What a Complete TOST Result Must Communicate
A TOST conclusion is not adequately reported by saying only that two treatments were “equivalent.” Readers need to know which quantity was assessed, what range counted as equivalent, what the data estimated, and how each of the two required tests turned out. Those details make the conclusion interpretable and allow it to be checked against the analysis.
The report should also identify the scale of the estimand. A treatment difference and a treatment ratio use different scales and margins; describing a ratio result as though it were a difference can change its meaning. The earlier material on equivalence confidence intervals and the TOST procedure establishes the decision rule. Here the focus is presenting that decision completely and consistently.
The estimand should be described in context, not just by a symbol. For example, “the mean difference in endpoint \(Y\), treatment minus reference, in the target population” names the comparison, direction, outcome, and population. For a ratio estimand, specify which treatment is in the numerator and which is in the denominator. State the endpoint and units, or say that the estimand is on a log scale when that is the scale used for inference.
Connect the Confidence Interval and the Two Tests
For a symmetric, t-based TOST, the component hypotheses are \(H_{0L}:\theta\leq L\) versus \(H_{aL}:\theta>L\), and \(H_{0U}:\theta\geq U\) versus \(H_{aU}:\theta<U\). Given an estimate \(\hat{\theta}\), standard error \(SE\), and degrees of freedom \(df\), the test statistics are
At one-sided significance level \(\alpha\), the lower test rejects when \(t_L>t_{1-\alpha,df}\), and the upper test rejects when \(t_U<-t_{1-\alpha,df}\). The overall TOST decision is equivalence only if both component null hypotheses are rejected. The component decisions should therefore be reported separately as well as summarized in the overall conclusion.
The corresponding two-sided confidence interval has confidence level \(100(1-2\alpha)\%\), provided the same estimate, standard error, degrees of freedom, and t-based model are used. For the interval \([\hat{\theta}-t_{1-\alpha,df}SE,\ \hat{\theta}+t_{1-\alpha,df}SE]\) to lie strictly inside \((L,U)\), its lower endpoint must exceed \(L\) and its upper endpoint must be less than \(U\). Rearranging those two inequalities gives the two rejection conditions above. This is why, for this procedure, the interval and test decisions agree.
This correspondence depends on using compatible calculations. The estimate and standard error must come from the stated estimand and analysis; the interval must use the critical value for the same \(\alpha\) and \(df\); and the margins must be expressed on that same scale. If a model or inferential method calls for a different interval construction, the report should name that method rather than imply that this particular t-interval identity applies.
State the population comparison, endpoint, direction, and units or transformation.
Report both margins, their units or scale, and the one-sided \(\alpha\).
Include the confidence level and enough precision to assess the margins.
Say whether the lower-bound and upper-bound null hypotheses were rejected.
Conclude equivalence only when both tests pass, and name the treatments or quantities being compared.
Worked Example: Reporting a Difference Estimand
Suppose a hypothetical study estimates the mean difference in a measured endpoint, treatment minus reference, in a target population. The prespecified equivalence margins are \(-5\) and \(5\) units, and the analysis gives \(\hat{\theta}=0.8\) units, \(SE=1.7\) units, and \(df=30\). Use \(\alpha=0.05\) for each one-sided test. Assume the stated t-based analysis is appropriate for this estimand and that its standard error and degrees of freedom are valid.
For \(df=30\), \(t_{0.95,30}\) is approximately \(1.697\). The corresponding 90% confidence interval is
Both interval endpoints are strictly within the margins \([-5,5]\) units. The component statistics are
The lower statistic exceeds \(1.697\), and the upper statistic is less than \(-1.697\). Thus, both component null hypotheses are rejected at the one-sided 0.05 level. A concise report is: “The estimated mean treatment-minus-reference difference was 0.8 units (90% CI: −2.08 to 3.68), with prespecified equivalence margins of −5 to 5 units. Both one-sided tests were significant at \(\alpha=0.05\); the results support equivalence for this estimand.” The conclusion is limited to the stated estimand and margins.
Reporting a Ratio on the Log Scale
In bioavailability and bioequivalence analyses, a ratio estimand is often analyzed after taking logarithms. In that case, the TOST calculations and confidence interval are performed on the log scale, where a ratio margin such as 0.80 to 1.25 corresponds to \(\log(0.80)\) and \(\log(1.25)\). For interpretation, exponentiate the estimate and both interval endpoints. The resulting estimate and interval are on the ratio scale. State clearly both the analysis scale and the scale used for the reported result.
Worked Example: Reporting a Log-Scale Ratio Result
Consider hypothetical log-scale data for a treatment-to-reference geometric mean ratio. Let \(\hat{\theta}=0.03053\), \(SE=0.05336\), and \(df=24\). Use margins \(\log(0.80)=-0.2231436\) and \(\log(1.25)=0.2231436\), with \(\alpha=0.05\). Assume the log-scale t-based inference is valid for the stated estimand. For \(df=24\), \(t_{0.95,24}\) is approximately \(1.711\).
