Hierarchy of models

The two 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
The blocks differ and factor affects Y
yijk  =  µ   +   βi   +   γj   +   εijk

Since we grouped the experimental units into blocks specifically because we believed that the blocks differed, there is little interest in considering a model without a block effect. However it is conventional to extend this hierarchy with a preliminary model with neither block nor factor effect.

No difference between blocks and factor does not affect Y
yijk  =  µ   +  εijk
The blocks differ , but factor does not affect Y
yijk  =  µ   +   βi  +   εijk
The blocks differ and factor affects Y
yijk  =  µ   +   βi   +   γj   +   εijk

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.

Clotting time of plasma

In this experiment, clotting times (mins) were found from plasma that was taken from eight subjects (blocks). Four sample of plasma were obtained from each subject and they were treated in four different ways before the clotting time was measured.

The hierarchy of three possible models is shown on the right. Initially the display alternates between the model with no terms (all fitted values are equal to the overall mean response) and the model allowing differences between the blocks but not the factors. The vertical red lines on the 3-dimensional scatterplot show how the fitted values change.

The explained sum of squares from adding blocks to the model is shown on the bottom right and is both:

  • the reduction in the residual sum of squares due to adding the term
  • the sum of the squared red lines

Clicking the lower arrow on the right flashes the display between the least squares fits of the models with only a term for blocks and one with terms for both blocks and the factor. The vertical red lines again show how the fitted values change due to adding the factor to the model.

The reduction in the residual sum of squares is again the sum of the squared red lines.