# Presentation: 547711

## Statistical Criteria for Establishing Bio-inequivalence among Drug Products 

- Qian H. Li, Lawrence Yu, Donald Schuirmann, Stella Machado, Yi Tsong
- CDER, FDA

## 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

## 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%,)

## 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.

## Testing for Bioequivalence

- Hypotheses of testing bioequivalence:
- H0: 	μ*T**/*μ*R* < 80% or μ*T**/*μ*R* >125%  vs.   HA: 	80%  μ*T**/*μ*R*  125%
- Perform two 1-sided tests
*H**01**: *μ*T**/*μ*R* < 80% vs. HA1: μ*T**/*μ*R*  80%
*H**02**: *μ*T**/*μ*R* >125% vs. HA2: μ*T**/*μ*R*  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

## 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 =0.05    (level of significance)

## 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.

## 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%

## Power vs. Variance for Bioequivalence

- 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.

## Power vs. Treatment Difference for Bioequivalence

## Test for Bio-inequivalence

- Hypotheses for testing bio-inequivalence:
- H0: 80%  μ*T**/*μ*R*  125%  vs.  HA: μ*T**/*μ*R* < 80% or μ*T**/*μ*R* >125%
- Perform 2 one-sided tests
- H01: μ*T**/*μ*R*  80%   vs. HA1: μ*T**/*μ*R* < 80%
- H02: μ*T**/*μ*R*  125% vs. HA2: μ*T**/*μ*R* >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.

## Power vs. Variance for Bio-inequivalence Test

- Rejection Region

- Rejection Region

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

## 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.

## 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.

## 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.

## Error Discussion

| Standard Error | n=10 |  | n=15 |  | n=20 |  |
| --- | --- | --- | --- | --- | --- | --- |
|  | Error Rate | S.D. | Error Rate | S.D. | Error Rate | S.D. |
| 0.106 | 0.050 | 0.237 | 0.050 | 0.290 | 0.050 | 0.335 |
| 0.166 | 0.051 | 0.371 | 0.050 | 0.455 | 0.050 | 0.525 |
| 0.226 | 0.052 | 0.505 | 0.051 | 0.619 | 0.051 | 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 |  | 0.100 |  | 0.100 |  |

## 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 0.226.
- 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.

## 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 μ*T**/*μ*R.*
- Bio-inequivalence holds if one of the three PK parameters is in the bio-inequivalence regions.

## 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?

## 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

## 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)

## 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.

## 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.

## 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?

## 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.