1. Qian H. Li, Lawrence Yu, Donald Schuirmann, Stella Machado, Yi Tsong
Statistical Criteria for Establishing Bio-inequivalence among Drug Products
Qian H. Li, Lawrence Yu, Donald Schuirmann, Stella Machado, Yi Tsong
CDER, FDA

2. Outline of the Presentation
Definition of bio-inequivalence
Explanation on failing to show bioequivalence
Explanation on a proposed statistical criterion for assessing bio-inequivalence using one PK parameter
Strategies for evaluating bio-inequivalence using three PK parameters

3. Definitions
Definitions in terms of the ratio of the two geometric means µT/µR, where µT and µR are the geometric means of the test and reference products, respectively
Bioequivalence interval [80%, 125%]
Bio-inequivalence regions (0, 80%) or (125%,)
80%
125%
µT/µR
Bioequivalence Interval
Bio-inequivalence Region

4. Failing to Show Bioequivalence
Is it appropriate to claim bio-inequivalence if a study fails to show bioequivalence?
Two drug products may be bioequivalent, however, they may fail to show bioequivalence due to
Inadequate power (primary reason);
Bias from inadequate statistical model.

5. Testing for Bioequivalence
Hypotheses of testing bioequivalence:
H0: µT/µR < 80% or µT/µR >125% vs. HA: 80%  µT/µR  125%
Perform two 1-sided tests
H01: µT/µR < 80% vs. HA1: µT/µR  80%
H02: µT/µR >125% vs. HA2: µT/µR  125%
Reject H0 if we reject both 1-sided null hypotheses, each at =0.05
Equivalent to a 2-sided 90% CI falling within [80%, 125%]
The type I error of rejecting H0 is controlled at 0.05

6. Error Associated with Claiming HA
To claim HA (bioequivalence), need to reject H0 (bio-inequivalence) in a statistical test with high confidence.
A rejection criterion is selected so that the error of wrongfully rejecting H0 and therefore wrongfully claiming HA is small.
Type I error, usually set at a level of = (level of significance)

7. Error Associated with Claiming H0 if You Don’t Claim HA
To claim H0 (bio-inequivalence), need to reject HA (bioequivalence) with high confidence.
Need to control the error of wrongfully rejecting HA to be small, which is equivalent to requiring large power to reject H0 in the bioequivalence test.
However, the power of the bioequivalence test may not be large for all values of T/R in HA.
Testing for bioequivalence focuses on controlling type I error for rejecting H0.
We may not have adequate power to claim bioequivalence when bioequivalence is true.

8. Example of Inadequate Power
For a regular 2-way crossover trial, assume that
Within-subject Variance (log scale) is 0.04
The ratio of the two geometric means deviates from 1 by no more than 5%
Wish to achieve 85% power
Require a sample size of 22
About 15% chance to see a failed bioequivalence study
If sample is kept at 22, power can be even lower
if Variance is higher
if the ratio deviates from 1 by more than 5%

9. Power vs. Variance for Bioequivalence
σt
Rejection Region
Rejection region: reject bio-inequivalence and claim bioequivalence if the estimated ratio is in this region
The width of the rejection region reflects the power, which depends on the variance of the estimate of T/R.

10. Power vs. Treatment Difference for Bioequivalence

11. Test for Bio-inequivalence
Hypotheses for testing bio-inequivalence:
H0: 80%  µT/µR  125% vs. HA: µT/µR < 80% or µT/µR >125%
Perform 2 one-sided tests
H01: µT/µR  80% vs. HA1: µT/µR < 80%
H02: µT/µR  125% vs. HA2: µT/µR >125%
The significance level for each one-sided test is controlled at 0.05
Reject bioequivalence when two-sided 90% CI are completely in either of the two bio-inequivalence regions
Under certain conditions, the type I error under H0 is controlled at or near the level of 0.05.
However, mathematically, the type I error can reach up to 0.1.

12. Power vs. Variance for Bio-inequivalence Test
Rejection Region
Rejection Region
σt
σt

13. Power vs. Treatment Difference for Bio-inequivalence Test

14. Error Discussion for Bio-inequivalence Test
The Type-I error for the bioequivalence test is controlled at the 0.05 level.
To be consistent, bio-inequivalence should also be tested at 0.05 level.
The error rate for testing bio-inequivalence
Where  = T/R, T is a test statistic that follows a t-distribution, and t is the critical point for the two one-sided tests.

