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chapter 8
Factorial and Mixed-Factorial
Analysis of Variance
Chapter Learning Objectives
After reading this chapter, you will be able to. . .
1. explain factorial and mixed-factorial designs.
2. relate sum of squares to factorial models.
3. compare, contrast, and identify various factorial designs.
4. demonstrate how to determine the main and interaction
effects in factorial designs using
multiple variables.
5. explain the combination of between- and within-group
variability to create mixed designs.
6. explain the use of partial-eta-squared (partial-h2) in ANOVA.
7. interpret results of factorial and mixed-factorial designs and
draw conclusions on these findings.
8. present relevant factorial and mixed results in APA format.
9. explain more complex design as a transition into advanced

statistical courses.
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suk85842_08_c08.indd 287 10/23/13 1:41 PM
CHAPTER 8Section 8.1 Factorial Analysis of Variance
Building on the concepts of Chapters 6 and 7 and the statistical
calculations of analysis of variance, we now consider more
complex between-group designs called factorial
ANOVA and a combination of between- and within-groups
designs known as mixed-
factorial ANOVA. The goal here is to explore the main effects,
which are the influence
of the independent variable on the dependent variable in testing
a hypothesis, and to
consider the combination of IVs influencing the DV known as
interaction effects. We will
continue to build on the magnitude of variance of the IV on a
DV, or effect size that was
introduced in Chapter 5 with Cohen’s d and in Chapters 6 and 7
with h2 and v2. Here we
will add another effect size measure, partial-h2, to the list of

effect size types.
The current chapter will also introduce even more complex
designs such as MANOVA
(multiple analysis of variance), ANCOVA (analysis of
covariance), and MANCOVA (mul-
tiple analysis of covariance). By the end, you will have a basic
understanding of factorial
designs and consider examples of these calculations using
statistical software.
8.1 Factorial Analysis of Variance
Before we consider factorial analysis of variance (ANOVA), we
first need to have a brief introduction to what are called
factorial designs. In the language of statistics, a fac-
tor is an independent variable, and a factorial ANOVA is one
that includes multiple IVs
(or factors) on one DV. Each of these relationships (i.e., an IV-
DV relationship) is called a
main effect.
As previously discussed, fluctuations in scores that are not
explained by the IV(s) in the
model emerge as error variance or unsystematic/unexplained
variance because it has not
been included in the experimental condition. Specifically, any
variability in the IV(s) that
are not related to the subjects’ DV becomes part of SS error
(SSerror) and then the MS within
(MSwith), which is a calculation of SSerror divided by the
degrees of freedom (df ).
Building on the concept of sum of squares, in the factorial
ANOVA, multiple IVs (or fac-
tors) can be included. As long as the researcher has data for

each variable, there is no theo-
retical limit to the number of factors. For each factor, a sum-of-
squares value is calculated
and divided by its degrees of freedom (df ) to produce a mean
square (MSbet). Each MSbet
is divided by the same MSwith (or error) value to produce F so
that there is a separate F for
each factor.
In addition, sometimes factors in combination affect the DV
differently than they do indi-
vidually. This is called an interaction. The F values are also
calculated for interactions
of these factors. For example, if a researcher wanted to examine
the impact of marital
status and college graduation on subjects’ optimism regarding
the economy, data would
be gathered on subjects’ marital status (married or not married)
and their college educa-
tion (graduated or did not graduate). Each of these combinations
of levels on their level
of optimism (the DV) would then be explored. Thus, the main
concept of using multiple
factor designs is to explore these interaction effects.
H1
TX_DC
BLF
TX
BL
BLL

suk85842_08_c08.indd 288 10/23/13 1:41 PM
As with all the ANOVAs discussed in prior chapters, we have
explored the concept of
variability and how that is broken down into each facet of
systematic variance (the vari-
ance that we know) and unsystematic variance (the variance that
we do not know or
error). Simply, the greater the ratio of systematic to
unsystematic variance, the higher the
F value and the stronger the overall ANOVA model. The
Factorial ANOVA model breaks
down the Total variability into between treatments (systematic
variance) and within
treatments (unsystematic/unexplained variance or error).
Subsequently, the between-
treatments portion is further portioned into factor A variability,
factor B variability, and
the interaction of these factors, A 3 B variability.
SS values, MS values, and F ratios would be calculated for all
variables that are similar to
between-group designs such as a one-way ANOVA:
• for the first factor (IV1)
• for a second factor (IV2)
• for the two factors (IVs) in combination or the interaction
The procedures involved in calculating a factorial ANOVA are
numerous and complex
(see Table 8.1). Happily they are no longer calculated by hand.

Software programs includ-
ing SPSS can perform an ANOVA with two factors (a two-way
ANOVA), three factors (a
three-way ANOVA), and so on. As you can see, the numbers in
each of these terms rep-
resent the factors or IVs, hence the term factorial designs.
Perhaps you have also noticed
that as you add factors the analysis becomes more multifaceted
because more interaction
effects between the factors will occur. This is not necessarily
more complicated, but simply
more information in the output to interpret.
Table 8.1: Formula of the factorial ANOVA model
Source SS df MS
Factor A SS(A) (a 2 1) MS(A) 5 SS(A)/
(a 2 1)
Factor B SS(B) (b 2 1) MS(B) 5 SS(B)/(b 2 1)
Interaction AB SS(AB) (a 2 1)(b 2 1) MS(AB) 5 SS(AB)/
(a 2 1)(b 2 1)
Error SSE (N 2 ab) SSE/(N 2 ab)
Total (Corrected) (N 2 1)
Source: NIST/SEMATECH e-Handbook of Statistical Methods,
http://www.itl.nist.gov/div898/handbook/, April, 2012.
Published by The
National Institute of Standards and Technology (NIST), an
agency of the U.S. Department of Commerce.
Factorial designs can be elegantly described using the number

of groups and levels of
each variable. For instance, the simplest two-way ANOVA will
have two factors with two
suk85842_08_c08.indd 289 10/23/13 1:41 PM
http://www.itl.nist.gov/div898/handbook/
The Research
Methods Knowledge
Base is an informative
website that taps more
into the design of
experiments, including
factorial designs. You can
learn more about factorial
designs by visiting the link
provided below.
http://www.social
researchmethods.net
/kb/expfact.php
Try It!
levels (or four conditions/cells). This is known as a 2 3 2
factorial
ANOVA. If we had two variables with three levels, then this
will be a
3 3 3 factorial design (nine conditions/cells). If we had three
variables
as in a three-way ANOVA, one with two levels, another with
five lev-
els, and the third with 10 levels, then this would be a 2 3 3 3 10

facto-
rial design (60 conditions/cells). You will commonly see this
notation
used in journal articles and textbooks.
The conditions or cells are calculated by simply multiplying the
levels
of each factor. Moreover, as you continue to add factors, it
becomes
more complex with main and interaction effects. So for instance
if
there was at least a sample size of 10 per cell for a 2 3 2 3 3
ANOVA
then that will be a sample size of n 5 120. This is a general rule
in cal-
culating appropriate sample sizes for factorial designs, albeit
not the
best approach. Instead GPower3 or SPSS Sample Power 3
should be
employed.
The Statistical Hypotheses in a Two-Way ANOVA
In the previous section, we discussed the simplest factorial
designs (i.e., a two-way
ANOVA where each factor has two levels). In this situation,
how would the hypotheses
read? As noted, a 2 3 2 ANOVA is essentially a combination of
two factors on a DV, so
there are three hypotheses that will be explored.
As an example, the simplest factorial design is a 2 3 2 ANOVA,
with two factors, each
with two levels: Gender (Female, Male) and Martial Status
(Married, Never Married) and
a dependent variable, work-life balance (WLB). The design will

have three null and three
alternative hypotheses. In fact, when you are performing a
factorial design, there will
always be at least three hypotheses consisting of two main
effects and one interaction effect.
For this example, the two main effects are:
H01: There is no statistically significant difference of Gender
on Work-Life
Balance
mmale_WLB 5 mfemale_WLB
Ha1: There is a statistically significant difference of Gender on
Work-Life
Balance
mmale_WLB ? mfemale_WLB
H02: There is no statistically significant difference of Marital
Status on
Work-Life Balance
mmarried_WLB 5 mnever married_WLB
Ha2: There is a statistically significant difference of Marital
Status on Work-
Life Balance
mmarried_WLB ? mnever married_WLB
suk85842_08_c08.indd 290 10/23/13 1:41 PM
http://www.socialresearchmethods.net/kb/expfact.php

The interaction hypothesis is
H03: There is no statistically significant interaction of Gender
and Marital
Status on Work-Life Balance
mmale/married_WLB 1 mmale/unmarried_WLB 5
mfemale/married_WLB 1 mfemale/unmarried_WLB
Ha3: There is a statistically significant interaction of Gender
and Marital
Status on Work-Life Balance
mmale/married 1 mmale/unmarried ? mfemale/married 1
mfemale/unmarried
Later in the chapter, we will test these hypotheses in performing
the appropriate analysis
using SPSS. See Example 1 in Section 8.6.
Effect Size Calculations for ANOVA
From our previous discussions of effect sizes, you will recall
that an effect size for a one-
way ANOVA is as such:
h2 5
SSeffect
SStotal
Formula 8.1

