Bioequivalence and Bioavailability Forum

Dear all,

A quick question about William design with 4 treatments (well, I guess it's applicable to other number of treatments as well...).

I always use the following William design because it's easy to remember due to the symmetry (always ABCD starting from 2 opposite corners).
ABCD
B  C
C  B
DCBA
The empty center can be AD/DA for row 2 and 3, respectively (or DA/AD). The similar sequences were also discussed in previous posts such as This one

This afternoon I was reviewing a protocol from a CRO with unbalanced Latin square design and trying to suggest the balanced William design instead. I suggested the following one. Let's call it X.
ABCD
BADC
CDAB
DCBA

While I was search the literature to support the argument and explain the reason, I came across an explanation which leads to the following sequences. Let's call it Y:
ABDC
BCAD
CDBA
DACB

For 4 treatments as shown in Y, there are 12 pair-wise comparisons:
- AB/BA -> 1 time each
- AC/CA -> 1 time each   
- AD/DA -> 1 time each
- BC/CB -> 1 time each
- BD/DB -> 1 time each
- CD/DC -> 1 time each


Note that each treatment is followed immediately after the another, i.e., there's no period separate them. This makes sense since if A has any effect on B, then the effect should be the greatest when B is followed immediately after A, instead of separated by another period (denoted by ∙) or two, e.g., A∙B, or A∙∙B

In comparison, X, the one I always used before gives:
- AB/BA -> 2 times each
- AC/CA -> 0 time each   
- AD/DA -> 1 time each   
- BC/CB -> 1 time each
- BD/DB -> 0 time each
- CD/DC -> 2 times each


Yes, there are A∙C/C∙A and B∙D/D∙B 2 times each, A∙∙D/D∙∙A and B∙∙C/C∙∙B 1 time each. So if we only consider 1 treatment appears after another without taking into consideration the period separating them, then both X and Y are equivalent (appear 2 times for each pair-wise comparison). But if we take the separating period between treatment pairs into consideration, X and Y are different, and in my opinion, Y is better.

So my question is, is there like a rank among different configurations of William design mentioned in the literature (e.g., one is better than another)? If so, why should we use the inferior one? In such case, X should never be used when the better Y is available.

Please let me know your opinions.