Two-Way Independent ANOVA
(GLM 3)
Slide 1
Aims
• Rationale of Factorial ANOVA
• Partitioning Variance
• Interaction Effects
–Interaction Graphs
–Interpretation
Slide 2
What is Two-Way
Independent ANOVA?
• Two Independent Variables
– Two-way = 2 Independent variables
– Three-way = 3 Independent variables
• Different participants in all conditions.
– Independent = ‘different participants’
• Several Independent Variables is known
as a factorial design.
Slide 3
Benefit of Factorial Designs
• We can look at how variables Interact.
• Interactions
– Show how the effects of one IV might depend on
the effects of another
– Are often more interesting than main effects.
• Examples
– Interaction between hangover and lecture topic on
sleeping during lectures.
• A hangover might have more effect on sleepiness during a
stats lecture than during a clinical one.
Slide 4
An Example
• Field (2009): Testing the effects of Alcohol and
Gender on ‘the beer-goggles effect’:
– IV 1 (Alcohol): None, 2 pints, 4 pints
– IV 2 (Gender): Male, Female
• Dependent Variable (DV) was an objective
measure of the attractiveness of the partner
selected at the end of the evening.
Slide 5
Alcohol
Gender
Slide 6
None
2 Pints
4 Pints
Female
Male
Female
Male
Female
Male
65
50
70
45
55
30
50
55
65
60
65
30
70
80
60
85
70
30
45
65
70
65
55
55
55
70
65
70
55
35
30
75
60
70
60
20
70
75
60
80
50
45
55
65
50
60
50
40
Total
485
535
500
535
460
285
Mean
60.625
66.875
62.50
66.875
57.50
35.625
Variance
24.55
106.70
42.86
156.70
50.00
117.41
SST (8967)
Variance between all scores
SSM
SSR
Variance explained by the
experimental manipulations
SSA
Effect of
Alcohol
SSB
Effect of
Gender
Error
Variance
SSA  B
Effect of
Interaction
Step 1: Calculate SST
65
50
70
45
55
30
50
55
65
60
65
30
70
80
60
85
70
30
45
65
70
65
55
55
55
70
65
70
55
35
30
75
60
70
60
20
70
75
60
80
50
45
55
65
50
60
50
40
Grand Mean = 58.33
2
SS T  sgrand
(N  1)
 190 .78 (48  1)
 8966 .66
Slide 8
Step 2: Calculate SSM
SSM 


ni xi  xgrand

2
SSM  8(60.625  58 .33)2  8(66.875  58 .33)2  8(62.5  58 .33)2
 8(66.875  58 .33)2  8(57.5  58 .33)2  8(35.625  58 .33)2
 8(2.295 )2  8 (8.545 )2  8(4.17)2  8 (8.545 )2  8(0.83)2  8 (22.705) 2
 42 .1362  584 .1362  139 .1112  584 .1362  5.5112  4124 .1362
 5479 .167
Slide 9
Step 2a: Calculate SSA
A1: Female
A2: Male
65
70
55
50
45
30
70
65
65
55
60
30
60
60
70
80
85
30
60
70
55
65
65
55
60
65
55
70
70
35
55
60
60
75
70
20
60
60
50
75
80
45
55
50
50
65
60
40
Mean Female = 60.21
Mean Male = 56.46


 24 (1.88)

2
2 (56.46  58 .33)2
SSGender

24
(60.21

58
.
33
)

24
SS 
n x x
A
i
2
i
grand
 24 (1.87) 2
 84 .8256  83 .9256
 168 .75
Slide 10
Step 2b: Calculate SSB
B1: None
B2: 2 Pints
B3: 4 Pints
65
50
70
45
55
30
70
55
65
60
65
30
60
80
60
85
70
30
60
65
70
65
55
55
60
70
65
70
55
35
55
75
60
70
60
20
60
75
60
80
50
45
55
65
50
60
50
40
Mean None = 63.75
Mean 2 Pints =
64.6875
 n x
Mean 4 Pints =
46.5625

