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:

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.

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

Soybean yield and trace elements

Soybeans are grown in plots with different levels of Mn and Cu applied.

Drag the red arrows in the analysis of variance table to add factor terms for Cu and Mn.

The sum of squares explained by differences between Mn levels is highly significant, but the p-value for differences between the Cu levels is 0.4219. We therefore conclude that: