Why a Larger Sample Size May Not Restore Power
A sample-size re-estimation rule can appear successful because it returns a larger number of participants, yet still fail to provide the planned power. The number is only a means to an operating characteristic: power is the probability of rejecting \(H_0\) under a specified alternative and the assumptions used to design the test. If a cap prevents reaching the recalculated target, if interim data imply little chance of eventual success, or if the nuisance-parameter estimate is wrong, the nominal update can be misleading.
The key diagnostic question is not simply, “Did the rule increase the sample size?” It is, “Under the updated design, the information actually attainable, and plausible parameter values, what power does the test have?” A recalculation is conditional on its inputs. It does not guarantee the advertised power when those inputs or operational constraints differ from what the calculation assumes.
Start With the Assumptions Behind the Calculation
For a simple reference calculation, suppose a continuous endpoint is analyzed by comparing two independent treatment groups with equal allocation. Assume normally distributed outcomes with a common, known standard deviation \(\sigma\), a fixed treatment difference \(\Delta>0\), no missing outcomes, and a two-sided test at level \(\alpha\). If \(n\) is the number of participants per group, the standardized test statistic has mean \(\Delta\sqrt{n/(2\sigma^2)}\) under the alternative and variance 1. With \(z_{1-\alpha/2}\) and \(z_{1-\beta}\) denoting standard normal quantiles, an approximate target is
Here \(1-\beta\) is the target power. The formula depends on the assumed effect and variance as well as the test and allocation. It does not account automatically for an enrollment cap, and plugging in an estimated nuisance parameter does not make that estimate correct. The calculated \(n\) is generally rounded up to a whole participant per group; the achieved power should then be checked at that rounded value.
Worked Example: Checking the Power After an Update
Consider a hypothetical two-arm trial with a continuous endpoint. The design assumes a common standard deviation of 10 units, a treatment difference of 5 units, a two-sided \(\alpha=0.05\), and 90% power. Assume the model conditions above hold and there is no cap. Find and check the updated sample size.
Using \(z_{0.975}=1.960\) and \(z_{0.90}=1.282\), the required number per group is
Rounding up gives \(n=85\) per group, or 170 participants in total. The noncentrality parameter at this rounded size is \(5\sqrt{85/(2(10^2))}=3.260\). The two-sided power is the probability of crossing either critical value:
Under the stated assumptions, the rounded sample size gives about 90.3% power, slightly above the 90% target. This check is meaningful only if the assumptions and the ability to enroll 170 participants are credible.
Caps and Misspecified Nuisance Parameters
An enrollment cap converts a calculated target into a constrained design. If the recalculation asks for more participants than can be enrolled, the relevant power is the power at the maximum attainable sample size—not at the unattainable target. Separately, the nuisance parameter used in the calculation may be wrong. For a continuous outcome, an underestimated \(\sigma\) reduces the calculated sample size because the formula depends on \(\sigma^2\). The resulting trial can be substantially underpowered if the true variability is larger.
These failures can occur together. A rule may correctly calculate the sample size for its input estimate, while that estimate is too optimistic; then an enrollment cap may prevent even a corrected calculation from being reached. Diagnostics should therefore report both the assumed values and the achievable values, and recalculate power under plausible nuisance-parameter scenarios.
Worked Example: An Optimistic Variance Estimate and a Binding Cap
In a hypothetical two-arm trial, suppose the target difference is 5 units, \(\alpha=0.05\), and the target power is 90%. A blinded re-estimation uses a standard deviation estimate of 8 units and returns a sample size. The actual standard deviation is 12 units, and enrollment cannot exceed 100 participants per group. Assume equal allocation and the same normal-model test as above. Calculate the nominal target and assess achievable power.
Using the estimate of 8, the calculation gives
so the rule would round up to 54 participants per group. But under the actual \(\sigma=12\), the target size would be
or 122 per group after rounding. That target exceeds the cap of 100. Even if the variance were known to be 12, the cap would leave the design below its 90% target. At the cap, the noncentrality is \(5\sqrt{100/(2(12^2))}=2.946\), giving two-sided power
Thus the maximum sample size has about 83.8% power under the actual variance, not 90%. At the nominal 54 per group, power under that same variance would be \( \Phi(2.165-1.960)+\Phi(-2.165-1.960)\approx0.581\), or about 58.1%. The update is not rescued by returning “54”: the estimate of variability is optimistic, and the cap is too low even for the correctly specified target.
Interim Data and Conditional Power
A sample-size formula based on nuisance parameters does not by itself reflect how promising the trial looks after interim outcomes have been observed. Conditional power is the probability, given the interim data and a specified future-effect assumption, that the final test will cross its efficacy boundary. It is not the same as unconditional power, which averages over all possible interim data and the resulting sample sizes under the full adaptive rule.
