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Total Means  SStot
The measure of an individual's score compared to the grand mean. sum((x11.x...)...(xii.x...)
df=n1

Between Group Means  SSbg
The measure of a group's mean compared to the grand mean. sum((x.1.x..)...(x.i.x...)

Within Group Means  SSwg
The measure of an individual's mean compared to the group's mean. sum((x11x.1)..(xiix.i)
df= nk
MS=SSwg/dfbg


Cohen's D values
 0.2  small
 0.5  medium
 0.8  large

Cohen's f values
 0.1  small
 0.25  medium
 0.4  large

Interaction Types
Disordinal: nonparallel with an intersection
Ordinal: nonparallel with no intersection
No interaction: parallel lines

SSwg for factorial ANOVA
SSwg=SSerror= Sum of the individual deviations from the cell means.
dfwg=dferror=Na*b=(n111)+(n121)+(n211)+(n221)
MSwg=MSerror=SSwg/dfwg

SSa*b
Sum of cell meanrow margin+column margin+grand mean)2 for all columns and rows.
dfa*b=(a1)(b1)
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=SSwgSSbl  how much error gets pulled out from the blocks

Repeated Measures ANOVA  MSerror
SSerror/dferror = SSerror/(N1)(k1)
where N= # of blocks and k=# of time points

Repeated Measures ANOVA  Fstat
F=MSbg/MSerror
df=(k1, (N1)(k1))

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)

orthogonal contrast
Nonindependent contrasts

Type II Error
Falsely rejecting the a true null hypothesis

