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:
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.