Jackson S.L. Research Methods and Statistics: A Critical Thinking Approach
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CHAPTER 10
IN REVIEW One-Way Repeated Measures ANOVA
CONCEPT DESCRIPTION
Null hypothesis (H
0
) The independent variable had no effect—the samples all represent the same population.
Alternative hypothesis (H
a
) The independent variable had an effect—at least one of the samples represents a different
population than the others.
F-ratio The ratio formed when the between variance is divided by the error variance.
Between variance An estimate of the variance of the treatment means about the grand mean; includes both
systematic variance and error variance.
Participant variance The variance due to individual differences; removed from the error variance.
Error variance An estimate of the variance within each condition in the experiment after variance due to
individual differences has been removed.
Eta-squared (
2
) A measure of effect size—the variability in the dependent variable attributable to the
independent variable.
Tukey’s post hoc test A test conducted to determine which conditions in a study with more than two groups
differ significantly from each other.
CRITICAL
THINKING
CHECK
10.4
1. Explain why a repeated measures ANOVA is statistically more
powerful than a simple randomized one-way ANOVA.
2. Identify other advantage(s) associated with using a within-
participants design.
Nonparametric Statistics for the
Multiple-Group Experiment
There are several nonparametric equivalents to the ANOVA. For a between-
participants design with more than two groups in which ordinal data have
been collected, the Kruskal-Wallis test is appropriate. Like the Wilcoxon
tests (discussed in the previous chapter), the Kruskal-Wallis is appropriate
when the data are ordinal or if the involved populations are not normally
distributed. The analogous test (more than two groups, ordinal data, and/
or skewed distributions) for use with a within-participants design is the
Friedman rank test. Each of these tests is beyond the scope of the present
text. However, coverage of these tests can be found in more comprehensive
statistics texts.
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Summary
In this chapter, we discussed designs that use more than two levels of an
independent variable. Advantages to such designs include being able to
compare more than two kinds of treatment, to use fewer participants, to
compare all treatments with a control group, and to use placebo groups.
In addition, we discussed the statistical analyses most appropriate for use
with these designs—most commonly, with interval-ratio data, an ANOVA. A
randomized one-way ANOVA is used for between-participants designs, and
a repeated measures one-way ANOVA for correlated-groups designs. Also
discussed were appropriate post hoc tests (Tukey’s HSD) and measures of
effect size (eta-squared). After completing this chapter, you should appreci-
ate the advantages of using more complicated designs and understand the
basic statistics used to analyze such designs.
Bonferroni adjustment
ANOVA (analysis of variance)
one-way randomized ANOVA
grand mean
error variance
within-groups variance
between-groups variance
F-ratio
total sum of squares
within-groups sum of squares
between-groups sum of squares
mean square
eta-squared (
2
)
post hoc test
Tukey’s honestly significant
difference (HSD)
one-way repeated measures
ANOVA
KEY TERMS
(Answers to odd-numbered exercises appear in
Appendix C.)
1. What is/are the advantage(s) of conducting a
study with three or more levels of the independ-
ent variable?
2. What is the difference between a randomized
ANOVA and a repeated measures ANOVA?
What does the term one-way mean with respect
to an ANOVA?
3. Explain between-groups variance and within-
groups variance.
4. If a researcher decides to use multiple compari-
sons in a study with three conditions, what is
the probability of a Type I error across these
comparisons? Use the Bonferroni adjustment to
determine the suggested alpha level.
5. If H
0
is true, what should the F-ratio equal or be
close to? If H
a
is supported, should the F-ratio be
greater than, less than, or equal to 1?
6. When should post hoc comparisons be
performed?
7. What information does eta-squared (
2
) provide?
8. Why is a repeated measures ANOVA statistically
more powerful than a randomized ANOVA?
9. A researcher conducts a study of the effects of
amount of sleep on creativity. The creativity
scores for four levels of sleep (2 hours, 4 hours,
6 hours, and 8 hours) for N 20 participants are
presented here.