The 90% confidence interval on the log scale is
Exponentiating gives an estimated ratio of \(\exp(0.03053)\approx1.031\), with a 90% confidence interval of \([\exp(-0.060769),\exp(0.121829)]\approx[0.9410,1.1296]\). This interval lies inside the ratio margins of 0.80 and 1.25.
The component statistics, calculated before rounding, are
Both statistics pass their respective one-sided critical-value tests; their one-sided p-values are each below 0.001. A suitable report is: “For the treatment-to-reference geometric mean ratio, the estimated ratio was 1.031 (90% CI: 0.9410 to 1.1296). The prespecified ratio margins were 0.80 to 1.25. At a one-sided \(\alpha=0.05\), both TOST components were significant (lower-bound \(t=4.754\); upper-bound \(t=-3.610\); both \(p<0.001\)); the results support equivalence for this ratio estimand.” Reporting the ratio and ratio-scale interval makes the result interpretable, while naming the log-scale analysis clarifies how inference was conducted.
Worked Example: Reporting a Result That Does Not Establish Equivalence
In another hypothetical ratio analysis, let the log-scale estimate be \(0.150\), with \(SE=0.060\) and \(df=18\). Use the same log margins, \(-0.2231436\) and \(0.2231436\), and \(\alpha=0.05\). Assume the same t-based conditions as in the previous example. The 90% critical value is approximately \(1.734\).
The log-scale interval is
The upper endpoint exceeds the upper log margin \(0.2231436\). On the ratio scale, the estimate is \(\exp(0.150)\approx1.162\), and the interval is approximately \([1.047,\ 1.289]\), which extends beyond 1.25. The upper component statistic is
Since \(-1.219\) is not less than \(-1.734\), the upper-bound null is not rejected; its one-sided p-value is approximately 0.12. The lower-bound test passes, but both tests are required. A correct report is: “The estimated treatment-to-reference ratio was 1.162 (90% CI: 1.047 to 1.289), against margins of 0.80 to 1.25. The lower-bound component passed, but the upper-bound component did not (\(t=-1.219\), \(p\approx0.12\)); equivalence was not established.” This result does not, by itself, demonstrate that the treatments are non-equivalent. It says the required evidence for equivalence was insufficient.
Reporting Choices, Alternatives, and Pitfalls
A confidence interval and two component p-values are complementary ways to show the TOST result. The interval communicates the range of values compatible with the estimate under the stated procedure and makes the margin comparison visible. The component tests show the two separate requirements and their evidence relative to the chosen test level. Reporting both can make the result easier to verify; if space is limited, the interval and explicit statement of both component decisions are usually more informative than an unsupported overall label.
A conventional test of a zero difference answers a different question. Failure to reject \(H_0:\theta=0\) does not show that the effect lies within equivalence margins. Likewise, a confidence interval that includes zero is not enough to establish equivalence. Report the TOST margins and decisions rather than substituting a nonsignificant difference test for the equivalence analysis.
For an analysis that requires equivalence on several endpoints, state which endpoints were required and whether the overall requirement was met. Do not imply that every endpoint passed if only some did. The interpretation of joint requirements and multiplicity is distinct from listing the result for each endpoint; any adjustment or joint decision rule should be described according to the analysis plan.
Before finalizing a report, check that the estimand, margins, estimate, interval, and component tests all refer to the same endpoint, treatment direction, analysis population, and scale. Give enough digits to make the relation to the margins clear, but do not let rounding create a false impression that an endpoint is inside a margin. State the confidence level and \(\alpha\), and use units for differences or an explicitly identified ratio scale for ratios.
Check Your Understanding
Answer each question using the reporting principles above. Brief solutions follow the questions.
- For a difference estimand, what information should accompany an estimate and confidence interval to make the TOST result interpretable?
- At one-sided \(\alpha=0.05\), what confidence level corresponds to the usual symmetric t-based TOST interval?
- A ratio analysis has a 90% interval of 0.83 to 1.18, with margins 0.80 to 1.25. If the interval and tests use the same valid analysis, what conclusion is supported?
- One TOST component is significant and the other is not. What should the overall report say?
Solutions: (1) Name the population estimand, endpoint, treatment direction, units or scale, both prespecified margins, the one-sided \(\alpha\), and the interval confidence level; report both component decisions and the overall conclusion. (2) The corresponding interval is 90%, because \(100(1-2(0.05))=90\%\). (3) The interval lies strictly within both margins, so both tests pass under the stated matching procedure and equivalence is supported for that ratio estimand. (4) Equivalence is not established, because TOST requires rejection of both component null hypotheses; report which component passed and which did not, without calling the treatments proven non-equivalent solely on that basis.