Nested structure of experimental units
In many experiments, the experimental units are structured in a two-level hierarchy — they are grouped into blocks of similar units.
Sometimes this structure can be extended with three of more levels. For example, an agricultural experiment may use 10 fields with 6 small plots of land selected within each. At the end of the growing season, 5 individual plants are analysed within each plot. The plots are nested within fields and the plants are nested within plots.
Analysis of variance
If the factor is varied at one level, its mean sum of squares should be compared to the sum of squares explained by the blocks (within the factor levels) at that level of the hierarchy to test for the effect of the factor.
Glycogen in rat livers
An experiment was conducted with 3 treatments given to 6 rats — each treatment was applied to two of the rats. The analysis was complicated by the fact that three preparations were taken from the liver of each rat, and two readings of glycogen content were taken from each preparation. The colours in the table below indicate the three treatments that were applied to the rats.
Glycogen | Glycogen | |||||
---|---|---|---|---|---|---|
Rat 1 | Preparation 1 | 131 130 | Rat 4 | Preparation 10 | 151 155 | |
Preparation 2 | 131 125 | Preparation 11 | 147 147 | |||
Preparation 3 | 136 142 | Preparation 12 | 162 152 | |||
Rat 2 | Preparation 4 | 150 148 | Rat 5 | Preparation 13 | 134 125 | |
Preparation 5 | 140 143 | Preparation 14 | 138 138 | |||
Preparation 6 | 160 150 | Preparation 15 | 135 136 | |||
Rat 3 | Preparation 7 | 157 145 | Rat 6 | Preparation 16 | 138 140 | |
Preparation 8 | 154 142 | Preparation 17 | 139 138 | |||
Preparation 9 | 147 153 | Preparation 18 | 134 127 |
The analysis of variance table initially describes the structure of the experimental units.
Since the treatments were varied at rat level — each rat got one treatment — the treatment sum of squares should be compared to the sum of squares between rats (within treatments) in the analysis of variance table.
Click Split rats to expand the sum of squares between rats and separate out the effect of the factor. Note that:
Each sum of squares is compared with the sum of squares at the lower level of the nesting hierarchy using an F ratio.
We therefore conclude that there is no evidence of any difference between the three factors.
We could also conclude that there is moderately strong evidence that the rats differ in their mean glycogen content (p-value = 0.0141), but very little evidence of a difference between the preparations (p-value = 0.0503).
Different experiment (artificial)
In order to illustrate how the structure of the data affects the analysis, consider another experiment in which the treatment can be varied at the level of the preparations instead of at rat level. All three treatments are therefore used within each rat in the table below.
Glycogen | Glycogen | |||||
---|---|---|---|---|---|---|
Rat 1 | Preparation 1 | 131 130 | Rat 4 | Preparation 10 | 151 155 | |
Preparation 2 | 131 125 | Preparation 11 | 147 147 | |||
Preparation 3 | 136 142 | Preparation 12 | 162 152 | |||
Rat 2 | Preparation 4 | 150 148 | Rat 5 | Preparation 13 | 134 125 | |
Preparation 5 | 140 143 | Preparation 14 | 138 138 | |||
Preparation 6 | 160 150 | Preparation 15 | 135 136 | |||
Rat 3 | Preparation 7 | 157 145 | Rat 6 | Preparation 16 | 138 140 | |
Preparation 8 | 154 142 | Preparation 17 | 139 138 | |||
Preparation 9 | 147 153 | Preparation 18 | 134 127 |
The analysis of variance table below is initially the same as that above since it describes the structure of the experimental units and not the treatments applied.
Click Split preparations to expand the sum of squares for the preparations into a sum of squares explained by the treatments and the 'residual' variation between preparations that is unexplained by the treatments.
Note that the treatment sum of squares should now be compared to the sum of squares between preparations (within treatments) in this experiment.