Reporting Equivalence Intervals on the Ratio Scale

Why Report a TOST Result on the Ratio Scale?

For a bioavailability comparison, a log-scale analysis can be convenient for estimation and testing, while the scientific result is often easier to read as a ratio. A report can therefore show both: the log-scale result used for TOST and its back-transformed ratio-scale presentation. The crucial requirement is consistency. The estimate, confidence interval, and equivalence limits must all be transformed and labelled coherently so the reader can see the same decision on either scale.

This tutorial focuses on that presentation step. It does not change the fitted model, the TOST tests, or the confidence interval chosen for the test. In the usual setup, the contrast is a log ratio: for example, the log of the test-to-reference geometric mean ratio. Exponentiating that contrast gives the ratio estimate. Because exponentiation is increasing, it also carries the interval endpoints and equivalence limits to the ratio scale without reversing their order.

Definition: Let \(\hat{\theta}\) be an estimated log ratio, with the TOST confidence interval \([l,u]\). The ratio estimate is \(\exp(\hat{\theta})\), and its back-transformed confidence interval is \([\exp(l),\exp(u)]\). If the positive ratio equivalence limits are \(M_L\) and \(M_U\), their log-scale limits are \(\ln(M_L)\) and \(\ln(M_U)\). Report the estimate and interval with the limits on the same scale.

The interval in this definition must be the interval used for the TOST decision. For example, when the TOST uses significance level \(\alpha=0.05\), the corresponding interval is conventionally the 90% confidence interval. Back-transforming a different confidence interval may produce a different inclusion result and would not report the decision from that TOST. The confidence level and the equivalence limits should therefore be stated, not left for the reader to infer.

Why the Decision Is Preserved

On the log scale, equivalence requires the full TOST interval to lie strictly between the transformed margins. On the ratio scale, it requires the full back-transformed interval to lie strictly between the ratio margins. The two statements are equivalent because the exponential function preserves order.

\ln(M_L)<l\leq u<\ln(M_U) \quad\Longleftrightarrow\quad M_L<\exp(l)\leq\exp(u)<M_U

This is the key reporting justification: if the interval and both margins are transformed consistently, the equivalence conclusion does not change. The use of a closed interval in the notation records its endpoints; the usual TOST rejection rule requires the confidence limits to be strictly inside the equivalence limits. If a limit equals a margin, do not describe it as strictly inside.

The result depends on a valid log-ratio analysis and a confidence interval appropriate to that TOST. In particular, the quantities must be positive before taking logarithms; the contrast must genuinely represent a log ratio; and the interval and margins must refer to the same comparison and estimand. Back-transformation cannot fix a misspecified analysis, an inappropriate margin, or an incorrectly selected confidence level.

Identify the tested contrast.
Confirm which quantity is in the numerator and denominator, such as test/reference, and retain that order in the reported ratio.

Keep the TOST interval and margins together.
Record the log estimate, both endpoints of the TOST confidence interval, and both log-scale equivalence limits.

Exponentiate consistently.
Exponentiate the estimate, each interval endpoint, and each margin. Do not exponentiate only the estimate or only the confidence limits.

Check and report the result.
Compare the back-transformed interval with the ratio limits, state the confidence level and units or dimensionless nature of the ratio, and give the TOST conclusion.

Worked Ratio-Scale Reports

Worked Example: An Interval Entirely Within the Ratio Limits

Suppose a hypothetical crossover analysis estimates a test-to-reference log ratio of \(\hat{\theta}=0.0400\). Its 90% TOST confidence interval is \([-0.0800,0.1600]\), and the log-scale equivalence limits are \([\ln(0.80),\ln(1.25)]\). Report the result on the ratio scale.

Exponentiate the estimate:

\exp(0.0400)=1.0408

Exponentiate each confidence limit:

\left[\exp(-0.0800),\exp(0.1600)\right] = [0.9231,1.1735]

The back-transformed margins are \(\exp(\ln(0.80))=0.80\) and \(\exp(\ln(1.25))=1.25\). Since \(0.80<0.9231\) and \(1.1735<1.25\), the complete 90% interval lies within the ratio limits. A suitable report is: “The estimated test-to-reference ratio was 1.041, with a 90% confidence interval of 0.923 to 1.174; the prespecified equivalence limits were 0.80 to 1.25. The TOST result supports equivalence.” The ratio is dimensionless; the names of the numerator and denominator make its direction clear.

Worked Example: Reporting Unequal Ratio Limits

Suppose a second hypothetical analysis gives \(\hat{\theta}=0.0050\), with a 90% TOST confidence interval \([-0.0700,0.0800]\). The positive ratio equivalence limits are 0.90 and 1.10. Show how to report the result and check whether it supports equivalence.

First transform the estimate and interval:

\exp(0.0050)=1.005

[\exp(-0.0700),\exp(0.0800)]=[0.9324,1.0833]

The ratio limits correspond to log limits \(\ln(0.90)=-0.1054\) and \(\ln(1.10)=0.09531\). The interval is strictly inside the ratio limits because \(0.90<0.9324\) and \(1.0833<1.10\). Thus the TOST decision supports equivalence. A concise report is: “The estimated test-to-reference ratio was 1.005 (90% confidence interval 0.932 to 1.083), within the prespecified limits of 0.90 to 1.10.”