2 (46.5625  58 .33)2
SSalc ohol  16(63.75  58 .33)2  16 (64.6875  58 .33)2  16
SSB 
 xgrand
i
i
 16 (5.42)  16 (6.3575)  16 (11 .7675 )2
2
2
 470 .0224  646 .6849  2215 .5849
 3332 .292
Slide 11
Step 2c: Calculate SSA  B
SSA B  SS M  SS A  SS B
SS A B  SS M  SS A  SS B
 5479 .167  168 .75  3332 .292
 1978 .125
Slide 12
Step 3: Calculate SSR
2
2
2
2
SSR  sgroup1
(n1  1)  sgroup2
(n2  1)  sgroup3
(n3  1)  sgroup
n (nn  1)
2
2
2
SSR  sgroup1
(n1  1)  sgroup2
(n2  1)  sgroup3
(n3  1)
2
2
2
 sgroup4
(n4  1)  sgroup5
(n5  1)  sgroup6
(n6  1)
 (24.55  7)  (106.7  7)  (42.86  7)
 (156.7  7)  (50  7)  (117.41  7)
 171 .85  746 .9  300  1096 .9  350  821 .87
 3487 .52
Slide 13
Summary Table
Slide 14
Interpretation: Main Effect Alcohol
There was a significant main effect of the amount of alcohol consumed at
the night-club, on the attractiveness of the mate that was selected, F(2, 42)
= 20.07, p < .001.
Slide 15
Interpretation: Main Effect
Gender
There was a nonsignificant main effect of gender on the attractiveness of
selected mates, F(1, 42) = 2.03, p = .161.
Slide 16
Interpretation: Interaction
There was a significant interaction between the amount of alcohol
consumed and the gender of the person selecting a mate, on the
attractiveness of the partner selected, F(2, 42) = 11.91, p < .001.
Slide 17
What is an Interaction?
Slide 18
Is there likely to be a significant
interaction effect?
Slide 19
Is there likely to be a significant
interaction effect?
Slide 20
Repeated-measures designs
(GLM 4)
Aims
• Rationale of Repeated Measures ANOVA
– One- and two-way
– Benefits
• Partitioning Variance
• Statistical Problems with Repeated
Measures Designs
– Sphericity
– Overcoming these problems
• Interpretation
Slide 22
Benefits of Repeated Measures
Designs
• Sensitivity
– Unsystematic variance is reduced.
– More sensitive to experimental effects.
• Economy
– Less participants are needed.
– But, be careful of fatigue.
Slide 23
An Example
• Are certain Bushtucker foods more revolting
than others?
• Four Foods tasted by 8 celebrities:
–
–
–
–
Stick Insect
Kangaroo Testicle
Fish Eyeball
Witchetty Grub
• Outcome:
– Time to retch (seconds).
Slide 24
The Data
Celebrity
Stick
Insect
Testicle
Fish Eye
Witchetty
Grub
Mean
Variance
Df
1
8
7
1
6
5.50
9.67
3
2
9
5
2
5
5.25
8.25
3
3
6
2
3
8
4.75
7.58
3
4
5
3
1
9
4.50
11.67
3
5
8
4
5
8
6.25
4.25
3
6
7
5
6
7
6.25
0.92
3
7
10
2
7
2
5.25
15.58
3
8
12
6
8
1
6.75
20.92
3
Mean
8.13
4.25
4.13
5.75
Grand Mean = 5.56
24
SST
Variance between all scores
SSW
SSBetween
Variance Within Individuals
SSM
Effect of Experiment
Slide 26
SSR
Error
Problems with Analyzing
Repeated Measures Designs
• Same participants in all conditions.
– Scores across conditions correlate.
– Violates assumption of independence
(lecture 2).
• Assumption of Sphericity.
– Crudely put: the correlation across
conditions should be the same.
– Adjust Degrees of Freedom.
Slide 27
The Assumption of Sphericity
• Basically means that the correlation
between treatment levels is the same.
• Actually, it assumes that variances in the
differences between conditions is equal.
• Measured using Mauchly’s test.
– P < .05, Sphericity is Violated.
– P > .05, Sphericity is met.
Slide 28
What is Sphericity?
Slide 29
Testicle Stick
Eye –
Stick
Witchetty
– Stick
Eye –
Testicle
Witchetty Witchetty
– Testicle
– Eye
1
-1
-7
-2
-6
-1
5
2
-4
-7
-4
-3
0
3
3
-4
-3
2
1
6
5
4
-2
-4
4
-2
6
8
5
-4
-3
0
1
4
3
6
-2
-1
0
1
2
1
7
-8
-3
-8
5
0
-5
8
-6
-4
-11
2
-5
-7
Variance
5.27
4.29
25.70
11.55
14.29
26.55
Estimates of Sphericity
• Three measures:
– Greenhouse-Geisser Estimate
~
– Huynh-Feldt Estimate

