Factorial ANOVA - California State University, Northridge

13 Pages·2008·0.11 MB·English

1/21/2008
Factorial ANOVA
Cal State Northridge
Ψ320
Andrew Ainsworth PhD
Topics in Factorial Designs
What is Factorial?
Assumptions
Analysis
Multiple Comparisons
–Main Effects
–Simple Effects
–Simple Comparisons
Effect Size estimates
Higher Order Analyses
Psy 320 -Cal State Northridge 2
Factorial?
Factorial – means that:
1. You have at least 2 IVs
2. And all levels of one variable occur in
combination with all levels of the other
variable(s).
Assumptions
– Same as one-way ANOVA but they
are tested within each cell
– i.e. Normality, Homogeneity and
Independence
Psy 320 -Cal State Northridge 3
1
1/21/2008
Simplest Form: 2 x 2 ANOVA
B
b b
1 2
a
1
A
a
2
Video Game
GTA NBA 2K7
Men
Gender
Women
Psy 320 -Cal State Northridge 4
Analysis
Performing a factorial analysis does the job of
three analyses in one
–Two one-way ANOVAs, one for each IV (called a
main effect)
–And a test of the interaction between the IVs
–Interaction? –the effect of one IV depends on the
level of another IV
•The variability that is left over after you assess each IV
•The 2 IVs together work to affect scores over and above
either of them independently
Psy320 -Cal State Northridge 5
Analysis
The between groups sums of squares
from 1-way ANOVA is further broken
down:
–Before SS = SS
bg effect
–Now SS = SS + SS + SS
bg A B AB
–In a two IV factorial design A, B and AxB
all differentiate between groups,
therefore they all add to the SS
bg
Psy 320 -Cal State Northridge 6
2
1/21/2008
Analysis
Total variability = (variability of A around
GM) + (variability of B around GM) +
(variability of each group mean {AB} around
GM) + (variability of each person’s score
around their group mean)
SS = SS + SS + SS + SS
Total A B AB error
∑(Y −Y )2=∑n (Y −Y )2+∑n (Y −Y )2
i GM a a GM b b GM
+∑nab(Yab−YGM)2−∑na(Ya−YGM)2−∑nb(Yb−YGM)2
+∑(Y −Y )2
i ab
Psy 320 -Cal State Northridge 7
Analysis
Degrees of Freedom
–df = #groups –1
A A
–df = #groups –1
B B
–df = (a –1)(b –1)
AB
–df = ab(n –1) = abn–ab= N –ab
error
–df = N –1 = a –1 + b –1 + (a –1)(b –1)
total
+ N –ab
Psy 320 -Cal State Northridge 8
Analysis
Breakdown of Breakdown of
sums of squares degrees of freedom
Psy 320 -Cal State Northridge 9
3
1/21/2008
Analysis
Mean square
–The mean squares are calculated the same
–SS/df = MS
–You just have more of them, MS , MS ,
A B
MS , and MS
AB WG
–This expands when you have more IVs
•One for each main effect, one for each
interaction (two-way, three-way, etc.)
Psy 320 -Cal State Northridge 10
Analysis
F-test
–Each effect and interaction is a separate
F-test
–Calculated the same way: MS /MS
effect WG
since MS is our error variance
WG
estimate
–You look up a separate F for each test
crit
using the df , df and tabled values
effect WG
Psy 320 -Cal State Northridge 11
Example
B: Vacation Length
A: Profession b1: 1 week b2: 2 weeks b3: 3 weeks
0 4 5
a1: Administrators 1 7 8
0 6 6
5 5 9
a2: Belly Dancers 7 6 8
6 7 8
5 9 3
a3: Politicians 6 9 3
8 9 2
∑ 2= 2+ 2+ + 2=
Psy 320 -Cal State Northridge 12
4
1/21/2008
Analysis
Sample data reconfigured into cell and
marginal means (with variances)
B: Vacation Length
A:Profession b1: 1 week b2: 2 weeks b3: 3 weeks Marginal A means
a1: Administrators Ysaa211bb11== 00..333333 Ysaa211bb22== 25..363637 Ysaa211bb33== 26..333333 Ya1= 4.111
a2: Belly Dancers Ysaa222bb11== 16 Ysaa222bb22== 16 Ysaa222bb33== 80..333333 Ya2= 6.778
a3: Politicians Ysaa233bb11== 26..333333 Ysaa233bb22== 90 Ysaa233bb33== 20..636373 Ya3= 6
Marginal B Means Yb1= 4.222 Yb2= 6.889 Yb3= 5.778 Y...= 5.630
∑Y2=1046
Psy 320 -Cal State Northridge 13
Example – Sums of Squares
SS =∑(Y −Y )2 =
total i GM
=(____−____)2+(____−____)2+(____−____)2+
(5−5.630)2+(7−5.630)2+(6−5.630)2+
(5−5.630)2+(6−5.630)2+(8−5.630)2+
(4−5.630)2+(7−5.630)2+(6−5.630)2+
+K+(3−5.630)2+(2−5.630)2 =190.296
Psy 320 -Cal State Northridge 14
Example – Sums of Squares
SS =∑n (Y −Y )2 =
A a a GM
=[___*(____−____)2]+[___*(____−____)2]+
[___*(6−5.630)2]=33.852
Psy 320 -Cal State Northridge 15
5
1/21/2008
Example – Sums of Squares
SS =∑n (Y −Y )2 =
B b b GM
=[___*(____−____)2]+[___*(____−____)2]+
[___*(5.778−5.630)2]=32.296
Psy 320 -Cal State Northridge 16
Example – Sums of Squares
SS =∑n (Y −Y )2−∑n (Y −Y )2−∑n (Y −Y )2=
AB ab ab GM a a GM b b GM
=[___*(____−____)2]+[___*(____−____)2]+
+[___*(____−____)2]+[___*(____−____)2]+
+[___*(____−____)2]+[___*(9−5.630)2]+
[___*(6.333−5.630)2]+[___*(8.333−5.630)2]+
+[___*(2.667−5.630)2]=170.296
=170.296−33.825−32.296=104.148
Psy 320 -Cal State Northridge 17
Example – Sums of Squares
SS =∑(Y −Y )2 =
Error i ab
=(____−____)2+(____−____)2+(____−____)2+
(____−____)2+(____−____)2+(____−____)2+
(5−6.333)2+(6−6.333)2+(8−6.333)2+
(4−5.667)2+(7−5.667)2+(6−5.667)2+
+K+(3−2.667)2+(2−2.667)2 =20
Psy 320 -Cal State Northridge 18
6
1/21/2008
Analysis – Computational
Marginal Totals – we look in the
margins of a data set when
computing main effects
Cell totals – we look at the cell totals
when computing interactions
In order to use the computational
formulas we need to compute both
marginal and cell totals
Psy 320 -Cal State Northridge 19
Analysis – Computational
Sample data reconfigured into cell
and marginal totals
B: Vacation Length
A: Profession b1: 1 week b2: 2 weeks b3: 3 weeks Marginal Sums for A
a1: Administrators 1 17 19 a1 = 37
a2: Belly Dancers 18 18 25 a2 = 61
a3: Politicians 19 27 8 a3 = 54
Marginal Sums for B b1 = 38 b2 = 62 b3 = 52 T = 152

