Explained sums of squares

Adding linear terms for a numerical factor X and another numerical factor Y both decrease the residual sum of squares. These reductions are explained sums of squares and can be shown in an analysis of variance table.

Similarly, generalising from a linear term for a factor to a term that models it as a categorical variable also reduces the residual sum of squares. This reduction is a sum of squares explained by curvature in the relationship.

These explained sums of squares all have degrees of freedom that are equal to the difference in the numbers of unknown parameters in the models.

Analysis of variance table

The analysis of variance table uses these sums of squares and adds columns with:

Mean sums of squares
Each sum of squares is divided by its degrees of freedom
F ratios
Each mean explained sum of squares is then divided by the mean residual sum of squares to form an F-ratio.
p-values
Finally the significance of each F-ratio is assessed by comparing its value to an F distribution to give a p-value.

Interpretation of p-values

The p-values are interpreted as follows:

p-value Interpretation
over 0.1 no evidence that the more complex model is needed
between 0.05 and 0.1 very weak evidence that the more complex model is needed
between 0.01 and 0.05 moderately strong evidence that the more complex model is needed
under 0.01 strong evidence that the more complex model is needed

Drilling thrust

The diagram below shows the analysis of variance table for the copper plate warping experiment. Since the two explanatory variables (copper percentage and temperature) are numerical, each variable can be included in the model as either a linear term (with 1 degree of freedom) or as a factor (with a total of 3 degrees of freedom for each factor).

Drag the red arrows to add linear and then categorical terms to the model. From the p-values, we conclude:

Since there is evidence of a nonlinear effect of drill speed, there is little point in trying to interpret the the p-value associated with the linear term in drill speed.