Transcript ANOVA with More than 1 IV - University of South Florida

ANOVA With More Than
One IV
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
Main Effects
N=30 per
cell
Workbook (Factor A)
Row
Means
Caffeine
(Factor B)
No
Yes
Yes
Caff
X =80
SD=5
Both
X =85
SD=5
82.5
No
Control
=75
X
SD=5
Book
=80
X
SD=5
77.5
Col Means
77.5
82.5
80
Main Effects and Interactions

Mean RM Test Score
86
84
With C affeine
82
80
Fac tor B

78
Without C affeine
76

74
No
Yes
Work book (Fac tor A)
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
Single Main Effect for B
S ingle Main E ffect
(Coffee only)
M ean Res pons e
25
B=2
20
A
15
B
2
1
1
10
2
20 20
10
10
B=1
5
0
1. 0
2. 0
Fac tor A
Single Main Effect for A
S ingle Main E ffect
M ean Res pons e
20
16
B=2
12
B=1
A
B
1
1 10
2 10
2
20
20
8
(Workbook only)
4
0
1. 0
2. 0
Fac tor A
Two Main Effects; Both A & B
Both workbook and coffee
Two Main E ffects
M ean Res pons e
35
30
B=2
25
A
20
15
B
1
1 10
2 20
2
20
30
B=1
10
5
0
1 .0
2 .0
Fac tor A
Interaction (1)
Interactions take many forms; all show lack of parallel
lines.
Interaction 1
M ean Res pons e
35
30
25
20
B=2
Coffee has no effect
without the workbook.
A
B
1
2
1
10
10
2
20
30
B=1
15
10
5
0
1 .0
2 .0
Fac tor A
Interaction (2)
Interaction 2
M ean Res pons e
25
20
B=2
A
15
B
10
1
1 10
2 20
2
20
10
B=1
People with workbook do better without
coffee; people without workbook do
better with coffee.
5
0
1 .0
2 .0
Fac tor A
Interaction (3)
Coffee always helps, but it helps more if you use
workbook.
Interaction 3
40
M ean Res pons e
35
30
B= 2
25
20
15
B= 1
10
5
0
1. 0
2. 0
Fac tor A
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.
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 %
Cells of the Design
Alcohol (Factor A)
Fatigue
None
(Factor B)
2 beers
.08 %
Cell 3
Tired
Cell 1
Cell 2
Rested
Cell 4
Cell 5:Rested, Cell 6
2 beers,
Porsche 911
DV – closed course driving performance ratings from instructors.
Factorial Example Results
Factorial Design
25
Dri v i ng Errors
20
Fati gued
15
10
Res ted
5
0
none
2 beers
Alc ohol Cons um pti on
Intox
Main Effects?
Interactions? Both main effects and the interaction
appear significant.
ANOVA Summary Table
Two Factor, Between Subjects Design
Source
SS
Df
MS
F
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
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
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
Test
In two-way ANOVA, the term interaction
means
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



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’.