What a Power Target Must Specify
A sample-size calculation for an equivalence study needs more than a desired power percentage. It needs a precise definition of success, a value for the treatment effect assumed true when planning, and a desired probability of success under that assumption. These choices determine the question that a later power calculation will answer; they do not, by themselves, determine a sample size.
For average bioequivalence, the effect is commonly expressed as the test-to-reference ratio of geometric means for a pharmacokinetic endpoint. Equivalence is assessed against prespecified lower and upper limits. A planning target is interpretable only when its limits, decision rule, assumed ratio, and power goal refer to the same endpoint and analysis.
Power is a probability calculated under specified assumptions about the true effect and the sampling distribution. It is not the probability that the treatments are equivalent, nor is it a guarantee that a particular study will succeed. A target such as 80% power means that, under the assumed true effect and the planning model, repeated studies of the planned design would meet the stated success rule about 80% of the time.
Define Success Before Choosing a Power Goal
Let \(R\) denote the test-to-reference geometric mean ratio, and let \(L\) and \(U\) be the prespecified lower and upper equivalence limits, with \(0<L<U\). On the log scale, write \(\theta=\log(R)\), \(l=\log(L)\), and \(u=\log(U)\). The equivalence region is \(L<R<U\), or equivalently \(l<\theta<u\).
In the usual two one-sided tests (TOST) formulation at one-sided significance level \(\alpha\), the null hypothesis is that the true effect is outside or on the limits, and the alternative is that it is inside them:
Success requires rejecting both one-sided null hypotheses. Under the usual matching confidence-interval procedure, that is equivalent to the corresponding two-sided \(100(1-2\alpha)\%\) confidence interval for \(\theta\) lying entirely inside \((l,u)\). For example, with \(\alpha=0.05\), the interval is a 90% confidence interval. Equivalently, on the ratio scale, its exponentiated limits must both lie strictly between \(L\) and \(U\). The specific analysis and interval must be prespecified; the equivalence decision is not based merely on whether the point estimate falls within the limits.
Worked Example: Writing the Decision Criterion
Suppose a hypothetical study uses the conventional ratio limits \(L=0.80\) and \(U=1.25\), with TOST at \(\alpha=0.05\). State the success rule on the log and ratio scales.
Step 1: Transform the limits. The log-scale limits are \(\log(0.80)=-0.2231\) and \(\log(1.25)=0.2231\), to four significant figures.
Step 2: State the interval rule. Success requires the 90% confidence interval for \(\theta\) to be wholly inside \((-0.2231,0.2231)\). Exponentiating the interval endpoints gives the equivalent rule that the 90% confidence interval for \(R\) is wholly inside \((0.80,1.25)\).
Interpretation. A point estimate of \(R=1.02\) alone would not establish success. The uncertainty interval must also satisfy the prespecified containment rule.
Specify the Assumed True Ratio
Power must be evaluated at a specified true effect. For this setting, the planning value is often expressed as an assumed true test-to-reference ratio \(R_0\), or as \(\theta_0=\log(R_0)\). It should represent a scientifically and operationally credible value for the intended product comparison, not simply the most convenient value for producing a desired sample size.
The location of \(R_0\) within the equivalence limits matters. If it is near the center of the limits, there is more room for sampling uncertainty while keeping the confidence interval inside. If it is closer to one limit, the decision is more sensitive to uncertainty on that side. Thus, the same margins, variability, design, and nominal power goal can imply different power at different assumed ratios.
Worked Example: Translating a Planning Ratio to the Log Scale
For a hypothetical endpoint, suppose the limits are 0.80 and 1.25 and the assumed true ratio is \(R_0=0.95\). Calculate the assumed log effect and its distances from the two log limits.
Assumed log effect: \(\theta_0=\log(0.95)=-0.05129\). The log limits are \(-0.2231\) and \(0.2231\), as calculated above.
Distance from the lower limit: \(-0.05129-(-0.2231)=0.1718\). Distance from the upper limit: \(0.2231-(-0.05129)=0.2744\).
The assumed effect is inside the equivalence region but closer to the lower limit on the log scale. This does not calculate power: a power calculation also requires information about sampling uncertainty and the study design. It does identify which boundary is nearer to the assumed true effect.
An assumed ratio outside the equivalence limits is generally not a sensible value at which to set a target for demonstrating equivalence. A study can, in principle, satisfy the decision rule by chance even when the true effect is outside the limits, but that probability is not a meaningful basis for claiming an adequately powered equivalence study. The planning assumption should reflect an effect for which equivalence is scientifically expected, while acknowledging uncertainty in that expectation.
State the Desired Probability and Its Scope
The desired probability of success is commonly called the target power, \(1-\beta\), where \(\beta\) is the probability of failing the decision rule under the specified planning assumptions. A target such as 80% or 90% is a design choice. It should be stated alongside the criterion and assumed true effect, not presented as a property of the treatment itself.
For one endpoint, a complete target might read: “At a true test-to-reference ratio of 0.95, plan for an 80% probability that the prespecified 90% confidence interval lies wholly within 0.80 to 1.25.” This sentence defines the decision event and the effect at which its probability is evaluated. It does not yet specify whether a particular design can achieve that probability.