Amount of Sleep (in hours)
2 4 6 8
3 4 10 10
5 7 11 13
6 8 13 10
4 3 9 9
2 2 10 10
CHAPTER EXERCISES
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CHAPTER 10
a. Complete the following ANOVA summary
table. (If your instructor wants you to calcu-
late the sums of squares, use the preceding
data to do so.)
Source df SS MS F
Between groups 187.75
Within groups 55.20
Total 242.95
b. Is F
obt
significant at .05?
c. Perform post hoc comparisons if necessary.
d. What conclusions can be drawn from the
F-ratio and the post hoc comparisons?
e. What is the effect size, and what does this
mean?
f. Graph the means.
10. In a study of the effects of stress on illness, a
researcher tallied the number of colds people
contracted during a 6-month period as a func-
tion of the amount of stress they reported during
that same time period. There were three stress
levels: minimal, moderate, and high stress. The
sums of squares appear in the following ANOVA
summary table. The mean for each condition and
the number of participants per condition are also
noted.
Source df SS MS F
Between groups 22.167
Within groups 14.750
Total 36.917
Stress Level Mean n
Minimal 3 4
Moderate 4 4
High 6 4
a. Complete the ANOVA summary table.
b. Is F
obt
significant at .05?
c. Perform post hoc comparisons if necessary.
d. What conclusions can be drawn from the
F-ratio and the post hoc comparisons?
e. What is the effect size, and what does this
mean?
f. Graph the means.
11. A researcher interested in the effects of exercise
on stress had participants exercise for 30, 60, or
90 minutes per day. The mean stress level on a
100-point stress scale (with 100 indicating high
stress) for each condition appears next, along
with the ANOVA summary table with the sums
of squares indicated.
Source df SS MS F
Between groups 4,689.27
Within groups 82,604.20
Total 87,293.47
Exercise Level Mean n
30 minutes 75.0 10
60 minutes 44.7 10
90 minutes 63.7 10
a. Complete the ANOVA summary table.
b. Is F
obt
significant at .05?
c. Perform post hoc comparisons if necessary.
d. What conclusions can be drawn from the
F-ratio and the post hoc comparisons?
e. What is the effect size, and what does this
mean?
f. Graph the means.
12. A researcher conducted an experiment on the
effects of a new “drug” on depression. The
researcher had a control group that received noth-
ing, a placebo group, and an experimental group
that received the “drug.” A depression inventory
that provided a measure of depression on a 50-
point scale was used (50 indicates that an individ-
ual is very high on the depression variable). The
ANOVA summary table appears next, along with
the mean depression score for each condition.
Source df SS MS F
Between groups 1,202.313
Within groups 2,118.00
Total 3,320.313
“Drug” Condition Mean n
Control 36.26 15
Placebo 33.33 15
“Drug” 24.13 15
a. Complete the ANOVA summary table.
b. Is F
obt
significant at .05?
c. Perform post hoc comparisons if necessary.
d. What conclusions can be drawn from the
F-ratio and the post hoc comparisons?
e. What is the effect size, and what does this
mean?
f. Graph the means.
13. A researcher is interested in the effects of practice
on accuracy in a signal-detection task. Participants
are tested with no practice, after 1 hour of prac-
tice, and after 2 hours of practice. Each person
participates in all three conditions. The following
data indicate how many signals each participant
detected accurately at each level of practice.
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Amount of Practice
Participant No Practice 1 Hour 2 Hours
1 3 4 6
2 4 5 5
3 2 3 4
4 1 3 5
5 3 6 7
6 3 4 6
7 2 3 4
Source df SS MS F
Participant 16.27
Between 25.81
Error 4.87
Total 46.95
a. Complete the ANOVA summary table. (If
your instructor wants you to calculate the
sums of squares, use the preceding data to
do so.)
b. Is F
obt
significant at .05?
c. Perform post hoc comparisons if necessary.
d. What conclusions can be drawn from the
F-ratio and the post hoc comparisons?
e. What is the effect size, and what does this
mean?
f. Graph the means.
14. A researcher has been hired by a pizzeria to
determine which type of crust customers prefer.
The restaurant offers three types of crust: hand-
tossed, thick, and thin. Following are the mean
number of 1-inch pieces of pizza eaten for each
condition from 10 participants who had the
opportunity to eat as many pieces with each type
of crust as they desired. The ANOVA summary
table also follows.
