ANOVA with Two Within-Subject Variables (2
of 3)
Degrees of Freedom
The degrees of freedom for Subjects is equal to the number
subjects minus one. In this example, this is N - 1 = 4 -1 =
3 since there are 4 subjects.
The degrees of freedom for each
main effect is equal to the number of levels of the
variable in question minus one. Therefore, since there are
two days, the effect of Days has 2 - 1 = 1 degrees of
freedom. There are three trials, so the effect of Trials has
3 - 1 = 2 degrees of freedom.
The degrees of freedom for the interaction
of the two variables is equal to the product of the degrees of freedom
for the main effects of these variables. Since Days has 1 df
and Trials has 2 df, the Days x Trials interaction has 1 x 2
= 2 df. The formulas are summarized below:
df Subjects = N - 1 = 4 - 1 = 3
df Days = d - 1 = 2 - 1 = 1
df error(Days) = (N -1) (d-1) = 3
df Trials = t - 1 = 3 - 1 = 2
df error(Trials)= (N -1) (t-1) = 6
df Days x Trials= (d-1) (t-1) = 2
df error (Days x Trials) = (N-1) (d-1) (t-1) = 6
df total = dtN - 1 = (2)(3)(4) - 1 = 23
d is the number of days; t is the number of trials (each
day); and N is the total number of subjects.