CHAPTER FOURTEEN
Two-Way Between-Groups
ANOVA
NOTE TO INSTRUCTORS
It is essential that students understand the
distinction
between
different
independent
variables and levels of independent variables. For
example, it would not be unusual for a student to
think that there are four independent variables
with a 2  2 ANOVA, when in fact there are only
two independent variables, each with two levels.
As you can probably imagine, if students do not
thoroughly
understand
the
distinction,
their
ability to perform most statistical calculations
will be severely hampered. Make sure you go
through
many
examples
with
students,
distinguishing between independent variables and
levels
of
one
independent
variable.
Another
important concept to stress is the distinction
between a main effect and an interaction. Students
often have a difficult time understanding what an
interaction actually tells the researcher and why
interactions are important. Therefore, a number of
supplementary
exercises
in
this
instructor’s
manual have been created to help students master
this distinction.
OUTLINE OF RESOURCES
I.
Two-Way ANOVA
 Discussion Question 14-1 (p. 126)
 Discussion Question 14-2 (p. 127)
 Classroom Activity 14-1: Two-Way Analysis of
Variance: Singers’ Height (p. 127)
 Discussion Question 14-3 (p. 127)
 Discussion Question 14-4 (p. 128)
Classroom Activity 14-2: Make It Your Own: Gender
Perception
(p. 128)

II.
Understanding Interactions
 Discussion Question 14-5
 Discussion Question 14-6
 Classroom Activity 14-3:
Visually (p. 130)
in ANOVA
(p. 128)
(p. 129)
Interpreting Effects
III. Conducting a Two-Way Between-Groups ANOVA
 Discussion Question 14-7 (p. 130)
 Classroom Activity 14-4: Make It Your Own:
Learning and Memory
(p. 131)
 Discussion Question 14-8 (p. 131)
 Classroom Activity 14-5: Affecting Memory (p.
131)
 Classroom Activity 14-6: Two-Way ANOVAs in
Context (p. 132)
IV.
Next Steps: Variations on ANOVA
 Discussion Question 14-9 (p. 133)
IV.
Handouts and Transparency Masters
 Handout 14-1: Interpreting Effects Visually (p.
134)
 Handout 14-2A: Affecting Memory (p. 136)
 Handout 14-2B: Affecting Memory (p. 137)
 Handout 14-2C: Affecting Memory (p. 138)
 Handout 14-3: Two-Way ANOVAs in Context (p. 139)
CHAPTER GUIDE
I.
Two-Way ANOVA
1. An interaction occurs in a factorial design
when two or more independent variables have an
effect in combination that we do not see when
we examine each independent variable on its
own.
2. A two-way ANOVA is a hypothesis test that
includes two nominal independent variables,
regardless of their numbers of levels, and an
interval dependent variable.
> Discussion Question 14-1
How does a one-way ANOVA hypothesis test differ from a twoway ANOVA hypothesis test?
Your students’ answers should include:
 A
one-way ANOVA includes one nominal independent
variable with more than two levels and an
interval dependent variable, whereas a two-way
ANOVA includes two nominal independent variables
with at least two levels and an interval
dependent variable.
3. Any ANOVA with at least two independent
variables can be called a factorial analysis of
variance (or multifactorial ANOVA). This term
refers to a statistical analysis used with one
interval
dependent
variable and at least two nominal independent
variables (also called factors).
4. A factor is another word used to describe an
independent variable in a study with more than
one independent variable.
5. Using a two-way ANOVA is advantageous over
conducting two separate one-way ANOVAs because,
with a two-way ANOVA, you have the opportunity
to explore interactions as well as the isolated
effect of the two independent variables on the
dependent variable.
> Discussion Question 14-2
Why is it better to conduct one two-way ANOVA rather than
two separate one-way ANOVAs?
Your students’ answers should include:
 Conducting
a two-way ANOVA allows the researcher
to compare the two independent variables under
the time, energy, and resources of one study.
 A
two-way ANOVA can study interactions between
the variables as well as the isolated effect of
the two independent variables on the dependent
variable.
