assumed effect for future participants, calculate conditional power at each prespecified candidate total, and select the smallest total meeting a target. This is an unblinded rule because it uses the separate treatment and control results. The final test is held fixed: its test statistic, direction, and critical value do not change when the sample size is updated.

Definition: Conditional power is the probability, given the interim data, that the final test will reject \(H_0\), under a specified model for future data and a specified value of the treatment effect. Plug-in conditional power uses the interim estimated treatment effect as that specified future effect. It is a model-based planning quantity, not the probability that the treatment effect is true or that \(H_0\) is false.

Model and Conditional-Power Formula

Assume a continuous outcome with independent observations in two arms and equal allocation. Each outcome is normally distributed with a common, known standard deviation \(\sigma\). Write \(\delta\) for the treatment-minus-control population mean difference. At the interim analysis, there are \(n_1\) participants per arm, with observed arm means \(\bar{x}_{T,1}\) and \(\bar{x}_{C,1}\). The observed difference is \(\hat{\delta}_1=\bar{x}_{T,1}-\bar{x}_{C,1}\), and its standardized interim statistic is \(Z_1=\hat{\delta}_1/(\sigma\sqrt{2/n_1})\).

Let \(N\) be a proposed final sample size per arm, where \(N>n_1\). The final analysis compares the overall arm means using the usual standardized difference. For a one-sided test at level \(\alpha=0.025\), in the prespecified positive direction, the fixed critical value is \(c=z_{1-\alpha}\approx1.960\); rejection occurs when \(Z_N>c\). Conditional power is the probability of that event given the interim data.

Let \(\delta^*\) be the assumed treatment difference among future participants. The future observations are independent of the interim observations under the model. Conditional on the interim data, the final statistic has the following normal distribution:

Z_N\mid D_1 \sim N\left( \sqrt{\frac{n_1}{N}}Z_1+ \frac{(N-n_1)\delta^*}{\sigma\sqrt{2N}}, \frac{N-n_1}{N} \right).

Here \(D_1\) denotes the interim data, and the second argument of \(N(\cdot,\cdot)\) is a variance. Denote the conditional mean by \(m_N\) and the conditional standard deviation by \(s_N=\sqrt{(N-n_1)/N}\). The conditional power is therefore:

CP(N\mid D_1,\delta^*)= 1-\Phi\left(\frac{c-m_N}{s_N}\right), \qquad m_N=\sqrt{\frac{n_1}{N}}Z_1+ \frac{(N-n_1)\delta^*}{\sigma\sqrt{2N}}.

For plug-in conditional power, set \(\delta^*=\hat{\delta}_1\). Both the interim statistic and the plug-in effect then come from the observed arm-specific data. The use of \(\hat{\delta}_1\) is a choice of assumption about future outcomes; it does not make that effect known.

Why the Formula Has This Form

The final difference in arm means is a weighted average of the interim difference and the difference among future participants. The interim part is already observed; the future part remains random. Standardizing this weighted average produces a conditional mean contribution from each part and leaves only the future observations contributing conditional variance.

Separate observed and future data.
With \(N-n_1\) additional participants per arm, the final difference is \(D_N=(n_1/N)\hat{\delta}_1+((N-n_1)/N)D_F\), where \(D_F\) is the future treatment-minus-control mean difference.

Specify the future distribution.
Under the model, \(D_F\mid D_1\sim N(\delta^*,2\sigma^2/(N-n_1))\). Independence means the interim data do not add uncertainty to this future difference.

Standardize the final difference.
Since \(Z_N=D_N/(\sigma\sqrt{2/N})\), its conditional mean is \(m_N\), and its conditional variance is \((N-n_1)/N\). The probability \(P(Z_N>c\mid D_1)\) gives the stated formula.

For the plug-in choice, the conditional mean can also be simplified. Since \(Z_1=\hat{\delta}_1\sqrt{n_1}/(\sigma\sqrt{2})\), substituting \(\delta^*=\hat{\delta}_1\) gives \(m_N=\hat{\delta}_1\sqrt{N}/(\sigma\sqrt{2})\). The conditional standard deviation still depends on how much information remains: it is \(\sqrt{(N-n_1)/N}\), not 1. The unsimplified formula is useful because it displays separately the observed interim contribution and the assumed future contribution.

If the prespecified fixed test is two-sided with critical value \(c\), rejection includes either tail. Its conditional power is \(1-\Phi((c-m_N)/s_N)+\Phi((-c-m_N)/s_N)\). The examples below use the one-sided positive-direction test, so their calculations use only the upper-tail formula.

Worked Example: Selecting the First Candidate That Meets the Target

Hypothetical data. After \(n_1=40\) participants per arm, the treatment mean is 54.5 units and the control mean is 50.0 units. The known common standard deviation is \(\sigma=10\) units. The positive-direction interim difference is \(4.5\) units. Candidate final totals per arm are 80, 90, 100, 110, and 120; the target plug-in conditional power is 0.960.

