Hierarchy of experimental units with three levels

In split plot experiments, the experimental units are often structured in a hierarchy with two levels (called blocks and units, or full plots and sub-plots), but the ideas can be extended to hierarchies with three or more levels. We will consider hierarchies with three levels of units here.

If no factors are allocated at the top level of the design, it is often called a split-plot design with blocking — each block contains two or more full plots and each of these contains sub-plots. In the next page, we will consider experiments in which a factor is allocated at the top level of the hierarchy when the design is usually called a split-split-plot designs (with full plots split into sub-plots and these split into sub-sub-plots).

Oats and manure

An experiment about the yield of oats was conducted using six blocks of land, each of which was split into three plots with one plot in each block used for each of varieties A1, A2 and A3 of oats. Each plot was divided into four subplots with one randomly given each of four levels of manure. The data are shown below.

    Manure applied     Manure applied
Block Variety   0.00     0.01     0.02     0.03     Block   0.00     0.01     0.02     0.03  
I A1
A2
A3
111
117
105
130
114
140
157
161
118
174
141
156
  I  74
 64
 70
 89
103
 89
 81
132
104
122
133
117
III A1
A2
A3
 61
 70
 96
 91
108
124
 97
126
121
100
149
144
  III  62
 80
 63
 90
 82
 70
100
 94
109
116
126
 99
IV A1
A2
A3
 68
 60
 89
 64
102
129
112
 89
132
 86
 96
124
  IV  53
 89
 97
 74
 82
 99
118
 86
119
113
104
121

The experimental units are structured in a 3-level hierarchy with blocks, plots nested within blocks, and sub-plots nested within the plots.

No factors are allocated at block level, but varieties are allocated at plot level and manure is applied at sub-plot level.

The anova table above initially describes the hierarchical structure of the experimental units (without taking account of variation explained by the two factors).

Click Split plots to separate the sum of squares explained by the three varieties from the whole-plot sum of squares. The significance of the varieties should be compared to the whole-plot sum of squares (within varieties) and we would conclude that there is moderately strong evidence that the varieties differ (p-value = 0.0124).

Click Split sub-plots to separate the sum of squares explained by the manure levels and the variety-manure interaction from the sub-plot sum of squares. From the p-value associated with the interaction (p-value = 0.9322) we can conclude that there is no evidence of interaction between the variety and manure level. We can also conclude that the manure almost certainly affects yield (p-value = 0.0000).