Classroom Activity 14-1
Two-Way Analysis of Variance: Singers’ Height
http://lib.stat.cmu.edu/DASL/Datafiles/Singers.htm
l
 Modify
the data set found at the above URL, and
create:
a new variable of voice (high voices are
sopranos and tenors while low voices are
altos and basses) and

a new variable of gender (females are altos
and sopranos while males are basses and
tenors).
 With
this modification you can perform a 2
(voice) by 2 (gender) analysis of variance. Are
deeper-voiced singers taller?

6. When we draw out the design of our study, we
call each box of the design a cell, which
depicts the unique combination of levels of the
independent variables in a factorial design.
7. Two-way ANOVAs are often notated by the number
of levels for each variable. For example, if we
were studying the effects of two independent
variables on the dependent variable and each
independent variable had two levels, we would
say that we were conducting a 2  2 ANOVA.
8. In addition, either the term between-groups or
within-groups is also included, depending on
the specific experimental design.
> Discussion Question 14-3
How would you interpret the phrase “4  2 between-groups
ANOVA”?
Your students’ answers should include:
 A 4  2 between-groups ANOVA is a hypothesis test
that includes two independent variables, one with
4 levels and one with 2 levels, and a dependent
variable.
9. A main effect occurs in a factorial design
when one of the independent variables has an
influence on the dependent variable.
10. An interaction occurs when the effect of one
independent variable on the dependent variable
depends on the particular level of the other
independent variables.
> Discussion Question 14-4
What is the difference between a main effect and an
interaction?
Your students’ answers should include:
 A main effect occurs when one of the independent
variables has an influence on the dependent
variable.

An interaction in a factorial design occurs when
the effect of one independent variable on the
dependent variable depends on the particular
level of the other independent variables.
Classroom Activity 14-2
Make It Your Own: Gender Perception
http://opl.apa.org/Experiments/About/AboutGenderPe
rception.aspx
This online experiment is a 3  2 design, so it is
appropriate only for larger classes or classes in
which
students
can
go
out
and
find
more
participants. The experiment was created by the
Online Psychology Laboratory. Explanations and
references are provided online.
11. In a two-way ANOVA, we will be calculating
three F statistics—one for each main effect and
one for the interaction. Each F statistic will
have its own between-groups sum of squares
(SS), degrees of freedom (df), mean square
(MS), and critical value. However, each F
statistic will share the MSwithin.
II.
Understanding interactions in ANOVA
1. There are two different kinds of interactions:
quantitative
interactions
and
qualitative
interactions.
2. A quantitative interaction is an interaction
in which one independent variable exhibits a
strengthening or weakening of its effect at one
or more levels of the other independent
variable, but the direction of the initial
effect does not change.
3. A qualitative interaction is a particular type
of quantitative interaction of two (or more)
independent variables in which one independent
variable reverses its effect depending on the
level of the other independent variable.
> Discussion Question 14-5
What would be an example of a quantitative interaction? A
qualitative interaction?
Your students’ answers should include:
 An example of a quantitative interaction would be
an overall increase in sales as a result of
advertising in the lifestyle section of the
newspaper versus advertising in the business
section of the same newspaper. However, when
advertising is placed in both the lifestyle and
the business sections of the newspaper, sales are
the highest.
 An
example of a qualitative interaction would be
a change in grades for men and women between the
first and second half of the semester, where
during the first half of the semester grades are
higher for men and during the second half of the
semester grades are higher for women. (See the
figure below.)
A Qualitative Interaction Example
4. Marginal means are the means of the rows and
the columns in a table that show the cells of a
study with a two-way ANOVA design.
5. The easiest way to understand the main effect
is to make a smaller table for each main
effect, with only the appropriate marginal
means. Separate tables allow us to focus on one
main effect at a time without being distracted
by the means in the cells.
6. When trying to examine interactions, it is
often useful to create a bar graph of the
findings. Once we have a bar graph, we can
connect each set of bars with a line.
