Post on 07-Jan-2016
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Practice
• Participants were asked about their current romantic relationship (dating, married, or cohabitating) and the number of children they have (none, one, or two). They also indicated how many arguments they had with their romantic partner during the past year.
• Use the data on the next page to complete the incomplete ANOVA table.
Source df SS MS F
Children 188.58
Relationship 356.04
AB 371.96
Within
Total 1971.78
None One Two
Dating 25 30 28
30 25 31
28 27 26
40 35 20
20 23 35
Married 11 31 23
18 20 28
26 22 35
15 23 27
14 19 21
Cohabitating 23 18 28
30 24 21
18 9 30
28 16 30
23 13 23
Number of children
Source df SS MS F
Children 2 188.58 94.29 3.22
Relationship 2 356.04 178.02 6.07*
AB 4 371.96 92.99 3.17*
Within 36 1055.20 29.31
Total 44 1971.78
F critical A (2, 36) = 3.27
F critical B (2, 36) = 3.27
F critical AB (4, 36) = 2.64
Remember
• You were asked to determine the effects of both college major (psychology, sociology, and biology) and gender (male and female) on class attendance
15.500 3 5.167 9.300 .002
1.389 1 1.389 2.500 .140
14.111 2 7.056 12.700 .001
8.778 2 4.389 7.900 .006
24.278 5 4.856 8.740 .001
6.667 12 .556
30.944 17 1.820
(Combined)
GENDER
MAJOR
Main Effects
GENDER *MAJOR
2-Way Interactions
Model
Residual
Total
DAYS
Sum ofSquares df
MeanSquare F Sig.
Unique Method
ANOVAa,b
DAYS by GENDER, MAJORa.
All effects entered simultaneouslyb.
0
0.5
1
1.5
2
2.5
3
3.5
4
female
male
Sociology Psychology Biology
Simple Effects
• The effect of one factor at one level on the other factor.– Use to “tease apart” a significant interaction
• You wonder if college major has an effect on class attendance for just females?
• You wonder if male and female psychology students differ in their class attendance?
0
0.5
1
1.5
2
2.5
3
3.5
4
female
male
Sociology Psychology Biology
0
0.5
1
1.5
2
2.5
3
3.5
4
female
male
Sociology Psychology Biology
0
0.5
1
1.5
2
2.5
3
3.5
4
female
male
Sociology Psychology Biology
0
0.5
1
1.5
2
2.5
3
3.5
4
female
male
Sociology Psychology Biology
0
0.5
1
1.5
2
2.5
3
3.5
4
female
male
Sociology Psychology Biology
0
0.5
1
1.5
2
2.5
3
3.5
4
female
male
Sociology Psychology Biology
Simple Effects
• Why not just do 5 new analyses– 2 ANOVAs– 3 Independent Samples t-tests
• You will lose df!
• Type I error problem still exists
0
0.5
1
1.5
2
2.5
3
3.5
4
female
male
Sociology Psychology Biology
Sociology Psychology Biology Mean
Female 2.00 1.00 1.00
3.00 .00 2.00
3.00 2.00 2.00
Mean1j 2.67 1.00 1.67 1.78
Males 4.00 2.00 1.00
3.00 4.00 .00
4.00 3.00 .00
Mean2j
Mean.j
3.67
3.17
3.00
2.00
0.33
1.00
2.33
2.06
Are these means different?
Simple Effects
• SS Major at Female
– Represents the SS deviations of the treatment means around the Female mean
– Its multiplied by n to give an estimate of the population variance (Central limit theorem)
– Same formula as SSbetween in the one-way
211__ .)( XXnSS jFemaleatMajor
Sociology Psychology Biology Mean
Female 2.00 1.00 1.00
3.00 .00 2.00
3.00 2.00 2.00
Mean1j 2.67 1.00 1.67 1.78
Males 4.00 2.00 1.00
3.00 4.00 .00
4.00 3.00 .00
Mean2j
Mean.j
3.67
3.17
3.00
2.00
0.33
1.00
2.33
2.06
SS M at Female = 3*((2.67 – 1.78)2 + (1.00 – 1.78)2 + (1.67 – 1.78)2 = 4.23
Note
4.222 2 2.111 3.800 .086
3.333 6 .556
7.556 8
BetweenGroups
WithinGroups
Total
DAYS
Sum ofSquares df
MeanSquare F Sig.
ANOVA
0
0.5
1
1.5
2
2.5
3
3.5
4
female
male
Sociology Psychology Biology
Sociology Psychology Biology Mean
Female 2.00 1.00 1.00
3.00 .00 2.00
3.00 2.00 2.00
Mean1j 2.67 1.00 1.67 1.78
Males 4.00 2.00 1.00
3.00 4.00 .00
4.00 3.00 .00
Mean2j
Mean.j
3.67
3.17
3.00
2.00
0.33
1.00
2.33
2.06
Are these means different?
