A simple unbalanced design
Balanced designs require very large numbers of blocks for some combinations of block size and number of treatments. However simple balanced designs using a small number of blocks sometimes exist. This page describes a class of simple designs for experiments in which the number of treatments is the square of the block size.
Consider an experiment for 25 treatments in blocks of size 5. The following group of blocks uses all treatments.
Treatments used | |||||
---|---|---|---|---|---|
Block 1 | 1 | 2 | 3 | 4 | 5 |
Block 2 | 6 | 7 | 8 | 9 | 10 |
Block 3 | 11 | 12 | 13 | 14 | 15 |
Block 4 | 16 | 17 | 18 | 19 | 20 |
Block 5 | 21 | 22 | 23 | 24 | 25 |
These five blocks are badly unbalanced — although treatments 1 and 2 can be validly compared (since they occur in the same block), a comparison of treatments 1 and 6 is confounded with the difference between blocks 1 and 2. They are however the basis of a balanced design called a lattice design.
Balanced lattice design
A lattice designs use multiple groups of r blocks of size r to compare r2 treatments. If (r + 1) groups of blocks are used, it is possible to arrange the treatments such that each pair of treatments occur together in exactly one block — such a design is balanced. An example is shown below for 25 treatments:
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 | |
Block 11 | 1 | 7 | 13 | 19 | 25 | Block 16 | 1 | 12 | 23 | 9 | 20 | |
Block 12 | 21 | 2 | 8 | 14 | 20 | Block 17 | 16 | 2 | 13 | 14 | 10 | |
Block 13 | 16 | 22 | 3 | 9 | 15 | Block 18 | 6 | 17 | 3 | 14 | 25 | |
Block 14 | 11 | 17 | 23 | 4 | 10 | Block 19 | 21 | 7 | 18 | 4 | 15 | |
Block 15 | 6 | 12 | 18 | 24 | 5 | Block 20 | 11 | 22 | 8 | 19 | 5 | |
Block 21 | 1 | 17 | 8 | 24 | 15 | Block 26 | 1 | 22 | 18 | 14 | 10 | |
Block 22 | 11 | 2 | 18 | 9 | 25 | Block 27 | 6 | 2 | 23 | 19 | 15 | |
Block 23 | 21 | 12 | 3 | 19 | 10 | Block 28 | 11 | 7 | 3 | 24 | 20 | |
Block 24 | 6 | 22 | 13 | 4 | 20 | Block 29 | 16 | 12 | 8 | 4 | 25 | |
Block 25 | 16 | 7 | 23 | 14 | 5 | Block 30 | 21 | 17 | 13 | 9 | 5 |
Since this design is balanced, every pair of treatments can be compared with equal precision.
Pig diets
An experiment was conducted to study the effects of nine feeding treatments on the growth rate of pigs. For a given breed, previous experience indicated that much of the variation in growth rate between pigs is caused by differences between litters, so litters of three pigs were used as the basis of blocks in the experiment. In each experimental block, two groups of three litter-mates were used and the experimental units were pairs of pigs, one from each litter. The response was the total weight gain (pounds per day) of the two pigs in an experimental unit. The experiment was laid out as a balanced lattice design and the results are shown below. (The treatments are shown in brackets.)
Block 1 | 2.20 (1) | 1.84 (2) | 2.18 (3) | Block 4 | 1.19 (1) | 1.20 (4) | 1.15 (7) | |
---|---|---|---|---|---|---|---|---|
Block 2 | 2.05 (4) | 0.85 (5) | 1.86 (6) | Block 5 | 2.26 (2) | 1.07 (5) | 1.46 (8) | |
Block 3 | 0.73 (7) | 1.60 (8) | 1.76 (9) | Block 6 | 2.12 (3) | 2.03 (6) | 1.63 (9) | |
Block 7 | 1.81 (1) | 1.16 (5) | 1.11 (9) | Block 10 | 1.77 (1) | 1.57 (6) | 1.43 (8) | |
Block 8 | 1.76 (2) | 2.16 (6) | 1.80 (7) | Block 11 | 1.50 (2) | 1.60 (4) | 1.42 (9) | |
Block 9 | 1.71 (3) | 1.57 (4) | 1.13 (8) | Block 12 | 2.04 (3) | 0.93 (5) | 1.78 (7) |
The analysis of variance table below can be used to test for differences between the diets.
From the p-value associated with the diets, we would conclude that the diets do affect weight gain.
In practice, when there are many treatments in an experiment, these treatments often have some internal structure and this can be used to ask more specific questions. For example, if the first three diets (1, 2 and 3) were high-protein diets and the others were low-protein ones, it would be of interest to ask whether there were differences between and within these groups of diets.
Click the checkbox Split diets under the anova table. From the resulting sums of squares and p-values, we would conclude that there is no evidence of differences between the low-protein diets. However there is strong evidence that the low- and high-protein diets differ and that there are differences between the high-protein diets.