General model
The general aim in experimental design and analysis is reduction of the unexplained variability in order to estimate more accurately the effect of the treatments of interest.
This is done by adding terms to the model to cover anything that is known about the structure of the experimental units. This structure may be built into the design of the experiment, such as the blocks in a randomised block experiment or may be information that is not used in the experimental design but is collected during the course of the experiment.
explained variation |
unexplained variation |
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yi = µ |
+ |
treatments <terms> |
+ |
blocks & covariates <terms> |
+ |
εi |
Variation in the response due to the experimental units may be modelled as the sum of terms, each based on measured characteristics of the units.
Terms based on numerical covariates
The earlier pages in this section showed how each numerical measurements about the experimental units (covariates) can be modelled by a linear term of the form:
... + γ xi + ...
Terms based on blocks or categorical covariates
Both blocks and categorical covariates are modelled in the same way. Experimental units in block j or for which the covariate has its j'th value are modelled by a term of the form:
... + δj + ...
Models may have several terms of these two forms to explain the structure in the experimental units.
Soybean yield and canker
A varietal trial was conducted on soybean to assess yield. Four varieties were tested, laid out as a randomised block design. Farmers traditionally used Variety 1, while 2, 3 and 4 were varieties developed under a new breeding programme. There was canker infestation throughout the experimental area, but it varied dramatically from plant to plant. It was believed that the varieties did not differ in their susceptibility to canker.
Yields were recorded in bushels/acre and canker infestation was classified as bad, moderate or mild.
Block | Traditional | New 1 | New 2 | New 3 | ||||||||
---|---|---|---|---|---|---|---|---|---|---|---|---|
Canker | Yield | Canker | Yield | Canker | Yield | Canker | Yield | |||||
1 2 3 4 |
severe severe mild mild |
20.3 17.1 26.7 25.3 |
moderate severe moderate mild |
28.3 20.7 26.0 34.1 |
mild severe mild mild |
26.7 14.7 25.0 29.0 |
severe severe moderate moderate |
25.1 20.1 24.9 19.8 |
In this experiment, canker infestation should be treated as a covariate — its effect on yield is not of primary interest but is expected to explain much of its variation.
The covariate Canker is not orthogonal to Blocks — blocks 1 and 2 have worse infestation than blocks 3 and 4 — so their explained sums of squares depend on the order of adding them to the model. This does not matter since we will only consider models that have both Canker and Blocks.
Drag the red line to add Varieties to the anova table. From the p-value associated with Varieties, we conclude that:
There is moderately strong evidence of differences between the average yields of the different varieties.
We can now consider splitting the explained sum of squares for Varieties to ask whether the new varieties differ from each other and from the traditional variety.
Again drag the red line to add:
From the p-values, we can conclude that: