Very different blocks
In some experiments, the blocks are so different that the effect of the factor varies between blocks.
The most general model for randomised block data has a separate parameter to allow an arbitrary response mean at each combination of block and factor level.
yijk = |
(explained by block-factor combination) µij |
+ |
(unexplained) εijk |
This is similar to the model for interaction between two factors, X and Z, and it is again useful to express it in the form,
yijk = µ |
+ |
(effect of block i) βi |
+ |
(main effect of factor level j) γj |
+ |
(block-factor interaction) δij |
+ |
(unexplained) εijk |
When written in this way, the sum of the terms (γj + δij) describes the effect of factor level j in block i. There are again redundant parameters, so we define all parameters that refer to the baseline block and factor level to be zero,
The δ-parameters are again called interaction terms and they describe how much the factor effects differ from block to block — if the factor effects are the same in all blocks, the δ-parameters will all be zero.
Graphical display of model
The diagram below again shows data from a randomised block experiment.
The diagram initially shows a model for the data in which the factor has the same effect within each block. Click Least squares then click the y-z rotation button the lines for the four blocks are parallel since the effect of changing the factor level is the same within each block.
Click the y-x-z rotation button then click the checkbox Interaction to add extra draggable arrows to the diagram allowing arbitrary response means for all combinations of block and factor level. Click Least squares then rotate the diagram to get a better feel for how changing the factor level has a different effect in different blocks.
Decreasing the level of the factor increases the response in blocks 1 and 3, but it decreases the response in blocks 2 and 4.