Splitting the treatment sum of squares

The type of analysis of variance table that was described in earlier sections of this chapter considered all treatments to comprise a single factor, so there was a single row of the table corresponding to the treatments.

Consider a randomised block experiment with two factors A and B and a single replicate. We will use yijk to denote the response measurement in block i that gets factor A at level j and factor B at level k. The diagram below shows the sequence of models implied by an analysis of variance with a single row for the treatments.

Neither blocks nor treatments
yijk  =  µ   +  εijk
Only blocks
yijk  =  µ   +   βi  +   εijk
Blocks and treatments
yijk  =  µ   +   βi   +   θjk  +   εijk

We can modify this sequence by expanding the yellow step to separately add terms for the individual factors and their interaction, as shown below.

Neither blocks nor treatments
yijk  =  µ   +  εijk
Only blocks
yijk  =  µ   +   βi  +   εijk
 
Blocks and X
yijk  =  µ   +   βi   +  γj +   εijk
  Blocks and Z
yijk  =  µ   +   βi   +  δk   +   εijk
 
Blocks, X and Z, but no interaction
yijk  =  µ   +   βi   +   γj   +  δk   +   εijk
Blocks and treatments
yijk  =  µ   +   βi   +  θjk  +   εijk

Each arrow in this new diagram corresponds to a reduction in the residual sum of squares (i.e. an explained sum of squares) due to adding one term. With equal replicates for all treatments, A and B are orthogonal so the order of adding their main effects does not affect their sums of squares. The diagram therefore shows how the anova table row for the treatment sum of squares and its degree of freedom can be split into three rows:

In this expanded anova table, the rows for blocks, residual and total remain the same.

Rice yield, nitrogen and variety

A randomised block experiment was conducted to assess how the yields of three varieties of rice were affected by different applications of nitrogen fertiliser. The fifteen treatments were the factorial combinations the following levels of rice and nitrogen and each was used once within each block.

Rice varieties   Nitrogen level
"6966"   0 kg/ha
"P1215936"   40 kg/ha
"Milfor 6(2)"    70 kg/ha
    100 kg/ha
    130 kg/ha

The analysis of variance table below initially contains a single row that can be used to test whether there are differences between the 15 treatments.

Click Split treatments to separately show the explained sums of squares for variety, nitrogen and their interaction. From the p-values associated with these sums of squares, we would conclude:

All terms are orthogonal (since there were equal replicates of all treatments in each blocks), so the effects of the two factors can be summarised in the following tables of mean yields.

Rice variety Mean yield
(t/ha)
  Nitrogen level Mean yield
(t/ha)
"6966" 4.769      0 kg/ha 3.483   
"P1215936" 5.042      40 kg/ha 4.761   
"Milfor 6(2)"  5.058      70 kg/ha 5.072   
      100 kg/ha 5.670   
      130 kg/ha 5.797   

It appears that the variety "6966" may have lower yield than the others, and that yield increases steadily with increases to the amount of nitrogen (within the range used in the experiment).