For a normal canonical information process, let \(t\) be the fraction of the projected final information already observed, \(z_1\) the interim standardized statistic, and \(\lambda\) the mean of the final standardized statistic under a specified alternative at that projected final information. Assume \(0<t<1\), independent normal increments, and a fixed alternative effect. Conditional on \(z_1\), the final statistic has mean \(\sqrt{t}z_1+\lambda(1-t)\) and variance \(1-t\). Therefore, for an upper efficacy boundary \(c\), conditional power is
This expression uses the remaining information, \(1-t\), and the observed interim statistic. It also requires a specified alternative represented by \(\lambda\); choosing a more favorable effect increases the calculated conditional power. A cap matters because the projected final information must be attainable. Conditional power under a hypothetical sample size above the cap does not describe the trial that can actually be completed.
Deriving the Conditional-Power Formula
The formula follows by separating the information already observed from the independent future increment. In the canonical model, the standardized final statistic can be written as \(Z_{\mathrm{final}}=\sqrt{t}z_1+\lambda(1-t)+\sqrt{1-t}\,\varepsilon\), conditional on the observed interim value, where \(\varepsilon\sim N(0,1)\). The first term carries forward the interim statistic; the second is the assumed mean contribution from the remaining information; the last is future random variation. Consequently, \(Z_{\mathrm{final}}\mid z_1\) is normal with the stated mean and variance. Standardizing the event \(Z_{\mathrm{final}}>c\) gives
which equals the displayed upper-tail probability. This calculation is conditional on a fixed projected final information and the specified alternative. If the rule may choose among several final sample sizes, each candidate must be assessed at its attainable information; the rule’s overall power requires averaging over interim outcomes and its sample-size decisions.
Worked Example: A Nominal Increase With Very Weak Conditional Power
Suppose, hypothetically, that an interim analysis occurs at half of the projected final information: \(t=0.5\). The observed interim statistic is \(z_1=-2.000\), the upper efficacy boundary is \(c=1.960\), and the alternative used for projection gives final noncentrality \(\lambda=5\sqrt{130/200}\). Treat the projected final information as the maximum the trial can attain. Calculate the conditional probability of crossing the upper boundary.
First, \(\lambda=5\sqrt{0.65}=4.031\). The conditional mean of the final statistic is
and its conditional standard deviation is \(\sqrt{1-0.5}=0.7071\). Thus
The conditional probability of crossing the upper boundary is about 2.734% under this specified alternative and projected information. Increasing the planned sample size cannot make this observed interim result disappear; any further increase would need to be evaluated at its attainable final information, and the calculation would still depend on the assumed effect. This is not a statement that the trial’s unconditional power is 2.734%: unconditional power averages across possible interim results and decisions.
Diagnostics, Alternatives, and Limits
A useful audit asks whether the update is internally coherent and whether it describes the design that can actually be run. Check the inputs, the rounding, and the cap before interpreting a recalculated number. Then verify power at the final feasible sample size under the design assumptions and under plausible departures from them. If interim outcomes inform the decision, inspect conditional power under a clinically meaningful alternative as well as the unconditional power of the complete rule.
- Cap check: Compare the required total sample size or event count with the maximum feasible amount. Recalculate power at the cap, not at the unachievable target.
- Nuisance-parameter check: Recompute power over plausible values of variance, event rate, or other relevant nuisance parameters. A single unstable or biased estimate should not be treated as known truth.
- Interim-prospects check: Calculate conditional power using the observed interim statistic and a clearly specified alternative. Very low conditional power is evidence that the projected continuation is unlikely to cross the boundary under that assumption.
- Whole-rule check: Evaluate the probability of success across possible interim data and resulting sample sizes. The power of one selected final sample size is not the operating characteristic of a rule that can select different sizes.
The main alternative to relying on a plug-in update is to evaluate a range of plausible nuisance values and feasible sample sizes, using analytic calculations where their assumptions hold and simulation when the rule is more complicated. Simulations should represent the interim timing, the re-estimation rule, the cap, and the final analysis. They can reveal shortfalls hidden by a single average estimate, although they cannot repair an incorrect model or an unrealistic set of parameter scenarios.
Conditional power also has limits. It changes with the alternative chosen, and a low value under one effect does not prove that the treatment has no effect. Nor does a high value guarantee success: future outcomes remain random. Finally, neither a favorable recalculated sample size nor a conditional-power calculation alone establishes that all inferential properties of an adaptive design are preserved. The full rule and final analysis must be assessed under their own assumptions.
Common Mistake
Check Your Understanding
Answer each question, then compare your reasoning with the brief solutions below.
- If a calculation requires 140 participants per group but the cap is 110, which sample size should be used to report attainable power?
- In the continuous-outcome formula, what happens to the required sample size if the assumed standard deviation is multiplied by 1.2 while all other inputs stay fixed?
- What does conditional power of 2.734% in the worked interim example mean, and what does it not mean?
- Why can a sample size that is correct for an estimated variance still fail to deliver target power?
Solutions. (1) Use 110 per group, because the trial cannot reach 140; calculate power at 110 under the specified assumptions. (2) The sample size is proportional to \(\sigma^2\), so it is multiplied by \(1.2^2=1.44\), before rounding. (3) Given the observed interim statistic and the specified alternative and projected information, there is a 2.734% chance of crossing the upper boundary. It is not the unconditional power of the adaptive trial and is not the probability that the null hypothesis is true. (4) The estimate may differ from the true nuisance parameter; for example, an underestimated standard deviation yields too small a sample size and lower actual power.