Name:___________________________
PSY 216
Assignment 14
a. The mean differences among the levels of one factor are referred to as the __main
effect_______________________________________________ of that factor.
b. A(n) __interaction_______________________________________________ between two factors occurs whenever the
mean differences between individual treatment conditions, or cells, are different from what would be
predicted from the overall main effects of the factors.
c. When the effect of one factor depends on the different levels of a second factor, then there is a(n)
__interaction_______________________________________________ between the factors.
d. When the results of a two-factor study are presented in a graph, the existence of nonparallel lines
(lines that cross or converge) indicates a(n) __interaction_______________________________________________
between the two factors.
e.
Problem 6 from the text
The following matrix present the results from an independent-measures, two-factor study with a sample of n = 10 participants
in each of the six treatment conditions. Note that one treatment mean is missing.
Factor A
a.
A1
A2
B1
M = 10
M = 20
Factor B
B2
M = 20
M = 30
B3
M = 40
What value for the missing mean would result in no main effect for factor A?
= (10 + 20 + 40) / 3 = 23.33
= (20 + 30 + X) / 3 = 23.33
X = 20
A1
A2
b. What value for the missing mean would result in no interaction?
A1,B2 A1,B3 X = 50
A1,B1
6.
A2,B1
A2,B2
A2,B3
= 10 – 20 = -10
= 20 – 30 = -10
= 40 – X = -10
Problem 7 from the text
For the data in the following graph:
a.
Is there a main effect for the treatment factor?
Likely Yes.
= (17 + 19 + 27) / 3 = 21
Treatment 2 = (13 + 10 + 3) / 3 = 8.67
There is a large difference between
Treatment 1
b.
Treatment 1 and
Is there a main effect for the age factor?
Likely No
8 Years = (17 + 13) / 2 = 15
9 Years = (19 + 10) / 2 = 14.5
10 Years = (27 + 3) / 2 = 15
There is a very small difference between
c.
Treatment 2
8 Years
and
9 Years
and
10 Years
Is there an interaction between age and treatment?
Likely Yes.
The lines are not parallel. The difference between Treatment 1, 8 years (17) and Treatment 2, 8 years (13) = 4 is much
smaller than the difference between Treatment 1, 10 years (27) and Treatment 2, 10 years (3) = 24
7. Problem 16 from the text
The Preview section for this chapter described a two-factor study examining performance under two audience conditions
(factor B) for high and low self-esteem participants (factor A). The following summary table presents possible results from
the analysis of that study. Assuming that the study used a separate sample of n = 15 participants in each treatment
condition (each cell), fill in the missing values in the table. (Hint: Start with the df values.)
Source
SS
df
Between treatments
67
_3_
_16_
_1_
_16_
F = _4_
Self-esteem
29
_1_
_29_
F = _7.25_
Interaction
_22_
_1_
_22_
F = 5.50
Within treatments
_224_
_56_
4
Total
_291_
_59_
Audience
MS
dfTotal = N – 1 ( 2 X 2 X 15) – 1 = 59
dfAudience = KAudience – 1 = 2 – 1 = 1
dfSelf-esteem = KSelf-esteem – 1 = 2 – 1 = 1
dfAudience X Self-esteem = dfAudience X dfSelf-esteem = 1 X 1 = 1
dfbetween treatment = dfAudience + dfSelf-esteem + dfAudience X Self-esteem = 1 + 1 + 1 = 3
dfwithin treatment = KAudience X KSelf-esteem X (n – 1) = 2 X 2 X (15 – 1) = 56 = dftotal – dfbetween treatments = 59 – 3 =
56
MSAudience X Self-esteem = FAudience x Self-esteem X MSWithin treatments = 5.50 X 4 = 22
SSWithin treatments = dfwithin treatment X MSwithin treatment = 56 X 4 = 224
SStotal = SSbetween treatment + SSwithin treatment = 67 + 224 = 291
SSAudience X Self-esteem = MSAudience X Self-esteem X dfAudience X Self-esteem = 22 X 1 = 22
SSAudience =SSbetween treatments – SSSelf-esteem – SSAudience X Self-esteem = 67 – 29 – 22 = 16
MSAudience = SSAudience / dfAudience = 16 / 1 = 16
MSSelf-esteem = SSSelf-esteem / dfSelf-esteem = 29 / 1 = 29
FAudience = MSAudience / MSWithin treatments = 16 / 4 = 4
FSelf-esteem = MSSelf-esteem / MSWithin treatments = 29 / 4 = 7.25
8. Enter the data from problem 19 into SPSS. Use SPSS to answer the following question: Do the data
indicate significant main effects and interaction? Give H0 and H1 for each hypothesis. Give α. Write a
sentence or two in APA format and include a table that summarizes the results of the analysis.
The following data are from a two-factor study examining the effects of three treatment conditions on males and females.