15. Error Discussion
The error rate depends on the distribution of the test statistics under the null: the location in the bioequivalence interval and variance.
For a given T, the maximum error rate occurs at the margins of the bioequivalence interval.
The maximum error increases as T increases. When T  , the maximum error rate could reach 0.10.

16. Error Discussion
For what kind of variance does the error rate start to inflate noticeably above 0.05?
Error calculations:
Assume 2-way balanced crossover trials
Using t distribution
Error is calculated at the margins
Standard error represents T, the standard deviation for the estimated treatment difference (log scale)
Standard deviation (S.D.) represents intra-subject variability between two drug products.

17. Error Discussion Standard Error n=10 n=15 n=20 Error Rate S.D. 0.106
0.050
0.237
0.290
0.335
0.166
0.051
0.371
0.455
0.525
0.226
0.052
0.505
0.619
0.715
0.436
0.060
0.975
0.058
1.194
0.056
1.379
0.766
0.070
1.713
0.067
2.098
0.066
2.422
1.366
0.080
3.054
0.078
3.741
0.077
4.320
3.106
0.090
6.932
0.089
8.506
0.088
9.822 
0.100

18. Error Discussion
For sample size n≥10, the type I error can be controlled around the level of 0.05 if the standard error (s.e.) is
For a balanced 2-way crossover trial, such variance is considered quite large (s.d. ≥0.505)
If bio-inequivalence study with sample size less than 10, it is possible that the type I error will be inflated with small s.e.
If a trial uses a parallel design, s.e. may be high enough to require adjusting the significance level.

19. Evaluating Three PK Endpoints
Three PK parameters (AUCt, AUC∞, Cmax) used to assess bioequivalence.
Bioequivalence of two drugs requires that all the three PK parameters should be equivalent in terms of the bioequivalence interval using ratios µT/µR.
Bio-inequivalence holds if one of the three PK parameters is in the bio-inequivalence regions.

20. Three PK Endpoints for Bioequivalence
The statistical criteria we use to demonstrate bioequivalence for the three PK parameters are to have all the 2-sided 90% CIs for the ratios to be within [80%, 125%].
What are the statistical criteria for bio-inequivalence?

21. Assessing Bio-inequivalence with Three PK Endpoints
Evaluate several strategies based on composite hypotheses:
H0_ineq: H10  H20  H30
vs.
HA_ineq: H1A  H2A  H3A
Hi0 are the equivalence intervals
HiA are the inequivalence regions
For i=1,2,3 (the three PK endpoints.)
Evaluate type I error and power under the composite hypotheses

22. Assessing Bio-inequivalence with Three PK Endpoints
Strategy I: At least one of the three PK parameters satisfies the bio-inequivalence criterion.
Pros: intuitive
Cons: May inflate type I error under H0_ineq if the three PK endpoints are not highly correlated (>0.99)

23. Assessing Bio-inequivalence with Three PK Endpoints
Strategy II: All three PK parameters must satisfy the bio-inequivalence criterion.
Pros: Can tightly control the type I error under all correlation structures for the three PK endpoints.
Cons: Usually does not provide adequate power for the alternatives that are of interest.

24. Assessing Bio-inequivalence with Three PK Endpoints
Strategy III: pre-specify one PK endpoint for bio-inequivalence testing.
For example, test AUCt only, ignore the other two.
Pros: Controls type I error.
Cons: may have very low power if AUCt is not the endpoint that is actually bio-inequivalent. Need a mechanism for pre-specification.

25. Assessing Bio-inequivalence with Three PK Endpoints
Other Strategies
1. - Require inequivalence to be shown for all three PK endpoints, but adjust the  levels for the individual tests while maintaining the overall level.
- the  levels for the individual tests may differ, and may be determined in a flexible manner.
- Currently under development in QMR.
2. Other?

26. Remarks Main focus of this presentation is power and error
Other statistical issues such as inadequate statistical models, study design, and conduct of the study may also bring in bias in bioequivalence and bio-inequivalence assessment.