The eta-squared (h2) calculation is straightforward as we have
only one IV on the DV.
This is certainly useful for one-way ANOVA designs but cannot
be used in multiple-factor
designs, as in a factorial ANOVA, as it cannot give effects
based on each individual fac-
tor’s variability nor error variability. A second drawback to h2
is that it tends to overes-
timate the population variance as the calculation only involves
the sums of squares as
seen in the formula. Therefore, the solution to addressing these
drawbacks in handling a
two-way, three-way, or more complex ANOVA that involves
factors simultaneously being
analyzed with the DV is to first separate out the effects of each
factor on the DV plus any
additional effects of error, hence the term partial-H2 as
compared to h2 in Chapter 6. In this
instance, partial-h2 for each IV-DV is calculated by
Partial h2 5
SSeffect
SSeffect 1 SSerror
Formula 8.2
Here the “partialing out” of the other variables is accomplished
by each factor’s indi-
vidual variance on the DV (SSeffect) and the error term
(SSerror). This statistic is conveniently
calculated using appropriate software (e.g., SPSS). In regards to
Cohen’s (1988) effect size
guidelines, small h2 5 .01, medium h2 5 .09, and large h2 5 .25.
These are a good barome-
ter to calculating effect sizes estimates using GPower3. The

partial-h2 effect size will range
from 0 to 1 with one drawback in that it cannot be summed
across the variables to get a
total effect size.
To address the second drawback of h2 is to calculate omega-
squared (V2), which was intro-
duced in Chapter 6. Omega is less biased in its calculation than
h2 and is always lower in
magnitude than its counterpart. In addition, it measures the
overall effect that you cannot
do with partial h2. The formula for this is
v2 5
SSeffect 2 1dfeffect2 1MSerror2
SStotal 1 MSerror
Formula 8.3
suk85842_08_c08.indd 291 10/23/13 1:41 PM
CHAPTER 8Section 8.2 Applying Factorial-ANOVA Research
Design
8.2 Applying Factorial-ANOVA Research Design
Factorial ANOVA can be used in an experiment where two
factors, each with two levels, have created four independent
conditions. This is a simple classic four-group design.
Here four experimental groups are compared for significant
differences. Therefore, such
an experimental design is a 2 3 2 independent-design ANOVA
where we have two factors
with one of the factors (Time in Instruction) being the two time

points (1 hour per week
and 4 hours per week) and the other factor being the Setting
(Pull-out and In-class). The
DV or outcome variable is a measurement of Exam Performance
(EP), which is the number
of correct items of the final exam. So think of this analysis as
executing two independent-
samples t-test simultaneously with the added benefit of
interaction effect between the
factors (i.e., Time in Instruction and Setting).
For this 2 3 2 independent-group ANOVA the hypotheses will
be
H01: There is no statistically significant difference of Time in
Instruction
groups on Exam Performance
m1hr_EP 5 m4hr_EP
Ha1: There is a statistically significant difference of Time in
Instruction
groups on Exam Performance
m1hr_EP ? m4hr_EP
H02: There is no statistically significant difference of Setting
groups on
Exam Performance
min-class_EP 5 mpull-out_EP
Ha2: There is a statistically significant difference of Setting
groups on Exam
Performance

min-class_EP ? mpull-out_EP
H03: There is no statistically significant interaction of Time in
Instruction
and Setting groups on Exam Performance
m1hr/in-class_EP 1 m1hr/pull-out_EP 5 m4hr/in-class_EP 1
m4hr/pull-out_EP
Ha3: There is a statistically significant interaction of Time in
Instruction and
Setting groups on Exam Performance
m1hr/in-class_EP 1 m1hr/pull-out_EP ? m4hr/in-class_EP 1
m4hr/pull-out_EP
Figure 8.1 illustrates this 2 3 2 factorial design as a comparison
of the four group condi-
tions on the dependent variable EP. In finding significant mean
differences in EP between
groups, the null hypotheses above can be rejected, and we can
conclude that support was
found for their respective alternative hypotheses.
suk85842_08_c08.indd 292 10/23/13 1:41 PM
CHAPTER 8Section 8.2 Applying Factorial-ANOVA Research
Design
Figure 8.1: The classic four-group design experiment
Source: Trochim, William M. The Research Methods
Knowledge Base, 2nd Edition.
www.socialresearchmethods.net/kb/

More Complex Designs
To help you understand more complex designs and additional
ANOVA
concepts, we will demonstrate a condition/cell breakdown of the
simplest three-way ANOVA, that is, the 2 3 2 3 2 ANOVA,
which is
three factors with two levels each. Labeling the three factors
IV1, IV2,
IV3, and the DV, the effects will be as such:
H1: Main effect: IV1 S DV
H4: Interaction effect: IV1 3 IV2 S DV
H7: Higher-order interaction effect: IV1 3 IV2 3 IV3 S DV
As you can see, there will be three main effects, and four
interac-
tion effects for this 2 3 2 3 2 factorial design. In addition, there
will
be a three-way interaction of factors on the dependent variable
in what is known as a
higher-order interaction effect. Now you can see that as more
factors are added, the num-
ber of higher-order interactions between factors will increase.

Group 1
average
In
-c
la
s
s
P
u
ll
-o
u
t
Group 2
average
Group 3
average
1 hour/week 4 hours/week
Group 4
average
Time In Instruction
S
e
tt

in
g
Levels:
Subdivisions
of Factors
Factors:
Major Independent Variables
There are many
applets on the
Internet to help in data
analysis on all of these
statistical concepts.
One useful website is
the Statistics Online
Computational Resource.
Along with applets, there
are other resources such
as e-books, activities, and
games to help you learn
statistics—visit the link
provided below:
http://www.socr.ucla
.edu/SOCR.html
Try It!
suk85842_08_c08.indd 293 10/23/13 1:41 PM
http://www.socr.ucla.edu/SOCR.html
http://www.socr.ucla.edu/SOCR.html

CHAPTER 8Section 8.3 Determining the Results’ Practical
Importance
To demonstrate conditions or cells by way of a simple example,
assume a psychologist
wants to study the effects of Gender (Male/Female),
Relationship Status (Married/Not
Married), and Age Range (Under 40/Over 40) on the subjects’
level of depression using
the Beck’s Depression Inventory. In this scenario, a 2 3 2 3 2
factorial design, there will
be eight conditions or cells:
Gender Relationship Status Age Range
Under 40 Over 40
Male Married C1 C2
Not Married C3 C4
Female Married C5 C6
Not Married C7 C8
C1: Male 3 married 3 under 40
C2: Male 3 Married 3 Over 40
C3: Male 3 Not Married 3 Under 40
C4: Male 3 Not Married 3 Over 40
C5: Female 3 Married 3 Under 40

C6: Female 3 Married 3 Over 40
C7: Female 3 Not Married 3 Under 40
C8: Female 3 Not Married 3 Over 40
Assumptions of Factorial Designs
As with all other parametric tests we have considered, the
assumptions of factorial designs
are the same in terms of linearity, homogeneity, and normality.
In addition, since factorial
designs are more complex compared to simpler designs, it
requires a larger sample size
as more factors are added. As we have mentioned, for more
cells, with n 5 10 per cell,
the number can increase with the addition of factors and factor
level. This is a precau-
tion in factorial designs that the researcher is able to ensure
sample sizes are appropriate.
Complex designs involving multiple factors will require larger
data sets, which may not
always be feasible, and a compromise in the sample size may
lead to erroneous conclu-
sions (i.e., the statistical conclusion validity).
8.3 Determining the Results’ Practical Importance
One of the main reasons for running factorial designs is to see
the interaction effects of factors on each other and their
influence on the DV. Figure 8.2 depicts how this can
be determined visually. A general rule of thumb when using an
ocular analysis (or eyeball
suk85842_08_c08.indd 294 10/23/13 1:41 PM

CHAPTER 8Section 8.3 Determining the Results’ Practical
Importance
test) is to see if the lines of each of the levels of treatments are
parallel to each other. Simply
put, parallel 5 no interaction effect, and nonparallel 5
interaction effect. Keep in mind
that this is a general rule of thumb. Relative to the sensitivity of
the graph, lines may seem
nonparallel, indicating an interaction effect; however, that may
not be the case. In any
event, these line graphs are used to corroborate the results based
on the ANOVA table’s F
and p-values.
Figure 8.2: Line graphs of interaction effects
Source: Yatani, K. (n.d.) Statistics for HCI research. Retrieved
from http://yatani.jp/HCIstats/images/img/interactions.png
Examples that follow discuss interaction effects of analysis of
real-world data.
N
o
M
a
in
E
ff

e
c
t
No Interaction Interaction
M
a
in
E
ff
e
c
t
No significant effects, no interactions A significant interaction
Factor A is significant
Factor B is significant
Factor A and B are significant
Significant main effects and
a significant interaction
suk85842_08_c08.indd 295 10/23/13 1:41 PM
http://yatani.jp/HCIstats/images/img/interactions.png
CHAPTER 8Section 8.4 Mixed-Factorial Analysis of Variance

