Williams Design is a special case of orthogonal latin squares design. It is a high-crossover design and typically used in Phase I studies. Due to the limitation of the # of subjects, we would like to achieve the balance and maximize the comparisons with the smallest # of subjects.
A Williams design possesses balance property and requires fewer sequences and periods. If the number of treatments (n) is an odd number, there will be 2 x n number of sequences. If the number of treatments (n) is an even number, there will be n number of sequences. The example below is a Williams Design with a 4 by 4 crossover (four treatments, four sequences, and also four periods).
Let A, B, C, and D stand for four different treatments, a Williams Design will be arranged as:
A D B C
B A C D
C B D A
D C A B
Notice that each treatment only occurs one time in one sequence, in one period. Furthermore, each treatment only follow another treatment one time. For example, treatment D following treatment B only one time in all sequences.
Several years ago, I wrote a paper on generating the randomization schedule using SAS. I illustrated an example for Williams Design.
There is a new paper by Wang et al specifically discussing about "The Construction of a Williams Design and Randomization in Cross-Over Clinical Trials using SAS"
Williams Design is deliberated in detail in the books "Design and Analysis of Clinical Trials" and Design and Analysis of Bioavailability and Bioequivalence Studies" by Chow and Liu
Williams Design is not purely used in Phase I or bioavailabity studies. I participated in a study with drug abuse area where a Williams design was used. It looks like that other people also uses Williams Design in drug abuse research.