- Difference squares for OBIBDs
- For each initial block, set up a k by k square, in which
the entries are the differences (in the appropriate number system),
between the k entries in the first treatment set and the k
entries in the second treatment set. The entries on the diagonal comprise
the differences between the first and second treatments at a plot, whereas
the off-diagonal differences comprise the differences between first
and second treatments within the block. These correspond (respectively)
to the two parts of the left-hand side of the matrix equation:
- nij- ni0n'j0
= 0, as given in the OBIBD definition
- Thus if s differences occur on the diagonal, not necessarily
an equal number of times, then they must occur k times as often
in the squares all told (i.e. including the diagonal).
For example, an OBIBD(9,24,8,3,2;2) is given by the initial blocks
(I,1; 0,7; 4,6), (1,5; 7,2; 6,3) (5,I; 2,0; 3,4), mod 8, where I is
an invariant treatment. The difference squares are:
| |
1 |
7 |
6 |
|
|
5 |
2 |
3 |
|
|
I |
0 |
4 |
| I |
-I |
-I |
-I |
|
1 |
4 |
1 |
3 |
|
5 |
I |
3 |
7 |
| 0 |
1 |
7 |
6 |
|
7 |
6 |
3 |
4 |
|
2 |
I |
6 |
2 |
| 4 |
5 |
3 |
2 |
|
6 |
7 |
4 |
5 |
|
3 |
I |
5 |
1 |
Each of the values 1-7, together with I and -I, occurs once on the
diagonal, and thrice overall.