Simple model

Consider a randomised block experiment for a single factor. We use the notation yijk to denote the k'th of the replicates for which the factor is at level j in block i. The simplest model for the response is given by the equation:



yijk  =  µ


 + 
(explained
by blocks
)
βi


 + 
(explained
by factor
)
γj


 + 

(unexplained)
εijk

where β1 = 0 and γ1 = 0. The parameters are interpreted as follows:

The β-parameters therefore describe the differences between the blocks whereas the γ-parameters capture the differences between the levels of factor.

It is important to understand this model since estimates of its parameters are often reported.

Relationship to model for two factors

Although the experimental design is totally different, the above model for randomised block data is identical in form to the no-interaction model that we used previously experiments with two factors (and no blocks).



yijk  =  µ


 + 
(explained
by factor B
)
βi


 + 
(explained
by factor C
)
γj


 + 

(unexplained)
εijk

The only difference is that the β-parameters now describe differences between the blocks instead of differences between levels of the second factor.

Graphical display of model

The diagram below shows data from a randomised block experiment against one axis for the blocks and another for the levels of the factor. The means from the best model (fitted by least squares) are shown by the coloured grid.

The blue value under the table is µ in the equation at the top of this page. The four red values are the parameters βi and represent the block effects. The three green values are the parameters γj and describe the differences between the factor levels.

Dragging the three red arrows for blocks 2, 3 and 4 adjusts the parameters β2, β3 and β4 in the model. Their values describe differences between their mean responses and that in block 1.

Similarly, the two red arrows for factor levels Low and Very low can be dragged to change γ2 and γ3. These parameters describe differences between their mean responses and that for treatment Regular.

(Clicking Least squares returns the parameter values to their least squares estimates.)