Analysis of variance

Each linear constraint on the model parameters increases the residual sum of squares. This corresponds to an explained sum of squares (explained by going in the opposite direction and releasing the constraint) and has 1 degree of freedom. A set of k such constraints corresponds to an explained sum of squares with k degrees of freedom.

The constraints define intermediate models between the full model (without constraints) and the model for which none of the treatments affect the response. The explained sums of squares corresponding to the constraints therefore effectively partitions of the treatments sum of squares in a standard analysis of variance table.

Effect of fertiliser on the yield of tomato plants

The analysis of variance table below initially shows a single entry for the sum of squares explained by the five treatments (four fertilisers and control).

Click Split treatments to see:

From the p-values associated with these sums of squares, we can conclude that there is only moderate evidence of a difference between the four fertiliser types (p = 0.0272) but it is almost certain that the fertilisers are different from the control group (p = 0.0000).

Order of applying constraints

The explained sums of squares in analysis of variance tables are always ordered — they give the changes to the residual sum of squares when sequentially adding terms to the model (from top to bottom) or when sequentially applying constraints (from bottom to top).

However with many common experimental designs, some of the rows can be exchanged. For example, in a factorial experiment with equal replicates for all factor combinations, the sums of squares explained by the two main effects can be written in either order. The terms are then called orthogonal.

In a similar way, the two sets of constraints described above are orthogonal, so their explained sums of squares can be written in either order.

Drag to reorder the sums of squares explained by the two sets of constraints. Note that the order does not matter since the constraints are orthogonal.