‘Two-at-a-Time’ approach with >3 treat­ments [Design Issues]

posted by Shuanghe  – Spain, 2022-06-12 21:02 (618 d 07:22 ago) – Posting: # 23053
Views: 2,002

Dear Helmut,

I vaguelly recall that for 4x4 William design, each row (sequence) and each column (period) should contain A, B, C, and D each exactly once. I think that there's a typo in the 3rd sequence of your first William design. Your 3rd and 4th column do not seem right. So 3rd period would have 2 Bs and 4th had 2 Ds. CABD should be CADB. So It would be ABCD/BDAC/CADB/DCBA. In such case, it would be balanced for pair-wise comparisons.

$$\small{\begin{array}{c|cccc}
s/p & \text{I} & \text{II} & \text{III} & \text{IV}\\\hline
1 & \text{A} & \text{B} & \text{C} & \text{D}\\
2 & \text{B} & \text{D} & \text{A} & \text{C}\\
3 & \text{C} & \text{A} & \text{D} & \text{B}\\
4 & \text{D} & \text{C} & \text{B} & \text{A}\\
\end{array}{\color{Blue}\mapsto}
\begin{array}{c|cccc}
s/p & \text{I} & \text{II} & \text{III} & \text{IV} \\\hline
1 & \text{A} & {\color{Red}\bullet} & {\color{Red}\bullet} & \text{D}\\
2 & {\color{Red}\bullet} & \text{D} & \text{A} & {\color{Red}\bullet}\\
3 & {\color{Red}\bullet} & \text{A} & \text{D} & {\color{Red}\bullet}\\
4 & \text{D} & {\color{Red}\bullet} & {\color{Red}\bullet} & \text{A}\\
\end{array}{\color{Blue}\wedge}
\begin{array}{c|cccc}
s/p & \text{I} & \text{II} & \text{III} & \text{IV}\\\hline
1 & {\color{Red}\bullet} & \text{B} & {\color{Red}\bullet} & \text{D}\\
2 & \text{B} & \text{D} & {\color{Red}\bullet} & {\color{Red}\bullet}\\
3 & {\color{Red}\bullet} & {\color{Red}\bullet} & \text{D} & \text{B}\\
4 & \text{D} & {\color{Red}\bullet} & \text{B} & {\color{Red}\bullet}\\
\end{array}{\color{Blue}\wedge}
\begin{array}{c|cccc}
s/p & \text{I} & \text{II} & \text{III} & \text{IV}\\\hline
1 & {\color{Red}\bullet} & {\color{Red}\bullet} & \text{C} & \text{D}\\
2 & {\color{Red}\bullet} & \text{D} & {\color{Red}\bullet} & \text{C}\\
3 & \text{C} & {\color{Red}\bullet} & \text{D} & {\color{Red}\bullet}\\
4 & \text{D} & \text{C} & {\color{Red}\bullet} & {\color{Red}\bullet}\\
\end{array}}$$

All the best,
Shuanghe

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