8.4 Mixed-Factorial Analysis of Variance
Thus far, we have discussed both within and between group
designs. Now we will describe the combination of both designs
simultaneously performed in what is termed
a mixed-factorial ANOVA. The simplest of such a design is a 2
3 2 mixed-design (four
conditions) where both the experiment and the control groups
are compared to each other
over two points in time or treatments (pre- and postmeasure, for
instance). Unlike the
2 3 2 independent four-group design in Section 8.2, the mixed
design uses each group
twice and, therefore, is also known as a two-group experimental
design. Think of this
analysis as executing both an independent-samples t-test and a
dependent-samples t-test
at the same time. Figure 8.3 gives an overview of the
calculation of df based on the total
sample size (N) and the number of treatment groups (k).
Figure 8.3: Mixed ANOVA df calculation
The sum of squares, mean squares, and calculation of F based
on the mean squares are cal-
culated with the formulas in Table 8.2. Again, these
calculations are seldom done by hand
with easy reliance on statistical programs. Calculations are
based on Yijk, an individual
score; Y.jk, a cell mean; Y.j., a mean for a level of factor A (the
between-subjects factor); Y..k,
a mean for factor B (pre- or posttest); Yi.., the mean for an
individual student; and Y. . ., the
grand mean.
Table 8.2: Mixed ANOVA model

Description SS (definitional formula) df MS F
A main effect
(between-subjects) SSA 5 1b2 1n2 a 1Y.j. 2 Y...2 2 a 2 1
SSA
dfA
MSA
MSS/A
Error term for
A main effects SSs/A 5 ba a 1Y.jk 2 Y.j.2 2 a(n 2 1)
SSs/A
dfs/A
B main effects
(within-subjects) SSB 5 1a2 1n2 a a 1Y..k 2 Y...2 2 b 2 1
SSB
dfB
MSB
MSB3S/A
dfbetween treatments
k � 1
dfwithin treatments
N � k
dftotal
N � 1

(continued)
suk85842_08_c08.indd 296 10/23/13 1:41 PM
Table 8.2: Mixed ANOVA model (continued)
Description SS (definitional formula) df MS F
Interaction
SSA3B 5 na a a 1Y.jk 2 Y.j. 2 Y..k 1 Y...2 2 (a 2 1)(b 2 1)
SSA3B
dfA3B
MSA3B
MSB3S/A
Error term for B
main effect and
interaction
SSB3S/A 5 a a a 1Yijk 2 Y.jk 2 Yi.. 2 Y.j.2 2 a(b 2 1)(n 2 1)
SSB3S/A
dfB3S/A
Total SST 5 c 1Yijk 2 Y...2 2 (a)(b)(n) 2 1
Source: Adapted from Myers, J.L., & Well, A.D., & Lorch,
R.F.,Jr. (2010). Research design and statistical analysis (3rd
Edition). Mahwah,
NJ: Erlbaum.

A Mixed-Factorial ANOVA Example
Consider the same scenario used above, but this time we will be
using two groups instead
of four groups. Each group (In-class and Pull-out) will be
subjected to two different con-
ditions/treatments (Time in Instruction, e.g., 1 hour/week and 4
hours/week). Conse-
quently, each of the two groups will be evaluated with an exam
at two different time
points.
This will be a 2 3 2 mixed-design ANOVA with the hypotheses
as follows:
H01: There is no statistically significant difference of Time in
Instruction on
Exam Performance
m1hr_EP 5 m4hr_EP
Ha1: There is a statistically significant difference of Time in
Instruction on
Exam Performance
m1hrEP ? m4hr_EP
H02: There is no statistically significant difference of Setting
groups on
Exam Performance
min-class_EP 5 mpull-out_EP
Ha2: There is a statistically significant difference of Setting
groups on Exam

Performance
min-class_EP ? mpull-out_EP
H03: There is no statistically significant interaction of Setting
groups based
on Time in Instruction on Exam Performance
m1hr/pull-out_EP 1 m1hr/pull-out_EP 5 m4hr/in-class_EP 1
m4hr/pull-out_EP
suk85842_08_c08.indd 297 10/23/13 1:41 PM
Ha3: There is a statistically significant interaction of Setting
groups based
on Time in Instruction on Exam Performance
m1hr/pull-out_EP 1 m1hr/pull-out_EP ? m4hr/in-class_EP 1
m4hr/pull-out_EP
Figure 8.4 illustrates this 2 3 2 mixed-factorial ANOVA design,
which is a comparison of
within group conditions (1 hour/week and 4 hours/week) as well
as difference between
groups (In-group and Pull-out). Within-group differences,
specifically that EP was sig-
nificantly different across groups having 1 hour/week versus 4
hour/week instruction,
allows us to reject H01: m1hr_EP 5 m4hr_EP and find support
for Ha1: m1hr_EP ? m4hr_EP or the
alternative hypothesis main effect.

Establishing significant mean differences of EP between group
Settings regardless
of time conditions such that H02: min-class_EP 5 mpull-out_EP
can be rejected in support for
Ha2: min-class_EP ? mpull-out_EP, which demonstrates the
second main effect.
Finally, establishing both within-group differences over
conditions (1 hour/week and
4 hours/week) and between-group differences (In-class and
Pull-out) allows us to reject
H03: m1hr/pull-out_EP 1 m1hr/pull-out_EP 5 m4hr/in-class_EP
1 m4hr/pull-out_EP and find support for
Ha3 : m1hr/pull-out_EP 1 m1hr/pull-out_EP ? m4hr/in-class_EP
1 m4hr/pull-out_EP.
Figure 8.4: The classic mixed-design experiment using two
groups
Source: Adapted from Trochim, William M. The Research
Methods Knowledge Base, 2nd Edition.
Group 1
average
In
-c
la
s
s
P
u

ll
-o
u
t
Group 2
average
Group 1
average
1 hour/week 4 hours/week
Group 2
average
Time In Instruction
S
e
tt
in
g
Levels:
Subdivisions
of Factors
Factors:
Major Independent Variables
suk85842_08_c08.indd 298 10/23/13 1:41 PM

CHAPTER 8Section 8.5 Presenting Results
Assumptions of Mixed Designs
For mixed designs, all of the aforementioned parametric
assumptions must be met includ-
ing normality, homogeneity, and linearity. Recall that sphericity
is the homogeneity of
differences between pairs of treatments across subjects. Ideally,
there should not be sig-
nificant differences. Testing for this entails the use of the
Mauchly’s W-test. The degree of
such differences or the level of sphericity is further quantified
by another measure called
epsilon (E). This measure ranges from 0 to 1 with 1 being the
highest measure that indi-
cates sphericity has been met. As e , 1, sphericity decreases,
which will lead to a violation
of sphericity based on the Mauchly’s W-test.
When there is a violation of sphericity, two adjustments are
made called the Greenhouse-
Geisser and Huynh-Feldt corrections, which were introduced in
Chapter 7. When using
SPSS these e-values will be calculated with the appropriate df
adjustments based on the
simple formula of multiplying the e-values by the df. For
instance, if
df 5 4
e 5 .8

Then,
Adjusted df 5 4 3 .8 5 3.2
In Chapter 7, we considered an example of a repeated-measures
design where decisions
based on the Mauchly’s W-test are made. In this chapter is an
example of performing this
analysis in SPSS and specifically examining sphericity with a
closer look at e-values.
8.5 Presenting Results
Each of the following examples presents steps for completing a
different type of ANOVA using SPSS.
SPSS Example 1: Steps for a Simple ANOVA
To start with the simplest ANOVA example using public data
from Pew Research (2010)
data set, the Changing American Family, a 2 3 2 factorial design
will be conducted using
gender (sex) and marital status (mar_status). Here sex has two
levels (male and female)
and mar_status has two as well (married, never married). We
will perform an analysis of
these two factors on participants work balance (How good a job
have you done balancing your
job, your marriage or partnership, and being a parent? Would
you say excellent, very good, good,
only fair, or poor?).
suk85842_08_c08.indd 299 10/23/13 1:41 PM

To perform the analysis, go to Analyze S General Linear Model
S Univariate. Input sex
and mar_status into the Fixed Factor(s) box and q28 into the
Dependent Variable box
(your screen should look like that in Figure 8.5). Next, click on
Plots and move sex into
Separate Lines and mar_status into Horizontal axis; then click
Add and Continue. Then
click Options and check Descriptive statistics and Estimates of
effect size. Then click
Continue and OK. Analysis results are presented in Figures 8.6
and 8.7.
Figure 8.5: SPSS steps for the 2 3 2 ANOVA
Source: Data from Pew Research: Social and Demographic
Trends. (2010). Changing American family. Retrieved from
http://www
.pewsocialtrends.org/category/datasets/
suk85842_08_c08.indd 300 10/23/13 1:41 PM
http://www.pewsocialtrends.org/category/datasets/
Figure 8.6: SPSS output for the 2 3 2 ANOVA
http://www

