Bioequivalence and Bioavailability Forum

Dear DLabes!

❝ As far as I have understood until now: The balance under 3. is only necessary if one is willing to adopt the view of first-order (simple) carry-over and incorporate this effect also in the analysis model. Only in that case the full 6-sequence design has its merits.

Not only that. To quote Senn¹:

Perfect balance with respect to sequences, is not, however, an absolute requirement for an analysis which allows for period effects as well as treatments: it is simply that other things being equal such designs are more efficient.

(my emphasis)

Jones and Kenward² give a comparison of different four-treatment designs (no carry-over in the model).
The full set of latin squares (sequences = treatments! = 1×2×3×4 = 24):
 1 A B C D    13 A D C B
 2 B C D A    14 D C B A
 3 C D A B    15 C B A D
 4 D A B C    16 B A D C
 5 A C B D    17 A C D B
 6 C B D A    18 C D B A
 7 B D A C    19 D B A C
 8 D A C B    20 B A C D
 9 A B D C    21 A D B C
10 B D C A    22 D B C A
11 D C A B    23 B C A D
12 C A B D    24 C A D B   Efficiency 100%

Orthogonal latin squares (balanced):
 1 A B C D     7 C B D A
 2 B A D C     8 D A C B
 3 C D A B     9 A C D B
 4 D C B A    10 B D C A
 5 A D C B    11 C A B D
 6 B C A D    12 D B A C   Efficiency 90.91%

Williams' design (balanced):
 1 A D B C
 2 B A C D
 3 C B D A
 4 D C A B   Efficiency 90.91%

One latin square:
 1 A B C D
 2 B C D A
 3 C D A B
 4 D A B C   Efficiency 18.18%

The Williams’ design shows the same efficiency as the orthogonal set (90.91%) - with only 4 as compared to 12 sequences. The simple latin square clearly is inferior to all others.

For the 6×3 Williams’ design they give an efficiency of 80% (unfortunately no value for a 3×3). There's also the routine³ XOEFFICIENCY for GenStat.

❝ Following the view of SENN¹⁾ that the model of simple carry-over is obsolet and not testing for it, especially in case of BE studies with appropriate wash-out (which view has found its way into the new EMEA DRAFT, one of the rare things I appreciate),…

True!

❝ … what are then the benefits of using balance with respect to condition 1.-3., the so called Williams designs?

See the quote of Senn above, and also this one:

It has been my habit in designing cross-over trials to use all six sequences for a three-period three-treatment design and to use a single Latin square for a four-treatment design.

❝ And why should regulators insist on it?

Do they?

❝ Has anybody factual experiences that studies were rejected if only balanced designs with respect to the conditions 1. and 2. (Latin square designs) were use?

No, but my experience is limited to 3×3 studies I’ve seen as a consultant (in my studies I never used this design). Any Williams’ design has the advantage that pairwise comparisons may be extracted (also recommended by Byron Jones in a personal communication at the BioInternational 2003) - which are also balanced (needed for nonparametric comparisons, which seems to be of historical interest in the EU…). I’ve seen small studies where due to dropouts the extracted 2×2 sets were extremely imbalanced - or even worse, didn't work any more at all.

❝ BTW which guidance state the necessity of the full 3x3x6 (treatment/period/sequence) design?

I didn’t check all national guidelines; the first one coming into my mind was ANVISA’s (2003).

1. S Senn
   Cross-over Trials in Clinical Research
   John Wiley & Sons, Chichester, pp162-163 (2^nd ed. 2002)
   
2. B Jones and MG Kenward
   Design and Analysis of Cross-over Trials
   Chapman & Hall/CRC, Boca Raton, Chapter 4 (2^nd ed. 2003)
   Parts of this chapter as a preview in GoogleBooks.
   
3. B Jones and PW Lane
   XOEFFICIENCY
   In RW Payne (ed.) GenStat - Release 6.1. New Features
   VSN International, Oxford (2002)