7. Perfectly parallel lines indicate the likely
absence of an interaction, but we almost never
see perfectly parallel lines emerging from
real-life data sets. Nonparallel lines may
indicate an interaction, but we need to conduct
an ANOVA and compare the F statistic with the
cutoff to be sure.
> Discussion Question 14-6
What is a quick, visual way to see if there is a
significant interaction?
Your students’ answers should include:
 Producing a bar graph of the findings is a quick,
visual way to see overall patterns; then we
connect each set of bars with a line. How close
to or divergent from parallel the lines are
indicates overall interaction patterns.
8. The core idea of an interaction is summed up
by the phrase “it depends.”
Classroom Activity 14-3
Interpreting Effects Visually
In
this
activity,
students
will
gain
more
confidence interpreting and understanding main
effects and interactions visually.
 Divide the class into small groups.
 Give each group copies of Handout 14-1, including
Handout Figures 14-1 and 14-2, found at the end
of this chapter. Handout 14-1 includes the
instructions for this exercise.
 Have
students label what variables they are
studying.
 Next, have them determine whether there is a main
effect and/or interaction.
 In
addition, have them develop ideas for an
experiment and what it would mean for the
experiment to reveal such effects.
III. Conducting a Two-Way Between-Groups ANOVA
1. When conducting a two-way ANOVA, we still use
the same six steps of hypothesis testing.
2. The assumptions for a two-way ANOVA are the
same as those for the one-way ANOVA. The sample
should be randomly selected, the populations
should
be
normally
distributed,
and
the
population variances should be equal.
3. In a two-way ANOVA, we will have three sets of
hypotheses for each of our F statistics. The
null hypothesis for the main effects is the
same as the one for the one-way ANOVA. For the
interaction, we state the null hypothesis in
words: the effect of one independent variable
does not depend on the levels of the other
independent variable. The research hypothesis
is that the effect of one independent variable
depends on the level of the other independent
variable.
> Discussion Question 14-7
What are the null hypotheses in a two-way ANOVA?
Your students’ answers should include:
 The three null hypotheses correspond to the three
elements of the ANOVA, the two main effects, and
the interaction.
 The null hypotheses for the main effects are that
there are no differences between groups.
 The
null hypothesis for the interaction is that
the effect of one independent variable is not
dependent on the levels of the other independent
variable.
Classroom Activity 14-4
Make It Your Own: Learning and Memory
http://opl.apa.org/Experiments/About/AboutLearning
andMemory.aspx
This online learning experiment studies traverse
patterning of arbitrary associations. The design
is a 2  3. The experiment was created by the
Online Psychology Laboratory. Explanations and
references are provided online.
4. The degrees of freedom for each independent
variable are the same as before. To calculate
the between-groups degrees of freedom for the
interaction, we multiply the degrees of freedom
for the two main effects. The within-groups
degrees of freedom are calculated the same as
those calculated for a one-way between-groups
ANOVA, by taking the sum of the degrees of
freedom in each of the cells.
5. The critical values are obtained the same way
as those obtained in the one-way ANOVA, except
there will be three cutoffs for each of the
three F statistics.
6. -Calculating the F statistic for each of the
main effects is the same as calculating the F
statistic in the one-way ANOVA. Only the F
statistic for the interaction is different. To
calculate the F statistic for the interaction,
obtain the between-groups sum of squares for
the interaction (SSaxr). The formula for SSaxr
is: SSaxr = SStotal – (SSa + SSr + SSwithin).
> Discussion Question 14-8
How do we calculate the between-groups sum of squares for
the interaction?
Your students’ answers should include:
 The
between-groups sum of squares for the
interaction is essentially what is left over when
the main effects and within-groups sum of squares
are accounted for.
 The formula for the between-groups sum of squares
for the interaction is: SSaxr = SSaxr = SStotal –
(SSa + SSr + SSwithin).
7. The last step is to compare the obtained F
statistic with the cutoff. This process is the
same as that used for the one-way ANOVA.
Classroom Activity 14-5
Affecting Memory
In
this
activity,
students
will
gain
an
opportunity to calculate a 2  2 ANOVA using a
hands-on activity.