Between-Subjects Factors
SEX [ENTER
RESPONDENT’S SEX:]
Marital Status
1
5
1
2 Female
Value
Label
Male
Married
Never married
N
365
706
93
434
Descriptive Statistics
Dependent Variable: Q.28 How good a job have you done

balancing your job, your (IF
MARITAL = 1: marriage/IF MARITAL = 2: partnership) and
being a parent? Would you
say excellent, very good, good, only fair, or poor?
Male
SEX [ENTER
RESPONDENT’S SEX:] Marital Status
Female
Total
Never married
Mean
Married
Total
NStd. Deviation
Never married
Married
Total
Never married
Married
Total

2.57
2.40
2.08
2.38
2.28
2.11
2.43
2.25
2.27
1.331
1.040
0.783
0.999
0.886
0.798
1.136
0.919
0.948

47
434
319
387
46
365
93
706
799
Tests of Between-Subjects Effects
Dependent Variable: Q.28 How good a job have you done
balancing your job, your (IF
MARITAL = 1: marriage/IF MARITAL = 2: partnership) and
being a parent? Would you
say excellent, very good, good, only fair, or poor?
Source df FMean Square Sig.
Partial Eta
Squared
1782.957
19.990a
7.072
.000

3.163
696.693
716.683
4826.000
Intercept
Corrected Model
sex
sex * mar_status
mar_status
Error
Total
Corrected Total
a. R Squared = 0.028 (Adjusted R Squared = 0.024)
1
1
1
3
1

795
798
799
1782.957
7.072
3.163
6.663
0.000
0.876
2034.541
8.070
3.609
7.604
0.000
0.000
0.005
0.058
0.000

0.987
0.719
0.010
0.005
0.028
0.000
Type III
Sum of Squares
suk85842_08_c08.indd 301 10/23/13 1:41 PM
Figure 8.7: SPSS output graph for the 2 3 2 ANOVA
http://www
SPSS Example 2: Steps for a Complex ANOVA
Next, we will take on a more complex ANOVA example, again
using public data from
Pew Research (2010) data set, the Changing American Family.

Keep in mind that when
working with real-world data, there may be issues of missing
values and small sample
sizes. The current examples demonstrate that these issues may
arise and that researchers
need to be realistic about the type of design they can perform,
as well as the statistical con-
clusions they can draw based on the quality of the data set. That
said, the design using this
data is a 2 3 5 3 7 factorial design that will be conducted using
gender (sex), age range
(ls2), and marital status (marital). Here sex has two levels, ls2
has five levels, and marital
has seven levels. We will perform an analysis of these three
factors on participants’ quality
of life (Please tell me how satisfied you are with your life
overall).
suk85842_08_c08.indd 302 10/23/13 1:41 PM
To perform the analysis, go to Analyze S General Linear Model
S Univariate. Input ls2,
sex, and marital into the Fixed Factor(s) box and q28 into the
Dependent Variables box.
On the right click on Plots and move sex into Separate Lines
and marital into Horizontal
axis; then click Add and Continue. Then, move sex into
Separate Lines and move ls2
into Horizontal axis (your screen should look like that in Figure
8.8). Then click on Post

Hoc and move marital and ls2 into Post Hoc Tests for; check
Tukey (your screen should
look like that in Figure 8.9). Click Continue. Then click Options
and check Descriptive
statistics and Estimates of effect size. Finally, click Continue
and OK. Analysis results
are presented in Figures 8.10, 8.11, and 8.12.
Figure 8.8: SPSS steps in a factorial ANOVA design
http://www
suk85842_08_c08.indd 303 10/23/13 1:41 PM
Figure 8.9: SPSS steps in a factorial ANOVA design
http://www
Figure 8.10: SPSS output in a factorial ANOVA design
Between-Subjects Factors
LS2. To make sure that our survey includes many
different kinds of people, I need to ask your age.

Just stop me when I get to the category that
includes your age, as of your last birthday.
Are you… (READ)
MARITAL Are you currently married,
living with a partner, divorced,
separated, widowed, or have
you never been married?
SEX [ENTER RESPONDENT'S SEX:]
Value
Label N
1
3
2
1
2
1
3
2
4
6
5

9
30 to 49
18 to 29
50 to 64, OR
Female
Male
Living with
a partner
Married
Divorced
Widowed
Separated
Never been
married
Don’t know/
refused (VOL.)
396
526
134
653

403
32
19
200
3
34
125
643
suk85842_08_c08.indd 304 10/23/13 1:41 PM
(continued)
Dependent Variable: Q.1 First, please tell me how satisfied you
are with your life overall—
would you say you are [READ IN ORDER]
Mean Std.
Deviation

N
Male
18 to 29
Total
Female
Male
30 to 49
Total
Female
LS2. To make sure that
our survey includes many
different kinds of people, I
need to ask your age. Just
stop me when I get to the
category that includes
your age, as of your last
birthday. Are you… (READ)
SEX [ENTER
RESPONDENT'S
SEX:]
MARITAL Are you
currently married,
living with a partner,
divorced, separated,
widowed, or have you

never been married?
Married 0.527 101.50
Living with a partner 0.577 41.50
Separated 0.000 11.00
Never been married 0.834 381.82
Total 0.769 531.72
Married 0.583 261.50
Divorced 0.707 21.50
Total 1.003 811.77
Married 0.561 361.50
Divorced 0.707 21.50
Total 0.915 1341.75
Married 0.583 1101.56
Divorced 0.632 112.00
Widowed 0.000 12.00
Total 0.629 1581.65
Married 0.539 1521.48
Divorced 0.701 281.75
Widowed 1.225 52.00
Don’t know/Refused
(VOL.)
3.00 0.000 1

Total 1.71 1.054 238
Married 1.52 0.559 262
Living with a partner 2.36 2.098 14
Divorced 1.82 0.683 39
Separated 2.50 2.673 8
Widowed 2.00 1.095 6
Never been married 1.97 1.150 66
(VOL.)
3.00 0.000 1
Total 1.68 0.908 396
suk85842_08_c08.indd 305 10/23/13 1:41 PM
(continued)
Descriptive Statistics (continued)
Mean Std.
Deviation
N
Male

50 to 64, OR
Total
Female
LS2. To make sure that
our survey includes many
different kinds of people, I
need to ask your age. Just
stop me when I get to the
category that includes
your age, as of your last
birthday. Are you… (READ)
SEX [ENTER
RESPONDENT'S
SEX:]
MARITAL Are you
currently married,
never been married?
Married 0.919 1341.66
Divorced 2.035 262.69
Widowed 0.000 12.00
Total 1.171 1921.85
Married 0.759 2111.59
Divorced 0.670 581.72

Widowed 1.645 252.04
(VOL.)
1.00 0.000 2
Total 1.67 0.856 334
Married 1.61 0.824 345
Divorced 2.02 1.326 84
Separated 2.11 0.928 9
Widowed 2.04 1.612 26
(VOL.)
1.00 0.000 2
Total 1.74 0.986 526
Male
Total
Total
Female
Married 0.776 2541.61
Divorced 1.758 372.49
Widowed 0.000 22.00

Total 0.945 4031.75
Married 0.671 3891.54
Divorced 0.673 881.73
Widowed 1.564 302.03
(VOL.)
1.67 1.155 3
Total 1.69 0.950 653
Married 1.57 0.715 643
Divorced 1.95 1.156 125
Separated 2.16 1.834 19
Widowed 2.03 1.513 32
(VOL.)
1.67 1.155 3
Total 1.72 0.948 1056
suk85842_08_c08.indd 306 10/23/13 1:41 PM
(continued)
Tests of Between-Subjects Effects

Mean
SquareSource
Type III
Sum of Squares F
Partial Eta
SquaredSig.df
1 285.925 336.806 0.000 0.248285.925Intercept
34 2.399 2.826 0.000 0.08681.583aCorrected Model
2 2.178 2.566 0.077 0.0054.357Is2
6 2.545 2.998 0.007 0.01715.270marital
1 0.064 0.075 0.784 0.0000.064sex
10 0.876 1.032 0.414 0.0108.757Is2 * marital
2 1.664 1.961 0.141 0.0043.329Is2 * sex
5 2.205 2.597 0.024 0.01311.025sex * marital
1021 0.849866.757Error
8 1.068 1.258 0.262 0.0108.542Is2 * sex * marital
10564061.000Total
1055948.340Corrected Total

a. R Squared = 0.086 (Adjusted R Squared = 0.056)
suk85842_08_c08.indd 307 10/23/13 1:41 PM
(continued)
http://www
Multiple Comparisons
Tukey HSD
Mean
Difference
(I-J)
Married
(I) MARITAL Are you
currently married,
never been married?