Treatments
II
7
2
9
M=6
T = 18
SS = 26
10
11
15
M = 12
T = 36
SS = 14
I
1
2
6
M=3
T=9
SS = 14
3
1
5
M=3
T=9
SS = 8
Male
Factor A:
Gender
Female
H0: μTx 1 = μTx 2 = μTx 3
H1: not H0
H0: μmales = μfemales
H1: not H0
H0: there is no interaction
H1: there is an interaction
α = .05
Descriptive Statistics
Dependent Variable:Dependent Variable
Treatment
gender
1.00
Male
3.0000
2.64575
3
Female
3.0000
2.00000
3
Total
3.0000
2.09762
6
Male
6.0000
3.60555
3
12.0000
2.64575
3
Total
9.0000
4.33590
6
Male
9.0000
2.00000
3
Female
15.0000
3.60555
3
Total
12.0000
4.19524
6
Male
6.0000
3.57071
9
10.0000
5.93717
9
2.00
Female
3.00
Total
Female
Mean
Std. Deviation
N
II
9
11
7
M=9
T = 27
SS = 8
16
18
11
M = 15
T = 45
SS = 26
Tmale = 54
N = 18
G = 144
2
ΣX = 1608
Tfemale = 90
Descriptive Statistics
Dependent Variable:Dependent Variable
Treatment
gender
1.00
Male
3.0000
2.64575
3
Female
3.0000
2.00000
3
Total
3.0000
2.09762
6
Male
6.0000
3.60555
3
12.0000
2.64575
3
Total
9.0000
4.33590
6
Male
9.0000
2.00000
3
Female
15.0000
3.60555
3
Total
12.0000
4.19524
6
Male
6.0000
3.57071
9
10.0000
5.93717
9
8.0000
5.17914
18
2.00
Mean
Female
3.00
Total
Female
Total
Std. Deviation
N
Tests of Between-Subjects Effects
Dependent Variable:Dependent Variable
Type III Sum of
Source
Partial Eta
Squares
df
Mean Square
F
Sig.
Squared
Corrected Model
360.000
a
5
72.000
9.000
.001
.789
Intercept
1152.000
1
1152.000
144.000
.000
.923
treatment
252.000
2
126.000
15.750
.000
.724
gender
72.000
1
72.000
9.000
.011
.429
treatment * gender
36.000
2
18.000
2.250
.148
.273
Error
96.000
12
8.000
Total
1608.000
18
456.000
17
Corrected Total
a. R Squared = .789 (Adjusted R Squared = .702)
1. Treatment
Dependent Variable:Dependent Variable
95% Confidence Interval
Treatment
Mean
Std. Error
Lower Bound
Upper Bound
1.00
3.000
1.155
.484
5.516
2.00
9.000
1.155
6.484
11.516
3.00
12.000
1.155
9.484
14.516
2. gender
Dependent Variable:Dependent Variable
95% Confidence Interval
gender
Male
Female
Mean
Std. Error
Lower Bound
Upper Bound
6.000
.943
3.946
8.054
10.000
.943
7.946
12.054
3. Treatment * gender
Dependent Variable:Dependent Variable
95% Confidence Interval
Treatment
gender
1.00
Male
3.000
1.633
-.558
6.558
Female
3.000
1.633
-.558
6.558
Male
6.000
1.633
2.442
9.558
12.000
1.633
8.442
15.558
9.000
1.633
5.442
12.558
15.000
1.633
11.442
18.558
2.00
Female
3.00
Mean
Male
Female
Std. Error
Lower Bound
Upper Bound
Multiple Comparisons
Dependent Variable
Tukey HSD
95% Confidence Interval
Mean
(I) Treatment
1.00
2.00
3.00
(J) Treatment
Difference (I-J)
Std. Error
Sig.
Lower Bound
Upper Bound
2.00
-6.0000
*
3.00
-9.0000
*
1.63299
.000
-13.3566
-4.6434
1.00
6.0000
*
1.63299
.008
1.6434
10.3566
3.00
-3.0000
1.63299
.199
-7.3566
1.3566
1.00
9.0000
*
1.63299
.000
4.6434
13.3566
2.00
3.0000
1.63299
.199
-1.3566
7.3566
1.63299
.008
-10.3566
-1.6434
Based on observed means.
The error term is Mean Square(Error) = 8.000.
*. The mean difference is significant at the .05 level.
Table 1 shows the means and standard deviations for all treatment conditions. The 2 X 3 analysis of
variance (ANOVA) revealed a significant main effect of treatment, F(2, 12) = 15.75, MSerror = 8.00, p =
.000, η2 = .724. Tukey multiple comparisons revealed that treatments I and II were reliably different, p = .008,
as were treatments I and III, p = .000. However, treatments II and III were not reliably different, p = .199. The
ANOVA revealed a significant main effect of gender, F(1, 12) = 9.00, p = .011, η2 = .429. The ANOVA failed to
reveal a significant interaction of treatment and gender, F(2, 12) = 2.25, p = .148, η2 = .273.