 First,
divide the class into two groups. One
group will receive Handout 14-2A listing a series
of common four-letter words. The second group
will receive Handout 14-2B containing a series of
nonsense four-letter words. Both handouts can be
found at the end of this chapter.
 Optional:
These
handout
can
be
used
as
transparencies. You may display the handouts on
an overhead projector by photocopying them onto
acetate, or you may use them with PowerPoint by
scanning them into your computer. To use this
optional method, you will most likely have to
administer the tests separately.
 Next, teach only half the class a mnemonic device
(such as the peg-word system or the method of
loci or some other device). The other half will
not learn any mnemonic technique.
 Now, have the students try to memorize their word
lists in a set amount of time.
 After the time is up, have students write down on
a blank sheet of paper the number of words that
they remember.
 Then, have them check their work to determine how
many words they remembered correctly.
 The
number of correct words will serve as the
dependent variable, and students will use this
information
to
calculate
a
two-way
ANOVA.
Students should receive Handout 14-2C, found at
the end of this chapter, to help them complete
the ANOVA.
Classroom Activity 14-6
Two-Way ANOVAs in Context
In this activity, students will read and interpret
findings
from
the
article:
Levav,
J.,
and
Fitzsimons, G. J. (2006). The role of ease of
representation. Psychological Science, 17(3), 207–
213. (To view or purchase this article, go
to
your
local
library
or
visit
Blackwell
Publishing
online
at
http://www.blackwellpublishing.com.)
 Have
students
identify
the
two-way
ANOVAs
discussed in the article.
 Then
have the students interpret what the
findings from the analyses mean using Handout 143, found at the end of this chapter.
8. We calculate R2 as our measure of effect size
for a two-way ANOVA just as we did for a oneway ANOVA.
9. The effect size for the main effects are
calculated the same way they are calculated for
the one-way ANOVA.
10. We must now also calculate the effect size for
the interaction, R2interaction. The effect size for
the interaction is calculated with the formula:
R2interaction = SSinteraction/(SStotal – SSrows – SScolumns).
IV.
Next Steps: Variations on ANOVA
1. There are four other more complicated designs
involving ANOVAs.
These include: a mixeddesign ANOVA, a multivariate analysis of
variance (MANOVA), an analysis of covariance
2.
3.
4.
5.
(ANCOVA), and a multivariate analysis of
covariance (MANCOVA).
A mixed-design ANOVA is used to analyze the
data from a study with at least two nominal
independent variables and an interval dependent
variable; at least one independent variable
must
be
within-groups
and
at
least
one
independent variable must be between-groups.
We can use a multivariate analysis of variance
(MANOVA), a form of ANOVA, when there is more
than one dependent variable.
Multivariate
refers to the number of dependent variables,
not the number of independent variables.
Sometimes we suspect that a third variable
might be affecting our dependent variable.
Such a variable would be known as a covariate,
or an interval variable that we suspect
associates, or covaries, with our independent
variable of interest. An analysis of covariance
(ANCOVA) is a type of ANOVA in which a
covariate is included so that statistical
findings reflect effects after an interval
variable has been statistically removed.
A
multivariate
analysis
of
covariance
(MANCOVA) combines aspects of the MANOVA with
the ANCOVA. Specifically, it is an ANOVA with
multiple dependent variables and the inclusion
of a covariate.
> Discussion Question 14-9
What are the differences between a MANOVA, an ANCOVA, and
a MANCOVA?
Your students’ answers should include:
 Since
multivariate refers to the number of
dependent variables; in a MANOVA, a form of an
ANOVA, there is more than one dependent variable.
 In
an ANCOVA, a covariate—an interval variable
that we suspect associates, or covaries, with our
independent variable of interest—is included so
that statistical findings reflect effects after
an interval variable has been statistically
removed.

A MANCOVA is an ANOVA with multiple dependent
variables and includes a covariate.
PLEASE NOTE: Due to formatting, the Handouts are only available in Adobe
PDF®.