# Bioequivalence and Bioavailability Forum

Back to the forum  2018-06-25 17:43 CEST (UTC+2h)
acfalcao

2007-09-16 22:34

Posting: # 1082
Views: 9,678

## Three-way crossover BABE Studies [Design Issues]

Dear All

My name is Amilcar Falcão and I am professor at the University of Coimbra, Portugal. I am looking for help regarding the execution and interpretation of three-way cross-over bioequivalence trials (for academic purposes).

I would like to have an example (explicit) where a three-way crossover study is appropriately analysed. Namely, I would like to know what kind of ANOVA should be performed and how the 90%CI should be calculated for the different combinations of study formulations (i.e. A vs B and A vs C). Perhaps this is a naive question, but in fact I am trying to look for it in literature and I have seen different approaches always without the adopted statistical pathway. Therefore, I would like to be sure regarding the correct way to deal with.

Best regards;

Amílcar Falcão
Laboratory of Pharmacology
Faculty of Pharmacy
University of Coimbra
Coimbra - Portugal
Helmut
Hero

Vienna, Austria,
2007-09-17 20:47

@ acfalcao
Posting: # 1088
Views: 9,117

## Three-way crossover example data set

Dear Amilcar!

» I would like to have an example (explicit) where a three-way crossover
» study is appropriately analysed.

You have to use a Williams' design (three period, six sequences); the topic was covered in previous threads.

A 6×3 design is needed in order to 'extract' two 2×2 tables, which are also balanced. Although the full 6×3 table will be used in the analysis of AUC and Cmax, you will need these 2×2s for the nonparametric analysis of Tmax (unfortunatelly there's no confidence interval based nonparametric method available for more than two formulations/periods). The asterisks `*` denote pseudo-sequences and pseudo-periods, e.g. `P1*` means only that the treatment was given in a period prior to `P2*` - irrespective of the true study period:
```+----+------------+  -->  +----+--------+  and  +----+--------+ |    | P1  P2  P3 |       |    | P1* P2*|       |    | P1* P2*| +----+------------+       +----+--------+       +----+--------+ | S1 | T   R1  R2 |       | S1*| T   R1 |       | S1*| T   R2 | | S2 | R1  R2  T  |       | S2*| R1  T  |       | S2*| R2  T  | | S3 | R2  T   R1 |       | S1*| T   R1 |       | S2*| R2  T  | | S4 | T   R2  R1 |       | S1*| T   R1 |       | S1*| T   R2 | | S5 | R1  T   R2 |       | S2*| R1  T  |       | S1*| T   R2 | | S6 | R2  R1  T  |       | S2*| R1  T  |       | S2*| R2  T  | +----+------------+       +----+--------+       +----+--------+                             ^   balanced          ^   balanced```
A common mistake is to design the study as a set of 3×3 latin squares, which will lead (especially if the sample size is small and in the case of drop outs) to extremely inbalanced data sets:
```+----+------------+  -->  +----+--------+  and  +----+--------+ |    | P1  P2  P3 |       |    | P1* P2*|       |    | P1* P2*| +----+------------+       +----+--------+       +----+--------+ | S1 | T   R1  R2 |       | S1*| T   R1 |       | S1*| T   R2 | | S2 | R1  R2  T  |       | S2*| R1  T  |       | S2*| R2  T  | | S3 | R2  T   R1 |       | S1*| T   R1 |       | S2*| R2  T  | +----+------------+       +----+--------+       +----+--------+                             ^ inbalanced          ^ inbalanced```

» Namely, I would like to know what kind of
» ANOVA should be performed and how the 90%CI should be calculated for the
» different combinations of study formulations (i.e. A vs B and A vs C).

For the design see Chapter 10 of
S-C Chow and J-p Liu
Design and Analysis of Bioavailability and Bioequivalence Studies
Marcel Dekker, New York (2nd ed. 2000), pp 302-332

For a detailed discussion of variance balanced designs see Chapter 4 of
B Jones and MG Kenward
Design and Analysis of Cross-over Trials
Chapman & Hall/CRC, Boca Raton (2nd ed. 2003), pp 151-204

For an example data set see Chapter 4 of
S Patterson and B Jones
Bioequivalence and Statistics in Clinical Pharmacology
Chapman & Hall/CRC, Boca Raton (2006), pp 79-132
Example 4.5 matches your design (one test formulation is compared to two reference formulations)
You may download zipped datasets and programs (SAS/S+) from the CRC's website. If you don't have access to commercial software, S+ code will run with open-source R with minor modifications.

