Joint Power for AUC and Cmax Requirements

Why Joint Power Is a Separate Planning Quantity

A bioequivalence study commonly evaluates both AUC and Cmax. A successful study must meet the equivalence criterion for each endpoint, so the relevant planning probability is the chance that both criteria are met. This is not generally the product of the two marginal powers: the endpoint estimates are measured in the same participants and may be correlated.

Marginal power answers, for example, how often AUC alone would pass its criterion under a specified design and true ratio. Joint power answers how often AUC and Cmax would pass in the same study. The difference matters when the two endpoints tend to move together or in opposite directions. Separate marginal power calculations, such as those developed for average bioequivalence, do not by themselves determine this joint probability.

Modeling the Two Equivalence Decisions

Work on the log scale. Let \(\theta_A\) and \(\theta_C\) denote the true log test-to-reference ratios for AUC and Cmax, and let \(\hat\theta_A\) and \(\hat\theta_C\) be their estimates. Write the corresponding prespecified log-scale equivalence limits as \(L_A,U_A\) and \(L_C,U_C\). For a two one-sided tests procedure at level \(\alpha\), an endpoint passes when its \(100(1-2\alpha)\%\) confidence interval lies entirely within its limits.

For planning, a useful approximation treats the standard errors \(s_A,s_C\) as fixed and the estimates as jointly normal. Let \(c\) be the critical value used for the interval (often a \(t\) critical value for the study’s degrees of freedom, or a normal critical value in a normal approximation). Then endpoint \(j\), where \(j=A\) or \(C\), passes exactly when \(L_j+c s_j < \hat\theta_j < U_j-c s_j\). If the lower bound is equal to or above the upper bound, that endpoint cannot pass under this fixed-standard-error rule.

Here \(\rho\) is the correlation between the endpoint estimates, not the correlation between the true ratios. Define standardized estimation errors \(Z_A=(\hat\theta_A-\theta_A)/s_A\) and \(Z_C=(\hat\theta_C-\theta_C)/s_C\). Under the approximation, both have standard normal marginal distributions and their correlation is \(\rho\). The success bounds become \(a_j=(L_j+c s_j-\theta_j)/s_j\) and \(b_j=(U_j-c s_j-\theta_j)/s_j\).

where \(\phi_\rho\) is the standard bivariate normal density with correlation \(\rho\). The integral is a probability over a rectangle. It can be evaluated by bivariate-normal software, numerical integration, or simulation. The marginal powers are only the two separate interval probabilities; they omit how often the two events occur together.

Worked Example: Joint Power With Correlated Endpoints

Derivation and Useful Checks

The probability expression follows directly from the two pass conditions. For endpoint \(j\), the confidence interval is \((\hat\theta_j-cs_j,\hat\theta_j+cs_j)\). It is contained in \((L_j,U_j)\) precisely when \(L_j+cs_j<\hat\theta_j<U_j-cs_j\). Subtracting \(\theta_j\) and dividing by \(s_j>0\) gives \(a_j<Z_j<b_j\). Both endpoints pass if and only if both inequalities hold, which is the rectangle event integrated in the formula above. This derivation also shows why an endpoint whose true ratio is close to a limit can have low power even when its standard error is modest: its standardized acceptance interval is shifted away from the center of the estimation-error distribution.

Two bounds can help check a calculation. If the marginal powers are \(p_A\) and \(p_C\), then \(\max(0,p_A+p_C-1)\leq P(\text{both pass})\leq\min(p_A,p_C)\). The upper bound follows because the intersection is contained in each event. The lower bound follows from \(P(A\cup C)\leq1\) and \(P(A\cap C)=P(A)+P(C)-P(A\cup C)\). These bounds are useful checks, but are usually too wide to plan a study.

Independence is a special case: if \(\rho=0\) in the bivariate normal model, the joint probability factors into the product of the marginal probabilities. A positive correlation does not always increase joint power, nor does a negative correlation always reduce it. The effect depends on where the true means lie relative to each endpoint’s acceptance interval. For centered, symmetric intervals, correlation of either sign can increase the chance that both standardized errors are simultaneously near zero; with shifted or asymmetric intervals, the direction and size of the change require calculation.

Simulation, Diagnostics, and Alternatives

For a more realistic power calculation, simulate the entire planned analysis. For each replicate, generate correlated endpoint estimates (or participant-level data) under the assumed true ratios and covariance structure, apply the specified model and endpoint-specific equivalence tests, and record whether both pass. With \(M\) simulated studies and \(K\) joint successes, estimated joint power is \(K/M\). Its approximate Monte Carlo standard error is \(\sqrt{\hat p(1-\hat p)/M}\), where \(\hat p=K/M\). This error describes simulation precision, not uncertainty in the assumptions or variance estimates used to generate the studies.

Before accepting a result, check that each marginal probability agrees with a separate endpoint calculation under the same assumptions, that the joint probability lies within the bounds above, and that the correlation matrix is valid (in the two-endpoint case, \(-1\leq\rho\leq1\)). Inspect whether the assumed correlation is supported by relevant within-subject data or a defensible planning value. Sensitivity analysis across plausible correlations is important when that input is uncertain. Also verify that the standard errors, critical values, log limits, true ratios, design, and analysis method match the proposed study.

The fixed-standard-error bivariate normal approach is an approximation. In finite samples, standard errors are estimated, critical values may be \(t\)-based, and the two endpoint test statistics may not have a simple bivariate normal distribution. If those features materially affect power, simulate data and apply the intended analysis to every replicate. A simulation is only as credible as its model: it should represent the planned design, endpoint covariance, variability, degrees of freedom, and any analysis rules relevant to success.

Joint power is distinct from type I error control. In a usual intersection-union bioequivalence claim, equivalence is concluded only when every required endpoint passes its own equivalence test. Requiring all component tests to pass at level \(\alpha\) controls the type I error for the overall claim without treating the probability of passing all tests as a multiplicity-adjusted confidence level. Joint power, in contrast, is the probability of passing all tests when the assumed true values are specified. Increasing the number or difficulty of required endpoint criteria can reduce joint power even when each marginal power is unchanged.

Common Pitfalls and Planning Choices

• Multiplying marginal powers automatically. The product is correct only when the two success events are independent. Otherwise calculate the joint probability from their dependence model.
• Assuming the sign of correlation determines the answer. The effect depends on the true means and the acceptance intervals, not just whether \(\rho\) is positive or negative.
• Using a correlation between raw measurements in place of estimate correlation. The required dependence is between the endpoint estimates under the planned design and analysis. It should be derived or justified for that setting.
• Confusing marginal power with the chance both endpoints pass. A study can have high power for each endpoint separately while having appreciably lower joint power.
• Treating a precise simulation as a well-supported plan. A small Monte Carlo standard error cannot repair an implausible covariance or variance assumption.

The main alternatives are to use the independence product when independence is justified, to report probability bounds when dependence is unknown, or to use a multivariate numerical or simulation method when dependence is modeled. Bounds are transparent but may not be informative enough for sample-size selection. A correlated calculation is more targeted, but its accuracy depends on credible assumptions. Report the assumed correlation and model alongside the joint-power estimate so the planning conclusion can be assessed and reproduced.