Analysis of variance table
Each explained sum of squares in the sum of squares table describes the reduction in residual sum of squares when a term is added to the model. Since small explained sums of squares result from adding terms that have little effect the fit of the model, they are used as the basis for testing whether the term is needed.
The sum of squares table is augmented with extra columns:
The resulting table is called an analysis of variance (anova) table.
Source of variation |
Sum of sqrs |
d.f. | Mean ssq | F-ratio | p-value |
---|---|---|---|---|---|
X | SSX | (gX - 1) | MSX | MSX / MSResid | (F distn) |
Z | SSZ | (gZ - 1) | MSZ | MSZ / MSResid | (F distn) |
Residual | SSResid | (n - gX - gZ - 1) | MSResid | ||
Total | SSTotal | n - 1 |
(Note that we are restricting attention in this section to experiments with the same number of replicates for each combination of factor levels and hence orthogonal factors. We therefore do not need to distinguish between the two orders of adding the factors X and Z. The rows for the two factors could be swapped and their p-values would be the same.)
Interpretation of p-values
Each p-value is used to test the null hypothesis that the corresponding model term is unnecessary. It is interpreted in a similar way to other hypothesis tests.
p-value | Interpretation |
---|---|
over 0.1 | no evidence that the term is needed |
between 0.05 and 0.1 | very weak evidence that the term is needed |
between 0.01 and 0.05 | moderately strong evidence that the term is needed |
under 0.01 | strong evidence that the term is needed |
Strength of asphaltic concrete
A civil engineer conducted an experiment to evaluate how different compaction methods and types of aggregate affect the strength of asphaltic concrete. Four compaction methods and two aggregate types were used.
Drag the red arrows in the analysis of variance table to add factor terms for Aggregate and Compaction.
The sum of squares explained by both factors are very highly significant. We can therefore conclude that:
Both the aggregate type and type of compaction of the concrete affect its strength.