Patterson/Jones give results in Table 4.12 (p 105) with:
T/R (% Test vs. Reference 1)
```+----------+-------+---------------+ | Endpoint |  PE   |     90% CI    | +----------+-------+---------------+ | AUC      | 116.2 | 109.0 , 123.9 | | Cmax     | 130.0 | 119.1 , 141.8 | +----------+-------+---------------+```
T/S (% Test vs. Reference 2)
```+----------+-------+---------------+ | Endpoint |  PE   |     90% CI    | +----------+-------+---------------+ | AUC      |  82.8 |  77.6 ,  88.3 | | Cmax     |  81.5 |  74.7 ,  89.0 | +----------+-------+---------------+```

WinNonlin 5.2 comes up with:
T/R (% Test vs. Reference 1)
```+----------+-------+---------------+ | Endpoint |  PE   |     90% CI    | +----------+-------+---------------+ | AUC      | 116.2 | 109.0 , 123.8 | | Cmax     | 129.7 | 118.4 , 141.5 | +----------+-------+---------------+```
T/S (% Test vs. Reference 2)
```+----------+-------+---------------+ | Endpoint |  PE   |     90% CI    | +----------+-------+---------------+ | AUC      |  82.6 |  77.5 ,  88.1 | | Cmax     |  81.2 |  74.3 ,  88.6 | +----------+-------+---------------+```

Results are slightly different (although both WinNonlin and SAS use GLM - not ANOVA; implementation, rounding, etc. is different - and of course 'proprietary information' and not documented). I assume Patterson/Jones' results were produced by SAS; I will check the results from their S+ code the next days.

» Perhaps this is a naive question,...

Not at all; little is published - and no worked examples at all.

There's another point which is a little bit tricky: multiplicity.
If you are testing one test (A) against two references (B, C), any impression of 'data dredging' must be avoided, e.g., calculation 90% CIs of A/B and A/C - and only picking out the best getting an approval.
Since in the EU the Innovator's product from any European country may be used as the reference, you may run into problems (A vs B is BE, whereas A vs C is not BE). It may be wise to use 95% CIs instead (overall Bonferroni-corrected patient's risk: alpha = 1-(1-0.05/k)k, where k is the number of simultaneous comparisons).
95% CI should also be applied in testing for dose proportionality of three dose levels (or 96.67% for four levels).
IMHO the only case where 90% CIs should be used is the comparison of two test formulations against one reference, and only one of them will be further used in the approval process.

Cheers,
Helmut Schütz

The quality of responses received is directly proportional to the quality of the question asked. ☼
Science Quotes
acfalcao

2007-09-17 23:02

@ Helmut
Posting: # 1089
Views: 8,365

## Three-way crossover example data set

Dear Helmut

Thank you very much for your help. The correct codification of "sequences/periods" is the key for this problem. Unfortunately the link to download the SAS files was not working. I hope it can be solve in a near future.

Best regards;

Amílcar

--
Edit: Full quote removed. Please see this post! [HS]
Helmut
Hero

Vienna, Austria,
2007-09-18 12:47

@ acfalcao
Posting: # 1092
Views: 8,532

## Three-way crossover example data set

Dear Amilcar!

» Thank you very much for your help. The correct codification of "sequences/periods" is the key for this problem. Unfortunately the link to download the SAS files was not working. I hope it can be solve in a near future.

Please try it again (it worked yesterday, and today as well): CRC's website
Having downloaded the file C5300.zip, unzip it keeping folder's structure; in folder [chapter4] you will find:
Example4.5.sas (the SAS code)
Example45.SSC (the S+ code)
exam45.dat (the raw data for S+)

As I get form the SAS-code a mixed model (degrees of freedom according to Kenward) was used, whereas WinNonlin uses Satterthwaite's method.
The S+ code does not calculate BE, but is used for the example's figures only.

