From Summary Statistics to a TOST Decision
A bioequivalence decision is not made by looking only at whether an estimated treatment ratio is close to one. It asks whether the data are sufficiently precise to rule out effects beyond both prespecified equivalence margins. When only summary statistics are supplied, the analysis still requires a sequence of checks: identify the estimate and its uncertainty from the same model, put them on the appropriate scale, conduct both one-sided tests, and verify the result with the matching confidence interval.
For a pharmacokinetic endpoint analyzed on the natural-log scale, let \(\theta\) denote the population mean treatment contrast, such as the mean log test-to-reference ratio. The corresponding population ratio is \(\exp(\theta)\). If the ratio-scale margins are \(L\) and \(U\), their log-scale values are \(\log(L)\) and \(\log(U)\). The familiar limits \(0.80\) and \(1.25\), for example, are reciprocal on the ratio scale and become symmetric log limits.
The summary inputs must be compatible. The estimate, \(SE\), and \(df\) should refer to the same treatment contrast, endpoint, population, model, and analysis. If the supplied estimate is a geometric mean ratio, take its logarithm before using a log-scale \(SE\). Do not pair a log-scale estimate with a ratio-scale standard error, or take the logarithm of an estimate that is already on the log scale.
A t-based calculation assumes that the reported contrast is appropriately modeled by the stated analysis and that its uncertainty is represented by the supplied \(SE\) and \(df\). The inferential justification commonly comes from normally distributed model errors or an adequate large-sample approximation, together with a correctly specified model and a valid study design. Summary statistics alone cannot verify those design and model conditions; they must be checked from the analysis description. If the supplied analysis instead specifies a normal reference distribution, use its stated critical values and tail probabilities rather than substituting t values.
Why the Matching Interval Is a Useful Check
For each test, the p-value is a tail probability under its corresponding boundary null hypothesis. With a t reference distribution having \(df\) degrees of freedom, \(F_{df}\) denotes its cumulative distribution function. The lower-margin test uses the upper tail, while the upper-margin test uses the lower tail:
Both p-values must meet the chosen test rule for equivalence. In the common t-based procedure, the matching two-sided confidence interval has confidence level \(100(1-2\alpha)\%\). Its critical value is the \(1-\alpha\) quantile of the t distribution, so at \(\alpha=0.05\) it is a 90% interval. Equivalence requires that this interval lie strictly between the two log margins. Exponentiating its endpoints gives the corresponding interval on the ratio scale.
The correspondence is direct: the lower endpoint exceeds \(\log(L)\) exactly when the lower one-sided test statistic exceeds \(t_{1-\alpha,df}\); the upper endpoint is below \(\log(U)\) exactly when the upper test statistic is less than \(-t_{1-\alpha,df}\). Thus a correctly calculated TOST decision and its matching interval should agree.
A Complete Calculation With Supplied Summary Statistics
Worked Example: A Ratio Estimate With Compatible Log-Scale Uncertainty
Suppose a hypothetical analysis reports a geometric mean ratio of approximately \(1.040\), a log-scale contrast \(\hat{\theta}=0.039216\), \(SE=0.05500\), and \(df=24\). The prespecified ratio margins are \(L=0.80\) and \(U=1.25\), and \(\alpha=0.05\). Assume the supplied \(SE\) and degrees of freedom come from the model that produced this contrast, and that its stated t-based inference is appropriate. Determine the TOST decision and check it using the interval.
1. Check the scale and margins. The reported contrast is already on the log scale, so it is used directly. The ratio margins become \(\log(0.80)=-0.2231436\) and \(\log(1.25)=0.2231436\). The point ratio implied by the contrast is \(\exp(0.039216)\approx1.0400\), consistent with the rounded supplied ratio.
2. Test the lower margin. Substituting the matching estimate, margin, and standard error gives
The p-value is the upper-tail probability with 24 degrees of freedom: \(p_L=1-F_{24}(4.77018)\approx0.00003726\), or about \(3.726\times10^{-5}\). This is below \(0.05\), so the lower-margin null hypothesis is rejected.
3. Test the upper margin.
The lower-tail probability is \(p_U=F_{24}(-3.34414)\approx0.00135\). It too is below \(0.05\), so the upper-margin null hypothesis is rejected. Both tests support equivalence.
4. Check the matching interval. For \(df=24\), \(t_{0.95,24}\approx1.71088\). The interval is
Both endpoints are strictly between \(-0.2231436\) and \(0.2231436\). Exponentiating the endpoints gives an approximate ratio-scale 90% confidence interval of \((0.9466,\ 1.1426)\), which lies within \((0.80,\ 1.25)\). The interval check agrees with both tests. Under the stated assumptions and analysis, the result supports average bioequivalence for this endpoint; it does not establish equivalence for other endpoints or populations.
Partial Support Is Not Equivalence
Worked Example: One Margin Is Not Ruled Out
Consider a second hypothetical analysis with \(\hat{\theta}=0.2050\), \(SE=0.07000\), and \(df=18\), using margins \(0.80\) and \(1.25\) and \(\alpha=0.05\). Assume again that these are compatible model-based summary statistics and that the t approximation is justified. Is equivalence established?