Table 1
Means and Standard Deviations for each Condition
Treatment I
M
sd
3.00
2.65
3.00
2.00
3.00
2.10
Males
Females
Marginal
Treatment II
M
sd
6.00
3.61
12.00
2.65
9.00
4.34
Treatment III
M
sd
9.00
2.00
15.00
3.61
12.00
4.20
Marginal
M
sd
6.00
3.57
10.00
5.94
Problem 19 by hand:
Factor B Treatments
II
I
Male
Female
Total
III
Total
1
2
6
7
2
9
9
11
7
ΣX = 9
ΣX2 = 41
SS = 41 – 92 / 3 = 14
M=3
3
1
5
ΣX = 18
ΣX2 = 134
SS =134 – 182 / 3 = 26
M=6
10
11
15
ΣX = 27
ΣX2 = 251
SS = 251 – 272 / 3 = 8
M=9
16
18
11
ΣX = 9 + 18 + 27 = 54
ΣX2 =41+134+251=426
SS = 426 – 542 / 9 = 102
M=6
ΣX = 9
ΣX2 = 35
SS = 35-92/3=8
M=3
ΣX = 9 + 9 = 18
ΣX2 = 41 + 35 = 76
SS = 76–182/6=22
M =3
ΣX = 36
ΣX2 = 446
SS = 446 – 362 / 3 = 14
M =12
ΣX = 18 + 36 = 54
ΣX2 = 134 + 446 = 580
SS = 580-542/6=94
M=9
ΣX = 45
ΣX2 = 701
SS = 701 – 452 / 3 =26
M = 15
ΣX = 27 + 45 = 72
ΣX2 = 251 + 701 = 952
SS = 952-722/6=88
M = 12
ΣX = 9 + 36 + 45 = 90
ΣX2 =35+446+701=1182
SS = 1182 – 902 / 9 = 282
M = 10
ΣX = 54 + 90 = 144
ΣX2 = 426 + 1182 = 1608
SSTotal = 1608-1442/18=456
SSBetween Treatments = n * SSM
n is the number of scores per condition
ΣM = 3 + 6 + 9 + 3 + 12 + 15 = 48
ΣM2 = 32 + 62 + 92 + 32 + 122 + 152 = 504
SSM = 504 – 482 / 6 = 120
SSBetween Treatments = 3 * 120 = 360
SSFactor A = nA * SSMA
ΣMA = 6 + 10 = 16
ΣMA2 = 62 + 102 = 136
SSMA = 136 – 162 / 2 = 8
SSFactor A =9 * 8 = 72
SSFactor B = nB * SSMB
ΣMB = 3 + 9 + 12 = 24
ΣMB2 = 32 + 92 + 122 = 234
SSMB = 234 – 242 / 3 = 42
SSFactor B =6 * 42 = 252
SSFactor A X Factor B = SSBetween Treatments – SSFactor A – SSFactor B = 360 – 72 – 252 = 36
SSWithin Treatment = Σ(SSeach condition) = 14 + 26 + 8 + 8 + 14 + 26 = 96
SSTotal = 456 = SSBetween Treatments + SSWithin Treatment = 360 + 96
dfFactor A = # levels of Factor A – 1 = 2 – 1 = 1
dfFactor B = # levels of Factor B – 1 = 3 – 1 = 2
dfFactor A X Factor B = SSFactor A * SSFactor B = 1 X 2 = 2
dfWithin Treatment = Σ(df for each condition) = (3 – 1) + (3 – 1) + (3 – 1) + (3 – 1) + (3 – 1) + (3 – 1) + (3 – 1) = 12
dfTotal = N – 1 = 18 – 1 = 17 = dfFactor A + dfFactor B + dfFactor A X Factor B + dfWithin Treatment = 1 + 2 + 2 + 12 = 17
MSFactor A = SSFactor A / dfFactor A = 72 / 1 = 72
MSFactor B = SSFactor B / dfFactor B = 252 / 2 = 126
MSFactor A X Factor B = SSFactor A X Factor B / dfFactor A X Factor B = 36 / 2 = 18
MSWithin Treatment = SSWithin Treatment / dfWithin Treatment = 96 / 12 = 8
FFactor A = MSFactor A / MSWithin Treatment = 72 / 8 = 9
FFactor B = MSFactor B / MSWithin Treatment = 126 / 8 = 15.75
FFactor A X Factor B = MSFactor A X Factor B / MSWithin Treatment = 18 / 8 = 2.25
η2Factor A = SSFactor A / (SSTotal – SSFactor B – SSFactor A X Factor B) = 72 / (456 – 252 – 36) = .43
η2Factor B = SSFactor B / (SSTotal – SSFactor A – SSFactor A X Factor B) = 252 / (456 – 72 – 36) = .72
η2Factor A X Factor B = SSFactor A X Factor B / (SSTotal – SSFactor A – SSFactor B) = 36 / (456 – 72 - 252) = .27