Why the Nearest Endpoint Matters
In a difference-scale bioequivalence analysis, the estimate can lie between the equivalence limits while still leaving too much uncertainty to establish equivalence. The decision depends not only on the estimate, but also on the confidence interval around it. When one endpoint lies close to a limit, that endpoint can determine the TOST result: moving it across the limit changes whether both one-sided tests reject.
This tutorial uses a hypothetical contrast between test and reference treatments, measured in ng·h/mL. The example has asymmetric, pre-specified equivalence limits. The calculations illustrate how to read the interval and verify the result from the two tests.
Assumptions and the Interval-Test Connection
The calculations below use an estimate \(\hat{\delta}\), its standard error \(SE\), and \(df\) degrees of freedom supplied by an appropriate model for the contrast. Assume that the contrast and its standard error are correctly specified, the observations or model errors meet the model's independence and distributional assumptions, and the reported degrees of freedom yield the stated \(t\)-reference distribution under the null boundary. In a conventional small-sample \(t\) analysis, this includes an appropriate normal-error model. The equivalence limits must also have been selected before examining the results and must be expressed in the same units and direction as \(\hat{\delta}\).
For each one-sided test, compare the statistic with a \(t\) distribution using the stated \(df\). The lower-bound test asks whether the difference is sufficiently above \(L\); the upper-bound test asks whether it is sufficiently below \(U\). Both tests must reject at significance level \(\alpha\). The corresponding interval has confidence level \(1-2\alpha\), so at \(\alpha=0.05\) it is a 90% two-sided \(t\) interval.
The connection follows directly from the test inequalities. Write \(t_{1-\alpha,df}\) for the \(1-\alpha\) quantile of the \(t\) distribution with \(df\) degrees of freedom. The lower test rejects when its statistic exceeds that critical value; rearranging gives a lower interval endpoint above \(L\). The upper test's rejection inequality rearranges to an upper endpoint below \(U\).
Thus the two tests reject exactly when the \(100(1-2\alpha)\%\) confidence interval, \(\hat{\delta}\pm t_{1-\alpha,df}SE\), lies wholly between \(L\) and \(U\). The statistic \(t_U\) is written as \((U-\hat{\delta})/SE\): its numerator is the distance from the estimate to the upper limit. Keeping the boundary labels tied to the limit being tested prevents a sign or label error.
Worked Example: A Difference Interval Just Inside the Lower Limit
Suppose a hypothetical analysis estimates the test-minus-reference difference as \(\hat{\delta}=-0.40\) ng·h/mL, with \(SE=0.20\) ng·h/mL and \(df=15\). The pre-specified limits are \(L=-0.80\) and \(U=0.10\) ng·h/mL. Use \(\alpha=0.05\).
Check the setup. The estimate, standard error, and both limits are differences in the same units and test-minus-reference direction. Assume the model conditions above hold and that the limits were set independently of these results. The required critical value is \(t_{0.95,15}\approx1.753\).
Construct the interval. Its confidence level is \(1-2(0.05)=0.90\). The margin of error is \(1.753(0.20)=0.3506\) ng·h/mL, so
The lower endpoint is 0.0494 ng·h/mL above the lower limit \(-0.80\); the upper endpoint is below \(0.10\). The interval is therefore inside both limits, but its lower endpoint is relatively close to one of them.
Verify both tests. For the lower limit, \(t_L=[-0.40-(-0.80)]/0.20=2.00\). Its one-sided p-value is \(P(T_{15}\geq2.00)\approx0.03201\). For the upper limit, \(t_U=[0.10-(-0.40)]/0.20=2.50\), with one-sided p-value \(P(T_{15}\geq2.50)\approx0.01225\). Both p-values are below 0.05, so both component null hypotheses are rejected. The TOST conclusion is equivalence within the stated difference limits, under the assumed model.
The lower endpoint is the limiting part of this result: it has less room before reaching its margin than the upper endpoint has before reaching its margin. The conclusion is not based on the estimate merely being closer to one limit than the other; it follows from the interval endpoints and both tests.
How a Small Change Can Alter the Decision
An endpoint near a limit is a useful signal to inspect the precision of the estimate. The next example keeps the estimate, limits, degrees of freedom, and model assumptions the same, but increases the standard error. This isolates the role of uncertainty. A changed standard error might arise from a different data set or a different precision estimate; it should not be changed just to obtain a preferred conclusion.
Worked Example: Greater Uncertainty Crosses One Limit
Keep \(\hat{\delta}=-0.40\) ng·h/mL, \(df=15\), \(L=-0.80\), and \(U=0.10\) ng·h/mL, but suppose \(SE=0.24\) ng·h/mL. Use the same 90% interval and one-sided \(\alpha=0.05\) tests.