The limits are not reciprocals: \(1/0.90\) is approximately 1.111, not 1.10. That does not invalidate the comparison. The correct procedure is to use the actual prespecified lower and upper limits, transform each one separately when working on the log scale, and report those same limits on the ratio scale.

Worked Example: A Back-Transformed Interval That Fails the Limits

Consider a hypothetical log-ratio estimate \(\hat{\theta}=-0.1000\), with 90% TOST confidence interval \([-0.2500,0.0500]\). The ratio limits are 0.80 and 1.25. Determine the ratio-scale conclusion.

The estimate and interval back-transform to:

\exp(-0.1000)=0.9048

[\exp(-0.2500),\exp(0.0500)]=[0.7788,1.051]

Although the point estimate, 0.9048, is between 0.80 and 1.25, the interval is not: its lower endpoint, 0.7788, is below 0.80. Therefore the full interval is not within the equivalence limits, and this TOST does not establish equivalence. The appropriate report is not “equivalent because the estimate is within the limits”; it is “the estimated ratio was 0.905 (90% confidence interval 0.779 to 1.051), which extends below the lower equivalence limit of 0.80.”

Choosing a Clear and Faithful Presentation

A ratio-scale report is most useful when it identifies the contrast and gives the estimate, confidence interval, and equivalence limits together. In bioavailability contexts, write “test/reference” or “reference/test” rather than just “ratio.” Reversing the comparison changes the estimate and the margins: the reciprocal of a ratio is not the same reported contrast. A report should also identify the confidence level, and should not imply that the interval is a range containing a stated probability of the true parameter.

The log-scale presentation and the ratio-scale presentation serve related but distinct purposes. The log scale displays the contrast used in the analysis and is convenient for expressing multiplicative effects. The ratio scale often makes the size and direction of the effect more interpretable. Neither should be substituted for the other without stating what has been done. If both are shown, label them explicitly and use the same confidence interval and equivalence limits in each version.

A raw difference-scale presentation is a different estimand. A difference of 0.05 and a ratio of 1.05 do not mean the same thing in general: their interpretation depends on the scale and the quantities being compared. Do not attach ratio limits to a difference interval, or describe a difference-scale analysis as a ratio result simply because the numerical values look similar.

Key takeaway: Report the TOST ratio estimate as \(\exp(\hat{\theta})\), the interval as \([\exp(l),\exp(u)]\), and the equivalence limits as \(M_L\) and \(M_U\). When all are transformed consistently, the ratio-scale inclusion check gives the same TOST decision as the log-scale check.

Common Reporting Pitfalls and Diagnostics

Checking only the point estimate. A point estimate inside the limits is not enough; the entire TOST confidence interval must be strictly inside them.
Mixing scales. Comparing log-scale interval endpoints with ratio limits is meaningless. Put both the interval and the limits on the same scale before checking inclusion.
Transforming inconsistently. Exponentiating the estimate and interval but leaving the margins on the log scale produces a mismatched report. Transform both margins too, or state all quantities on the log scale.
Omitting the comparison direction. “Ratio = 1.04” is incomplete if the reader cannot tell whether it means test/reference or reference/test.
Rounding before the decision check. Displayed values can conceal whether a limit is crossed. Make the decision using unrounded values from the analysis, then report enough digits for the presentation to remain clear.
Changing the confidence level. The reported interval must be the interval associated with the TOST significance level. A familiar 95% interval is not automatically the interval used for a 5% TOST.

A useful diagnostic is to verify the decision twice: check whether \(l\) and \(u\) lie within the log margins, then check whether \(\exp(l)\) and \(\exp(u)\) lie within the ratio margins. Disagreement signals a transcription, scale, endpoint-order, or rounding error. Since exponentiation preserves order, a correctly transformed result cannot give opposite decisions.

Check Your Understanding

Answer each question, then compare your reasoning with the solutions.

A log-ratio estimate is 0.0200 and its TOST interval is \([-0.0600,0.1000]\). What are the ratio estimate and back-transformed interval to three significant figures?
Why must the equivalence limits be reported on the same scale as the confidence interval used for the inclusion check?
A ratio estimate is 0.98, and its confidence interval is 0.79 to 1.08. The limits are 0.80 to 1.25. Does the estimate being inside the limits establish equivalence?
A report gives a ratio interval of 0.92 to 1.15 but lists limits as \(\ln(0.80)\) to \(\ln(1.25)\). What is wrong with this presentation?
What information should accompany a reported ratio estimate so its direction is unambiguous?

Solutions. (1) The estimate is \(\exp(0.0200)=1.02\). The interval is \([\exp(-0.0600),\exp(0.1000)]=[0.942,1.11]\). (2) Inclusion is meaningful only when the interval and limits are expressed on the same scale; exponentiating both preserves their order and the decision. (3) No. The interval extends below the lower limit because \(0.79<0.80\), so it is not wholly within the limits. (4) The interval is on the ratio scale but the limits are on the log scale. Report the limits as 0.80 and 1.25, or transform the interval to the log scale. (5) State which quantity is in the numerator and which is in the denominator, such as test/reference.