Sample Size for Partial AUC Equivalence Criteria

Why Partial-AUC Planning Needs Its Own Variability Calculation

A partial area under the concentration-time curve (partial AUC, or pAUC) summarizes exposure over a specified part of the profile. It can be important when the timing of exposure matters, not just total exposure. A study that must meet a pAUC equivalence criterion therefore needs enough evaluable participants for that endpoint—not merely enough for conventional AUC and Cmax criteria. The relevant sample size depends on the pAUC contrast’s expected value, its within-subject variability, the equivalence limits, and whether success is required on other endpoints too.

For prespecified times \(t_1\) and \(t_2\), define an individual’s partial area as the concentration-time integral over that window. In practice it is commonly calculated using a specified numerical integration rule applied to observed concentrations. For positive pAUC values, the treatment contrast is usually formed on the log scale. The time window and calculation rule must be fixed in the protocol: changing either can change both the estimated contrast and its variability.

If the ratio limits are 0.80 and 1.25, then \(L=\log(0.80)\approx-0.223144\) and \(U=\log(1.25)\approx0.223144\). They are exactly symmetric because \(0.80=1/1.25\). These are common limits, not a universal rule for every pAUC endpoint; the applicable margins must be justified and prespecified. A log-scale contrast is dimensionless even though each pAUC has concentration-times-time units.

Model, Standard Error, and Equivalence Success

For basic planning, assume the subject-level contrasts are independent and identically distributed as \(D_{p,i}\sim N(\theta_p,\sigma_p^2)\). This assumes that the model adequately accounts for the crossover design and that there is no unmodelled pattern of variability across subjects. In an analysis using a fitted treatment, period, and sequence model, the corresponding residual variance and degrees of freedom should be used instead of applying a simple paired analysis mechanically.

With \(n\) evaluable subjects, the estimated mean contrast has standard error \(SE_p=s_p/\sqrt n\), where \(s_p\) estimates \(\sigma_p\). At significance level \(\alpha\), the two one-sided tests procedure (TOST) declares equivalence when the two-sided \(100(1-2\alpha)\%\) confidence interval is inside both limits. For a paired analysis with \(df=n-1\), its endpoints are \(\bar D_p\pm t_{1-\alpha,df}SE_p\). Equivalently, the sample mean must lie in a narrower interval than the equivalence limits:

If \(L+cSE_p\ge U-cSE_p\), there is no possible sample mean that can produce a successful confidence interval at that standard error. This condition matters when planning: an arithmetic expression that subtracts two normal cumulative probabilities must not be used as a positive power when the success region is empty.

For a transparent approximation, treat the planned standard error as fixed at \(\sigma_p/\sqrt n\), and approximate \(\bar D_p\) by a normal variable with mean \(\theta_p\) and that standard error. The resulting approximate power is

Here \(\Phi\) is the standard normal cumulative distribution function. The approximation uses a \(t\) critical value but does not account for the randomness of the estimated standard deviation; it is useful for transparent initial planning, not a substitute for exact power when precision is important. Under the normal paired model, exact TOST power can be obtained by integrating over the sampling distribution of the sample variance. If \(Q=(n-1)S_p^2/\sigma_p^2\), then \(Q\sim\chi^2_{n-1}\), independently of \(\bar D_p\). For each value \(q\), use \(SE_p(q)=\sigma_p\sqrt{q/[(n-1)n]}\), apply the piecewise probability above with \(SE_p(q)\), and average that conditional probability over the chi-square distribution. This includes the possibility that the interval is too wide to fit inside the margins.

A Planning Calculation

When the expected contrast is centered at zero and the margins are symmetric, a convenient initial calculation follows directly from the approximate success event. Let the limits be \(-\Delta\) and \(+\Delta\). Then success requires \(|\bar D_p|<\Delta-cSE_p\), which is possible only if \(cSE_p<\Delta\). Under the fixed-standard-error normal approximation,

For a target power \(1-\beta\), solving this approximation gives the initial planning relation

This relation is implicit if \(c=t_{1-\alpha,n-1}\), because the critical value depends on the unknown sample size. It is therefore a starting value: check nearby integer sample sizes with the appropriate critical value, then use exact power or a design-appropriate calculation for final planning. The formula also assumes the expected true contrast is zero. If it is not, use the general power expression rather than the centered formula.

