Why Substitute the Interim Estimate?
Conditional power asks how likely the trial is to cross its final efficacy boundary, given the data observed so far and a specified assumption about the treatment effect. The assumption is consequential. Conditional power under the design alternative keeps the effect fixed at the value used to plan the study. Plug-in conditional power instead substitutes an interim treatment-effect estimate for that planning value. It answers a different, data-dependent question: what would the conditional probability be if the effect during the remaining information were equal to the effect estimated so far?
That distinction is useful when assessing how persuasive the interim evidence is for continued progress toward the target. But the plug-in value is not a direct measurement of the true effect, nor a probability that the treatment works. The interim estimate is uncertain, and using it as if it were the effect in future data is an assumption made for this calculation.
Model and Calculation
The formula below uses the canonical normal model for a one-sided efficacy test. It assumes that information is known or treated as fixed, that the test statistic has the standard normal canonical form under a fixed effect \(\delta\), and that information accumulates from \(I_1\) to \(I_2\). In this model, \(Z_1\) and \(Z_2\) have unit variances and correlation \(\sqrt{I_1/I_2}\); their means under effect \(\delta\) are \(\delta\sqrt{I_1}\) and \(\delta\sqrt{I_2}\), respectively. The final decision is \(Z_2\geq c\).
Under these assumptions, the conditional distribution of the final statistic has mean \(\sqrt{I_1/I_2}\,z_1+\delta(I_2-I_1)/\sqrt{I_2}\) and variance \((I_2-I_1)/I_2\). Thus the conditional standard deviation is \(\sqrt{(I_2-I_1)/I_2}\), and the upper-tail probability is:
For conditional power under the design alternative, set \(\delta=\delta_0\), where \(\delta_0\) is the prespecified design effect. For plug-in conditional power, set \(\delta=\hat{\delta}_1\). In the canonical model with a compatible interim effect estimator, \(\hat{\delta}_1=z_1/\sqrt{I_1}\). If an interim estimate comes from a different estimator or model, it must first be expressed on the model's effect scale; substituting an incompatible estimate does not produce a meaningful plug-in calculation.
The two terms in the conditional mean play different roles. The term \(\sqrt{I_1/I_2}\,z_1\) carries forward the interim evidence. The term involving \(\delta\) describes the assumed mean contribution from information still to be collected. Plugging in \(\hat{\delta}_1\) changes that future-increment assumption; it does not remove or replace the observed interim statistic.
Worked Example: An Interim Estimate Above the Design Effect
Suppose a hypothetical one-sided trial has \(I_1=25\), \(I_2=100\), observed \(z_1=1.20\), final boundary \(c=1.96\), and design effect \(\delta_0=0.20\). Use the canonical estimate \(\hat{\delta}_1=z_1/\sqrt{I_1}\). Information is positive and increases from interim to final; the fixed-boundary normal model is being assumed, and the estimate is on the effect scale used by that model.
First, \(\hat{\delta}_1=1.20/\sqrt{25}=0.2400\). The conditional standard deviation is \(\sqrt{(100-25)/100}=\sqrt{0.75}=0.866025\). Also, \(\sqrt{25/100}\,z_1=0.5(1.20)=0.6000\), and \((I_2-I_1)/\sqrt{I_2}=75/10=7.5\).
Under the design effect, the conditional mean is \(0.6000+0.20(7.5)=2.100\). Therefore:
Under the interim estimate, the conditional mean is \(0.6000+0.2400(7.5)=2.400\), so:
The plug-in value is higher because the estimated effect, 0.2400, exceeds the design effect, 0.20. These are model-based conditional probabilities of crossing the fixed final boundary, not probabilities that either effect is true.
When the Interim Estimate Is Smaller
The direction of the difference follows from the formula. Holding \(z_1\), \(I_1\), \(I_2\), and \(c\) fixed, the conditional mean has a positive coefficient \((I_2-I_1)/\sqrt{I_2}\) on \(\delta\). Because \(\Phi\) is increasing, a smaller assumed effect produces a smaller conditional power. This does not mean the conditional mean must lie below the boundary: it may be above or below it. The correct conclusion is only that lowering the assumed effect lowers the probability of crossing the fixed boundary.
Worked Example: A Plug-In Value Below the Design-Alternative Value
Consider another hypothetical calculation with \(I_1=25\), \(I_2=100\), \(z_1=0.50\), \(c=1.96\), and design effect \(\delta_0=0.20\). Here \(\hat{\delta}_1=0.50/5=0.1000\). As before, the conditional standard deviation is \(\sqrt{0.75}=0.866025\), and the interim contribution to the conditional mean is \(0.5(0.50)=0.2500\). These values satisfy the same information and boundary assumptions as the previous calculation.
For the design effect, the conditional mean is \(0.2500+0.20(7.5)=1.750\). Hence:
For the plug-in effect, the mean is \(0.2500+0.1000(7.5)=1.000\). Therefore:
The estimate is below the design effect, so the plug-in conditional power is lower. Both calculations retain the same interim evidence, \(z_1=0.50\); they differ in the assumed effect for the remaining information.
