Blocks of experimental units
When the experimental units are grouped in blocks, the model for the response should contain a term explaining differences between the blocks. If there are covariates, terms for them can also be used. For example, if there are blocks, a single numerical covariate and one factor, the following model may be used.
yij = µ |
+ |
explained by blocks αi |
+ |
explained by factor βi |
+ |
explained by covariate γ xi |
+ |
εij |
Both the blocks and covariate should be included in the model before testing the effect of the factor.
Pruning sweet cherry trees
The cherry tree example at the start of the previous page was incompletely described there. Four different pruning systems were used in the experiment and it was laid out as a randomised block design with a row of four trees in each block. There were six blocks oriented perpendicular to the prevailing wind and the yields from the full experiment were:
Pruning system | |||||||||||
---|---|---|---|---|---|---|---|---|---|---|---|
1 | 2 | 3 | 4 | ||||||||
Block | Area | Yield | Area | Yield | Area | Yield | Area | Yield | |||
1 2 3 4 5 6 |
140 200 144 127 173 225 |
2.1 3.5 1.8 3.6 3.8 5.0 |
139 222 209 179 108 210 |
3.4 5.9 3.8 5.6 2.7 5.0 |
203 231 228 114 86 89 |
4.3 5.0 5.0 3.6 2.3 1.8 |
110 95 171 199 215 191 |
4.1 4.1 4.9 7.1 6.5 6.2 |
An analysis of variance table is shown below.
Terms for the blocks and covariate (the initial cross-sectional area of the trees) should be included in all models — they describe the structure of the experimental units.
Drag the red arrow down to add a term for the pruning system. Since its p-value is reported as "0.0000", we conclude that:
It is almost certain that the four pruning systems result in different mean yields.
Reporting the results
Since the treatments are orthogonal to the blocks in the experiment, the raw treatment means do not need to be adjusted for differences between the blocks — all treatments are used once within each block. However the cross-sectional areas of the trees are not orthogonal to the treatments — some pruning systems were used (by chance) more for smaller trees than others. The raw treatment means should therefore not be used to summarise differences between the treatments.
In the full model with blocks covariate and the pruning system, the parameter estimates for the pruning systems are honest descriptions of the differences between pruning system 1 and the others.
Pruning system |
Parameter estimate |
Adjusted mean |
---|---|---|
1 | 0 | 3.273 |
2 | 0.878 | 4.151 |
3 | 0.588 | 3.862 |
4 | 2.290 | 5.564 |
The parameter estimates are differences in mean yield between the pruning systems and the first (baseline) system. It is usually easier for readers to understand adjusted mean yields. These are the predicted yields for an average block and average value of the covariate — a tree with cross-sectional area 167 cm2. The final column in the table above gives these adjusted means.