Deriving the Log-Scale Expansion of Equivalence Limits

From Reference Variability to Equivalence Limits

Conventional equivalence limits are fixed on the test-to-reference ratio scale, commonly written as 0.80 to 1.25. Because exposure measures are analyzed on a logarithmic scale, these limits correspond to equal distances below and above zero: \(\ln(0.80)=-\ln(1.25)\). A variability-based rule changes the size of that log-scale distance according to the within-subject variability of the reference product.

The key mathematical step is simple but consequential: define a symmetric margin on the log scale, then exponentiate each endpoint. The resulting ratio-scale limits are reciprocals of one another, but they are generally not equally distant from one. This tutorial derives that relationship and clarifies what the bounds mean. Deriving bounds alone does not specify every part of a regulatory bioequivalence procedure or establish a complete decision rule.

Definition: Let \(\theta=\mu_T-\mu_R\) be the difference between the population mean log exposure for test and reference, and let \(\exp(\theta)\) be the corresponding test-to-reference geometric mean ratio. Let \(\sigma_{wR}\) denote the within-subject standard deviation for the reference on the log scale. A proportional variability-based rule sets a log-scale margin \(M=c\sigma_{wR}\), where \(c>0\) is a specified scaling coefficient. Its symmetric log-scale limits are \(-M\) and \(M\).

This formulation assumes the comparison is expressed as a difference of mean log exposures and the variability measure is the within-subject reference standard deviation on the same log scale. The rule also presumes a specified, nonnegative scaling coefficient and a nonnegative variability value. If \(\sigma_{wR}=0\), the proportional margin is zero; practical procedures must define how such boundary cases are handled. The rule is about how limits are constructed, not how \(\sigma_{wR}\) is estimated from replicate observations.

A scaling coefficient can be set by choosing an anchor variability \(\sigma_0>0\) at which the scaled margin is required to equal a chosen conventional log margin \(\Delta>0\). At the anchor, \(c\sigma_0=\Delta\), so \(c=\Delta/\sigma_0\). Once \(c\) is fixed, increasing reference variability increases the margin linearly on the log scale. The particular anchor and coefficient are choices of the rule; they are not consequences of the algebra.

Deriving the Ratio-Scale Limits

Start with the symmetric log-scale limits \(-M\) and \(M\), where \(M=c\sigma_{wR}\). The exponential function is strictly increasing, so exponentiating preserves the order of the endpoints. Since the test-to-reference ratio corresponding to \(\theta\) is \(\exp(\theta)\), the ratio-scale limits are the exponentials of the log-scale limits.

-M \leq \theta \leq M \quad\Longrightarrow\quad \exp(-M) \leq \exp(\theta) \leq \exp(M).

Substituting \(M=c\sigma_{wR}\) gives the general expanded limits:

L=\exp(-c\sigma_{wR}),\qquad U=\exp(c\sigma_{wR}).

These limits are reciprocal because \(LU=\exp(-c\sigma_{wR})\exp(c\sigma_{wR})=1\). Their product is one, so they are symmetric in multiplicative terms around a ratio of one. They are not usually symmetric in additive distance from one: the distances are \(1-L\) below one and \(U-1\) above one. For a positive margin \(M\), \(U-1>1-L\), because \(\exp(M)+\exp(-M)>2\). Thus “symmetric limits” refers to the log scale, not equal numerical distances on the ratio scale.

The conventional limits illustrate the same transformation. If \(\Delta=\ln(1.25)\), then \(\exp(-\Delta)=1/1.25=0.80\) and \(\exp(\Delta)=1.25\). A variability-based rule replaces the fixed margin \(\Delta\) with \(c\sigma_{wR}\). When the reference variability is at the chosen anchor, the two constructions coincide by design; at other values, the scaled limits differ from the fixed limits.

Derivation: A proportional rule requires the log margin to satisfy \(M\propto\sigma_{wR}\), so write \(M=c\sigma_{wR}\) for a fixed coefficient \(c\). Symmetry around zero gives endpoints \(-c\sigma_{wR}\) and \(c\sigma_{wR}\). Since the ratio is \(\exp(\theta)\), monotonicity of the exponential maps those endpoints to \(\exp(-c\sigma_{wR})\) and \(\exp(c\sigma_{wR})\). Their product is one, establishing reciprocal ratio limits.

Worked Constructions

Worked Example: Deriving a Scaling Coefficient From an Anchor

Suppose a hypothetical scaling rule is anchored so that a reference log-scale SD of \(\sigma_0=0.294\) corresponds to the conventional upper ratio limit 1.25. Derive the coefficient \(c\). This example illustrates the algebra; the anchor is stipulated for the example.

The conventional log margin is \(\Delta=\ln(1.25)\). Numerically, \(\Delta\approx0.2231\). At the anchor, the proportional margin must satisfy \(c\sigma_0=\Delta\). Therefore,

c=\frac{\Delta}{\sigma_0} =\frac{\ln(1.25)}{0.294} =\frac{0.2231}{0.294} \approx 0.7590.

The coefficient is dimensionless because both the log margin and the log-scale SD are on the same scale. At \(\sigma_{wR}=0.294\), the margin is \(0.7590(0.294)\approx0.2231\); exponentiating its positive endpoint gives \(\exp(0.2231)\approx1.25\), as required by the anchor. The lower endpoint is \(\exp(-0.2231)\approx0.8000\).

Worked Example: Exponentiating a Scaled Margin

For a hypothetical rule, take \(c=0.750\) and a reference within-subject log-scale SD of \(\sigma_{wR}=0.400\). Calculate the log margin and both ratio-scale limits.