--
Edit: Wrong link corrected in both posts. [HS]

Cheers,
Helmut Schütz

The quality of responses received is directly proportional to the quality of the question asked. ☼
Science Quotes
Nirali
Regular

India,
2007-09-21 06:49

@ Helmut
Posting: # 1106
Views: 8,373

## Three-way crossover (WinNonlin)

Dear Helmut,

Which hypothesis Winnonlin will use while running BE for 2 test (T1 & T2) vs a reference (R)?
Would it be "MEAN(T1)=MEAN(T2)=MEAN(R)"?
In threeway crossover study, does we try to compare Equivalence of all three product together?

Thanks & Regards,
Nirali
Helmut
Hero

Vienna, Austria,
2007-09-21 13:04

@ Nirali
Posting: # 1109
Views: 8,384

## Three-way crossover (WinNonlin)

Dear Nirali,

» Which hypothesis Winnonlin will use while running BE for 2 test (T1 & T2) vs a reference (R)?
» Would it be "MEAN(T1)=MEAN(T2)=MEAN(R)"?
» In threeway crossover study, does we try to compare Equivalence of all three product together?

Comparisons are based on paired differences; IMHO a simultaneous test of T1=T2=R is not possible.
WinNonlin always uses one reference in a run, i.e., T1=R (one test), and T2=R (the second test).
If you are interested in T1=T2 you have to start a second run (defining T2 as the reference), and will obtain two tests (T1=T2 and R=T2) within.

BTW, why are you interested in T1=T2?

Cheers,
Helmut Schütz

The quality of responses received is directly proportional to the quality of the question asked. ☼
Science Quotes
Nirali
Regular

India,
2007-09-25 08:05

@ Helmut
Posting: # 1127
Views: 8,335

## Three-way crossover (WinNonlin)

Dear Helmut,

Actully, I was under impression that WinNonlin is using the hypo:
"MEAN(Test1)=MEAN(Test2)=MEAN(Ref)".

(A) suppose we have two REFERENCES and single TEST [R1, R2, T]for threeway comparison: if we select R1 as REFERENCE, WNL will calculate BE with hypo:
(1) R1=T and (2) R1=R2.

same way,
(B) for two TEST and single REFERENCE [T1, T2, R]: if we select R as standard, hypo will be: (1) R=T1 and (2)R=T2

pls do correnct me if i am wrong to understand.

Thanks & Regards,
Nirali
Helmut
Hero

Vienna, Austria,
2007-09-25 13:15

@ Nirali
Posting: # 1132
Views: 8,442

## Three-way crossover (WinNonlin)

Dear Nirali!

» Actully, I was under impression that WinNonlin is using the hypo:
» "MEAN(Test1)=MEAN(Test2)=MEAN(Ref)".

Again, we obtain only pairwise comparisons, not simultaneous ones.

» (A) suppose we have two REFERENCES and single TEST [R1, R2, T]for threeway comparison: if we select R1 as REFERENCE, WNL will calculate BE with hypo:
» (1) R1=T and (2) R1=R2.

Not quite, but very close.
If you run the BE wizard (R1=reference), you will get
(1a) T vs. R1, and
(1b) R2 vs. R1.
You have to run the BE wizard again (this time with R2 as reference), getting
(2a) T vs. R2, and
(2b) R1 vs. R2.
In order to get comparisons of T=R1 and T=R2, you have to run the wizard twice (results 1a and 2a), omitting 1b/2b. Of course PE and CL of 2b=1/1b.

» same way,
» (B) for two TEST and single REFERENCE [T1, T2, R]: if we select R as standard, hypo will be: (1) R=T1 and (2)R=T2

Yes this is correct; in this case only one run of the BE wizard is necessary, getting:
(1a) T1 vs. R, and
(1b) T2 vs. R.

Cheers,
Helmut Schütz

The quality of responses received is directly proportional to the quality of the question asked. ☼
Science Quotes
Irene_I
Junior

Indonesia,
2018-06-07 11:09

@ Helmut
Posting: # 18861
Views: 619

## Three-way crossover example data set

Dear Helmut,

» Please try it again (it worked yesterday, and today as well): CRC's website
» Having downloaded the file C5300.zip, unzip it keeping folder's structure; in folder [chapter4] you will find:
» Example4.5.sas (the SAS code)
» Example45.SSC (the S+ code)
» exam45.dat (the raw data for S+)

I know that this topic was discussed years ago, but I think I really need to download these files for my further understanding about this topic. I would really appreciate if you could help me with this. Thank you.

