Why Back-Transform the Limits?
In a bioequivalence analysis, the treatment comparison is often estimated on the natural-log scale. The parameter on that scale is a difference of log means; exponentiating it gives a ratio on the original scale. A confidence interval calculated on the log scale must therefore be transformed at both endpoints to express the range of ratios supported by that interval.
This is not just a change in presentation. The exponential function is nonlinear, so the transformed interval is generally not symmetric around the transformed point estimate. The endpoints must be transformed separately, and their order must be preserved. The transformation does not repair an invalid confidence interval: it carries the interval’s inferential properties to the ratio scale if the original interval is valid.
The interval \([L,U]\) may be a TOST confidence interval, with its confidence level chosen for the equivalence procedure. The specific confidence level and the method used to calculate the log-scale endpoints come from the analysis; back-transformation does not change either one. The earlier material on the confidence-interval rule for TOST establishes how the interval is used for an equivalence decision. This tutorial focuses on converting its limits to the ratio scale.
Transform Each Endpoint
Suppose the confidence interval for \(\theta\) is \([L,U]\). The lower limit on the ratio scale is \(\exp(L)\), and the upper limit is \(\exp(U)\). Because the exponential function is strictly increasing, \(L\leq U\) implies \(\exp(L)\leq\exp(U)\). Thus the transformed lower limit remains the lower limit; there is no reversal of endpoints.
The same monotonicity also explains why the transformation preserves the interval’s coverage. If the original interval has coverage \(1-\alpha\), then its defining event is \(L\leq\theta\leq U\). Applying the increasing function \(\exp\) to all three terms gives the equivalent event \(\exp(L)\leq\exp(\theta)\leq\exp(U)\). Since \(\exp(\theta)=R\), the transformed interval contains \(R\) on exactly the same samples for which the original interval contains \(\theta\).
This argument is conditional on having a valid interval for \(\theta\). For example, if a model-based interval uses an estimated standard error and a \(t\) critical value, its validity depends on the assumptions and analysis that justify that interval. Exponentiating the result does not make those assumptions more or less plausible. It simply transfers the interval from the log scale to the ratio scale.
Worked Example: Back-Transforming a Symmetric Log-Scale Interval
Suppose a hypothetical crossover analysis estimates a log treatment-to-reference ratio of \(0.035\), with standard error \(0.045\) and \(df=18\). For a 90% confidence interval, use the \(t\) critical value \(t_{0.95,18}=1.734\). Assume the model-based interval is appropriate for these data and that the estimate, standard error, and degrees of freedom have been calculated correctly.
First calculate the log-scale margin of error:
The log-scale limits are therefore:
Exponentiate each endpoint:
The back-transformed 90% confidence interval is approximately \([0.9579,1.1197]\). Its order is correct because \(-0.04303<0.11303\) and the exponential function is increasing. As a check, the exponentiated point estimate is \(\exp(0.035)\approx1.0356\), which lies between the transformed limits. The ratio-scale interval is not symmetric around \(1.0356\): the log-scale interval was symmetric, but exponentiation is nonlinear.
Asymmetric Limits and Ratio Margins
Not every log-scale interval is symmetric around its point estimate. An interval may be asymmetric because of the method used to construct it, or because the reported limits have been rounded. The endpoint rule remains the same: exponentiate the actual lower and upper limits separately. Do not infer one transformed limit from the other, and do not assume that the point estimate is the arithmetic midpoint of the transformed limits.
Worked Example: Transforming Asymmetric Log-Scale Limits
Suppose a hypothetical analysis provides a 90% confidence interval for the log ratio with limits \(-0.080\) and \(0.050\). Assume that this interval was obtained by a method appropriate for the analysis, and that the values shown are the limits to be transformed.
Transform the lower endpoint:
Transform the upper endpoint:
The ratio-scale interval is approximately \([0.9231,1.0513]\). The original limits were ordered, \(-0.080<0.050\), and the transformed limits remain ordered, \(0.9231<1.0513\). The result should not be forced into a symmetric form around a point estimate: the transformation is defined by the two given endpoints, not by symmetry.
