Witte & Witte, 9e
Chapter 18
Page 1 of 6 Pages
Chapter 18: Analysis of Variance (Two Factors)
Exercise 1
Several 2 × 2 tables of means are presented below. Assuming that any mean differences
are statistically significant, identify the main effect(s) and/or interaction that are depicted
in each table.
Table A
Locus of Control
Internal
External
Gender
Male
5
2
Female
5
2
Table B
Locus of Control
Internal
External
Gender
Male
5
5
Female
5
5
Table C
Locus of Control
Internal
External
Gender
Male
5
5
Female
2
2
Table D
Locus of Control
Internal
External
Gender
Male
5
2
Female
5
2
Table E
Locus of Control
Internal
External
Gender
Male
2
5
Female
2
2
Table F
Locus of Control
Internal
External
Gender
Male
5
3
Female
2
6
Table G
Locus of Control
Internal
External
Gender
Male
5
2
Female
2
5
1
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Chapter 18
Page 2 of 6 Pages
Answers:
Table A. Main effect of gender, no main effect of locus of control, no interaction
Table B. No main effect of gender, no main effect of locus of control, no interaction
Table C. Main effect of gender, no main effect of locus of control, no interaction
Table D. No main effect of gender, main effect of locus of control, no interaction
Table E. Main effect of gender, main effect of locus of control, interaction
Table F. No main effect of gender, main effect of locus of control, interaction
Table G. No main effect of gender, no main effect of locus of control, interaction
Exercise 2
A researcher is investigating the effects of cognitive style and picture color on memory
for picture content. Cognitive style has two levels: Analytical and Global. Picture color
also has two levels: Realistic color and black-and-white. Present the null and alternative
hypotheses for the two main effects and for the interaction.
Answers:
Main effect of cognitive style:
H0: No main effect due to cognitive style (or A = G)
H1: H0 is not true.
Main effect of picture color:
H0: No main effect due to picture color (or RC = BW)
H1: H0 is not true.
Interaction of cognitive style and picture color:
H0: No interaction
H1: H0 is not true.
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Witte & Witte, 9e
Chapter 18
Page 3 of 6 Pages
Exercise 3
The researchers mentioned in Exercise 2 measured the participants’ memory for picture
content with a recall test. The recall test scores are shown in the table below. Use this
information to answer these questions.
OUTCOME OF TWO-FACTOR EXPERIMENT
COGNITIVE STYLE x PICTURE COLOR
(Memory Test Score)
COGNITIVE STYLE
Analytical
Global
Column mean
a.
b.
c.
d.
e.
f.
g.
h.
i.
j.
k.
l.
Realistic Color
12
11
11
8
13
10
10
9
10.5
ROW MEAN
10.875
8.5
9.6875
Calculate dftotal.
Calculate dfcolumn.
Calculate dfrow.
Calculate dfinteraction.
Calculate dfwithin.
Identify the critical F value for the picture color main effect using a significance
level of .05. Refer to Table C in your textbook.
Identify the critical F value for the cognitive style main effect using a significance
level of .05.
Identify the critical F value for the picture color by cognitive style interaction
using a significance level of .05.
Calculate the mean for the Analytical/Realistic Color cell.
Calculate the mean for the Global/Realistic Color cell.
Calculate the mean for the Analytical/Black-and-White cell.
Calculate the mean for the Global/Black-and-White cell.
Answers:
a.
b.
c.
d.
PICTURE COLOR
Black-and-White
13
11
11
10
9
7
6
4
8.875
dftotal = N – 1 = 16 – 1 = 15
dfcolumn = 2 – 1 = 1
dfrow = 2 – 1 = 1
dfinteraction = (2 – 1) (2 – 1) = 1
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Witte & Witte, 9e
Chapter 18
e.
f.
g.
h.
i.
j.
k.
l.
Page 4 of 6 Pages
dfwithin = N – (2)(2) = 16 – 4 = 12
Critical F = 4.75
Critical F = 4.75
Critical F = 4.75
10.5
10.5
11.25
6.5
Exercise 4
The ANOVA summary table for the Exercise 3 data is presented below. Refer to this
information and to the Exercise 3 information to answer the questions. Evaluate
statistical significance with alpha equal to .05. Note that p-values are included in the
summary table.
