Bioequivalence and Bioavailability Forum

Dear Helmut, dear beman!

Let me throw in my two cents.

❝ ❝ Using Wiliams-Desing i need 10 sequences, but i have only 12 subjects, so it is not possible to have a balanced design.

Right, but: The Williams design with the additional balance that each treatment precedes each other the same number is only 'optimal' if some sort of carry-over is incorporated in the analysis model, namely first order carry-over. If not it not necessary to have such a design.
First order carry-over is criticised by Senn (1). If one follows his reasoning it is superflous. Additionally the EMA guidances recommends us not to statistically test for carry-over.

Thus I would suggest you to go with any Latin square for 5 treatments.

❝ I’m traveling, so I’m referring to my (bad) memory. If I recall it correctly Jones & Kenward¹ stated that it’s not possible to construct a fully variance balanced Williams’ design with five treatments.

Right recall but only for a single 5x5 Latin square. Chapter 4.2 "... As already noted earlier, for odd t no single sequentially counterbalanced square exists for t = 3, 5 and 7, although single squares have been found by computer search for t = 9, 15, 21 and 27 ..."
(sequentially counterbalanced square = Williams)

@beman: The example mentioned by Helmut of an ‘almost balanced one’ (in the sense of balancedness of Williams design) is
Subj.  Period
      1 2 3 4 5
  1   A B C D E
  2   B D E C A
  3   C E B A D
  4   D C A E B
  5   E A D B C
a good candidate for a 5x5 Latin square, I think.

❝ It’s not a good idea to already start with an imbalanced design. Drop-outs will do that for you anyway.

Full ACK.
The number of subjects = 12 is not a "carved in stone number" (© by Helmut) for pilot studies also it is used often, regardless if appropriate.
Thus choose 15 to have a 'balanced' design if using 5 sequences.

❝ What you could do is some variant of an incomplete block design (also saving periods)...

@BEman
See literature (2) for that suggestion of Helmut.
I think for a pilot study with more or less "proof of concept" character (whatever concept here means ) the lower power of those designs don't matter.

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(1)S. Senn
Cross-over Trials in Clinical Research
John Wiley & Sons, Chichester, pp162-163 (2nd ed. 2002)
Chapter 10

(2)S-C Chow and J-P Liu
Design and Analysis of Bioavailability and Bioequivalence Studies
Chapman & Hall / CRC, Boca Raton, 3rd Ed. (2009)
Chapter 2.5.5