Calculate the interval. The half-width is \(1.753(0.24)=0.4207\) ng·h/mL. Therefore,
The upper endpoint remains below \(U=0.10\), but the lower endpoint is below \(L=-0.80\). The lower-bound test confirms this: \(t_L=[-0.40-(-0.80)]/0.24=1.667\), which is less than \(t_{0.95,15}=1.753\). Thus it does not reject \(H_{0L}\) at 0.05. The TOST procedure does not establish equivalence because both tests have not rejected.
This is a different inferential result from the first example even though the estimate is unchanged. Greater uncertainty widened the interval enough to cross the lower limit. The relevant diagnostic is the endpoint's position relative to its margin, not whether the point estimate itself falls inside the limits.
Checking the Result and Comparing Approaches
A practical check is to calculate the interval and the two statistics independently, then verify that the conclusions agree. Here are the expected relationships:
Define the contrast, such as test minus reference, and check that both equivalence limits use that same definition and unit.
Check the model-based estimate, standard error, degrees of freedom, confidence level, and one-sided \(\alpha\).
Compare the lower interval endpoint with \(L\) and the upper endpoint with \(U\); then calculate \(t_L\) and \(t_U\) or their p-values.
The interval should support equivalence exactly when both one-sided tests reject at the specified level, subject to the same model and unrounded calculations.
The confidence-interval view gives a compact picture of uncertainty relative to both limits. The two-test view identifies which boundary is limiting and gives a p-value for each boundary. They are equivalent ways to implement TOST when the same standard error, degrees of freedom, and significance level are used. Looking only at the point estimate is not a substitute: it omits uncertainty. A conventional test of whether the difference equals zero also answers a different question, since TOST assesses whether the difference is sufficiently bounded by the specified margins.
For the usual intersection-union logic, the overall equivalence claim is made only when both component tests reject. If the true difference is outside the equivalence range, at least one component null is true; requiring both rejections keeps the probability of falsely claiming equivalence at or below the component level \(\alpha\). This does not mean that each test can be ignored: a strong result at one boundary cannot compensate for failure at the other.
The method has limits. Its conclusion depends on the contrast being appropriate, the model-based \(SE\) and \(df\) being valid, and the margins being scientifically justified and pre-specified. A confidence interval close to a margin also signals limited separation from that boundary: a modest change in precision or estimate can change the decision. Report enough digits during calculation to assess the endpoint accurately, and state the final limits, interval, units, and decision so that the reader can see which boundary governed the result.
Worked Example: Locating the More Restrictive Boundary
In a further hypothetical analysis, let \(\hat{\delta}=-0.36\) ng·h/mL, \(SE=0.20\) ng·h/mL, \(df=15\), and retain \(L=-0.80\), \(U=0.10\) ng·h/mL. Assume the same model conditions and \(\alpha=0.05\).
Calculate the interval. Using \(t_{0.95,15}=1.753\), its half-width is \(1.753(0.20)=0.3506\). The 90% interval is \(-0.36\pm0.3506=[-0.7106,-0.00939]\) ng·h/mL. Its lower endpoint is 0.0894 above \(L\), and its upper endpoint is 0.1094 below \(U\); both are within the margins.
Check the tests. The lower statistic is \((-0.36+0.80)/0.20=2.20\), and the upper statistic is \((0.10+0.36)/0.20=2.30\). Each exceeds 1.753, so both one-sided tests reject at 0.05 and support equivalence. The lower boundary is closer in interval-distance terms, but the result has more room at that boundary than in the first example.
This comparison shows why a statement such as “the estimate is near the lower margin” is not enough to characterize a result. The position of the interval endpoint, relative to its own margin, reflects both the estimate and its uncertainty.
Check Your Understanding
Use a 90% interval and one-sided \(\alpha=0.05\) unless a question says otherwise. Assume model conditions hold.
- For \(L=-1.0\), \(U=0.5\), \(\hat{\delta}=-0.20\), and \(SE=0.30\), with \(df=15\), calculate the interval and decide whether both limits are cleared.
- In the first worked example, which component test is closer to its critical value, and what does that indicate about the more restrictive boundary?
- Suppose an estimate is inside both equivalence limits, but its interval crosses the upper limit. Can TOST establish equivalence? Explain.
- Why must the difference direction and units be checked before comparing an interval with the margins?
Solutions. (1) The half-width is \(1.753(0.30)=0.5259\), giving \([-0.7259,\,0.3259]\). This lies inside \((-1.0,0.5)\), so both tests reject and equivalence is supported under the assumptions. (2) In the first example, the lower statistic is 2.00 and the upper statistic is 2.50; the lower test is closer to the critical value 1.753, so the lower boundary is more restrictive. (3) No. The upper-bound test has not rejected if the interval reaches or crosses that limit; both component tests must reject. (4) Reversing the contrast changes its sign, and comparing quantities in different units is invalid. Limits and interval must describe the same contrast in the same units.