Simple Effects
• SS Major at Male
– Same formula as SSbetween in the one-way
222__ .)( XXnSS jMaleatMajor
Sociology Psychology Biology Mean
Female 2.00 1.00 1.00
3.00 .00 2.00
3.00 2.00 2.00
Mean1j 2.67 1.00 1.67 1.78
Males 4.00 2.00 1.00
3.00 4.00 .00
4.00 3.00 .00
Mean2j
Mean.j
3.67
3.17
3.00
2.00
0.33
1.00
2.33
2.06
SS M at Male = 3*((3.67 – 2.33)2 + (3.00 – 2.33)2 + (0.33 – 2.33)2 = 18.71
18.667 2 9.333 16.800 .003
3.333 6 .556
22.000 8
BetweenGroups
WithinGroups
Total
DAYS
Sum ofSquares df
MeanSquare F Sig.
ANOVA
0
0.5
1
1.5
2
2.5
3
3.5
4
female
male
Sociology Psychology Biology
Sociology Psychology Biology Mean
Female 2.00 1.00 1.00
3.00 .00 2.00
3.00 2.00 2.00
Mean1j 2.67 1.00 1.67 1.78
Males 4.00 2.00 1.00
3.00 4.00 .00
4.00 3.00 .00
Mean2j
Mean.j
3.67
3.17
3.00
2.00
0.33
1.00
2.33
2.06
Are these means different?
Simple Effects
• SS Gender at Socio
– Same formula as SSbetween in the one-way
211__ ).( XXnSS iSocioatGender
Sociology Psychology Biology Mean
Female 2.00 1.00 1.00
3.00 .00 2.00
3.00 2.00 2.00
Mean1j 2.67 1.00 1.67 1.78
Males 4.00 2.00 1.00
3.00 4.00 .00
4.00 3.00 .00
Mean2j
Mean.j
3.67
3.17
3.00
2.00
0.33
1.00
2.33
2.06
SS G at Socio = 3*((2.67 – 3.17)2 + (3.67 – 3.17)2 = 1.5
1.500 1 1.500 4.500 .101
1.333 4 .333
2.833 5
BetweenGroups
WithinGroups
Total
DAYS
Sum ofSquares df
MeanSquare F Sig.
ANOVA
0
0.5
1
1.5
2
2.5
3
3.5
4
female
male
Sociology Psychology Biology
Sociology Psychology Biology Mean
Female 2.00 1.00 1.00
3.00 .00 2.00
3.00 2.00 2.00
Mean1j 2.67 1.00 1.67 1.78
Males 4.00 2.00 1.00
3.00 4.00 .00
4.00 3.00 .00
Mean2j
Mean.j
3.67
3.17
3.00
2.00
0.33
1.00
2.33
2.06
Are these means different?
Simple Effects
• SS Gender at Psych
– Same formula as SSbetween in the one-way
222__ ).( XXnSS iPsychatGender
Sociology Psychology Biology Mean
Female 2.00 1.00 1.00
3.00 .00 2.00
3.00 2.00 2.00
Mean1j 2.67 1.00 1.67 1.78
Males 4.00 2.00 1.00
3.00 4.00 .00
4.00 3.00 .00
Mean2j
Mean.j
3.67
3.17
3.00
2.00
0.33
1.00
2.33
2.06
SS G at Psych = 3*((1.00 – 2.00)2 + (3.00 – 2.00)2 = 6.00
6.000 1 6.000 6.000 .070
4.000 4 1.000
10.000 5
BetweenGroups
WithinGroups
Total
DAYS
Sum ofSquares df
MeanSquare F Sig.
ANOVA
0
0.5
1
1.5
2
2.5
3
3.5
4
female
male
Sociology Psychology Biology
Sociology Psychology Biology Mean
Female 2.00 1.00 1.00
3.00 .00 2.00
3.00 2.00 2.00
Mean1j 2.67 1.00 1.67 1.78
Males 4.00 2.00 1.00
3.00 4.00 .00
4.00 3.00 .00
Mean2j
Mean.j
3.67
3.17
3.00
2.00
0.33
1.00
2.33
2.06
Are these means different?