Estimating and Checking pAUC Variability

The needed planning input is the variability of the within-subject log pAUC contrast, not the between-subject spread of pAUC values in one treatment arm. In the basic paired model, estimate it as the sample standard deviation of the \(D_{p,i}\) values. If variability is supplied as a coefficient of variation for an approximately lognormal pAUC, the corresponding log-scale variance is \(\log(1+CV^2)\); however, a marginal CV for one treatment does not automatically equal the variance of paired log differences. The paired covariance matters. Use a variance estimate that matches the planned design and analysis.

For example, a trapezoidal pAUC from 0 to 2 hours with concentrations at 0, 1, and 2 hours is the sum of the two trapezoids. If concentrations are 0, 5, and 8 mg/L, the area is \([(0+5)/2](1)+[(5+8)/2](1)=9.00\) mg·h/L. Repeating the calculation under test treatment for each subject gives paired treatment and reference areas, from which the log contrasts are formed. The pAUC calculation, including sampling times and any interpolation, is part of the endpoint definition, not an incidental data-processing choice.

Joint Success With AUC and Cmax

If study success requires equivalence for pAUC and conventional exposure endpoints such as AUC and Cmax, overall success means that all required confidence intervals satisfy their criteria. The probability of that intersection is joint power. It is not generally equal to the smallest endpoint power, and it is not generally the product of the marginal powers. The endpoint estimates come from the same subjects and can be correlated.

For planning, define the vector of estimated log contrasts and its covariance matrix. A multivariate-normal approximation calculates the probability that every estimate falls within its endpoint-specific success interval, with each interval narrowed by its standard error and critical value. This approach requires planning values for each endpoint’s mean and variance and the correlation matrix of the endpoint estimates. A product of marginal powers is justified only under independence; when correlation is unknown, report that limitation or use a conservative planning strategy rather than silently assuming independence. The earlier material on joint power for AUC and Cmax provides the general joint-endpoint framework; here pAUC must be included with its own variance and covariance inputs.

Diagnostics, Alternatives, and Common Pitfalls

Before accepting a proposed sample size, check that the endpoint definition, analysis model, and variance input describe the same quantity. In particular, confirm that the variance is for paired log pAUC contrasts, that the pAUC window and integration method are fixed, and that the expected treatment ratio is plausible. Check whether the success interval is nonempty at the planned standard error. Examine the sample-size result across a range of plausible variances: sample size rises approximately with \(\sigma_p^2\), so modest underestimation can materially reduce power.

The normal approximation is convenient and interpretable, while exact TOST power accounts for the random sample variance under the normal paired model. For fitted crossover models, small samples, complex covariance structures, or several correlated endpoints, simulation from the planned analysis model can better represent the actual decision rule. Simulation must reproduce the design, endpoint correlation, missing-data assumptions, and analysis; it does not repair unrealistic inputs. None of these methods removes the need to justify the pAUC margins or the expected contrast.

• Mistake: treating the marginal CV as the paired-difference variability. The correction is to use the variance of the within-subject log contrasts, which incorporates the test-reference covariance.
• Mistake: subtracting normal CDF values after the success interval has vanished. If \(L+cSE_p\ge U-cSE_p\), set the approximate power to zero; the CDF difference is only valid when the success region is nonempty.
• Mistake: sizing only for pAUC when success requires several endpoints. Plan for the probability of meeting all criteria, accounting for endpoint dependence where possible.
• Mistake: reporting the initial normal-approximation integer as an exact answer. Recheck nearby sample sizes with the intended critical values and, when warranted, exact or simulation-based power.