Requiring Equivalence Across Multiple Exposure Endpoints

Why Multiple Endpoints Require a Joint Decision

A bioequivalence assessment may involve more than one exposure endpoint. For example, a protocol may require equivalence for both area under the concentration-time curve (AUC) and maximum concentration (\(C_{\max}\)). If the scientific claim is that every required endpoint is equivalent, then a favorable result for one endpoint cannot make up for an unfavorable result for another. The joint conclusion is therefore built from the separate endpoint decisions.

This tutorial assumes that the endpoints, their estimands, their equivalence limits, and the analysis methods have been specified before examining the results. For each endpoint, the estimate and its standard error must come from a suitable analysis, and the assumptions supporting that analysis must be reasonable. The particular model may depend on the study design; here, the model-based estimates and standard errors are treated as supplied by an appropriate analysis.

Definition: For endpoint \(j\), let \(\theta_j\) be the parameter being assessed and let \(L_j\) and \(U_j\) be its lower and upper equivalence limits, with \(L_j<U_j\). Equivalence for that endpoint means \(L_j<\theta_j<U_j\). If the required claim is equivalence for every endpoint in a prespecified set, the joint claim is that all of these endpoint-specific inequalities hold.

Separate TOST Decisions and the Joint Rule

For each endpoint, a two one-sided tests procedure (TOST) evaluates whether the parameter is above the lower limit and below the upper limit. Its hypotheses for endpoint \(j\) are

H_{0j}: \theta_j \leq L_j \ \text{or}\ \theta_j \geq U_j \qquad\text{versus}\qquad H_{aj}: L_j < \theta_j < U_j.

The endpoint passes only when both one-sided null hypotheses are rejected at the specified significance level \(\alpha\). Equivalently, using the usual TOST confidence-interval procedure, its \(100(1-2\alpha)\%\) confidence interval must lie strictly inside the endpoint’s equivalence limits. For example, at \(\alpha=0.05\), the corresponding interval is a 90% confidence interval. If the interval touches or crosses either limit, that endpoint does not pass.

When ratios are analyzed on the logarithmic scale, the parameter and limits must be expressed on the same scale for the TOST. A confidence interval calculated for a log ratio can then be exponentiated to give the interval for the ratio. The comparison with ratio limits is made on that original ratio scale. Differences, ratios, or other endpoint parameters must not be mixed with limits defined for a different scale.

Joint decision rule: If equivalence is required for every prespecified endpoint, conclude joint equivalence only when every endpoint passes its own TOST. If even one required endpoint fails, joint equivalence has not been demonstrated.

The logic can be justified directly. The global alternative is the intersection of the endpoint alternatives: all endpoints are equivalent. Its null is that at least one endpoint is not equivalent. If that global null is true, at least one endpoint-specific null is true. A false joint conclusion requires rejecting the null for that true component as part of rejecting every endpoint null. Thus the all-pass rule protects the joint claim through the component decisions; it does not require endpoint estimates to be independent. Dependence can affect the estimates and their uncertainty, but it does not change the logical requirement that every endpoint pass.

Worked Decisions for Multiple Exposure Endpoints

Worked Example: Both AUC and \(C_{\max}\) Pass

Suppose a hypothetical crossover study requires equivalence for AUC and \(C_{\max}\), each expressed as a test-to-reference geometric mean ratio. Both endpoints have prespecified limits of 0.80 and 1.25. Assume the analyses are appropriate, the model assumptions are adequate, and the model-based standard errors and degrees of freedom below are valid. Use \(\alpha=0.05\), so a 90% confidence interval is compared with the limits.

For AUC, suppose the estimated log ratio is \(\ln(1.04)=0.03922\), with \(SE=0.0450\) and \(df=23\). The 90% \(t\)-critical value is approximately 1.714. The interval on the log scale is

0.03922 \pm 1.714(0.0450) =0.03922\pm0.07713 =(-0.03791,\ 0.11635).

Exponentiating the two endpoints gives the ratio interval \((e^{-0.03791},e^{0.11635})=(0.9628,1.1234)\). It lies strictly within \((0.80,1.25)\), so AUC passes.

For \(C_{\max}\), suppose the estimated log ratio is \(\ln(1.08)=0.07696\), with \(SE=0.0500\) and the same \(df=23\). Its interval is

0.07696\pm1.714(0.0500) =0.07696\pm0.08570 =(-0.00874,\ 0.16266).

After exponentiation, this is \((0.9913,1.1766)\), also strictly within \((0.80,1.25)\). Both required endpoints pass, so the joint equivalence requirement is met for these endpoints under the stated assumptions.

Worked Example: One Endpoint Prevents the Joint Conclusion

Consider a second hypothetical assessment with two required ratio endpoints and limits of 0.80 and 1.25 for each. Assume the endpoint analyses and their standard errors are valid. The AUC result is a 90% confidence interval of \((0.93,1.10)\), which lies within the limits and passes.

For \(C_{\max}\), suppose the estimated ratio is 0.84, so the log estimate is \(\ln(0.84)=-0.17435\). Let \(SE=0.0650\), \(df=18\), and use the 90% \(t\)-critical value 1.734. Then

-0.17435\pm1.734(0.0650) =-0.17435\pm0.11271 =(-0.28706,\ -0.06164).

