Signature
SE = sqrt(2 * MSE / n)
| Inputs | Definition | Unit |
|---|---|---|
MSE | Residual mean square from the analysis of variance of the log-transformed measure, estimating the within-subject variance | squared natural log units |
n | Subjects completing both periods, divided equally between the two sequences | count of subjects |
SE | Standard error of the test minus reference difference in log means | natural log units |
|---|
Function
Average bioequivalence and generic price function
For an ANDA, bioequivalence to the reference listed drug is shown when the 90% confidence interval for the ratio of geometric means of a pharmacokinetic measure, such as AUC or Cmax, lies within 80.00% to 125.00% after rounding. The interval is computed on the log scale and exponentiated. For economic models, the approval of generic competitors is then translated into a post-entry price.
Implementations
Excel
Standard error from the residual mean square
With the residual mean square in ResidualMS and the number of subjects in Subjects, Excel returns the standard error; the second formula gives the within-subject coefficient of variation.
=SQRT(2*ResidualMS/Subjects); =SQRT(EXP(ResidualMS)-1)
Assumptions
Balanced sequences and no carryover
The two sequences have equal numbers of subjects and there is no carryover from the first period. With unequal sequences the factor 2 divided by n becomes one half of the sum of 1 over n_1 and 1 over n_2.
Common within-subject variance
Test and reference share the same within-subject variance. Replicate designs estimate them separately, as used for highly variable and narrow therapeutic index drugs.
Worked examples
Residual mean square of 0.0432 in 24 subjects
A residual mean square of 0.0432 on the log scale, a within-subject coefficient of variation of about 21%, gives a standard error of 0.06 with 24 subjects, the value used in the passing example. The figures are illustrative.
MSE = 0.0432; n = 24; SE = 0.06
Residual mean square of 0.0588 in 24 subjects
A residual mean square of 0.0588 gives a standard error of 0.07, the value used in the failing example.
MSE = 0.0588; n = 24; SE = 0.07
Common errors
Using the between-subject variance
In a crossover each subject is their own control, so the between-subject variance cancels from the difference. Using the total variance gives a standard error that is too large and a sample size that is too high.
Sources
Textbook analysis of the standard two-period crossover
Chow SC, Liu JP. Design and Analysis of Bioavailability and Bioequivalence Studies. 3rd ed. Boca Raton: Chapman and Hall/CRC; 2008. Chapters on the standard 2 x 2 crossover design: variance of the estimated formulation effect from the intra-subject mean square, and confidence intervals for average bioequivalence on the log scale.
Canonical Identity
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