Source df SS MS F
Participant 2.75
Between 180.05
Error 21.65
Total 204.45
Crust Type Mean n
Hand-tossed 2.73 10
Thick 4.20 10
Thin 8.50 10
a. Complete the ANOVA summary table.
b. Is F
obt
significant at .05?
c. Perform post hoc comparisons if necessary.
d. What conclusions can be drawn from the
F-ratio and the post hoc comparisons?
e. What is the effect size, and what does this
mean?
f. Graph the means.
15. A researcher is interested in whether massed or
spaced studying has a greater impact on grades
in a course. The researcher has her class study
for 6 hours all in one day for one exam (massed
study condition). She has them study for 2 hours
each day for 3 days for another exam (3-day
spaced condition). Last, she has them study for
1 hour a day for 6 days for a third exam (6-day
spaced condition). The mean exam score (out of
a possible 100 points) for each condition appears
next, along with the ANOVA summary table.
Source df SS MS F
Participant 136.96
Between 3,350.96
Error 499.03
Total 3,986.95
Study Condition Mean n
Massed 69.13 15
3-day spaced 79.33 15
6-day spaced 90.27 15
a. Complete the ANOVA summary table.
b. Is F
obt
significant at .05?
c. Perform post hoc comparisons if necessary.
d. What conclusions can be drawn from the
F-ratio and the post hoc comparisons?
e. What is the effect size, and what does this
mean?
f. Graph the means.
10.1
1. The probability of a Type I error is 26.5%: [1
(1 .05)
6
] [1 (.95)
6
] [1 .735] 26.5%.
With the Bonferroni adjustment, the alpha level
is .008 for each comparison.
10.2
1. Both the within-groups and between-groups
variances are moderate to small. This should
lead to an F-ratio of approximately 1.
CRITICAL THINKING CHECK ANSWERS
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CHAPTER 10
Check your knowledge of the content and key terms
in this chapter with a practice quiz and interactive
flashcards at http://academic.cengage.com/
psychology/jackson, or, for step-by-step practice and
information, check out the Statistics and Research
Methods Workshops at http://academic.cengage
.com/ psychology/workshops.
WEB RESOURCES
For hands-on experience using the research methods
described in this chapter, see Chapters 5 “One-Way
Designs I” and 6 “One-Way Designs II” in Research
Methods Laboratory Manual for Psychology, 2nd ed.,
by William Langston (Wadsworth, 2005) and Lab
10 “Three-Group Designs” in Doing Research: A
Lab Manual for Psychology, by Jane F. Gaultney
(Wadsworth, 2007).
LAB RESOURCES
10.3
1. The F-ratio 1.95/0.26 7.5 suggests that the
independent variable had an effect on the
dependent variable.
2. ANOVA Summary Table
Source df SS MS F
Between 2 4,689.27 2,344.64 0.766
Within 27 82,604.20 3,059.41
Total 29 87,293.47
The resulting F-ratio is less than 1 and thus not
significant. Although stress levels differ across
some of the groups, the difference is not large
enough to be significant.
10.4
1. A repeated measures ANOVA is statistically
more powerful because the within-groups vari-
ance is divided into two sources of variance: that
due to individual differences (participant) and
that left over (error). Only the error variance is
used to calculate the F-ratio. We therefore divide
by a smaller number, thus resulting in a larger
F-ratio and a greater chance that it will be
significant.
2. The within-participants design also has the
advantages of requiring fewer participants,
assuring equivalency of groups, and usually
requiring less time to conduct the study. See
Chapter 8 for a review of this information.
For hands-on experience using statistical soft-
ware to complete the analyses described in this
chapter, see Chapter 8 “Single-Factor ANOVA”
and Exercises 8.1 “The ANOVA Model,” 8.2 “The
Single-Factor ANOVA: Related versus Independent
Samples,” and 8.3 “Unplanned Comparisons”
in The Excel Statistics Companion Version 2.0 by
Kenneth M. Rosenberg (Wadsworth, 2007).