Exponentiating gives \((e^{-0.28706},e^{-0.06164})=(0.7505,0.9402)\). The lower endpoint, 0.7505, is below the lower limit 0.80, so \(C_{\max}\) fails. AUC passes, but the joint requirement does not: equivalence has not been demonstrated for every required endpoint. This conclusion does not say that \(C_{\max}\) is proven non-equivalent; it says the TOST criterion for that endpoint was not met.

Worked Example: Endpoint-Specific Limits and a Complete Joint Check

Suppose a hypothetical protocol requires equivalence for AUC and \(C_{\max}\), but specifies different ratio limits based on the endpoint’s scientific and regulatory context: \((0.80,1.25)\) for AUC and \((0.75,1.33)\) for \(C_{\max}\). These hypothetical limits are assumed to have been justified and fixed in advance. Assume valid log-scale model estimates, \(df=28\), and a 90% interval, for which \(t_{0.95,28}\) is approximately 1.701.

For AUC, let the log estimate be 0.01980 and \(SE=0.0350\). The half-width is \(1.701(0.0350)=0.05954\), giving the log interval \((-0.03974,0.07934)\). Exponentiating gives \((0.9610,1.0826)\), which is within the AUC limits.

For \(C_{\max}\), let the log estimate be 0.11333 and \(SE=0.0600\). The half-width is \(1.701(0.0600)=0.10206\), so the log interval is \((0.01127,0.21539)\). Exponentiating gives \((1.0113,1.2403)\), which lies within the \(C_{\max}\) limits of \((0.75,1.33)\). Each endpoint passes against its own specified limits; therefore, the joint requirement passes. Comparing both intervals with the AUC limits instead would be an error because the protocol defined endpoint-specific limits.

Assumptions, Diagnostics, and Failure Modes

A joint decision is only as sound as the endpoint analyses feeding it. For each endpoint, check that the parameter being estimated matches the prespecified estimand, that its confidence interval uses the appropriate standard error and degrees of freedom, and that the model assumptions are plausible for the study design and data. On a log scale, confirm that the estimate, interval, and equivalence limits are all interpreted consistently after transformation. A calculation can be arithmetically correct yet answer the wrong question if, for example, it uses a difference estimate with ratio limits.

Also verify that the full set of required endpoints has been carried through to the decision. Endpoint definitions, margins, transformations, analysis populations, and handling of missing or unusable data should be prespecified and applied consistently. A joint claim cannot be supported by reporting only the endpoints that passed. If an endpoint cannot be estimated reliably, that is not evidence that it passed; the joint conclusion remains unsupported unless the planned analysis provides a valid basis for its decision.

The endpoint estimates may be correlated, especially when they are measured on the same participants. That correlation does not permit one endpoint to compensate for another under an all-endpoints requirement. Nor should endpoint estimates be averaged into a single number unless a separately justified composite estimand was defined; a composite answers a different scientific question. Endpoint-by-endpoint TOST is appropriate when the claim itself is that each specified exposure measure meets its own criterion.

Common Pitfall and Interpretation

A common mistake is to reason, “AUC passed by a wide margin, so it offsets the \(C_{\max}\) interval that crosses its limit.” This is incorrect because the required alternative is a conjunction: AUC must be equivalent and \(C_{\max}\) must be equivalent. The correction is to record a separate TOST decision for each prespecified endpoint and then apply the all-pass rule.

Likewise, a failed endpoint should not automatically be described as proof of a meaningful exposure difference. A confidence interval that crosses a limit can reflect an estimate outside the acceptable range, insufficient precision, or both. The appropriate statement is that equivalence was not demonstrated for that endpoint under the specified test and data. The joint requirement is consequently not met, even if every other endpoint passes.

This procedure does not determine which endpoints ought to be required, what their margins should be, or whether the underlying analyses are scientifically appropriate. Those choices need justification before results are known. Its role is narrower: given a prespecified endpoint set and valid endpoint-specific TOST results, it gives the correct combined conclusion when success requires equivalence for all of them.

Check Your Understanding

For each question, state the endpoint decisions and the joint conclusion. The solutions are provided after the questions.

Three required endpoints have confidence intervals wholly within their respective equivalence limits. What is the joint conclusion?
Two endpoints are required. One passes, while the other interval crosses its lower equivalence limit. What can be concluded about the joint requirement, and what cannot be concluded about the failed endpoint?
Why is it invalid to use a particularly favorable result for one endpoint to offset another endpoint that fails?
A ratio endpoint has limits \((0.80,1.25)\), but the analyst compares its log-scale interval directly with those untransformed ratio limits. What is wrong, and how should the comparison be made?
A required endpoint is not estimable under the planned analysis because of unusable data. May it be counted as a pass because no interval crossed a margin?

Solutions. (1) The joint requirement passes because every required endpoint passes its own TOST. (2) The joint requirement is not met. The failed endpoint has not demonstrated equivalence, but failure to demonstrate equivalence is not by itself proof of non-equivalence. (3) The claim requires all endpoint-specific alternatives to hold; a favorable result for one cannot satisfy a different endpoint’s criterion. (4) The scales are inconsistent. Exponentiate the log-scale interval and compare the resulting ratio interval with \((0.80,1.25)\), or compare the log interval with the logarithms of both limits. (5) No. An endpoint without a valid decision is not a pass, so the all-endpoints requirement has not been demonstrated.