(J) MARITAL Are you
currently married,
never been married?
95% Confidence
Interval
Std.
Error
Lower
Bound
Upper
Bound
Sig.
Living with a partner
Divorced
Separated
Widowed
Never been married
(VOL.)
Married
Divorced
Separated
Widowed

Never been married
(VOL.)
Divorced
Married
Separated
Widowed
Never been married
(VOL.)
Separated
Married
Divorced
Widowed
Never been married
(VOL.)
Widowed
Married
Divorced
Separated
Never been married
(VOL.)
Never been married

Married
Divorced
Separated
Widowed
(VOL.)
(VOL.)
Married
Divorced
Separated
Widowed
Never been married
�0.43
�0.38*
�0.59
�0.46
�0.34*
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0.43
0.05
�0.16
�0.03
0.09
0.33

0.38*
�0.05
�0.21
�0.08
0.04
0.29
0.59
0.16
0.21
0.13
0.25
0.49
0.46
0.03
0.08
�0.13
0.12
0.36
0.34*
�0.09
�0.04
�0.25
�0.12
0.24
0.10
�0.33

�0.29
�0.49
�0.36
�0.24
0.162
0.090
0.214
0.167
0.075
0.533
0.162
0.178
0.264
0.227
0.171
0.555
0.090
0.178
0.227
0.183
0.105
0.538
0.214
0.264
0.227
0.267
0.221
0.572

0.167
0.227
0.183
0.267
0.175
0.556
0.075
0.171
0.105
0.221
0.175
0.536
0.533
0.555
0.538
0.572
0.556
0.536
0.108
0.000
0.087
0.081
0.000
1.000
0.108
1.000
0.997
1.000

0.998
0.997
0.000
1.000
0.971
0.999
1.000
0.998
0.087
0.997
0.971
0.999
0.922
0.978
0.081
1.000
0.999
0.999
0.993
0.995
0.000
0.998
1.000
0.922
0.993
0.999

1.000
0.997
0.998
0.978
0.995
0.999
�0.91
�0.65
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�0.96
�0.56
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0.12
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0.91
0.57
0.62
0.64
0.59
1.97
0.65
0.48
0.46
0.46
0.35
1.88
1.22
0.94
0.88
0.91
0.90
2.18
0.96
0.70
0.62
0.66
0.64
2.01
0.56
0.41
0.27
0.41

0.40
1.83
1.67
1.31
1.30
1.20
1.28
1.34
Based on observed means.
The error term is Mean Square(Error) = 0.849.
* The mean difference is significant at the 0.05 level.
suk85842_08_c08.indd 308 10/23/13 1:41 PM
Figure 8.11: SPSS output graph in a factorial ANOVA design
http://www
Figure 8.12: SPSS output graph in a factorial ANOVA design
http://www

suk85842_08_c08.indd 309 10/23/13 1:41 PM
SPSS Example 3: Steps for a Mixed-Factorial ANOVA
The last example is a mixed-factorial ANOVA where a
researcher would like to compare
groups in a longitudinal study. As discussed earlier in the
chapter, this involves both a
between-group and within-group design simultaneously. Using
the data set available
from the Statistical Consulting Group at the University of
California Los Angeles, Stress_
Data_dataset (Figure 8.13), an occupational therapist wishes to
evaluate the effectiveness
of two stress treatments over time. After each stress treatment
the therapist gives a reliable
and valid stress test to her patients that indicates higher stress
based on a larger numerical
value. In addition, she has a control group that is not going
through any stress treatments.
Based on this scenario and the dataset provided, this will be a 3
3 3 mixed-ANOVA design
with three measurement times of patients stress levels
(measurement_time) and compari-
son of the three groups (treatment_1, treatment_2, and control).

• Go to Analyze S General Linear Model.
• Click Repeated Measures.
• Type in Measurement_Time in the Within-Subject Factor
Name box, 3 in the
Number of Levels box, and Stress_Level in the Measure Name
box (your screen
should look like that in Figure 8.14).
• Click Define.
• As shown in Figure 8.15, put in the three measurement times
(Time_1, Time_2,
and Time_3) in simultaneous order in the Within-Subjects
Variables box.
• Move group into the Between-Subjects Factor(s) box (your
screen should look
like that in Figure 8.15).
• On the right, click Plots and move Measurement_Time into the
Horizontal axis.
• Click Add.
• Do the same for group, and then put Measurement_Time into
the Horizontal
axis, and group into Separate Lines (your screen should look
like that in Figure
8.16).
• Click Add.
• Click Options and move Measurement_Time into Display
Means for.
• Click Compare Main Effects and select Sidak from the
dropdown box just below.
• Click on Post Hoc and move group into the Post Hoc Tests for
and check Dun-

nett and Dunnett’s C (a commonly used post hoc test when
comparing treatment
groups to control groups).
• Click Options.
• Click Descriptive statistics, Estimates of effect size, and
Homogeneity tests.
• Click Continue.
• Click OK. Analysis results are presented in Figures 8.17, 8.18,
8.19, and 8.20.
suk85842_08_c08.indd 310 10/23/13 1:41 PM
Figure 8.13: SPSS steps for a mixed-factorial design
Source: Based on data from Introduction to SAS (2007). UCLA:
Statistical Consulting Group. Retrieved from
http://www.ats.ucla.edu
/stat/spss/dae/manova1.htm
suk85842_08_c08.indd 311 10/23/13 1:41 PM
http://www.ats.ucla.edu/stat/spss/dae/manova1.htm

from http://www.ats.ucla.edu/stat/spss/dae/manova1.htm
from http://www.ats.ucla.edu/stat/spss/dae/manova1.htm
suk85842_08_c08.indd 312 10/23/13 1:41 PM
Figure 8.17: SPSS output for a mixed-factorial design
Between-Subject Factors
Value Label N

treatment 2
treatment 1
control
11
11
11
group
1.00
2.00
3.00
Mean Std. Deviation N
treatment_2
treatment_1
control
11
11
11

group
Time_1
18.1182
15.5273
15.3455
3.90380
2.07562
3.13827
Total 3316.3303 3.29246
treatment_2
treatment_1
control
11
11
11
Time_2
6.1909
5.5818
5.3727

1.89971
2.43426
1.75903
Total 335.7152 2.01760
treatment_2
treatment_1
control
11
11
11
Time_3
8.6818
5.1091
5.6364
4.86309
2.53119
3.54691
Total 336.4758 3.98513

Mauchly’s Test of Sphericitya
Measure: StressLevel
Mauchly’s WWithin Subject
Effect
Approx.
Chi-Square
0.750 0.500Measurement_Time
Tests the null hypothesis that the error covariance matrix of the
orthonormalized transformed dependent
variables is proportional to an identity matrix.
a. Design: Intercept + group
Within Subjects Design: Measurement_Time
b. May be used to adjust the degrees of freedom for the
averaged tests of significance. Corrected tests are
displayed in the Tests of Within-Subjects Effects table.
0.532 18.298 2 0.000 0.681
Epsilonbdf Sig.
Greenhouse-
Geisser
Huynh-
Feldt
Lower
Bound
suk85842_08_c08.indd 313 10/23/13 1:41 PM

(continued)
Tests of Within-Subjects Effects
Source
Measurement_Time
Measurement_Time *
group
Error(Measurement_Time)
F
102.730
102.730
102.730
102.730
0.640
0.640
0.640
0.640

Sig.
0.000
0.000
0.000
0.000
0.636
0.579
0.594
0.535
Partial
Eta
Squared
0.774
0.774
0.774
0.774
0.041
0.041
0.041

0.041
Type III
Sum of
Squares
2314.092
2314.092
2314.092
2314.092
28.815
28.815
28.815
28.815
675.776
675.776
675.776
675.776
df
2
1.362

1.500
1.000
4
2.725
3.000
2.000
60
40.874
45.005
30.000
Mean
Square
1157.046
1698.457
1542.544
2314.092
7.204
10.575

9.604
14.407
11.263
16.533
15.015
22.526
Sphericity Assumed
Greenhouse-Geisser
Huynh-Feldt
Lower-Bound
Sphericity Assumed
Greenhouse-Geisser
Huynh-Feldt
Lower-Bound
Sphericity Assumed
Greenhouse-Geisser
Huynh-Feldt
Lower-Bound

Tests of Within-Subjects Contrasts
Source
Measurement_Time
Measurement_Time
Measurement_Time *
group
Error(Measurement_Time)
F
84.639
198.017
0.075
3.615
Sig.
0.000
0.000
0.928
0.039
Partial
Eta

Squared
0.738
0.868
0.005
0.194
Type III
Sum of
Squares
1602.349
711.743
2.825
25.989
567.945
107.831
df
1
1
2
2