Equivalence limits can also be expressed on either scale. If the ratio-scale limits are \(R_L\) and \(R_U\), where both are positive and \(R_L<R_U\), the corresponding log-scale limits are \(\ln(R_L)\) and \(\ln(R_U)\). Since the logarithm is increasing on positive values, their order is preserved too. For a ratio-scale interval, checking that both limits fall within the ratio margins is equivalent to checking the corresponding log-scale limits against their log-transformed margins.
Worked Example: Checking Transformed Limits Against Ratio Margins
Suppose a hypothetical TOST analysis has a 90% confidence interval on the log scale of \([-0.200,0.180]\). The ratio-scale equivalence margins are \(0.80\) and \(1.25\). Assume the confidence interval and margins are appropriate for the planned analysis.
Back-transform the confidence limits:
The transformed interval is approximately \([0.8187,1.1972]\). Both limits lie inside the ratio-scale margins because \(0.80<0.8187\) and \(1.1972<1.25\). As a direct check on the scale conversion, the margins correspond to log limits:
The log interval also lies inside these limits: \(-0.2231<-0.200\) and \(0.180<0.2231\). The two checks agree because logarithm and exponential are increasing inverse functions. This example illustrates the scale conversion; it does not replace the analysis needed to justify the model-based interval.
Practical Checks and Common Errors
A few simple checks catch most back-transformation mistakes. First, confirm that the quantity being exponentiated is a natural-log ratio. Exponentiating a difference on another scale, a ratio that has not been logged, or a value calculated using a different logarithm base will not produce the intended ratio limits. Second, confirm that both endpoints were used. Transforming only the point estimate gives a point estimate on the ratio scale, not a confidence interval.
A third check is order: if \(L\leq U\), the reported limits must satisfy \(\exp(L)\leq\exp(U)\). If they do not, there is a calculation or labelling error. Finally, check the transformed point estimate, when available: it should fall between the transformed limits if the original point estimate fell between the log limits. These are arithmetic checks, not tests of whether the original model or interval was appropriate.
A direct ratio-scale interval can be a reasonable alternative when a method is specifically constructed to estimate uncertainty on that scale. It is not automatically interchangeable with a back-transformed log-scale interval: the analysis, estimand, and assumptions must support the interval being used. In the usual log-scale analysis, endpoint transformation is transparent and preserves the original interval’s coverage by monotonicity. An additive interval formed directly around the ratio estimate, by contrast, may not reproduce the uncertainty represented by the log-scale analysis.
Back-transformation also has limits. It cannot correct a poorly chosen model, an inappropriate standard error, invalid degrees of freedom, or a confidence level selected incorrectly for the intended procedure. Nor should rounded intermediate values be treated as exact: when precision matters near a margin, use unrounded endpoints for the transformation and comparison, then round the displayed result appropriately.
Identify its lower limit \(L\) and upper limit \(U\), and confirm that the interval is for the natural-log ratio.
Calculate \(\exp(L)\) and \(\exp(U)\) using adequate precision.
Verify that the transformed lower limit is no greater than the upper limit and that both values are ratios on the intended scale.
State the confidence level and interpret the transformed interval as an interval for the ratio, without implying that exponentiation validates the source analysis.
Check Your Understanding
Answer each question, then compare with the solutions below.
- A log-scale confidence interval is \([-0.10,0.06]\). What are its back-transformed limits, to four significant figures?
- Why can the transformed interval have unequal distances from the exponentiated point estimate to its two endpoints?
- A log-scale interval has limits \(0.12\) and \(-0.04\), in that order. What should be checked before reporting its back-transformed limits?
- Ratio-scale margins are \(0.90\) and \(1.10\). Give their corresponding natural-log limits to four significant figures.
Solutions. (1) \(\exp(-0.10)\approx0.9048\) and \(\exp(0.06)\approx1.062\), so the interval is approximately \([0.9048,1.062]\). (2) Exponentiation is nonlinear, so symmetry around the log estimate does not generally become symmetry around its exponential. Transform both endpoints rather than adding and subtracting a common distance. (3) The purported lower limit exceeds the upper limit. Check whether they were reversed, mislabeled, or incorrectly calculated before back-transforming; a valid ordered pair must satisfy \(L\leq U\). (4) \(\ln(0.90)\approx-0.1054\) and \(\ln(1.10)\approx0.09531\). Because logarithm is increasing, the lower and upper margins retain their order.