Best Regards,

Irene I
Helmut
Hero

Vienna, Austria,
2018-06-07 13:04

@ Irene_I
Posting: # 18862
Views: 617

## Three-way crossover example data set

Hi Irene,

» I know that this topic was discussed years ago, but I think I really need to download these files for my further understanding about this topic. I would really appreciate if you could help me with this.

The files of the 2005 edition are gone. However, here is the link for the 2016 edition.

Cheers,
Helmut Schütz

The quality of responses received is directly proportional to the quality of the question asked. ☼
Science Quotes
Irene_I
Junior

Indonesia,
2018-06-08 11:09

@ Helmut
Posting: # 18869
Views: 590

## Three-way crossover example data set

Dear Helmut,

Thanks a lot! It is really a great help to me..

Regards,

Irene I
Irene_I
Junior

Indonesia,
2018-06-12 09:21

@ Helmut
Posting: # 18886
Views: 445

## Three-way crossover example data set

Dear Helmut,

After I tried to run the SAS code (Example 4.5), I got the CI, but there is only one Residual value. I guess I suppose to calculate the intra-subject CV from this value, right? So, My question is, Do we only get one Intra-subject CV from three-comparison of the treatment (T-R, S-T, S-R) while we got three CI (T-R, S-T, S-R)?

I am looking forward for your help.
Thank you very much.

Best Regards,

Irene I
Helmut
Hero

Vienna, Austria,
2018-06-12 12:42

@ Irene_I
Posting: # 18888
Views: 433

## Leave-One-Out (IBD)

Hi Irene,

» After I tried to run the SAS code (Example 4.5), I got the CI, but there is only one Residual value. I guess I suppose to calculate the intra-subject CV from this value, right? So, My question is, Do we only get one Intra-subject CV from three-comparison of the treatment (T-R, S-T, S-R) while we got three CI (T-R, S-T, S-R)?

I’m not equipped with  … Hence, my understanding of the code is limited.
You are right, in the model you get only one (pooled) estimate of the variance. That’s not a good idea, since it will “work” only if intra-subject variances would be identical and the treatment differences at least very similar. Otherwise, the treatment estimates will be biased and the type I error is not controlled. See also this post.
At the 2nd Conference of the Global Bioequivalence Harmonisation Initiative (Rockville, September 2016) Pina D’Angelo gave a presentation “Testing for Bioequivalence in Higher‐Order Crossover Designs: Two‐at‐a‐Time Principle Versus Pooled ANOVA” showing exactly that. Here some of her slides:

Purpose

• To determine which method of statistical analysis
is more appropriate to conclude bioequivalence in
higher‐order crossover studies:
1. The two‐at‐a‐time principle using two separate
incomplete block design ANOVAs
2. A pooled approach using one ANOVA and a common
error term for the two contrasts

Introduction

Statistical Concerns:

1. Different means (point estimates) between formulations
2. Different variances between formulations
• If either situations exist, which method of analysis
reduces bias the most: two‐at‐a‐time principle or
pooled ANOVA?

Simulated Data: Summary

• When all three treatments have similar means and there is
homogeneity of variances, both methods give very similar
results.
• When treatment means differ but there is homogeneity of
variances, both methods give very similar results. With higher
variability, the power is slightly increased when using the pooled
ANOVA method.
• When treatment means are similar but variances are not
homogeneous, the two‐at‐a‐time method gives higher power to
detect BE for the treatment with lower variability
• When treatment means differ and variances are not
homogeneous, the two‐at‐a‐time method increases power to
detect BE for the treatment with lower variability. Moreover,
type I error is higher when using the pooled ANOVA method.

