Repeated measures ANOVA - degrees of freedom

#1
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
 
#3

fed2

Active Member
#6
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.