Simple Effects
• SS Gender at Bio
– Same formula as SSbetween in the one-way
233__ ).( XXnSS iBioatGender
Sociology Psychology Biology Mean
Female 2.00 1.00 1.00
3.00 .00 2.00
3.00 2.00 2.00
Mean1j 2.67 1.00 1.67 1.78
Males 4.00 2.00 1.00
3.00 4.00 .00
4.00 3.00 .00
Mean2j
Mean.j
3.67
3.17
3.00
2.00
0.33
1.00
2.33
2.06
SS G at Bio = 3*((1.67 – 1.00)2 + (0.33 – 1.00)2 = 2.69
2.667 1 2.667 8.000 .047
1.333 4 .333
4.000 5
BetweenGroups
WithinGroups
Total
DAYS
Sum ofSquares df
MeanSquare F Sig.
ANOVA
Overall ANOVA
Source df SS MS F
Gender 1 1.36 1.36 2.47
Major 2 14.16 7.08 12.87*
G x M 2 8.83 4.42 8.03*
Within 12 6.59 .55
Total 17 30.94
Simple Effects
Source df SS MS F
Gender
G at Socio
G at Psych
G at Bio
Major
M at Female
M at Male
Within 12 6.59 .55
Simple Effects
Source df SS MS F
Gender
G at Socio
G at Psych
G at Bio
1.50
6.00
2.69
Major
M at Female
M at Male
4.23
18.71
Within 12 6.59 .55
Simple Effects
Source df SS MS F
Gender
G at Socio
G at Psych
G at Bio
1
1
1
1.50
6.00
2.69
Major
M at Female
M at Male
2
2
4.23
18.71
Within 12 6.59 .55
dfA (Gender) = a – 1dfB (Major) = b – 1
Simple Effects
Source df SS MS F
Gender
G at Socio
G at Psych
G at Bio
1
1
1
1.50
6.00
2.69
1.50
6.00
2.69
Major
M at Female
M at Male
2
2
4.23
18.71
2.12
9.36
Within 12 6.59 .55
Compute MS
Simple Effects
Source df SS MS F
Gender
G at Socio
G at Psych
G at Bio
1
1
1
1.50
6.00
2.69
1.50
6.00
2.69
2.72
10.91
4.89
Major
M at Female
M at Male
2
2
4.23
18.71
2.12
9.36
3.85
17.02
Within 12 6.59 .55
Compute F
Simple Effects
Source df SS MS F
Gender
G at Socio
G at Psych
G at Bio
1
1
1
1.50
6.00
2.69
1.50
6.00
2.69
2.72
10.91*
4.89*
Major
M at Female
M at Male
2
2
4.23
18.71
2.12
9.36
3.85
17.02*
Within 12 6.59 .55
F critical (1, 12) = 4.75
F critical B (2, 12) = 3.89
0
0.5
1
1.5
2
2.5
3
3.5
4
female
male
Sociology Psychology Biology
Practice
• ANOVA is from exercise 13.5
• 1) Interpret the ANOVA results
• 2) Plot the cell means
• 3) Compute an entire set of simple effects
Tests of Between-Subjects Effects
Dependent Variable: time
916.578a 8 114.572 3.909 .002
26402.222 1 26402.222 900.758 .000
356.044 2 178.022 6.074 .005
188.578 2 94.289 3.217 .052
371.956 4 92.989 3.172 .025
1055.200 36 29.311
28374.000 45
1971.778 44
SourceCorrected Model
Intercept
area
delay
area * delay
Error
Total
Corrected Total
Type III Sumof Squares df Mean Square F Sig.
R Squared = .465 (Adjusted R Squared = .346)a.
Neutral Area A Area B Mean50 28.6 16.8 24.4 23.27
100 28 23 16 22.33150 28 26.8 26.4 27.07
Mean 28.2 22.2 22.27 24.22
Tests of Between-Subjects Effects
Dependent Variable: time
916.578a 8 114.572 3.909 .002
26402.222 1 26402.222 900.758 .000
356.044 2 178.022 6.074 .005
188.578 2 94.289 3.217 .052
371.956 4 92.989 3.172 .025
1055.200 36 29.311
28374.000 45
1971.778 44
SourceCorrected Model
Intercept
area
delay
area * delay
Error
Total
Corrected Total
Type III Sumof Squares df Mean Square F Sig.
R Squared = .465 (Adjusted R Squared = .346)a.
0
5
10
15
20
25
30
35
50 100 150
Neutral
Area A
Area B
0
5
10
15
20
25
30
35
Neutral Area A Area B
50
100
150
Source df SS MS F
Delay
D at Neutral
D at Area A
D at Area B
2
2
2
1.20
254.80
304.53
0.60
127.40
152.26
.02
4.34*
5.19*
Area
A at D 50
A at D 100
A at D 150
2
2
2
357.74
363.33
6.93
178.87
181.67
3.47
6.10*
6.19*
.12
Within 36 1055.20 29.31
SPSS Problem
• 13.1• Compute a factorial ANOVA• Conduct a complete set of simple effects (you
can do this by hand)
• Handout