Internal structure of factors in more complex experiments

We have introduced analysis of the internal structure of factors with many levels through constraints in the context of a completely randomised experiment. Similar methods can be used to analyse the internal structure of a factor with several levels that is used in factorial experiments with several factors and in randomised block experiments.

Analysis again partitions the sum of squares explained by the factor into sums of squares into sums of squares explained by various constraints on the model parameters.

Randomised block experiment with several treatments

We now illustrate how this partition of the treatment sum of squares can be performed in a randomised block experiment. Assuming no interaction between the factor and blocks, the model can be written as:


yijk  =  µ

 + 
(block effect)
βi

 + 
(effect of factor, X)
γj

 + 
(unexplained)
εijk

where β1 = 0 and γ1 = 0. Constraints about the internal structure of the factor levels for X are expressed as constraints on the values of the γ-parameters.

The example below illustrates for a randomised block experiment in which the treatments can be partitioned into two groups.

Wheat yield in randomised block experiment

This experiment is similar to the one on the previous page, except that the experimental units were grouped into five blocks of five plots and the five wheat varieties were randomly allocated within each block.

In the diagram, the different colours correspond to the different blocks. Note that the different coloured lines are parallel since we are assuming that there is no interaction between varieties and blocks.

The three checkboxes again correspond to constraints on the values of parameters for the five varieties.

Analysis of variance

The analysis of variance table below initially shows a single explained sum of squares corresponding to the five varieties.

Click Split varieties to split the explained sum of squares into three components defined by the three sets of constraints described above. These sums of squares can again be used to test for differences between the group A and group B varieties, for differences within the group A varieties, and for differences within the group B varieties.

Again there is fairly weak evidence of differences between the group B varieties but very strong evidence between group A varieties and between group A and group B.