– Lower-bound Estimate
ˆ
• Multiply df by these estimates to correct for
the effect of Sphericity.
• G-G is conservative, and H-F liberal.
Slide 30
Correcting for Sphericity
Df = 3, 21
Mauchly's Test of Sphericity
Measure: MEASURE_1
Epsilon
Within Subjects Effect
Animal
Mauchly's W
.136
Approx.
Chi-Square
11.406
df
5
Sig.
.047
Greenhous e
-Geiss er
.533
Huynh-Feldt
.666
Lower-bound
.333
Tests the null hypothesis that the error covariance matrix of the orthonormalized trans formed dependent variables is
proportional to an identity matrix.
Slide 31
Output
Tests of Within-Subjects Effects
Measure: MEASURE_1
Source
Animal
Error(Animal)
Slide 32
Sphericity Assumed
Greenhous e-Geisser
Huynh-Feldt
Lower-bound
Sphericity Assumed
Greenhous e-Geisser
Huynh-Feldt
Lower-bound
Type III Sum
of Squares
83.125
83.125
83.125
83.125
153.375
153.375
153.375
153.375
df
3
1.599
1.997
1.000
21
11.190
13.981
7.000
Mean Square
27.708
52.001
41.619
83.125
7.304
13.707
10.970
21.911
F
3.794
3.794
3.794
3.794
Sig.
.026
.063
.048
.092
Interpretation: Main Effect
Bushtucker Trials
10
9
Time to Retch (s)
8
7
6
5
4
3
2
1
0
Stick Insect
Kangaroo Testicle
Animal
Slide 33
Fish Eye
Witchetty Grub
Post Hoc Tests
• Compare each mean against all others (t-tests).
• In general terms they use a stricter criterion to
accept an effect as significant.
– Hence, control the familywise error rate.
– Simplest example is the Bonferroni method:
Bonferroni  
Slide 34

Number of Tests
What is Two-Way Repeated
Measures ANOVA?
• Two Independent Variables
– Two-way = 2 IVs
– Three-Way = 3 IVs
• The same participants in all conditions.
– Repeated Measures = ‘same participants’
– A.k.a. ‘within-subjects’
Slide 35
An Example
• Field (2009): Effects of advertising on
evaluations of different drink types.
– IV 1 (Drink): Beer, Wine, Water
– IV 2 (Imagery): Positive, negative, neutral
– Dependent Variable (DV): Evaluation of
product from -100 dislike very much to
+100 like very much)
Slide 36
SST
Variance between all participants
SSR
SSM
BetweenParticipant
Variance
Within-Particpant Variance
Variance explained by the
experimental manipulations
SSA
SSA  B
Effect of
Drink
Effect of
Imagery
Effect of
Interaction
SSRA
SSRB
SSRA  B
Error for
Drink
Slide 37
SSB
Error for
Imagery
Error for
Interaction
Running the Analysis: Naming
factors
Slide 38
Defining Variables
Slide 39
Defining Variables
Slide 40
Contrasts
Slide 41
Plots
Slide 42
Getting Means
Slide 43
Output: Sphericity
Slide 44
Output: Main ANOVA
Slide 45
Main Effect of Drink
F(1.15, 21.93) = 5.11, p < .05
Slide 46
Main Effect of Imagery
F(1.50, 28.40) = 122.57, p < .001
Slide 47
Drink by Dose Interaction
F(4, 76) = 17.16, p < .001
Slide 48
Contrasts