STATISTICAL SOFTWARE RESOURCES
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CHAPTER 10 SUMMARY AND REVIEW: EXPERIMENTAL DESIGNS WITH MORE THAN TWO
LEVELS OF AN INDEPENDENT VARIABLE
In this chapter, designs using more than two levels of
an independent variable were discussed. Advantages
to such designs include being able to compare more
than two kinds of treatment, using fewer participants,
comparing all treatments to a control group, and
using placebo groups. In addition, the chapter dis-
cussed statistical analyses most appropriate for use
with these designs—most commonly with interval-
ratio data, an ANOVA. A randomized one-way
ANOVA would be used for between-participants
designs, and a repeated-measures one-way ANOVA
for correlated-groups designs. Also discussed were
appropriate post hoc tests (Tukey’s HSD) and meas-
ures of effect size (eta-squared). After completing
this chapter, you should appreciate the advantages
of using more complicated designs and possess an
understanding of the basic statistics used to analyze
such designs.
Chapter 10
■
Study Guide
CHAPTER TEN REVIEW EXERCISES
(Answers to exercises appear in Appendix C.)
FILL-IN SELF TEST
Answer the following questions. If you have trouble
answering any of the questions, restudy the relevant
material before going on to the multiple-choice self
test.
1. The
provides a means of setting
a more stringent alpha level for multiple tests to
minimize Type I errors.
2. A
is an inert substance that par-
ticipants believe is a treatment.
3. A(n)
is an inferential statistical
test for comparing the means of three or more
groups.
4. The mean performance across all participants is
represented by the
.
5. The variance is an estimate of
the effect of the independent variable, confounds,
and error variance.
6. The sum of squared deviations of each score from
the grand mean is the
.
7. When we divide an SS score by its degrees of
freedom, we have calculated a
.
8.
is an inferential statistic for
measuring effect size with an ANOVA.
9. For an ANOVA, we use
to
compare all possible pairs of groups to determine
which ones differ significantly from each other.
10. The ANOVA for use with one independent
variable and a correlated-groups design is the
.
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MULTIPLE-CHOICE SELF TEST
Select the single best answer for each of the following
questions. If you have trouble answering any of the
questions, restudy the relevant material.
1. The F-ratio is determined by dividing
by .
a. error variance; systematic variance
b. between-groups variance; within-groups vari-
ance
c. within-groups variance; between-groups vari-
ance
d. systematic variance; error variance
2. If the between-groups variance is large, then we
have observed:
a. experimenter effects.
b. large systematic variance.
c. large error variance.
d. possibly both b and c.
3. The larger the F-ratio, the greater the chance
that:
a. a mistake has been made in the computation.
b. there are large systematic effects present.
c. the experimental manipulation probably did
not have the predicted effects.
d. the between-groups variation is no larger than
would be expected by chance alone and no
larger than the within groups variance.
4. One reason to use an ANOVA over a t test is to
reduce the risk of:
a. a Type II error.
b. a Type I error.
c. confounds.
d. error variance.
5. If the null hypothesis for an ANOVA is false, then
the F-ratio should be:
a. greater than 1.00.
b. a negative number.
c. 0.00.
d. 1.00.
6. In a randomized ANOVA, if there are four groups
with 15 participants in each group, then the df for
the F-ratio is equal to:
a. 60.
b. 59.
c. 3, 56.
d. 3, 57.
7. For an F-ratio with df (3, 20), the F
cv
for .05
would be:
a. 3.10.
b. 4.94.
c. 8.66.
d. 5.53.
8. If a researcher reported an F-ratio with df
(2, 21) for a randomized one-way ANOVA, then
there were
conditions in the experiment
and
total participants.
a. 2; 21
b. 3; 23
c. 2; 24
d. 3; 24
9. Systematic variance and error variance comprise
the
variance.
a within-groups
b. total
c. between-groups
d. participant
10. If a randomized one-way ANOVA produced
MS
between
25 and MS
within
5, then the F-ratio
would be:
a 25/5 5.
b. 5/25 .20.
c. 25/30 .83.
d. 30/5 6.
