Subsets of a fully balanced lattice design
In a fully balanced experiment for r2 treatments, r2(r + 1) experimental units are required.
Number of treatments, r2 | ||||||
---|---|---|---|---|---|---|
9 | 16 | 25 | 36 | 49 | ||
Number of blocks | r(r + 1) | 12 | 20 | 30 | 42 | 56 |
Number of experimental units | r2(r + 1) | 36 | 80 | 150 | 252 | 392 |
If there are many treatments, the number of experimental units soon becomes impractical.
Simple lattice design
We explained on the previous page that a single group of r blocks does not allow us to compare all pairs of treatments — some comparisons are confounded with differences between blocks. However two groups of r blocks do allow an assessment of the differences between all pairs of treatments. This unbalanced design is called a simple lattice design.
For example, a fully balanced lattice design for 25 treatments would involve 30 blocks (as shown on the previous page). The first two groups of blocks are:
Block 1 | 1 | 2 | 3 | 4 | 5 | Block 6 | 1 | 6 | 11 | 16 | 21 | |
---|---|---|---|---|---|---|---|---|---|---|---|---|
Block 2 | 6 | 7 | 8 | 9 | 10 | Block 7 | 2 | 7 | 12 | 17 | 22 | |
Block 3 | 11 | 12 | 13 | 14 | 15 | Block 8 | 3 | 8 | 13 | 18 | 23 | |
Block 4 | 16 | 17 | 18 | 19 | 20 | Block 9 | 4 | 9 | 14 | 19 | 24 | |
Block 5 | 21 | 22 | 23 | 24 | 25 | Block 10 | 5 | 10 | 15 | 20 | 25 |
Some comparisons between treatments can be made directly from a single block:
Other comparisons require the use of two blocks:
Although all comparisons cannot be made with equal accuracy, all treatments can be compared.
More than two groups of blocks
A full balanced lattice design for r2 treatments uses (r + 1) groups of r blocks. A simple lattice design uses 2 such groups. If resources allow, it is possible to use intermediate numbers. Analysis of variance is the same however many groups of blocks are used.
Yield of soybean varieties
The table below shows yields of soybeans (bushels per acre, minus 30) from a lattice experiment on 25 varieties developed in a soybean breeding programme in North Carolina. (The varieties used are shown in brackets after the yields.)
Block 1 | 6 (1) | 7 (2) | 5 (3) | 8 (4) | 6 (5) |
---|---|---|---|---|---|
Block 2 | 16 (6) | 12 (7) | 12 (8) | 13 (9) | 8 (10) |
Block 3 | 17 (11) | 7 (12) | 7 (13) | 9 (14) | 14 (15) |
Block 4 | 18 (16) | 16 (17) | 13 (18) | 13 (19) | 14 (20) |
Block 5 | 14 (21) | 15 (22) | 11 (23) | 14 (24) | 14 (25) |
Block 6 | 24 (1) | 13 (6) | 24 (11) | 11 (16) | 8 (21) |
Block 7 | 21 (2) | 11 (7) | 14 (12) | 11 (17) | 23 (22) |
Block 8 | 16 (3) | 4 (8) | 12 (13) | 12 (18) | 12 (23) |
Block 9 | 17 (4) | 10 (9) | 30 (14) | 9 (19) | 23 (24) |
Block 10 | 15 (5) | 15 (10) | 22 (15) | 16 (20) | 19 (25) |
Block 11 | 13 (1) | 26 (2) | 9 (3) | 13 (4) | 11 (5) |
Block 12 | 15 (6) | 18 (7) | 22 (8) | 11 (9) | 15 (10) |
Block 13 | 19 (11) | 10 (12) | 10 (13) | 10 (14) | 16 (15) |
Block 14 | 21 (16) | 16 (17) | 17 (18) | 4 (19) | 17 (20) |
Block 15 | 15 (21) | 12 (22) | 13 (23) | 20 (24) | 8 (25) |
Block 16 | 16 (1) | 7 (6) | 20 (11) | 13 (16) | 21 (21) |
Block 17 | 15 (2) | 10 (7) | 11 (12) | 7 (17) | 14 (22) |
Block 18 | 7 (3) | 11 (8) | 15 (13) | 15 (18) | 16 (23) |
Block 19 | 19 (4) | 14 (9) | 20 (14) | 6 (19) | 16 (24) |
Block 20 | 17 (5) | 18 (10) | 20 (15) | 15 (20) | 14 (25) |
The first ten blocks would be a simple lattice design, so we will initially analyse only these data. The analysis of variance table below can be used to test whether there are any differences between the 25 varieties.
From the p-value associated with the varieties, 0.0564, we would conclude that there is very little evidence from the experiment that the varieties have different yields.
In this experiment, there was internal structure to the blocks — the groups of five blocks consisted of adjacent blocks. Click Split blocks to split the blocks sum of squares into two components — a sum of squares due to differences between the 'super-blocks' and a sum of squares between blocks within each group of five. In the above analysis of variance table, the treatments sum of squares and p-value is unaffected but we will describe further analysis that uses this split in the next chapter.
The second group of ten blocks is simply a repeat of the design used for the first group. It could be argued that a better design would have used two different groups from the six groups of blocks in a balanced lattice design — different pairs of treatments would have been estimated with closer accuracies — but a repeat of the same structure can still be analysed.
Select All twenty blocks from the pop-up menu. With the full data set, each treatment is repeated in four blocks (rather than two) and the resulting test for differences is therefore much more powerful.
It is almost certain that the 25 varieties have different mean yields.