Hierarchy of models
The most important special cases of the model are shown in the diagram below.
The blocks differ, but factor does not affect Y yijk = µ + βi + εijk |
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The blocks differ and factor affects Y yijk = µ + βi + γj + εijk |
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The factor affects Y in different ways within different blocks yijk = µ + βi + γj + δij + εijk |
There is little interest in a model without a block effect but the model with neither block nor factor terms is conventionally added at the top of this hierarchy.
Explained sums of squares
Each addition of a term to the model allows it to reduce the residual sum of squares (when the model is fitted by least squares). As in the 2-factor model, these reductions are called the sum of squares explained by blocks and explained by the factor.
Each explained sums of squares can again be interpreted as the sum of squared differences between the fitted values from the two models.
Randomised block example
The diagram below shows data from a randomised block experiment.
The arrows on the right represent the addition of the block effect, main factor effect and block-factor interaction to the model. When any arrow is clicked, the red lines on the 3-dimensional scatterplot show the changes in the fitted values that result from adding the term. The coloured grid alternates between the least squares fits of the two models.
The explained sum of squares shown on the bottom right can be interpreted as either:
Click the different arrows on the right to display the three explained sums of squares.