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.