The interim standard error is \(10\sqrt{2/40}=2.236\) units, so \(Z_1=4.5/2.236=2.012\). For each candidate use \(\delta^*=4.5\), \(c=1.960\), \(m_N=\sqrt{40/N}(2.012)+(N-40)(4.5)/(10\sqrt{2N})\), and \(s_N=\sqrt{(N-40)/N}\).

For \(N=80\), \(m_{80}=\sqrt{40/80}(2.012)+40(4.5)/(10\sqrt{160})=1.423+1.423=2.846\), and \(s_{80}=\sqrt{40/80}=0.7071\). Thus \(CP=1-\Phi((1.960-2.846)/0.7071)=1-\Phi(-1.253)\approx0.895\), below target.

For \(N=90\), \(m_{90}=\sqrt{40/90}(2.012)+50(4.5)/(10\sqrt{180})=1.341+1.677=3.019\), and \(s_{90}=\sqrt{50/90}=0.7454\). Thus \(CP=1-\Phi((1.960-3.019)/0.7454)=1-\Phi(-1.421)\approx0.922\), also below target.

Evaluating the remaining candidates gives \(CP(100)\approx0.943\), \(CP(110)\approx0.958\), and \(CP(120)\approx0.969\). The first candidate reaching 0.960 is therefore \(N=120\) per arm. The rule selects 120 per arm, an increase of 80 participants in total, or 40 additional participants per arm beyond the original candidate of 80 per arm.

The selection is determined by the prespecified candidate set, not by rounding a continuous sample-size solution. The candidate at 110 is close to the target but remains below it.

Applying an Increase Rule

A conditional-power calculation becomes an operational rule only after the candidate totals and target are specified. For example, let \(\mathcal{N}\) be an ordered set of allowable final sample sizes per arm, each greater than the current interim size. For target \(\gamma\), the rule selects the smallest \(N\in\mathcal{N}\) for which \(CP(N\mid D_1,\hat{\delta}_1)\geq\gamma\). If none qualifies, the protocol must say what happens, such as selecting a prespecified maximum or making no increase. That fallback should not be improvised after seeing the data.

The interim sample is included in \(N\). If there are currently \(n_1\) participants per arm and the chosen total is \(N\) per arm, the number to recruit additionally is \(N-n_1\) per arm, not \(N\). This distinction matters in both protocol specifications and operational reports.

Worked Example: Computing From Arm Means and Checking a Target

Hypothetical data. At an interim analysis there are 50 participants per arm. The treatment and control means are 8.0 and 4.0 points, respectively, and the known common standard deviation is 12 points. Candidate final totals per arm are 100, 125, and 150. The target conditional power is 0.800.

The observed difference is \(\hat{\delta}_1=8.0-4.0=4.0\) points. Its standard error is \(12\sqrt{2/50}=2.400\) points, giving \(Z_1=4.0/2.400=1.667\). For \(N=100\), \(m_{100}=\sqrt{50/100}(1.667)+50(4.0)/(12\sqrt{200})=1.179+1.179=2.357\); \(s_{100}=\sqrt{50/100}=0.7071\). Hence \(CP(100)=1-\Phi((1.960-2.357)/0.7071)\approx0.713\), which does not meet target.

For \(N=125\), \(m_{125}=\sqrt{50/125}(1.667)+75(4.0)/(12\sqrt{250})=1.054+1.581=2.635\); \(s_{125}=\sqrt{75/125}=0.7746\). Thus \(CP(125)=1-\Phi((1.960-2.635)/0.7746)=1-\Phi(-0.8716)\approx0.808\). This is the first candidate to meet the target, so the rule selects 125 per arm, requiring 75 additional participants per arm.

The assumptions used here are explicit: equal allocation, a common known standard deviation, independent normally distributed outcomes, and the observed 4.0-point difference as the assumed future effect. If that effect assumption changes, the conditional-power values can change even though the interim data do not.

Worked Example: Retaining the Original Sample Size

Hypothetical data. After 30 participants per arm, the treatment mean is 52 units and the control mean is 48 units. Suppose the known common standard deviation is 8 units, the planned final total is 60 per arm, and the fixed test and target are the same as above. The observed difference is \(4\) units and the interim standard error is \(8\sqrt{2/30}=2.066\) units, so \(Z_1=1.936\).

For \(N=60\), \(m_{60}=\sqrt{30/60}(1.936)+30(4)/(8\sqrt{120})=1.369+1.369=2.739\), and \(s_{60}=\sqrt{30/60}=0.7071\). Therefore \(CP(60)=1-\Phi((1.960-2.739)/0.7071)=1-\Phi(-1.101)\approx0.865\). Since 0.865 exceeds the target 0.800, the rule retains the planned 60 participants per arm; it does not increase the total.

