Crossover bioequivalence
Evaluate equivalence of a positive exposure endpoint between test and reference formulations in a balanced crossover.
Concepts and reasoning
Similar exposure is an interval question
Bioequivalence asks whether exposure from a test formulation is sufficiently close to that from a reference under a defined design and acceptance criterion. It is not established by a nonsignificant superiority test or by a point estimate close to one.
Exposure endpoints such as AUC and Cmax are positive and often vary multiplicatively. A log transformation turns a treatment ratio into a difference, making additive statistical models useful. Exponentiating the treatment contrast returns a geometric mean ratio, test/reference.
In a two-period crossover, each participant receives both formulations in a randomized sequence. This controls stable between-person differences because participants contribute their own comparison. Period effects, adequate washout, carryover, sequence allocation, and missing observations still matter. Two measurements on a participant are not independent.
Equivalence has a prespecified scope
The entire confidence interval must fit inside justified limits. The frequently encountered range 0.80 to 1.25 is reciprocal on the ratio scale and symmetric on the log scale. It is not a universal rule for every product, endpoint, or design.
ICH M13A as published by FDA concerns immediate-release solid oral dosage forms and describes the applicable design and analysis context. Specialized products and designs can require different criteria. The educational calculations below illustrate the statistical logic, not product-specific study recommendations.
Mathematical foundations
From log contrast to geometric mean ratio
Let θ be the expected log-exposure contrast between test and reference. Then the geometric mean ratio is
A model-based estimate θ̂ with standard error SE and suitable residual degrees of freedom ν gives a 90% interval
The t critical value accounts for estimated residual variation. A normal critical value is only a large-sample approximation.
Two one-sided tests at alpha 0.05 test against the lower and upper nonequivalence boundaries. Both must reject. This is equivalent to containing the 90% confidence interval within the equivalence limits for the corresponding procedure. A 90% interval here is not an arbitrary relaxation of uncertainty; it follows from the intersection of the two one-sided tests.
With limits 0.80 and 1.25, the log boundaries are approximately −0.2231 and +0.2231. Ratio-scale distance is asymmetric in ordinary percentage points but symmetric after logging.
Worked example
Worked example: a close point estimate can still fail
Suppose a fitted crossover model gives θ̂ = log(1.04) = 0.03922 and SE = 0.05. Use 1.645 for a large-sample illustration of a 90% interval:
Exponentiating gives approximately 0.958 to 1.129. The point estimate is 1.040, and the whole interval lies inside 0.80 to 1.25.
Now keep the point estimate but increase SE to 0.15. The approximate interval becomes exp(0.03922 ± 0.24675), or 0.813 to 1.331. The upper limit exceeds 1.25, so equivalence has not been demonstrated.
| Geometric mean ratio | Approximate 90% interval | Within 0.80–1.25? |
|---|---|---|
| 1.040 | 0.958–1.129 | Yes |
| 1.040 | 0.813–1.331 | No |
The second result does not prove that the true ratio is outside the limits. It says the data are not sufficiently precise, relative to the estimate and criterion, to establish equivalence.
The geometric mean ratio is also not generally the ratio of arithmetic sample means. For lognormal-like exposure distributions these summarize different features. Use the model's log contrast and specify its adjustment for period and other design terms.
Interpreting the results
Equivalence requires the entire interval to fit
A geometric mean ratio compares test with reference exposure on a multiplicative scale. A value of 1.05 means the estimated test exposure is 5% higher. Equivalence depends on whether the full appropriately constructed interval lies inside the prespecified lower and upper ratio limits, not whether the point estimate is close to one.
Two one-sided tests evaluate the lower and upper margins separately. At one-sided alpha 0.05, their joint decision corresponds to a two-sided 90% confidence interval. A nonsignificant difference test does not establish equivalence, because a wide interval can include both no difference and unacceptable differences.
An unsuccessful equivalence result may reflect imprecision, a genuine formulation difference, or both. Report the ratio, interval, limits, design, and analyzed endpoint. Equivalence for one exposure measure does not automatically establish every endpoint or the full product-specific requirement.
Assumptions and limitations
A crossover depends on valid within-person comparisons
Each participant receives both formulations in the intended sequence with adequate separation between periods. Residual treatment from the first period, changes in condition, or inconsistent dosing can compromise the second-period comparison. The model's period adjustment does not automatically remove carryover.
Log-transformed exposure should be positive and reasonably compatible with the residual model. A common residual variance is assumed by the simple analysis. Sequence assignment, complete periods, and consistent endpoint definitions matter as much as the final ratio.
Limits and endpoints should be determined for the scientific and regulatory context before analysis. A conventional interval does not cover every narrow-therapeutic-index, highly variable, replicate, or special formulation setting. The M13A guidance describes its specific scope for immediate-release oral products.
Diagnostics and sensitivity analysis
Audit treatment order and the log-scale model
Verify that each subject has one observation from each period and that treatment order agrees with the randomized sequence. Investigate missing or invalid exposure values using the prespecified analysis plan. A balance requirement should not be satisfied by selectively dropping inconvenient participants.
Inspect within-person log differences, period patterns, outliers, and residual spread. Review dosing, washout, sampling, and assay records when unusual observations occur. A carryover concern is a design issue, not something a favorable confidence interval can dismiss.
Compare the unrounded interval with the prespecified limits. Three-decimal display rounding can obscure a boundary decision, so use retained precision when reviewing borderline results. Report both one-sided tests and the interval, and distinguish exploratory sensitivity analyses from the primary analysis population.
References and other material
Read the methodological references and supporting tutorials below. The examples in this lesson are original teaching calculations, not findings from these sources.
- FDA / ICH M13A: bioequivalence — Study-design and analysis guidance for immediate-release solid oral dosage forms.