Psy 320 -Cal State Northridge 20
Analysis – Computational
Formulas for SS
∑(∑a)2 T2
SS = −
A bn abn
∑(∑b)2 T2
SS = −
B an abn
∑(∑ab)2 ∑(∑a)2 ∑(∑b)2 T2
SS = − − +
AB n bn an abn
∑(∑ab)2
SS =∑Y2−
error n
SS =∑Y2−T2
T abn 21
7
1/21/2008
Analysis – Computational
Example
∑(∑a)2 T2
SS = −
A bn abn
___2+___2+542 ___2
SS = − =____−____=33.85
A 3(3) 3(3)(3)
∑(∑b)2 T2
SS = − =
B an abn
382+___2+___2 ___2
SS = − =888−855.7=32.30
B 3(3) 3(3)(3)
Psy 320 -Cal State Northridge 22
Analysis – Computational
Example
∑(∑ab)2 ∑(∑a)2 ∑(∑b)2 T2
SS = − − +
AB n bn an abn
___2+___2+___2+182+182+252+192+272+82
SS =
AB 3
372+612+542 382+622+522 1522
− − +
3(3) 3(3) 3(3)(3)
=____−889.55−888+855.7=104.15
Psy 320 -Cal State Northridge 23
Analysis – Computational
Example
∑(∑ab)2
SS =∑Y2−
error n
12+172+192+182+182+252+192+272+82
SS =_____−
error 3
=____−1026=20
SS =∑Y2− T2
T abn
1522
SS =1046− =1046−855.7=190.30
T 3(3)(3)
Psy 320 -Cal State Northridge 24
8
1/21/2008
Analysis – Computational
Example
df =a−1=3−1=2
A
df =b−1=3−1=2
B
df =(a−1)(b−1)=(3−1)(3−1)=2(2)=4
AB
df =abn−ab=27−9=18
Error
df =abn−1=27−1=26
total
Psy 320 -Cal State Northridge 25
Analysis
Example
Tests of Between-Subjects Effects
Dependent Variable: ENJOY
Type III Sum
Source of Squares df Mean Square F Sig.
PROFESSION 33.852 2 16.926 15.233 .000
LENGTH OF STAY 32.296 2 16.148 14.533 .000
PROFESSION * LENGTH 104.148 4 26.037 23.433 .000
WITHIN GROUPS 20.000 18 1.111
TOTAL 190.296 26
The MS is also the pooled (average)
WG
variance across the cells, since all n are equal:
(.333+2.333+2.333+1+1+.333+2.333+0+.333)/9 = 1.111
Psy 320 -Cal State Northridge 26
Analysis
F (2,18)=3.55
crit
F (4,18)=2.93
crit
Since 15.25 > 3.55, the effect for
profession is significant
Since 14.55 > 3.55, the effect for
length is significant
Since 23.46 > 2.93, the effect for
profession * length is significant
Psy 320 -Cal State Northridge 27
9
1/21/2008
Effect Size Revisited
Eta Squared is calculated for each
effect
SS
η2 = effect
effect SS
total
Omega Squared also for each effect
SS −(k −1)MS
ω2 = Effect Effect WG
Effect SS +MS
T WG
Psy 320 -Cal State Northridge 28
Effect Size Example
Effect Size for Profession
SS 33.852
η2 = Profession = =.178
Profession SS 190.296
total
SS −(k −1)MS
ω2 = Profession Profession WG =
Profession SS +MS
T WG
33.853−[(3−1)*1.111]
ω2 = =.165
Profession 190.296+1.111
Psy 320 -Cal State Northridge 29
Multiple Comparisons
If a main effect is significant and has more
than 2 levels, than you need to do marginal
comparisons
If the interaction is significant
–You should break the interaction down by
performing a simple effect analysisof A at
each level of B (The effect of A at B , A at B , A
1 2
at B , etc.) and vice versa
3
–If any of them are significant and if A has more
than 2 levels, follow up with simple comparisons
Psy 320 -Cal State Northridge 30
10