Residual and explained sums of squares
Each additional main effect or interaction term that is added to the model gives extra flexibility, allowing the residual sum of squares to be reduced. These reductions are the sums of squares that are explained by the terms and they are presented in a sum of squares table with their degrees of freedom.
In a factorial experiment with equal replicates for all factor combinations, the factors and their interactions are orthogonal (uncorrelated) so the order of adding terms in the sum of squares table does not affect their sums of squares.
Analysis of variance (anova) table
A column of mean sums of squares (sums of squares divided by their degrees of freedom) is added to the sum of squares table. The mean explained sums of squares are compared to the mean residual sum of squares with F ratios.
Large explained sums of squares correspond to significant terms.
A p-value for each term shows whether its F ratio is larger than could be expected by chance.
The p-values are used to decide which terms are unimportant and can be dropped from the full model.
Note however that we should only consider hierarchical models, so terms must not be dropped if higher-order interactions involving them are still in the model.
Soft drink bottling
The diagram below initially shows the best-fitting model with all interactions for the soft drink bottling data.
The 3-factor interaction term has a high p-value and is not significant so it can be dropped from the model. (Click the checkbox to the left of the term to delete it.)
There is similarly no evidence of interactions between Line speed and the other two factors, so drop them from the full model too. This leaves a model with all main effects (all factors affect mean Fill height deviation) and an interaction between Pressure and Carbonation.
Click the y-z rotation button. The purple lines show the mean Fill height deviation for low Line speed — examine them to help understand the nature of the interaction between Pressure and Carbonation. (The green lines for high Line speed have the same pattern.)
Increasing pressure has less effect on the mean response at 10% carbonation than at higher carbonation levels.
Wear of coated fabrics
An experiment was conducted to assess the durability of coated fabric subjected to standard abrasive tests. A factorial experiment was conducted with two different fillers (F1 and F2) in three different proportions (25%, 50% and 75%), either with or without surface treatment of the fabric. Two replicate fabric specimens were tested for each of the 12 treatment combinations in a completely randomised design. The response is the weight loss (mg) of the fabric specimens in the abrasion test.
For this data set, neither the 3-factor interaction nor the 2-factor interaction between Surface treatment and Percentage filler are significant. Use the checkboxes to delete these two terms from the model.
This model has two 2-factor interaction terms so it is a little harder to understand than the best model for the soft drink data. Click the y-z rotation button to help understand the nature of this model.
Changing from filler F1 to F2
decreases abrasion loss if there is no surface treatment, but makes
little difference if the fabric is surface treated.
Percentage filler has little effect on abrasion loss for filler F2
but more effect for filler F1.