## Williams design 3-way [Design Issues]

Dear Helmut,

I have a question on this topic that you mentioned a long time ago:

❝ 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

According to what you said, 6 sequences had to be done with a 3-way study to get balanced sequences on the two 2x2 tablets. But, with following approach, balanced sequences would also be achieved:

ABC
BAC
CAB
CBA

It's possible? What problems would it have? It is not williams square design, right?

Best regards,