Testing a Log-Ratio Estimate Against Equivalence Limits

From Ratio Limits to Two Log-Scale Tests

In a bioequivalence analysis, the effect of interest may be a ratio, such as a test-to-reference ratio. The equivalence limits are then stated as a lower and upper ratio bound. Testing on the log scale puts this ratio comparison into the usual one-parameter testing framework: the ratio becomes a log-ratio estimate, and both ratio bounds become log-scale limits. The task here is to use that estimate and its standard error to test both limits, then require both tests to support equivalence.

Let \(R\) denote the population test-to-reference ratio, and define the log-ratio parameter as \(\theta=\log(R)\), using the natural logarithm. Suppose the prespecified ratio margins are \(R_L\) and \(R_U\), with \(0<R_L<R_U\). Define the log-scale limits as \(\ell=\log(R_L)\) and \(u=\log(R_U)\). Because the logarithm is increasing, the ratio is within its margins exactly when \(\ell<\theta<u\). A model-based analysis supplies an estimate \(\hat{\theta}\) and its standard error \(SE(\hat{\theta})\).

These are two one-sided tests (TOST). Their direction matters: evidence for the lower test must point above \(\ell\), while evidence for the upper test must point below \(u\). This is not a test of whether the log-ratio differs from zero. A ratio of one corresponds to \(\theta=0\), but equivalence is judged against the specified limits, which need not be symmetric around zero.

Calculating the Two Tests

For the calculations below, assume the sampling distribution of \(\hat{\theta}\) is adequately represented by a normal distribution with mean \(\theta\) and standard error \(SE(\hat{\theta})\). The normal reference distribution is specified for these examples; an analysis must use the reference distribution justified for its model and design. The standard error must quantify uncertainty in the same log-ratio contrast being tested.

At the lower boundary, standardize the distance from \(\hat{\theta}\) to \(\ell\). At the upper boundary, standardize the distance from \(\hat{\theta}\) to \(u\). Let \(\Phi\) be the cumulative distribution function of a standard normal random variable. The corresponding one-sided p-values are:

The lower test is an upper-tail test, so its p-value is the probability, under the boundary value, of a standard normal statistic at least as large as \(z_L\). The upper test is a lower-tail test, so its p-value is the probability of a statistic at most \(z_U\). At level \(\alpha\), equivalence requires \(p_L<\alpha\) and \(p_U<\alpha\). Equivalently, both test statistics must exceed or fall below their respective one-sided critical values in the directions specified by the alternatives.

Why Testing the Boundaries Controls the Composite Nulls

Each null hypothesis includes a range of parameter values, not just one boundary point. For the lower test, the null is \(\theta\leq\ell\), and rejection requires a large estimate. Under the normal model, the probability of such a large estimate is greatest at \(\theta=\ell\); moving \(\theta\) farther below \(\ell\) shifts the estimate distribution left and makes rejection less likely. For the upper test, the null is \(\theta\geq u\), and rejection requires a small estimate. Its rejection probability is greatest at \(\theta=u\); moving \(\theta\) above \(u\) shifts the distribution right and makes rejection less likely. Thus, testing at each boundary controls the error probability over the corresponding composite null.

The overall null for equivalence is that at least one margin is not met: \(\theta\leq\ell\) or \(\theta\geq u\). The alternative is that both are met: \(\ell<\theta<u\). Because the overall null is rejected only when both component nulls are rejected, the two tests form an intersection-union procedure. Each is conducted at level \(\alpha\); equivalence is not declared by choosing whichever test happens to pass.

Worked Calculations

Assumptions, Diagnostics, and Alternatives

A numerical TOST calculation is meaningful only if its inputs and model match the question. In a crossover analysis, for example, the estimate must represent the intended test-to-reference contrast on the log scale, and its standard error must reflect the fitted model and design. The ratio margins must be prespecified and strictly positive so their logarithms exist. The estimate, standard error, and limits must all use the same contrast, scale, and units. A standard error from a different contrast cannot be substituted just because it is available.

Check the model and its output before interpreting the p-values. Confirm the direction of the contrast (test/reference rather than reference/test), that the estimate and standard error are finite, and that the reported standard error is on the log scale. Verify the margin conversion: since the logarithm is increasing, the lower ratio limit maps to the lower log limit and the upper ratio limit maps to the upper log limit. A reversed contrast changes the interpretation and requires consistent reciprocal margins; it is not repaired by changing the sign of the estimate alone.

A useful arithmetic check is that \(z_L\) and \(z_U\) use the same positive standard error and the same \(\hat{\theta}\), but subtract different boundaries. A sign error can reverse a one-sided p-value: if the estimate is well above \(\ell\), \(z_L\) should be positive and \(p_L\) small; if it is well below \(u\), \(z_U\) should be negative and \(p_U\) small. If these directional checks fail, revisit the hypotheses and signs before making a decision.

An alternative presentation uses a confidence interval and asks whether its endpoints lie within the equivalence limits; the corresponding confidence level for TOST is tied to the one-sided significance level. The two-test calculation is particularly transparent when an estimate, standard error, and reference distribution are supplied, because it displays which boundary controls the decision. Neither presentation fixes a misspecified model, an inappropriate standard error, or margins chosen after examining results. A nonsignificant test against a ratio of one is also not a substitute: failure to detect a difference does not show that both equivalence limits have been met.

Common Mistakes and Practical Limits

• Using only one test. Rejecting the lower-bound null does not show the estimate is sufficiently below the upper limit. Both p-values must be below \(\alpha\).
• Testing the wrong value. The TOST boundaries are \(\ell\) and \(u\), not automatically zero. Zero is the log of a ratio of one and is not itself an equivalence limit unless specified as one.
• Using ratio limits directly in a log-scale statistic. Convert each positive ratio margin with the natural logarithm before subtracting it from \(\hat{\theta}\).
• Reversing a tail. The lower test uses \(1-\Phi(z_L)\); the upper test uses \(\Phi(z_U)\). A small statistic is not automatically evidence for either alternative.
• Treating a p-value as the probability that equivalence is true. Each p-value is a tail probability calculated under its boundary null and the assumed reference distribution; neither is a posterior probability for a hypothesis.

The test is only as reliable as its estimand, model, and uncertainty calculation. In particular, the normal-reference computations above are conditional on that reference distribution being justified. If the analysis calls for another reference distribution, its test statistics and tail probabilities must be calculated accordingly rather than borrowing the normal p-values shown here.