The log margins remain \(-0.2231436\) and \(0.2231436\). The lower test gives
This is strong evidence against the lower-margin null. But the upper test gives
The upper-margin null is not rejected. The tests therefore do not establish equivalence, regardless of the result for the lower margin.
The matching interval uses \(t_{0.95,18}\approx1.73406\):
The lower endpoint exceeds the lower margin, but the upper endpoint exceeds \(\log(1.25)=0.2231436\). On the ratio scale the interval is approximately \((1.0872,\ 1.3860)\), which extends above the upper limit. This is a direct interval check of the same inconclusive-for-equivalence result.
Boundary Checks, Diagnostics, and Reporting
Worked Example: An Interval Endpoint Exactly at a Margin
For an approximate boundary check, suppose a hypothetical analysis has \(\hat{\theta}=0\), \(SE=0.1000\), \(df=24\), and \(\alpha=0.05\). Set the log margins to \(-0.171088\) and \(0.171088\), using the rounded 90% critical distance \(1.71088(0.1000)\). The corresponding ratio limits are approximately \(0.84275\) and \(1.1866\). Assume the same t-based model conditions as above.
Using the displayed rounded critical value, the 90% confidence interval is \(0\pm1.71088(0.1000)=(-0.171088,\ 0.171088)\): each endpoint matches a displayed margin. The lower test statistic is \(1.71088\), and the upper statistic is \(-1.71088\); each is slightly short of the exact critical value \(t_{0.95,24}\approx1.71088208\), so each one-sided p-value is slightly greater than \(0.05\) and rounds to \(0.0500\). Under the strict rejection rule the result does not establish equivalence. Exact equality is a boundary case, not a result to resolve by rounding. If the displayed values are rounded, use the unrounded estimate, \(SE\), margins, and critical value to make the decision.
A practical workflow is to record the scale and provenance of every input before calculating. Then calculate each test separately, keeping the tail direction explicit, and compare the interval from the same \(SE\), \(df\), and \(\alpha\). If one test fails, the conclusion is not equivalence, even if the point estimate is inside the margins or the other test has a very small p-value.
Several discrepancies are diagnostic warnings rather than reasons to choose whichever result is more favorable. A TOST p-value pair that disagrees with the matching interval can indicate a wrong tail, a mistaken confidence level, a ratio/log-scale mix-up, or mismatched degrees of freedom or standard error. Also check that an estimated ratio was logged exactly once, that the interval uses \(t_{1-\alpha,df}\) rather than a two-sided \(1-\alpha\) critical value, and that the margins were set independently of the observed result.
The interval is often the clearest primary presentation because it displays the estimate, precision, and both margins together. TOST p-values make the two component tests explicit. They are not competing procedures when the same model and confidence level are used: they encode the same decision. A test of whether the treatment contrast differs from zero, or whether a ratio differs from one, answers a different question. Failure to detect a difference does not demonstrate that effects outside the equivalence margins have been ruled out.
A correctly executed summary-statistics calculation still has limits. It does not validate the study design, endpoint handling, model specification, or the suitability of the chosen equivalence margins. Nor does success for one endpoint automatically imply success for multiple exposure endpoints; the required overall decision depends on the prespecified analysis plan. Report the endpoint and scale, estimate, \(SE\), \(df\), margins, interval and confidence level, both p-values, and the decision. Avoid implying that equivalence is a probability statement about the true parameter.
Check Your Understanding
For each question, give the decision and the reason. Solutions are included so that the scale, tail, and interval checks can be compared.
- A ratio estimate is \(1.10\), and its supplied standard error is explicitly on the log scale. What estimate should be used in a log-scale TOST?
- At \(\alpha=0.05\), what confidence level corresponds to the conventional t-based interval for a TOST?
- Both TOST tests have p-values below \(0.05\), but a reported matching interval crosses the upper margin. What should be checked?
- One one-sided p-value is \(0.03\) and the other is \(0.08\). Can equivalence be concluded?
- An unrounded interval endpoint equals an equivalence margin exactly. Does the strict TOST rejection rule establish equivalence at that boundary?
Solutions. (1) Use \(\log(1.10)\approx0.09531\), not \(1.10\), because the standard error is on the log scale and the contrast must be on that same scale. (2) A \(100(1-2\alpha)\%=90\%\) interval, using the \(0.95\) t quantile for its critical value. (3) Reconcile the calculations before reporting a decision: check that the interval uses the same estimate, \(SE\), \(df\), and margins; confirm the log transformation, tail directions, and 90% critical value. Correctly matched results should agree. (4) No. Both tests must reject their respective null hypotheses; the \(0.08\) result does not pass the \(0.05\) criterion. (5) No under the strict rejection rule: equality gives a test statistic at its critical value and a p-value equal to \(0.05\), rather than a statistic strictly beyond the critical value. Do not use rounded output to disguise a boundary.