ANOVA
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![Page 1: ANOVA](https://reader035.fdocuments.net/reader035/viewer/2022080223/55cf9143550346f57b8c1424/html5/thumbnails/1.jpg)
ANOVA With More Than One IV
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2-way ANOVA
So far, 1-Way ANOVA, but can have 2 or more IVs. IVs aka Factors.
Example: Study aids for exam IV 1: workbook or not IV 2: 1 cup of coffee or not
Workbook (Factor A)
Caffeine (Factor B)
No Yes
Yes Caffeine only
Both
No Neither (Control)
Workbook only
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Main Effects N=30 per cell
Workbook (Factor A) Row Means
Caffeine
(Factor B)
No Yes
Yes Caff
=80
SD=5
Both
=85
SD=5
82.5
No Control
=75
SD=5
Book
=80
SD=5
77.5
Col Means 77.5 82.5 80
X X
X X
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Main Effects and Interactions
Main effects seen by row and column means; Slopes and breaks.
Interactions seen by lack of parallel lines.
Interactions are a main reason to use multiple IVs
Workbook (Factor A)
86
84
82
80
78
76
74
Mea
n R
M T
est S
core
No Yes
Without Caffeine
With Caffeine
Factor B
![Page 5: ANOVA](https://reader035.fdocuments.net/reader035/viewer/2022080223/55cf9143550346f57b8c1424/html5/thumbnails/5.jpg)
Single Main Effect for B
2.0 1.0
Factor A
25
20
15
10
5
0
Mea
n R
espo
nseSingle Main Effect
B=1
B=2
A
1 2
B1
2
10 10
20 20
(Coffee only)
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Single Main Effect for A
2.0 1.0
Factor A
20
16
12
8
4
0
Mea
n R
espo
nse
Single Main Effect
B=1
B=2
A
1 2
B1
2
10 20
10 20
(Workbook only)
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Two Main Effects; Both A & B
2.0 1.0
Factor A
35
30
25
20
15
10
5
0
Mea
n R
espo
nse
Two Main Effects
B=1
B=2
A
1 2
B1
2
10 20
20 30
Both workbook and coffee
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Interaction (1)
2.0 1.0
Factor A
35
30
25
20
15
10
5
0
Mean R
esponse
Interaction 1
B=1
B=2
A
1 2
B1
2
10 20
10 30
Interactions take many forms; all show lack of parallel lines.
Coffee has no effect without the workbook.
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Interaction (2)
2.0 1.0
Factor A
25
20
15
10
5
0
Mea
n R
espo
nse
Interaction 2
B=1
B=2
A
1 2
B1
2
10 20
20 10
People with workbook do better without coffee; people without workbook do better with coffee.
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Interaction (3)
2.0 1.0
Factor A
40
35
30
25
20
15
10
5
0
Mea
n R
espo
nse
Interaction 3
B = 1
B = 2
Coffee always helps, but it helps more if you use workbook.
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Labeling Factorial Designs
Levels – each IV is referred to by its number of levels, e.g., 2X2, 3X2, 4X3 designs. Two by two factorial ANOVA.
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Example Factorial Design (1)
Effects of fatigue and alcohol consumption on driving performance.
Fatigue Rested (8 hrs sleep then awake 4 hrs) Fatigued (24 hrs no sleep)
Alcohol consumption None (control) 2 beers Blood alcohol .08 %
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Cells of the Design
Alcohol (Factor A)
Fatigue (Factor B)
None 2 beers .08 %
Tired Cell 1 Cell 2 Cell 3
Rested Cell 4 Cell 5:Rested, 2 beers, Porsche 911
Cell 6
DV – closed course driving performance ratings from instructors.
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Factorial Example Results
Intox2 beersnoneAlcohol Consumption
25
20
15
10
5
0
Driv
ing
Err
ors
Factorial Design
Rested
Fatigued
Main Effects?Interactions? Both main effects and the interaction
appear significant.
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ANOVA Summary Table
SourceSource SSSS DfDf MSMS FF
A SSA a-1 SSA/dfA MSA/MSError
B SSB b-1 SSB/dfB MSB/MSError
AxB SSAxB (a-1)(b-1) SSAxB/dfAxB MSAxB/MSError
Error SSError ab(n-1)or
N-ab
SSError/dfError
Total SSTotal N-1
Two Factor, Between Subjects Design
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Review
In a 3 X 3 ANOVA How many IVs are there? How many df does factor A have How many df does the interaction have
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Test
We can see the main effect for a variable if we examine means of the dependent variable while ________
Considering the joint effects of both variables Examining a single value of a second factor Examining each cell Ignoring the other variable
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Test
In two-way ANOVA, the term interaction means
Both IVs have an impact on the DV The effect of one IV depends on the value of the
other IV The on IV has no effect unless the other IV has a
certain value There is a crossover – a graph of two lines
shows an ‘X’.