Worked Example: Ratio Confidence Interval Crossing One Limit

Why a Crossed Limit Identifies a Failed Test

For a ratio-scale treatment comparison, the equivalence region is usually specified by a lower ratio limit \(L\) and an upper ratio limit \(U\), where \(0<L<U\). A confidence interval that extends past one of these limits does not establish equivalence, even if the point estimate is close to 1. The location of the crossed endpoint tells us which one-sided test in the two one-sided tests procedure (TOST) fails.

The calculations are performed on the log scale because a ratio becomes a difference after taking logarithms. Let \(\theta\) denote the population ratio of interest, such as a treatment-to-reference geometric mean ratio, and let \(\eta=\log(\theta)\). Write \(\hat{\eta}\) for the estimate of \(\eta\), with standard error \(SE\). The ratio limits \(L\) and \(U\) correspond to log-scale limits \(\log(L)\) and \(\log(U)\).

For the common limits \(L=0.80\) and \(U=1.25\), the log limits are approximately \(-0.22314\) and \(0.22314\). They are symmetric around zero on the log scale, although they are not equally far from 1 as additive differences on the ratio scale. Exponentiation is increasing, so a log-scale interval crosses a log limit if and only if its back-transformed ratio interval crosses the corresponding ratio limit.

How the Interval and the Two Tests Correspond

Suppose the estimate follows a normal model with known \(SE\), or a \(t\) model with an appropriate degrees of freedom \(df\). Let \(c\) be the relevant one-sided critical value: \(z_{1-\alpha}\) for a standard normal test, or \(t_{1-\alpha,df}\) for a \(t\) test. The two one-sided statistics are

The lower-bound test rejects when \(T_L>c\); the upper-bound test rejects when \(T_U<-c\). Equivalently, for the matching \(100(1-2\alpha)\%\) two-sided confidence interval, both endpoints must fall strictly inside the log-scale equivalence limits.

The connection follows directly from rearranging the test inequalities. The lower test rejects exactly when

Thus, if the lower confidence endpoint is below \(\log(L)\), the lower-bound test does not reject. For the upper test,

So if the upper confidence endpoint is above \(\log(U)\), the upper-bound test does not reject. Exponentiating the interval endpoints gives the same checks on the ratio scale. This relationship assumes the interval and tests use the same estimate, standard error, degrees of freedom or reference distribution, and significance level.

Check the Corresponding Boundary, Not Just the Estimate

A confidence interval can fail the equivalence criterion at either boundary. Crossing the lower limit points to the lower-bound test; crossing the upper limit points to the upper-bound test. The test direction matters: the lower test looks for evidence that the parameter is above the lower limit, whereas the upper test looks for evidence that it is below the upper limit. A statistic's sign alone does not determine the decision; compare it with the appropriate critical value.

Assumptions, Diagnostics, and Alternatives

These calculations are only as reliable as the estimate and uncertainty model that produced them. The contrast must estimate the intended log ratio, the standard error must represent its sampling variability, and the stated reference distribution and degrees of freedom must be appropriate. For a \(t\)-based calculation, the standard error and degrees of freedom must come from the model or analysis method supporting that \(t\) inference. The equivalence limits should be specified independently of the observed results.

Useful checks include confirming that both endpoints were calculated from the same \(\hat{\eta}\) and \(SE\), that the confidence level matches the TOST significance level, and that the ratio interval was obtained by exponentiating—not by adding—the log-scale endpoints. Because exponentiation is monotone, the ratio interval must preserve endpoint order. A reversed interval or a reported endpoint that does not exponentiate from its stated log value signals a calculation or transcription error.

The confidence-interval presentation and the two-test presentation are alternatives for communicating the same decision when they use matching assumptions and settings. The interval makes the estimated range and the crossed boundary visible; the tests name the one-sided null hypothesis that was not rejected and can provide a p-value. Either way, crossing one limit means the evidence does not establish that the full interval lies within the prespecified equivalence region. It does not, by itself, show that the ratio equals 1 or that the treatments differ by a particular amount.

• Pitfall: “The point estimate is inside 0.80 to 1.25, so equivalence has been shown.” Correction: Equivalence requires both one-sided tests to reject; check the interval endpoints, not only the point estimate.
• Pitfall: “The interval crosses 1, so the upper-bound test fails.” Correction: Crossing 1 is not the test criterion. Compare the endpoint with 0.80 and 1.25 and identify which equivalence limit it crosses.
• Pitfall: “A statistic of \(-0.51\) is negative, so the upper test rejects.” Correction: The upper test rejects only when its statistic is sufficiently negative—below the negative critical value.