11. One advantage of a correlated-groups design is
that the effects of have been removed.
a. individual differences
b. experimenter effects
c. subject bias effects
d. measurement error
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SELF-TEST PROBLEMS
1. Calculate Tukey’s HSD and eta-squared for the
following ANOVA.
Anova Summary Table
Source df SS MS F
Participants 9 25
Between 2 150
Error 18 100
Total 29
2. The following ANOVA table corresponds to an
experiment on pain reliever effectiveness. Three
types of pain reliever are used (aspirin, aceta-
minophen, and ibuprofen), and effectiveness is
rated on a 0-10 scale. The scores for the six par-
ticipants in each group follow:
Aspirin: 4, 6, 4, 4, 3, 5
Acetaminophen: 6, 4, 6, 7, 3, 5
Ibuprofen: 7, 6, 5, 8, 6, 5
The sums of squares are provided in the follow-
ing table. However, for practice, see if you can
correctly calculate them by hand.
Anova Summary Table
Source df SS MS F
Participants 9.12
Between 10.19
Error 13.90
Total
a. Complete the ANOVA Summary Table pre-
sented here.
b. Is F
obt
significant at .05?
c. Perform post hoc comparisons if necessary.
d. What conclusions can be drawn from the
F-ratio and the post hoc comparisons?
e. What is the effect size and what does this
mean?
f. Graph the means.
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Complex Experimental
Designs
11
CHAPTER
Using Designs with More Than One Independent Variable
Factorial Notation and Factorial Designs
Main Effects and Interaction Effects
Possible Outcomes of a 2 2 Factorial Design
Statistical Analysis of Complex Designs
Two-Way Randomized ANOVA: What It Is and What It Does
Calculations for the Two-Way Randomized ANOVA • Interpreting the Two-Way
Randomized ANOVA • Assumptions of the Two-Way Randomized ANOVA • Post Hoc
Tests and Effect Size
Two-Way Repeated Measures ANOVA and Mixed ANOVAs
Beyond the Two-Way ANOVA
Summary
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Learning Objectives
• Explain factorial notation and the advantages of factorial designs.
• Identify main effects and interaction effects based on looking at graphs.
• Draw graphs for factorial designs based on matrices of means.
• Explain what a two-way randomized ANOVA is and what it does.
• Calculate a two-way randomized ANOVA.
• Interpret a two-way randomized ANOVA.
• Explain what a two-way repeated measures ANOVA is.
I
n Chapter 10, we discussed designs with more than two levels of an
independent variable. In this chapter, we will look at more complex
designs—those with more than one independent variable. These are
usually referred to as factorial designs, indicating that more than one factor
or variable is being manipulated in the study. We will discuss the advantages
of such designs over simpler designs. In addition, we will explain how to
interpret the findings (called main effects and interaction effects) from such
designs. Last, we will consider the statistical analysis of such designs.
Using Designs with More Than One
Independent Variable
Remember the study discussed in Chapter 10 on the effects of rehearsal on
memory. Participants used one of three types of rehearsal (rote, imagery, or
story) to determine their effects on the number of words recalled. Imagine
that upon further analysis of the data, we discovered that participants
recalled concrete words (for example, desk, bike, tree) better than abstract
words (for example, love, truth, honesty) in one rehearsal condition but not in
another. Such a result is called an interaction between variables; this concept
will be discussed in more detail later in the chapter. One advantage of using
factorial designs is that they allow us to assess how variables interact. In the
real world, it would be unusual to find that a certain behavior is produced
by only one variable; behavior is usually contingent on many variables
operating together in an interactive way. Designing experiments with more
than one independent variable allows researchers to assess how multiple
variables may affect behavior.
Factorial Notation and Factorial Designs
A factorial design, then, is one with more than one factor or independent
variable. A complete factorial design is one in which all levels of each inde-
pendent variable are paired with all levels of every other independent
variable. An incomplete factorial design also has more than one independent
variable, but all levels of each variable are not paired with all levels of every
other variable. The design illustrated in this chapter is a complete factorial
design.
factorial design A design
with more than one independent
variable.
factorial design A design
with more than one independent
variable.
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