## Degrees of Freedom For a Factorial ANOVA

### 2001-04-15

A categorical independent variable is called a **factor**. The
categories are called the **levels** of the factor.

**Replications** are experiment observations made under the
same conditions, that is, under the same combination of factor
levels.

An experimental design is said to be **balanced** if each
combination of factor levels is replicated the same number of times.

For the main effect of a factor, the degrees of freedom is the
number of levels of the factor minus 1. To understand this
intuitively, note that if there are I levels, there are I - 1
comparisons between the levels.

For an interaction between factors, the degrees of freedom is the
product of the degrees of freedom for the corresponding main effects.
To understand this intuitively, note that if there are two factors,
with I levels of the first factor and J levels of the second factor,
there are I - 1 comparisons of the levels of the first factor at each
of the J - 1 comparisons of the levels of the second factor.

For error, think of the error mean square as a pooled
s^{2}. If there were 2 replications at each combination of
the 2 factors, you would have s^{2} on 2-1 degrees of freedom
at each combination. Hence, pooling over all factor combinations,

error df = (2-1)*(number of combinations).

If there were 3 replications

error df = (3-1)*(number of combinations).

Then you should find that all the degrees of freedom add up to the
total which is the total number of observations minus 1.

*Consider a balanced 2-factor design.* Suppose there
are I levels of the first factor and J levels of the second factor,
and K replications at each of the IJ combinations of the two factors.
Then there are IJK observations and

(IJK - 1) = (I - 1) + (J - 1) + (I -1)(J - 1) + IJ(K -
1)

is the breakdown of the total degrees of freedom into: main effect
of the first factor, main effect of the second factor, interaction
between the two factors, and error.

For example, suppose you have Factor A at 4 levels, Factor B at 3
levels, and 3 replications of every combination of Factor A and
Factor B. Then I = 4, J = 3, K = 3 and there are 36 observations. The
SV and DF columns of the ANOVA will be:

SV DF
Factor A 3
Factor B 2
Interaction AxB 6
Error 24
Total 35