Reading the Calculation and Checking Its Inputs
A plug-in calculation is only as relevant as its inputs and model. Before interpreting it, check that the information levels are on the same scale, \(I_2>I_1\), and the boundary corresponds to the intended final analysis. Check that the interim estimate is an effect estimate for the treatment contrast being tested, not a raw outcome mean or a statistic on a different scale. Finally, distinguish an estimate used to describe the interim trend from the prespecified design effect used to plan power.
Here is a further calculation where the estimate is obtained directly from a canonical interim statistic. Suppose hypothetical values are \(I_1=40\), \(I_2=90\), \(z_1=1.40\), \(c=2.00\), and \(\delta_0=0.18\). The canonical interim estimate is \(1.40/\sqrt{40}\approx0.2214\). The conditional standard deviation is \(\sqrt{50/90}\approx0.7454\), and the interim contribution to the conditional mean is \(\sqrt{40/90}(1.40)\approx0.9333\). The effect coefficient is \(50/\sqrt{90}\approx5.2705\).
Worked Example: Comparing Both Assumptions at a Different Information Level
The stated information increases from 40 to 90, and we assume the same canonical normal model and fixed upper final boundary. Under the design effect, the conditional mean is \(0.9333+0.18(5.2705)\approx1.8820\), giving:
Under the plug-in estimate, the effect contribution is approximately \(0.2214(5.2705)=1.1667\), so the conditional mean is approximately \(0.9333+1.1667=2.100\). Thus:
The estimated effect exceeds the design effect, and the plug-in calculation consequently gives a larger crossing probability. The result is conditional on the observed interim statistic and on the stated assumptions; it is not a forecast that accounts for uncertainty about the effect estimate itself.
Interpretation, Alternatives, and Limitations
The principal alternative is conditional power under a prespecified effect, which retains the design alternative regardless of the interim estimate. That quantity is useful for asking how likely success would be if the effect used in planning were the effect governing future increments. Plug-in conditional power instead asks what the calculation gives when that effect assumption is replaced by the observed estimate. Neither quantity is universally preferable; they answer different questions, and reporting both can make the role of the assumption transparent.
Plug-in conditional power is sensitive to random variation in the interim estimate. A noisy or unusually high estimate can make the calculation look encouraging; a low estimate can make it look discouraging. The calculation does not average over uncertainty in the true effect, shrink an extreme estimate, or account for the chance that the estimate would differ at another interim analysis. A predictive probability that explicitly averages over parameter uncertainty is a different quantity and requires additional modeling assumptions.
Also, conditional power is not itself a rule for changing sample size, continuing the trial, or stopping for efficacy or futility. If a design uses a data-dependent sample-size or decision rule, its operating characteristics—including Type I error and power—must be assessed for the complete procedure. A plug-in probability alone does not establish that a proposed adaptation preserves error rates. The formula here also presumes a fixed final boundary and a canonical normal model; a design with additional future looks, a different boundary structure, or a materially different variance model needs a calculation matched to that design.
A practical diagnostic is to recompute both versions with the same \(z_1\), \(I_1\), \(I_2\), and \(c\), changing only \(\delta_0\) to \(\hat{\delta}_1\). If \(\hat{\delta}_1>\delta_0\), plug-in conditional power should be greater; if \(\hat{\delta}_1<\delta_0\), it should be smaller, under this one-sided upper-boundary model. A reversed ordering signals a likely sign, scale, or arithmetic error. This check does not validate the model assumptions, but it can catch an inconsistent calculation.
Check Your Understanding
Use the canonical normal model and a fixed upper final boundary unless a question says otherwise. Solutions follow each question.
- In one sentence, distinguish conditional power under a design effect from plug-in conditional power.
Solution: Design-effect conditional power uses the prespecified effect \(\delta_0\) for future increments, whereas plug-in conditional power replaces it with the interim estimate \(\hat{\delta}_1\). - For fixed interim statistic and information, what happens to upper-boundary conditional power when the assumed effect is reduced? Does the conditional mean have to be below the boundary?
Solution: The conditional mean decreases, so the upper-tail probability decreases. No: the mean may remain above the boundary; the direction of change alone does not determine its final position. - Suppose \(I_1=16\), \(I_2=64\), and \(z_1=1.00\). What is the canonical interim estimate \(\hat{\delta}_1\)?
Solution: \(\hat{\delta}_1=z_1/\sqrt{I_1}=1.00/4=0.2500\), on the effect scale of the canonical model. - Why is plug-in conditional power not a probability that the treatment effect equals the interim estimate?
Solution: It is a conditional probability of crossing a statistical boundary calculated under the assumption that the effect for future increments equals the estimate. It does not assign a probability to parameter values or account for uncertainty in the estimate. - If a proposed design changes sample size according to plug-in conditional power, does calculating that probability alone prove Type I error is controlled?
Solution: No. The full adaptive procedure must be evaluated for its operating characteristics; the plug-in calculation by itself does not establish Type I error control.