Three-way crossover example data set [Design Issues]
❝ 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 (unfortunately 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 imbalanced 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 |
+----+------------+ +----+--------+ +----+--------+
^ imbalanced ^ imbalanced
❝ 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
Chow S-C, Liu J-p.
Design and Analysis of Bioavailability and Bioequivalence Studies. New York: Marcel Dekker; 2nd ed. 2001, p. 302–32.
Jones B, Kenward MG.
Design and Analysis of Cross-over Trials. Boca Raton: Chapman & Hall/CRC; 2nd ed. 2003, p. 151–204.
Patterson S, Jones B.
Bioequivalence and Statistics in Clinical Pharmacology. Boca Raton: Chapman & Hall/CRC; 2006, p. 79–132.
You may download zipped datasets and programs (SAS/S+) from 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: α = 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.
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Science Quotes
Complete thread:
- Three-way crossover BABE Studies acfalcao 2007-09-16 22:34 [Design Issues]
- Three-way crossover example data setHelmut 2007-09-17 20:47
- Three-way crossover example data set acfalcao 2007-09-17 23:02
- Three-way crossover example data set Helmut 2007-09-18 12:47
- Three-way crossover (WinNonlin) Nirali 2007-09-21 06:49
- Three-way crossover (WinNonlin) Helmut 2007-09-21 13:04
- Three-way crossover (WinNonlin) Nirali 2007-09-25 08:05
- Three-way crossover (WinNonlin) Helmut 2007-09-25 13:15
- Three-way crossover (WinNonlin) Nirali 2007-09-25 08:05
- Three-way crossover (WinNonlin) Helmut 2007-09-21 13:04
- Three-way crossover example data set Irene_I 2018-06-07 11:09
- Three-way crossover example data set Helmut 2018-06-07 13:04
- Three-way crossover example data set Irene_I 2018-06-08 11:09
- Three-way crossover example data set Irene_I 2018-06-12 09:21
- Leave-One-Out (IBD) Helmut 2018-06-12 12:42
- Leave-One-Out (IBD) Irene_I 2018-06-13 05:02
- Impact of pooled variance (bias, CI) Helmut 2018-06-13 15:02
- carry (over?) d_labes 2018-06-13 15:31
- Leave-One-Out (IBD) Irene_I 2018-06-13 05:02
- Leave-One-Out (IBD) Helmut 2018-06-12 12:42
- Three-way crossover example data set Helmut 2018-06-07 13:04
- Three-way crossover (WinNonlin) Nirali 2007-09-21 06:49
- Three-way crossover example data set Helmut 2007-09-18 12:47
- Pseudo-periods ElAlumno 2019-03-14 23:40
- Pseudo-periods Helmut 2019-03-15 00:47
- Two‐at‐a‐Time analysis in R ElAlumno 2019-03-22 21:59
- fixed & mixed (dammit!) and a request to SASians Helmut 2019-03-23 01:03
- Pooled vs IBD T-R in SAS from non-SASian mittyri 2019-03-23 23:13
- Pooled vs IBD T-R in SAS from non-SASian Helmut 2019-03-23 23:36
- mixed in R mittyri 2019-03-23 23:50
- mixed in R (EMA B ≠ FDA) Helmut 2019-03-24 01:23
- Pooled vs IBD T-R in SAS from non-SASian mittyri 2019-03-23 23:13
- fixed & mixed (dammit!) and a request to SASians Helmut 2019-03-23 01:03
- Two‐at‐a‐Time analysis in R ElAlumno 2019-03-22 21:59
- Pseudo-periods Helmut 2019-03-15 00:47
- Williams design 3-way Brus 2021-06-09 16:48
- Williams design 3-way Helmut 2021-06-09 17:48
- Williams design 3-way Brus 2021-06-10 14:33
- Williams design 3-way Helmut 2021-06-10 15:03
- Williams design 3-way vezz 2021-06-10 17:23
- Williams design 3-way Helmut 2021-06-10 20:14
- Williams design 3-way vezz 2021-06-11 09:11
- Williams design 3-way Brus 2021-06-11 12:19
- Williams design 3-way Relaxation 2021-06-11 12:42
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- Williams design 3-way vezz 2021-06-11 16:56
- Williams design 3-way vezz 2021-06-11 09:11
- Williams design 3-way Helmut 2021-06-10 20:14
- Williams design 3-way vezz 2021-06-10 17:23
- Williams design 3-way Helmut 2021-06-10 15:03
- Williams design 3-way Brus 2021-06-10 14:33
- Williams design 3-way Helmut 2021-06-09 17:48
- Three-way crossover example data set acfalcao 2007-09-17 23:02
- Three-way crossover example data setHelmut 2007-09-17 20:47