ANOVA Summary Table (Picture Color and Cognitive Style)
SOURCE
SS
df
MS
F
p-value
10.5625
1 10.5625 3.5455
0.0842
Picture Color
22.5625
1 22.5625 7.5734
0.0175
Cognitive Style
22.5625
1 22.5625 7.5734
0.0175
Interaction
35.75
12
2.9792
Within
91.4375
15
Total
a. Is the picture color main effect statistically significant? If so, which color was
associated with better memory test performance?
b. Is the cognitive style main effective statistically significant? If so, which
cognitive style was associated with better memory test performance?
c. Is the interaction statistically significant? If so, which picture color is associated
with better test performance for each of the two cognitive styles?
d. Calculate the proportion of variance explained by cognitive style.
e. Calculate the proportion of variance explained by the interaction.
Answers:
a. The picture color main effect is not statistically significant.
b. The cognitive style main effect is statistically significant. Analytical students had
better memory performance than global students.
c. The cognitive style by picture color interaction is statistically significant. It
appears that realistic color pictures are better for global students than black-andwhite pictures, whereas picture color does not have much influence on the
performance of analytical students.
d.  p2 (cognitive style) = .39
e.  p2 (cognitive style  picture color ) = .39
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Witte & Witte, 9e
Chapter 18
Page 5 of 6 Pages
Exercise 5
Researchers carried out an investigation on the effects of the author’s gender and the
rater’s gender on the perceived quality of the article. Author’s gender had three levels:
Male, female, and no gender specified. An equal number of male and female graduate
student volunteers were given exactly the same article to read and were asked to assign a
quality score. One-third of the participants of each gender (n = 8) were told the author
was John Davis, one-third were told the author was Jane Davis, and one-third were given
no author name information. A table of means and the ANOVA summary table are
shown below. Use this information to answer the questions in this exercise.
Table of Means
AUTHOR'S GENDER
Male (John Davis)
Female (Jane Davis)
No name
RATER'S GENDER
Male
Female
83.875
85.875
71.875
82.625
82.000
83.375
ANOVA Summary Table (Rater’s Gender and Author’s Gender)
SOURCE
SS
df
MS
F
p-value
266.0208
1
266.0208
6.6411
0.0136
Rater’s Gender
493.2917
2 246.6458 6.1574
0.0045
Author’s Gender
219.7917
2 109.8958 2.7435
0.0759
Interaction
1682.3750
42 40.0565
Within
2661.4792
47
Total
a. Identify statistically significant pairwise differences among the three levels of
author’s gender using Tukey’s HSD test and a significance level of .05. First,
refer to Table G to identify the value of q that will be used in the computations.
b. Calculate HSD.
c. Find all possible absolute differences between pairs of means.
d. Identify statistically significant pairs of means.
e. Calculate the standardized effect size of any significant difference(s), Cohen’s d,
using formula 16.10 on page 355 of your textbook.
Answers:
a. q = 3.44
b. HSD  q
MS wi
40.0565
 3.44
 5.44
n
16
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Chapter 18
Page 6 of 6 Pages
c.
Male Author
(John Davis)
Male Author
(John Davis)
Female Author
(Jane Davis)
No Author’s
Name Given
Female Author
(Jane Davis)
No Author’s
Name Given
84.875  77.25  7.625
84.875  82.6875  2.1875
77.25  82.6875  5.4375
d. The male author versus the female author comparison is statistically significant.
The students gave a higher quality score to the male author.
The female author versus the no author comparison just misses being statistically
significant. Recall that we used 40 df to identify the q value to use in the HSD
calculations because 42 df was not provided in Table G. Because q values
decrease as df values increase, we might want to obtain a more precise value for
q. There is a website that does in fact provide this information. It is:
http://faculty.vassar.edu/lowry/tabs.html#q. The calculator available at that site
returns a value of 3.44 for 42 df. So, even with a more precise value, we would
declare the difference to be not statistically significant.
e. The standardized effect size for the male author versus female author comparison
is 1.20.
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