30
30
Mean
Square
1602.349
711.743
1.413
12.995
18.932
3.594
Linear
Quadratic
Linear
Quadratic
Linear
Quadratic
Tests of Within-Subjects Effects
Transformed Variable: Average

Source
Intercept
group
Error
F
1571.235
9.650
Sig.
0.000
0.001
Partial
Eta
Squared
0.981
0.391
Type III
Sum of
Squares
8948.055
109.914

170.848
df
1
2
30
Mean
Square
8948.055
54.957
5.695
suk85842_08_c08.indd 314 10/23/13 1:41 PM
(continued)
Figure 8.18: SPSS output graph in a mixed-factorial design

http://www.ats.ucla.edu/
stat/spss/dae/manova1.htm
Multiple Comparisons
(I) group (J) group Mean
Difference
(I-J)
Std. Error Sig.
0.001
0.996
0.56986
0.60904
0.56986
0.58290
0.60904
0.58290
0.58749
0.58749
2.2576*

2.2121*
�2.2576*
�0.0455
�2.2121*
0.0455
2.2121*
�0.0455
treatment_2
control
treatment_1
control
treatment_1
treatment_2
control
control
treatment_1
treatment_2
treatment 1

treatment 2
control
Dunnett C
Dunnett t
(2-sided)b
95% Confidence
Interval
0.6954
0.5426
�3.8197
�1.6434
�3.8817
�1.5524
0.8487
�1.4089
Lower
Bound
3.8197
3.8817

�0.6954
1.5524
�0.5426
1.6434
3.5755
1.3180
Upper
Bound
Based on observed means.
The error term is Mean Square(Error) = 1.898.
*. The mean difference is significant at the .05 level.
b. Dunnett t-tests treat one group as a control, and compare all
other groups against it.
suk85842_08_c08.indd 315 10/23/13 1:41 PM

suk85842_08_c08.indd 316 10/23/13 1:41 PM
CHAPTER 8Section 8.6 Interpreting Results
8.6 Interpreting Results
Refer to the most recent edition of the APA manual for specific
detail on formatting statistics; Table 8.3 may be used as a quick
guide in presenting the statistics covered
in this chapter.
Table 8.3: Guide to APA formatting of F statistic results
Abbreviation or term Description
F F test statistic score
Partial-h2 An effect size based on part of the factorial design
for one factor
W Mauchly’s test of sphericity

SS Sum of squares
MS Mean square
e Epsilon
M Box’s test of equality of covariance matrices
Source: Publication Manual of the American Psychological
Association, 6th edition. © 2009 American Psychological
Association,
pp. 119–122.
Using the results from SPSS Example 1 of the 2 3 2 ANOVA,
we present the results
(Figure 8.6), in the following way:
• There is a significant main effect of sex on work balance F(1,
795) 5 8.070,
p , .05, partial-h2 5 .01, specifically males (M 5 2.40, SD 5
1.04) reported a
significantly higher mean than females’ scores (M 5 2.11, SD 5
.798) in regards
to work balance.
• There is no significant main effect of marital status on work
balance
F(1, 795) 5 3.609, p 5 .058, partial-h2 5 .005, specifically
married (M 5 2.25,
SD 5 .919) reported no significant differences than never
married (M 5 2.11,
SD 5 .798) in regards to work balance.
• There is no interaction effect between sex and marital status in
regards to the

work balance F(1, 795) 5 .000, p 5 .987, partial-h2 5 .000. This
is also corrobo-
rated in the interaction line graph, shown in Figure 8.7, where
both gender lines
are parallel to each other across marital status on the x-axis.
Using the results from SPSS Example 2 of the 2 3 5 3 7
ANOVA, we present the results
(Figure 8.10), in the following way:
• There is no significant main effect of age on quality of life
F(2, 1021) 5 2.566,
p 5 .077, partial-h2 5 .005
• There is no significant main effect of sex on quality of life
F(1, 1021) 5 .075,
p 5 .784, partial-h2 5 .000, specifically males (M 5 1.95, SD 5
1.156) reported a
significantly higher mean than females (M 5 1.69, SD 5 .940) in
regards to qual-
ity of life.
• There is a significant main effect of marital status on quality
of life F(1, 1021)
5 2.998, p 5 .007, partial-h2 5 .017, specifically divorce scores
(M 5 1.95,
SD 5 .940) reported a significantly higher mean than married
(M 5 1.57,
SD 5 .715) in regards to quality of life. Additionally never been
married scores
suk85842_08_c08.indd 317 10/23/13 1:41 PM
CHAPTER 8Section 8.7 Other Factorial Designs

(M 5 1.91, SD 5 1.028) reported a significantly higher mean
than married
(M 5 1.57, SD 5 .715).
• There is no interaction effect between the age and sex on
quality of life
F(1, 1021) 5 1.961, p 5 .141, partial-h2 5 .004.
• There is no interaction effect between the age and marital on
quality of life
F(1, 1021) 5 1.032, p 5 .414, partial-h2 5 .010.
• There is an interaction effect between the sex and marital on
quality of life
F(1, 1021) 5 2.597, p , .05, partial-h2 5 .013. This is
corroborated in the interac-
tion line graph, shown in Figure 8.12, where both gender lines
are nonparallel to
each other across marital status on the x-axis. Looking at this
graph a bit closer,
you can see substantially lower quality of life scores for males
living with a part-
ner than women. The opposite is true for divorced males whose
scores represent
a higher quality of life than divorced females.
• There is no higher-order interaction effect between the age,
sex, and marital on
quality of life F(1, 1021) 5 1.258, p 5 .262, partial-h2 5 .010.
Using the results from SPSS Example 3, Figure 8.17, we could
present the results in the
following way:
• According to Mauchly’s test of Sphericity, W 5 .532, x2 5

18.30, p , .05; there-
fore, a violation of sphericity has occurred. Therefore, using the
Huynh-Feldt,
the e 5 .750 adjustment, there is a significant difference in
measurement times
of stress within groups F(1.5, 45.00) 5 102.73, p , .05, partial-
h2 5 .774, specifi-
cally Time_1 (M 5 16.33) is significantly different from Time_2
(M 5 5.715) and
Time_3 (M 5 6.476).
• There were also overall between-group differences, F(2, 30) 5
9.65, p , .05,
partial-h2 5 .391, specifically Treatment_1 (M 5 10.99) was
significantly different
than Treatment_2 (M 5 8.74) and the control group (M 5 8.78).
• There were no interaction effects for the independent
variables, F(3, 45.00) 5 .640,
p , .535, partial-h2 5 .041.
8.7 Other Factorial Designs
As mentioned, the addition of variables to factorial designs will
become more intri-cate, yet interesting with various interactions
in the model, which is usually the core
interest of researchers. This makes the analysis of theoretical
models, involving multiple
variables that may include between-, within-, and mixed-
designs, possible. Some of these
more complex designs, found in advanced (multivariate)
statistical textbooks, are worth
mentioning for your own knowledge, understanding, and
perhaps a general love of sta-
tistical techniques.

Multivariate Analysis of Variance (MANOVA)
Building upon factorial designs discussed thus far, we have
started with one-way,
two-way, and repeated-measures ANOVAs, and by now it
should be clear at this point
that there is a commonality among all of these in the use of only
one dependent vari-
able. That said, factorial designs could become even more
complex with the addition
suk85842_08_c08.indd 318 10/23/13 1:41 PM
CHAPTER 8Section 8.7 Other Factorial Designs
of multiple DVs, which are known as a multivariate analysis of
variance (MANOVA).
The use of multiple DVs adds to increased complexity in
regards to interaction effects
and the use of multiple factors acting together accounting for
systematic variance. As
such, the MANOVA is a way to control for family-wise rate
error (FWER) as discussed in
Chapter 6, where, as in this case, running several separate
ANOVAs will inflate the type I
error. To combat this issue, the omnibus MANOVA test is used
with consequential follow-
up ANOVAs and post hoc tests.
When executing the MANOVA—as seen in Figure 8.21—the
general modus operandi is to
perform the overall MANOVA first with all of the variables
(IVs and DVs) tested together,
then followed by subsequent ANOVAs, where each of the DVs

are tested separately. Con-
sequently, if these ANOVAs are significant, then post hoc tests
are involved to identify
specific mean difference. If this is done by manual calculations,
it will take a painfully
long time, but with the use of software such as SPSS, R, or
SAS, it will take just a few
minutes to execute. That said, MANOVAs are quite commonly
used for dissertations and
reported in journal articles when testing theoretical models that
involve many variables.
To reiterate, the elegance of the MANOVA is that the variables
can be testing simultane-
ously to explore main and interaction effects using multiple IVs
and DVs.
Figure 8.21: General modus operandi for executing the
MANOVA
Analysis of Covariance (ANCOVA) and Multivariate Analysis
of
Covariance (MANCOVA)
This section on multivariate analysis takes a different approach
to the addition of vari-
ables. As background context, commonly in a laboratory setting
as in biology, chemistry,
and physics labs, control of unforeseen or nuisance variables
that can affect the outcome
of the experiment is imperative and standard in these
disciplines. By controlling the envi-
ronment (e.g., the lights, sounds, equipment), the experimenter
will eliminate chances of
these variables affecting the outcome. This control, which will
ultimately lead to what is
called internal validity, can also be consistent across

