Reading Power Curves and Sample Size Tables

From a Power Calculation to a Design Decision

A power curve or sample-size table is a summary of calculations, not a guarantee that a study will succeed. It reports how power changes with design inputs under a specified model. Reading it well means checking what each number represents, locating the design sizes that meet the target, and asking whether the assumptions behind the curve are plausible for the study being planned.

For an average bioequivalence study, power is the probability of concluding equivalence when the assumed true treatment difference and variability apply, and the study is analyzed using the specified procedure. The curve depends on more than sample size: it also depends on the equivalence limits, the assumed true test-to-reference ratio, within-subject variability, the significance level, and the analysis method. A table without these inputs is not enough to establish that a proposed design is adequate.

A power curve plots calculated power against a design quantity, usually the number of evaluable subjects. A sample-size table gives selected points from such a relationship, often alongside the inputs used to calculate them. The vertical coordinate answers “what is the power at this size?” The first size meeting a target answers “what is the minimum size under these assumptions?” Those are different questions, and neither answer should be read without the assumptions.

How the Power Curve Is Constructed

For a transparent illustration, consider a two-period crossover analyzed on the log scale. Let \(\delta\) be the true mean test-minus-reference log difference, and let \(\hat{\delta}\) estimate it from \(n\) independent evaluable subjects. Suppose, for this illustrative calculation, that \(\hat{\delta}\sim N(\delta,\sigma_d^2/n)\), where \(\sigma_d\) is the known standard deviation of within-subject log differences. Assume fixed equivalence limits \(-M\) and \(M\), a one-sided significance level \(\alpha\), and the usual two one-sided tests (TOST). The normal model and known variance make the calculation a planning approximation; an analysis that estimates variance or uses a \(t\) distribution can yield different values.

Under TOST, equivalence is concluded only if both one-sided tests reject their null hypotheses. In this normal approximation, that is equivalent to the \(100(1-2\alpha)\%\) confidence interval for \(\delta\) lying entirely within the equivalence limits. Writing \(SE=\sigma_d/\sqrt{n}\) and \(z_{1-\alpha}\) for the \(1-\alpha\) standard normal quantile, the decision condition is:

The two bounds account for the fact that both tests must pass. With \(\hat{\delta}\sim N(\delta,SE^2)\), the probability that the estimate falls between those bounds is:

Here \(\Phi\) is the standard normal cumulative distribution function. This expression is a direct justification for reading the curve: each plotted value is the probability of landing in the interval of estimates that would lead to an equivalence conclusion, under the assumed sampling distribution. If the planning value is \(\delta=0\), the expression simplifies by symmetry to \(2\Phi(M/SE-z_{1-\alpha})-1\).

Reading Feasible Regions and Integer Thresholds

On a graph, draw or imagine a horizontal line at the target power. Points on or above that line meet the power criterion; points below it do not. The corresponding sample sizes form a feasible region only with respect to that criterion and the assumptions used to calculate the curve. A design can meet the power target and still fail another constraint, such as a required allocation pattern or a limit on the total number of subjects.

Sample size is discrete: a study cannot enroll 7.4 subjects. Therefore, a smooth curve should not be interpreted as though every point on the horizontal axis were an available design. If a calculation gives a continuous threshold, the operational choice must be an allowable integer size, and power should be recalculated or read at that integer. Further integer constraints can create additional steps. For example, a protocol requiring equal allocation to two sequences permits even totals only. That is a protocol constraint, not a general rule that crossover sample sizes must always be even.

When the Assumed Treatment Ratio Is Not Centered

The power at a given size also depends on the assumed true difference. A true ratio of 1 corresponds to \(\delta=0\) on the log scale, but the planning value may instead reflect an expected ratio away from 1. In that case, the acceptance interval remains centered on zero while the sampling distribution is centered on \(\delta\). Power can be lower, and the simple symmetric expression for \(\delta=0\) no longer applies.

Diagnostics, Alternatives, and Limitations

Before using a curve or table, check that its labels and inputs agree with the design question. In particular, verify whether \(n\) denotes subjects, complete subject profiles, or another unit; whether the calculation is for one endpoint or a joint requirement; and whether the reported power uses the exact analysis planned for the study. Check the equivalence limits, significance level, assumed ratio, variability input, and whether the tabulated sample sizes obey any allocation rules.

Inspect neighboring points rather than relying only on the row that appears to meet the target. Under a fixed model with the true difference inside the equivalence limits, power generally increases as the standard error falls with increasing \(n\). But do not assume every curve must increase throughout its range: if the assumed true difference lies outside the equivalence limits, increasing precision can eventually make an equivalence conclusion less likely. A surprising jump, reversal, or irregularity should prompt checks of the calculation method, the inputs, and whether the design changes between rows.

A normal approximation is convenient for showing how \(n\), variability, and the assumed difference affect power. An exact power calculation for average bioequivalence, covered in the earlier material on exact power, is more appropriate when the planned method’s finite-sample distribution and estimated variance matter. A normal approximation can be useful as a cross-check, but it should not silently replace the planned exact method. If two tables disagree, first compare their assumptions and calculation methods; then use the method appropriate to the study’s analysis plan.

A power curve is conditional, not a statement that the target will be achieved in the realized study. It does not account for uncertainty in the planning inputs unless that uncertainty is explicitly incorporated, and it does not describe the probability that the assumed treatment ratio or variability is correct. A value such as 80.9% is not an 80.9% probability that the study will succeed after its data are observed. It is the calculated probability of an equivalence conclusion over repeated studies under the specified assumptions and procedure.

A plan that barely meets a target deserves particular attention. Rounding the displayed power, changing an input slightly, or imposing an integer design constraint can change whether it qualifies. Report the unrounded result when available, state the actual selected integer size, and make clear which assumptions make it feasible. This is more informative than presenting a single rounded sample size as though it were independent of the planning scenario.