When two or more endpoints must all pass, distinguish endpoint-specific power from the probability of overall success. If \(A\) is success for one endpoint and \(B\) is success for another, overall success is the event \(A\cap B\), and its probability is \(P(A\cap B)\). It is not generally equal to either marginal probability or their product; the product applies only under independence.
Worked Example: Defining Power for Two Required Endpoints
A hypothetical study requires both AUC and \(C_{\max}\) to meet their prespecified equivalence criteria. The planning team wants a 90% probability that both endpoints pass. What must the target say, and why are two separate 90% endpoint targets not enough to establish it?
Let \(A\) denote AUC success and \(C\) denote \(C_{\max}\) success. The requested target is \(P(A\cap C)=0.90\), evaluated under the specified assumed true ratios and planning model for both endpoints. This is a joint power target.
If, for illustration only, the endpoint success events were independent and each had probability 0.90, then \(P(A\cap C)=P(A)P(C)=0.90\times0.90=0.81\). Thus, even under independence, 90% power for each endpoint would give 81% joint power, not 90%. In practice the events need not be independent, so their dependence must be handled by the chosen planning method; marginal targets alone do not determine the joint probability.
To make this target complete, the team must state the success criterion and assumed true ratio for each endpoint, as well as that the desired 90% concerns both passing. Those values define the target; the calculation of whether a proposed design achieves it is a separate step.
Why the Confidence-Interval Rule Matches TOST
The equivalence criterion can be justified directly from the two one-sided tests. Let \(\hat{\theta}\) be the estimated log ratio and \(SE\) its standard error. For a symmetric interval using the appropriate critical value \(c\), the matching confidence interval is \(\hat{\theta}\pm cSE\). At one-sided level \(\alpha\), the lower-bound test rejects when \(\hat{\theta}-cSE>l\), and the upper-bound test rejects when \(\hat{\theta}+cSE<u\).
The two inequalities say exactly that the interval’s lower endpoint is above the lower equivalence limit and its upper endpoint is below the upper limit. With \(c\) chosen for a two-sided \(100(1-2\alpha)\%\) interval, this proves the stated correspondence under the same model and standard-error calculation. It also shows why a point estimate inside the margins is insufficient: neither inequality follows from \(l<\hat{\theta}<u\) alone.
Common Mistakes, Diagnostics, and Limits
- Giving a power percentage without defining success. “90% power” is incomplete if the margins, analysis rule, or required endpoints are unspecified. Write the decision event explicitly.
- Using the null value as the assumed effect by default. A ratio of 1.00 may be a reasonable planning assumption, but it is a scientific assumption, not a requirement. Check whether a different plausible ratio is expected and how close it is to either limit.
- Treating an in-range point estimate as success. Equivalence requires the complete prespecified confidence interval to fit within the limits. A narrow decision rule cannot be replaced by a visual check of the point estimate.
- Confusing power with the probability equivalence is true. Power is the probability of the specified decision under a specified true effect and model. It is not a posterior probability about the hypothesis.
- Mixing scales or endpoints. Confirm that the ratio limits, assumed ratio, and estimated effect refer to the same endpoint and that log-scale limits are obtained by taking logarithms of the ratio limits.
- Calling marginal power “overall power.” If every endpoint must pass, define success as their intersection and target its probability. Separate endpoint probabilities do not, without dependence information, determine that probability.
The target is conditional on the assumptions used in planning. If the true ratio differs from the assumed value, or if the planned model and variability assumptions do not adequately describe the data, the achieved probability may differ from the target. A single target also cannot summarize every possible effect value: power varies across the equivalence region. Sensitivity to plausible alternative ratios is therefore useful, but it does not change the need to state one primary planning target.
The main alternative to describing success through TOST is to state it as confidence-interval containment. Under the matching interval construction, these are equivalent descriptions of the same decision, not competing evidence standards. By contrast, testing whether the ratio differs from 1.00 answers a different question: failure to detect a difference does not establish that the effect lies within the equivalence limits.
Check Your Understanding
Answer each question, then compare with the solutions below.
- With TOST at \(\alpha=0.05\), what confidence-interval level corresponds to the usual interval-containment rule?
- For limits 0.80 and 1.25 and assumed ratio 1.10, calculate the assumed log effect and its distances from the two log limits.
- Why does a point estimate inside the equivalence limits not, by itself, establish study success?
- If two required endpoint success events are independent and each has probability 0.90, what is the probability that both succeed?
- Explain why “80% power” alone is not a complete power target for an equivalence study.
Solutions. (1) A 90% confidence interval, because \(100(1-2\alpha)=90\%\). (2) \(\log(1.10)=0.09531\). The lower log limit is \(-0.2231\) and the upper is \(0.2231\); the distances are \(0.09531-(-0.2231)=0.3184\) and \(0.2231-0.09531=0.1278\), respectively. The assumed effect is closer to the upper limit. (3) Success requires the full prespecified confidence interval to lie within both limits. (4) Under the stated independence assumption, \(0.90\times0.90=0.81\), or 81%. (5) It does not identify the success criterion, the assumed true effect at which the probability is evaluated, or whether the target concerns one endpoint or all required endpoints.