Anova table

The total sum of squares can be split into three component sums of squares.

These sums of squares are usually laid out in an analysis of variance table (or simply anova table).

The anova table adds a few extra columns:

Degrees of freedom
The degrees of freedom for block and treatment are the number of blocks or treatments minus one. The residual degrees of freedom is the total degrees of freedom minus the block and treatment degrees of freedom.
Mean sums of squares
These are the sums of squares divided by their degrees of freedom.
F-ratios
For the blocks and treatments, the F-ratio divides the mean sum of squares by the mean residual sum of squares.

Tests

The treatment sum of squares depends on the differences between the treatment means and the overall mean. There is strongest evidence of differences between the treatments if the treatment sum of squares is large, relative to the residual sum of squares.

The F-ratio for the treatments contains this information in a form that can be used for a formal hypothesis test for whether the mean response varies between treatments.

If there are no differences between the treatments, the F-ratio would be expected to be around 1. Larger values of F suggest that the treatments differ.

The F-ratio for the blocks holds evidence about whether the blocks differ, but it is the F-ratio for treatments that is usually of most interest.

P-values and tests

A hypothesis test for whether there are differences between the treatments is based on the F-ratio for treatments. It asks whether the F-ratio for treatments is unusually high by comparing the F-ratio to a type of standard distribution called an F distribution — the p-value for treatments is the probability of getting such a high F-ratio if all treatments were really identical.

This p-value is often added to the anova table. It is interpreted in a similar way to other p-values:

p-value > 0.1
No evidence of a difference between treatments
0.1 < p-value < 0.05
Very mild evidence of a difference between treatments
0.05 < p-value < 0.01
Moderately strong evidence of a difference between treatments
p-value < 0.01
Strong evidence of a difference between treatments

A similar p-value can be used to test whether there are any differences between the blocks.

Examples

In practice, computer software will produce the anova table for you, so you only need to interpret the p-value associated with the treatments.

The diagram below shows the analysis of variance tables for a few data sets and interprets their p-values. Use the pop-up menu to change data sets.