experiments and with each partici-
pant. Subsequently, high internal validity will lead to a strong
conclusion on a factor’s
influence on a DV that is called statistical conclusion validity.
On the other hand, psychological and business research involves
collecting data in a field
setting as in an organization, on the street, or in some other
public venue. Such method-
ologies make it very difficult and almost impossible to control
all environmental factors
that can affect the field experiment. Because of inherent
limitations, known as nuisance
variables or confounding variables, internal validity that can
affect the IV-DV relation-
ship and hence the statistical conclusion validity is
compromised. To combat these extra
variables, the researcher must think about what these variables
are and then how to mea-
sure them. If they are measured, they can be added to the
analysis as a covariate to gauge
MANOVA ANOVA Post hoc test
suk85842_08_c08.indd 319 10/23/13 1:41 PM
CHAPTER 8Key Terms
their level of influence on the IV-DV relationship. One of the
common ways to do this is
by running an analysis of covariance (ANCOVA). By being
knowledgeable about con-
founding influences or covariates, the researcher can decide
whether they may have had

a covarying influence with the IV on the DV. If these covariates
are discovered to have an
influence, then limitations of the study are duly reported as
best-practice research.
As in other types of ANOVAs, including ANCOVA, the
commonality is one DV. Conse-
quently, if there are multiple DVs, then a multiple analysis of
covariance (MANCOVA)
may be used.
Summary
The chapter provided a wrap-up of the comparison of between-
group designs that started
with t-tests in Chapter 5 to ANOVAs in Chapter 6; and then
comparison of groups over
time and treatments, as in within-groups designs from Chapter
7. The current chapter
explored factorial designs including the analysis of multiple IVs
on a DV as well as mixed
designs that are both between-group and within-group designs
performed simultaneously
(Objective 1). Breakdowns of the sum of squares variances were
shown for both types of
designs as a way to compare and contrast both models
(Objectives 2, 3, and 5). Main
and interaction effects were explained as an important
component to factorial designs
using multiple variables (Objective 4). New methods of effect
sizes such as partial-h2
(Objective 6) were discussed and calculated. In addition, we
presented the SPSS steps of
these analyses, interpreted, and reported the results in proper
APA formatting (Objectives
7 and 8). We ended the chapter with a brief overview of even
more complex multivariate

designs that include ANCOVA, MANOVA, and MANCOVA
(Objective 9).
Key Terms
analysis of covariance (ANCOVA) The
analysis used for the influence of a factor
(IV) or multiple factors on a single depen-
dent variable while exploring the influence
of covariate variables and their influence
on the IV-DV relationship.
covariate A distinction used for a third
variable that covaries with the independent
variable to affect the dependent variable.
epsilon (E) A measure of sphericity that
ranges from 0 to 1 with 1 being the maxi-
mum level. Values of e , 1 may lead to a
violation of sphericity that is detected using
the Mauchly’s W-test.
factorial designs Research design involv-
ing one (one-way ANOVA) or multiple
(e.g., two-way or three-way ANOVA) fac-
tors on a dependent variable or multiple
dependent variables (MANOVA).
suk85842_08_c08.indd 320 10/23/13 1:41 PM
CHAPTER 8Chapter Exercises
higher-order interaction effect The inter-
action of three or more factors on the

dependent variable. As more factors are
added, more higher-order interactions will
occur.
interaction The influence of one factor (IV)
on another factor manifested in their com-
bined influence on the dependent variable.
main effects The influence of a factor (IV)
on the dependent variable in a factorial
design.
multivariate analysis of covariance
(MANCOVA) The analysis used for the
influence of a factor (IV) or multiple fac-
tors on multiple dependent variables while
exploring the influence of covariate vari-
ables and their influence on each IV-DV
relationship.
multivariate analysis of variance
(MANOVA) The analysis used for the
influence of a factor (IV) or multiple factors
on a multiple dependent variable.
omega-squared (V2) Measures the overall
effects size and is less biased than e2 in its
calculation and is always lower in magni-
tude than e2.
partial-H2 A measure of effect size where a
partial measurement is given for each fac-
tor on the dependent variable, that is, each
IV-DV effect size for the factorial design.
These values cannot be summed to give a
total effect size but rather omega-squared

(v2) must be calculated.
Chapter Exercises
Review Questions
The answers to the odd-numbered items can be found in the
answers appendix.
Again, using the data set available from the Statistical
Consulting Group at the University
of California Los Angeles, Drug_Treatment_dataset (see Figure
8.22), perform a mixed-
methods ANOVA. Assume that you as a researcher want to
evaluate the effects two treat-
ment (Drug_1 and Drug_2) using a Treatment and two Control
(control_1 and control_2)
groups and answer the following questions.
1. What type of factorial design is this? Be specific (e.g., 2 3 2
factorial design).
2. What are the assumptions prior to running a factorial design?
3. List all the hypotheses for this factorial design.
After performing your factorial design analysis in Excel, SPSS,
or other capable software
program, answer the following questions.
suk85842_08_c08.indd 321 10/23/13 1:41 PM
4. What are the main effect results? Are they significant? How

do you know?
5. What are the interaction results? Are they significant? How
do you know?
6. Report the effect sizes for each of the main and interaction
effects. What do these
sizes convey in regards to Cohen’s (1988) values?
7. Based on your analysis, write your overall conclusions in
proper APA formatting.
Figure 8.22: Dataset for the Review Questions factorial analysis
Source: Based data from Introduction to SAS (2007). UCLA:
http://www.ats.ucla.edu/stat
/spss/dae/manova1.htm
suk85842_08_c08.indd 322 10/23/13 1:41 PM
Analyzing the Research
Review the article abstracts provided below. You can then
access the full articles via your
university’s online library portal to answer the critical thinking
questions. Answers can be
found in the answers appendix.
Using Factorial Design for a Cognitive Strategies Study

Forys, K. L., & Dahlquist, L. M. (2007). The influence of
preferred coping style and cogni-
tive strategy on laboratory-induced pain. Health Psychology,
26(1), 22–29.
Article Abstract
Objective: To evaluate the effects of matching an individual’s
coping style (low, mixed, or
high monitoring) to an appropriate cognitive strategy
(distraction or sensation monitor-
ing) to improve pain management. Design: This study used a
split-plot factorial design in
a laboratory setting. Main Outcome Measures: Main outcomes
were pain threshold, pain
tolerance, pain intensity, pain affect, and anxiety. Results: The
results of the 2 3 3 3 3
(Experimental Condition 3 Coping Style 3 Trial) analysis of
variance (ANOVA) inter-
action were significant for pain threshold scores, F(4, 178) 5
2.95, p , .01. Low moni-
tors in the matched distraction trial had higher pain threshold
scores than during base-
line, t(15) 5 22.68, p , .017, and the mismatched sensation
monitoring trial, t(15) 5 2.80,
p , .014. High monitors’ pain threshold scores were higher than
baseline only during the
matched sensation monitoring trial, t(27) 5 22.75, p , .010. The
results of the 2 3 3 3 3
ANOVA interaction were not significant for pain tolerance
scores; however, when the
mixed monitors were excluded, the 3-way interaction was
significant, F(2, 124) 5 3.48,
p , .05. The results were nonsignificant for pain intensity, pain
affect, and anxiety.
Conclusion: Results demonstrate that matching coping style to

the appropriate cognitive
strategy is important for improving pain threshold and pain
tolerance; however, match-
ing did not reduce pain intensity, pain affect, or anxiety. Future
studies should explore
the explanation for differential responses of high and low
monitors and should test these
hypotheses in a clinical setting.
Critical Thinking Questions
1. What is the factorial design in this study?
2. How many independent variables are used in this study?
3. Why does this study use a factorial analysis of variance?
4. In the study, it was determined there was a significant Coping
Style 3 Experimental
Condition 3 Trial interaction for pain threshold scores. What
does this interaction
show?
Using Mixed-Factorial ANOVA for a Performance Patterns
Study
Oliver, R., & Williams, R. L. (2006). Performance patterns of
high, medium, and low per-
formers during and following a reward versus non-reward
contingency phase.
School Psychology Quarterly, 21(2), 119–147.
suk85842_08_c08.indd 323 10/23/13 1:41 PM

Article Abstract
Three contingency conditions were applied to the math
performance of 4th and 5th grad-
ers: bonus credit for accurately solving math problems, bonus
credit for completing math
problems, and no bonus credit for accurately answering or
completing math problems.
Mixed ANOVAs were used in tracking the performance of high,
medium, and low per-
formers during the experimental phase across a mandatory
follow-up phase and a choice
follow-up phase. The two reward contingencies produced
generally higher performance
than the non-reward contingency (control condition) in the
experimental phase, but all
performance levels did better in the mandatory follow-up phase
after the non-reward
condition than after either reward contingency. Plus, high
performers did substantially
better in the choice phase following a non-reward contingency
than following either
reward contingency, most especially following the accuracy
contingency. The pattern of
results generally points to an overjustification effect for
contingent bonus credit, with this
effect more attributable to a perception of control than a
perception of competency.
Critical Thinking Questions
1. What are the independent variables used in the study that
made them choose to
run a mixed-factorial ANOVA?