First calculate the margin:

M=c\sigma_{wR}=0.750(0.400)=0.300.

The symmetric log-scale endpoints are \(-0.300\) and \(0.300\). Exponentiating gives

L=\exp(-0.300)\approx0.7408,\qquad U=\exp(0.300)\approx1.350.

The limits are reciprocal: \(0.7408(1.350)\approx1.000\), allowing for rounding. Their additive distances from one are approximately \(1-0.7408=0.2592\) below and \(1.350-1=0.350\) above. Those unequal distances do not contradict the symmetry, which is on the log scale.

Worked Example: Checking a Ratio Against the Derived Bounds

Suppose a hypothetical rule has \(c=0.800\), and the reference within-subject log-scale SD is \(0.250\). A test-to-reference geometric mean ratio is 1.15. Calculate the bounds and check whether this ratio lies between them. This comparison concerns the point value only; it is not, by itself, a complete bioequivalence analysis.

The log margin is \(M=0.800(0.250)=0.200\). The ratio limits are

L=\exp(-0.200)\approx0.8187,\qquad U=\exp(0.200)\approx1.221.

The observed ratio satisfies \(0.8187<1.15<1.221\), so it lies within these constructed bounds. Equivalently, its log is \(\ln(1.15)\approx0.1398\), and \(-0.200<0.1398<0.200\). This equivalence of checks follows because the logarithm is also strictly increasing. Whether an inferential procedure supports a bioequivalence conclusion requires more than comparing a point estimate with limits.

Assumptions, Interpretation, and Diagnostics

The derivation relies on a few conditions that are easy to overlook. The parameter being bounded must be the mean difference on the log scale, so its exponentiation is the test-to-reference geometric mean ratio. The variability input must be a within-subject reference SD on that same log scale, not a standard deviation of untransformed concentrations or a between-subject SD. Finally, the scaling coefficient and any anchor must be defined independently of the observed test-to-reference effect; choosing them after seeing the result would make the limits outcome-dependent.

A useful diagnostic is to check the construction in both scales. On the log scale, the two endpoints should have equal magnitude and opposite signs. On the ratio scale, the lower and upper limits should multiply to one. If either check fails beyond rounding error, likely causes include a sign error, an incorrect transformation, or inconsistent units or definitions for the variability input. For a positive margin, the upper limit should exceed one and the lower limit should be below one.

The rule also has mathematical and practical limits. If variability increases, the proportional log margin grows, and exponentiation makes the ratio limits widen multiplicatively. That property is exactly what scaling specifies; it does not establish that wider limits are appropriate for every drug, endpoint, or clinical context. A regulatory approach may prescribe a particular coefficient, a cap, or additional requirements. Those details are not implied by the general derivation here.

Approach	How the margin is set	Scale relationship
Fixed equivalence limits	A prespecified log margin \(\Delta\) does not change with \(\sigma_{wR}\).	Limits are \(\exp(-\Delta)\) and \(\exp(\Delta)\).
Proportional variability-based limits	The log margin is \(c\sigma_{wR}\).	Limits are \(\exp(-c\sigma_{wR})\) and \(\exp(c\sigma_{wR})\).
Direct additive ratio-scale margins	Bounds are set as equal numerical distances below and above one.	These bounds are generally not reciprocal and do not follow from a symmetric log-scale margin.

The fixed-limit approach is a useful contrast: it holds the log margin constant regardless of reference variability. The scaled approach changes that margin according to the chosen variability rule. Directly adding and subtracting a number on the ratio scale is different again; it does not preserve the multiplicative symmetry of log-scale limits. These approaches should not be treated as interchangeable merely because each produces lower and upper bounds.

Common Mistakes

Assuming equal distances from one. The bounds are symmetric on the log scale, not in additive ratio-scale distance. Check reciprocity by multiplying them.
Exponentiating the same positive endpoint twice. The lower endpoint comes from the negative log margin: \(L=\exp(-M)\), not \(\exp(M)\).
Using the wrong variability quantity. The rule uses the specified within-subject reference SD on the log scale; substituting another variance measure changes the rule.
Treating bounds as a complete decision. Bounds describe the allowed range under a specified rule. A point estimate inside them does not alone demonstrate that the required inferential criterion is met.
Calling the coefficient universal. The coefficient follows from the rule or its chosen anchor. The algebra does not select a regulatory coefficient or justify a particular policy.

Key takeaway: A proportional scaling rule gives the log margin \(M=c\sigma_{wR}\). Its ratio-scale limits are \(\exp(-M)\) and \(\exp(M)\): reciprocal bounds around one, with unequal additive distances from one.

Check Your Understanding

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

For \(c=0.700\) and \(\sigma_{wR}=0.300\), calculate the log margin and ratio-scale limits.
Why are limits \(\exp(-M)\) and \(\exp(M)\) reciprocal?
A proposed lower and upper limit are 0.82 and 1.20. Can they arise from a symmetric log-scale margin centered at zero? Explain.
State one reason why a point estimate inside variability-based limits is not, on its own, a complete bioequivalence conclusion.

Solutions. (1) \(M=0.700(0.300)=0.210\); the limits are \(\exp(-0.210)\approx0.8106\) and \(\exp(0.210)\approx1.234\). (2) Their product is \(\exp(-M)\exp(M)=1\). (3) No. A symmetric log-scale margin produces reciprocal limits, but \(0.82(1.20)=0.984\), not 1; rounding cannot explain this size of discrepancy. (4) A point estimate does not include the uncertainty assessment required by an inferential procedure, and the applicable procedure may impose additional conditions.