Closing Remarks

• Using a two‐at‐a‐time principle for statistical analysis of a
higher‐order pilot study will have more value for decision-
making on which multiple tests lots will be selected for use
in a pivotal study based on the pilot study results. The
intra‐subject variability of a specific test‐to‐reference
comparison can be determined using the two‐at‐a‐time
principle, which may be an important factor in selecting a
test product considering test products are generally
formulated to show different characteristics for testing in a
pilot study.
(my emphasis)

If you are interested in all pairwise comparisons, generate three data sets (values or T&R, S&R, T&S, i.e., exclude all values of the respective other treatment S, T, R) whilst keeping the codes for sequence and period. This will give you data sets which represent an IBD (incomplete block design). Run the usual model on them.
This approach is also recommended in the EMA’s BE-GL:

In studies with more than two treatment arms (e.g. a three period study including two references, one from EU and another from USA […], the analysis for each comparison should be conducted excluding the data from the treatments that are not relevant for the comparison in question.

(my emphasis)
Example 4.5 (Phoenix/WinNonlin 8, mixed effects, Satterthwaite’s degrees of freedom):
```                           PE (%)    90% CI (%)       s²w    CVw (%) pooled model T vs. R  AUC  116.15 (108.97, 123.81)  0.043954  21.20              T vs. R  Cmax 129.65 (118.84, 141.45)  0.084569  29.71              S vs. R  AUC  140.63 (131.93, 149.90)  0.043954  21.20              S vs. R  Cmax 159.75 (146.42, 174.30)  0.084569  29.71              T vs. S  AUC   82.60 ( 77.46,  88.07)  0.043954  21.20              T vs. S  Cmax  81.16 ( 74.35,  88.56)  0.084569  29.71 IBD models   T vs. R  AUC  116.05 (108.92, 123.65)  0.042525  20.84              T vs. R  Cmax 129.54 (119.26, 140.71)  0.074744  27.86              S vs. R  AUC  141.09 (131.39, 151.51)  0.053706  23.49              S vs. R  Cmax 160.48 (145.20, 177.36)  0.109492  34.02              T vs. S  AUC   83.12 ( 78.37,  88.15)  0.035788  19.09              T vs. S  Cmax  81.63 ( 75.33,  88.46)  0.069372  26.80```
It’s clear that the variances are not identical. The ones of T/R are smaller than the ones of S/R. If we apply the pooled model the CIs of T/R will be wider than in the IBD model and the CIs of S/R narrower.

BTW, I don’t understand what the purpose of the `carry` variable in Byron’s code is.

Cheers,
Helmut Schütz

The quality of responses received is directly proportional to the quality of the question asked. ☼
Science Quotes
Irene_I
Junior

Indonesia,
2018-06-13 05:02

@ Helmut
Posting: # 18896
Views: 379

Dear Helmut,

Best Regards,

Irene I
Helmut
Hero

Vienna, Austria,
2018-06-13 15:02

@ Irene_I
Posting: # 18898
Views: 327

## Impact of pooled variance (bias, CI)

Hi Irene,

» Thank you for your explanation.

You are welcome. A picture tells more than a thousand words (AUC only).

The PE and its CI of T/R are similar. The small bias is caused by the fact that the average PE of S/R and T/S (√141.09×83.12=108.29) is close to the T/R with 116.05. The other comparisons show negative biases. The CI of S/R is narrower in the pooled model and the CI of T/S is wider.
We don’t want to go there.

Cheers,
Helmut Schütz

The quality of responses received is directly proportional to the quality of the question asked. ☼
Science Quotes
d_labes
Hero

Berlin, Germany,
2018-06-13 15:31

@ Helmut
Posting: # 18899
Views: 302

## carry (over?)

Dear Helmut,

» BTW, I don’t understand what the purpose of the `carry` variable in Byron’s code is.

As far as I see it: It serves for nothing because it is not used except in the class statement of `Proc Mixed`.
The class statement serves the purpose of declaring factors even if a variable is numeric. Nothing more.

The `carry` variable seems to me a left over from old times where carry-over was an issue.

Regards,

Detlew
Bioequivalence and Bioavailability Forum |  Admin contact
18,428 posts in 3,913 threads, 1,176 registered users;
online 23 (1 registered, 22 guests [including 14 identified bots]).

Lack of clarity is always a sign of dishonesty.    Celia Green

The BIOEQUIVALENCE / BIOAVAILABILITY FORUM is hosted by
Ing. Helmut Schütz