If 80 per arm were also an allowable candidate, its plug-in conditional power would be about 0.936: \(m_{80}=4\sqrt{80}/(8\sqrt{2})=3.162\), \(s_{80}=\sqrt{50/80}=0.7906\), and \(1-\Phi((1.960-3.162)/0.7906)\approx0.936\). That higher value is not a reason to select 80 when the rule calls for the smallest candidate meeting target.

Assumptions, Diagnostics, and Limitations

The calculation is appropriate only to the design and model specified. In particular, the formula assumes equal allocation, independent interim and future participants, a common known \(\sigma\), and a normally distributed endpoint (or a setting where this normal approximation is justified). The final statistic must be the stated standardized difference and the direction and critical value must remain fixed. Candidate totals must exceed \(n_1\), so that there are future observations and \(s_N>0\).

Before using a recommendation, check that both arm-specific sample sizes and outcome summaries are correct, that the treatment contrast has the stated direction, and that the interim data are sufficiently complete for the model. Examine whether variability, allocation, missingness, or outcome distribution depart materially from the assumptions. A conditional-power number based on an inappropriate standard error or an incorrectly coded arm label can be numerically precise and still be misleading. With an estimated rather than known standard deviation, this normal formula is an approximation; the analysis plan should specify how that estimate is obtained and how its uncertainty is handled.

The plug-in rule is especially sensitive to the interim difference. Random variation can make \(\hat{\delta}_1\) larger or smaller than the eventual treatment effect, and using it as the future effect carries that interim fluctuation forward. Unblinded access also creates operational and inferential concerns: access should be restricted to the designated decision-making group, and the adaptation should be prespecified. As discussed in the earlier material on type I error with nuisance-parameter re-estimation, retaining a familiar final critical value does not by itself establish that the adaptive procedure has the intended type I error. The entire decision rule and final analysis must be evaluated together.

Conditional power is not a guarantee of success. It is conditional on the observed interim data and on the assumed future effect and model. Nor does a high value prove that the treatment works. It is a planning probability under stated assumptions. The introductory material on blinded and unblinded sample size re-estimation distinguishes the information available under those approaches; here, the key consequence is that arm-specific results allow direct use of the observed effect but require careful control of access and interpretation.

Common Mistakes

Using the wrong interim contribution. For \(n_1=40\) and \(N=90\), \(\sqrt{40/90}(2.012)\approx1.341\), not 1.643. Substitute each factor separately and check the arithmetic before calculating the tail probability.
Confusing total and additional sample size. A selected final total of 120 per arm after 40 per arm means 80 more per arm, not 120 more.
Treating conditional power as an observed-data-only fact. It depends on a specified future-effect assumption. Plug-in conditional power uses the interim difference, but another assumption would produce another probability.
Choosing a candidate that is merely close to the target. If the rule requires at least 0.960, a value of 0.958 does not qualify. Apply the threshold to unrounded calculations and then report suitably rounded values.
Assuming an unchanged critical value guarantees unchanged error rates. A fixed final test component is not the same as a fixed overall adaptive procedure; operating characteristics must be assessed for the whole rule.

Key takeaway: Calculate conditional power from the arm-specific interim difference and the specified future-effect assumption, then apply the prespecified candidate rule exactly. Report the selected final total per arm, the additional recruitment required, and the assumptions underlying the probability.

Check Your Understanding

For each question, state the reasoning as well as the result. Solutions follow.

With 25 participants per arm, an interim difference of 3 units, and known \(\sigma=6\) units, what is the interim standard error and \(Z_1\)?
In the normal equal-allocation model, why is the conditional variance of \(Z_N\) equal to \((N-n_1)/N\), rather than 1?
Suppose a target is 0.90 and successive candidate conditional powers are 0.82, 0.898, and 0.914. Which candidate qualifies under a smallest-candidate rule?
After 35 participants per arm, the selected final total is 100 per arm. How many additional participants per arm are needed?
Explain why a plug-in conditional power of 0.95 does not mean there is a 95% probability that the treatment effect is positive.

Solutions. (1) \(SE=6\sqrt{2/25}=1.697\) units, so \(Z_1=3/1.697=1.768\). (2) The interim contribution is fixed after conditioning on \(D_1\); only the future mean difference varies. Its variance, after standardizing the final statistic, is \((N-n_1)/N\). (3) The third candidate qualifies: 0.914 is the first value at least 0.90. (4) \(100-35=65\) additional participants per arm. (5) The 0.95 is the probability of final rejection conditional on the interim data, model, and plug-in future-effect assumption. It is not a probability statement about the unknown treatment effect or the truth of a hypothesis.