Sum of cell mean-row margin+column margin+grand mean)2 for all columns and rows.
dfa*b=(a-1)(b-1)
MSa*b=SSa*b/dfa*b
Mathematical Model for factorial ANOVA
yijk= μ +α j + β k +φ jk + eijk
μ = overall average in the population j
α= effect of being in jth achievement levelj
μ +α = population average for the jth achievement levelk
β= effect of being in kth CAI methodk μ + β = population mean for kth CAI methodjk
φ = effect of having the jkth combination of achievement level and CAI methodj k jk
μ +α + β +φ = population mean for the jkth combination ofachievement level and CAI method ijk
e = individual error
Key to the factorial ANOVA equation
yijk= μ +α j + β k +φ jk + eijk
α=SSa
β= SSb
μ +α + β +φ = SSa*b
Full Regression SS
Type III or unique SS
Adjust each effect for all other effects in the design to obtain its unique contribution (nothing is being counted twice)
Experimental SS
Type II estimates main effects adjusting for the other main effects, but ignoring the interaction. Estimates the interaction adjusting for main effects. SS for A and B are too big. A and B are not unique, but A*B is.
Hierarchical SS
Type I
Uses theory or previous research to establish order for the effects, Adjusts each effect only for those preceding it in order.
Repeated Measures ANOVA - SSbg
Sum of the mean for time group and the grand mean to the jth group
Repeated Measures ANOVA - SSwg
Block score minus average score at each time point
Repeated Measures ANOVA - SSbl
Row block minus grand mean
Repeated Measures ANOVA - SSerror
SSerror=SSwg-SSbl - how much error gets pulled out from the blocks
Repeated Measures ANOVA - MSerror
SSerror/dferror = SSerror/(N-1)(k-1)
where N= # of blocks and k=# of time points
Repeated Measures ANOVA - F-stat
F=MSbg/MSerror
df=(k-1, (N-1)(k-1))
Repeated Measures ANOVA - Sphericity
E is less than or equal to 1. The closer E is to 1, the smaller the violation of the sphericity assumption.
If close to or equal to one, do not adjust F values
Nominal alpha
Level of significance - type 1 error rate (If assumptions are not violated)
Actual alpha
Percentage of making a type 1 error when at least 1 assumption has been violated. Test statistic is robust if actual alpha is very close to nominal alpha.
Experimental alpha
The alpha level set for the whole study (usually 0.05)