# Repeated measures ANOVA - degrees of freedom

#### camila jones

##### New Member
I'm studying repeated measures ANOVAs, found this example online and can't understand how are the df calculated. What would be the mathematical expression for calculating both df values in this example?

gender (2 levels) and treatment (3 levels) are between-subject variables and phase (3 levels) and hour (5 levels) are within-subject variables.

Code:
data(obk.long, package = "afex")

# estimate mixed ANOVA on the full design:
aov_car(value ~ treatment * gender + Error(id/(phase*hour)),
data = obk.long, observed = "gender")

# the three calls return the same ANOVA table:
# Anova Table (Type 3 tests)
#
# Response: value
#                         Effect          df   MSE         F  ges p.value
# 1                    treatment       2, 10 22.81    3.94 + .198    .055
# 2                       gender       1, 10 22.81    3.66 + .115    .085
# 3             treatment:gender       2, 10 22.81      2.86 .179    .104
# 4                        phase 1.60, 15.99  5.02 16.13 *** .151   <.001
# 5              treatment: phase 3.20, 15.99  5.02    4.85 * .097    .013
# 6                 gender: phase 1.60, 15.99  5.02      0.28 .003    .709
# 7       treatment:gender: phase 3.20, 15.99  5.02      0.64 .014    .612
# 8                         hour 1.84, 18.41  3.39 16.69 *** .125   <.001
# 9               treatment:hour 3.68, 18.41  3.39      0.09 .002    .979
# 10                 gender:hour 1.84, 18.41  3.39      0.45 .004    .628
# 11       treatment:gender:hour 3.68, 18.41  3.39      0.62 .011    .641
# 12                  phase:hour 3.60, 35.96  2.67      1.18 .015    .335
# 13        treatment: phase:hour 7.19, 35.96  2.67      0.35 .009    .930
# 14           gender: phase:hour 3.60, 35.96  2.67      0.93 .012    .449
# 15 treatment:gender: phase:hour 7.19, 35.96  2.67      0.74 .019    .646
# ---
# Signif. codes:  0 ‘***’ 0.001 ‘**’ 0.01 ‘*’ 0.05 ‘+’ 0.1 ‘ ’ 1

#### fed2

##### Active Member
according tot he docs it is 'split plot'.https://www.stat.purdue.edu/~zhanghao/STAT514/handout/SplitPlot.pdf

The treatment is (#treatments - 1), gender is 1 (number of biological genders - 1), and interaction is (#treatments - 1) * number of biological genders - 1).

The reason 'why' these are the degrees of freedom have to do with the expected values of various sums of squares.

#### camila jones

##### New Member
Are you able to explain the df2 = 10 explicitly ?

#### fed2

##### Active Member
10 = (16 )- (2 + 2 + 1) - 1 = n - k - 1, like in a two-way non-repeated measures. ie (n-1) - any used up by the treatment, gender and interaction terms.

denominator degrees of freedom in this type of setting is weird and debatable.