2. If the study was worried about violating the assumption of
sphericity, what test
should be run to test for this?
3. If there were a main effect, what would this mean? Rewrite
the nonsignificant
main effect to make it significant. F(5, 68) 5 .87, p 5 .51.
suk85842_08_c08.indd 324 10/23/13 1:41 PM
Article
Leadership and the more-
important-than-average effect:
Overestimation of group goals
and the justification of
unethical behavior
Crystal L. Hoyt
University of Richmond
Terry L. Price
Alyson E. Emrick
Abstract
This research investigates the empirical assumptions behind the
claim that leaders exaggerate the

importance of their group’s goals more so than non-leaders and
that they may use these beliefs to
justify deviating from generally accepted moral requirements
when doing so is necessary for goal
achievement. We tested these biased thought processes across
three studies. The results from
these three studies established the more-important-than-average
effect, both for real and illusory
groups. Participants claimed that their group goals are more
important than the goals of others,
and this effect was stronger for leaders than for non-leading
group members. In Study 3, we
demonstrated the justification bias and connected this bias to
beliefs about the importance of
group goals. Participants indicated that they would be more
justified than others in engaging in
unethical behaviors to attain their group’s goals; leaders
reported being more justified in such
deviations than non-leaders; and the more highly leaders
evaluated their group’s goals, the greater
justification bias they reported.
Keywords
self-enhancement bias, more-important-than-average effect,
group goals, leadership, unethical

behavior, ethics, morality
Corresponding author:
Crystal L. Hoyt, University of Richmond, USA
Email: [email protected]
Leadership
6(4) 391–407
! The Author(s) 2010
Reprints and permissions:
sagepub.co.uk/journalsPermissions.nav
DOI: 10.1177/1742715010379309
lea.sagepub.com
Introduction
Social psychologists have recently shown an increased interest
in understanding ethics and
morality (Haidt, 2008), issues that have long concerned those in
the field of philosophy. For
more than a decade now, philosophers working in ethical theory
have also paid greater
attention to experimental work in social psychology (Doris,
1998, 2002; Flanagan, 1991;
Glover, 2000; Harman, 1999, 2003). Advocates of this approach
to ethics claim that moral

theorizing must be appropriately informed by research on well-
established psychological
phenomena. How people think about morality, as well as how
they are motivated by
what they take to be its requirements, has important
implications for what we can legiti-
mately expect in terms of ethical behavior. Doris (1998, 2002),
for example, uses studies on
helping behavior and obedience to authority in his critique of
virtue theories of ethics.
Contrary to the situationist perspective that dominates social
psychology, advocates of
virtue ethics assume stable behavioral dispositions. According
to virtue ethicists’ empirically
informed critics, social psychological findings give us reason to
question the virtue ethicist’s
claim that people can rely upon personal traits to behave
morally across situations.
Social psychologists’ mounting interest in empirical ethics can
thus provide data that either
support or undermine the ethical thinking of philosophers. One
such empirical claim is central
to the philosophical argument that leaders fail ethically not only
because they believe they can
get away with immorality but also because they believe that
their goals are sufficiently impor-
tant to justify deviating from moral requirements (Price, 2006).
This argument is based on a
conceptual distinction between understanding the content of a
moral requirement—for exam-
ple, that lying is generally wrong—and understanding the scope
of that requirement, that is,
whether the requirement applies in a particular case (Hampton,
1989; Price, 2006). Given this
distinction, leaders can accept a general moral requirement but

believe that they are justified in
deviating from it because they think too highly of their group
goals. In short, leaders can
believe that their rule-breaking behavior was not wrong after
all. This theory of ethical failures
in leadership lends itself to social psychological research
because it makes the empirical
assumption that leaders will be inclined to overestimate the
importance of their goals and
that these biases are connected to beliefs about justification.
In the social psychological literature, studies on self-
enhancement phenomena typically
focus on individuals’ perceptions of their own traits and
behaviors relative to the traits and
behaviors of others. Focusing the studies in this way may be
important for understanding
psychological mechanisms that explain why leaders sometimes
behave immorally. Leaders’
beliefs that they are particularly virtuous or that their behavior
is uncharacteristically ethical
can compete with the view that they are capable of doing
something immoral. However,
ethical reasoning involves more than a leader’s views of his or
her own traits and behaviors.
For one thing, it involves the leader’s views about the
importance of his or her group goals.
The goal of this paper is to demonstrate how an extension of
self-enhancement phenomena
can help us understand the way people, and leaders in
particular, think about morality. We
empirically address the following questions: Do people think
that their group goals are more
important than average? In other words, is there a more-
important-than-average effect
(MITA) for people’s beliefs about their goals? Moreover, are

these biases stronger for
leaders than for non-leading group members? Finally, is there
any connection between
these biases and beliefs about justification for engaging in
unethical behavior in the service
of their group’s goals? That is, do leaders who exaggerate the
importance of their groups’
goals also believe that they are more justified than average?
392 Leadership 6(4)
The more-important-than-average effect
The self-enhancement bias, which results from comparative
judgments with others at the
individual level, is variously referred to as the better-than-
average effect, the above-average-
effect, the uniqueness bias, and the Lake Wobegon effect.
According to Alicke and Govorun
(2005: 85), our inclination to see ourselves in an
uncharacteristically positive light is ‘one of
social psychology’s chestnuts’, having been confirmed ‘in
numerous studies, with diverse
populations, on multiple dimensions, and with various
measurement techniques’.
Goethals, Messick and Allison also note the pervasive
differential between how we view
others and how we view ourselves:
The uniqueness bias reflects our tendency to see ourselves as
somewhat better than average, a
tendency that has been observed in a wide variety of domains
including vulnerability to major

life events, driving abilities, responses to victimization,
perceptions of fairness, and goodness.
(1991: 19)
Although the better-than-average effect is considered to be ‘one
of the most robust of all
self-enhancement phenomena’, (Alicke and Govorun, 2005: 85)
it has not been demonstrated
to apply to the goals of the group to which one belongs. In this
context, to say that one’s
goals are better than average means that they are more
important than average. Yet there are
good reasons to expect that the general effect extends to group
goals. These reasons appeal
to the sources and limitations of the effect itself. Self-
evaluation biases have been shown to
stem from both non-motivated (e.g. information-processing
limitations) and motivated
sources (e.g. to see oneself in the best possible light; Chambers
and Windschitl, 2004). For
example, the effect is stronger for moral qualities and behaviors
than it is for non-moral
traits such as intelligence – precisely because the latter
desirable traits, unlike the former, are
easily tested against reality (Allison et al., 1989). The
uniqueness bias is accordingly limited
when there is low motivation to see oneself as better than others
or when the behavior can
easily be verified (Goethals et al., 1991). Because people’s
beliefs about the importance of
their group goals are desirable but not readily verifiable, the
corresponding ratings of impor-
tance are likely to display self-enhancement biases.
Although the better-than-average effect has been empirically
investigated as an indivi-

dual-level phenomenon, there are a number of reasons to
suggest that this effect likely
extends beyond the self to aspects of meaningful groups to
which people belong. The
highly influential and robust line of research on social identity
theory provides strong sup-
port for this contention. The part of an individual’s self-concept
that derives from member-
ship in social groups is referred to as a social identity (Tajfel,
1982; Tajfel and Turner, 1979).
Considerable research on social identity theory has
demonstrated that these social identities
result in a number of cognitive biases that favor the ingroup and
disadvantage the outgroup.
For example, ingroup favoritism refers to the tendency of
people to view their own group
more positively than other groups (Tajfel and Turner, 1979); the
outgroup homogeneity bias
denotes the tendency of people to oversimplify perceptions of
outgroup members and have
more diversified perceptions of ingroup members (Park and
Rothbart, 1982); and the group
serving bias suggests that people make dispositional attributions
for their ingroup members’
positive behaviors but situational attributions for their negative
behaviors, and vice versa for
members of the outgroup (Heine and Lehman, 1997).
The proposed MITA effect is consistent with the conclusion of
the ingroup bias
literature that people tend to view the ingroup more positively
than other groups
Hoyt et al. 393

(Brewer, 1979; Tajfel, 1982). For example, Sherif and Sherif
(1953) found that group mem-
bers evaluate their group’s products more positively than other
groups’ products, and Price
(2000) found that people made more optimistic judgments about
their team members’ per-
formance than about the performance of non-team members. Not
only do people prefer
their own meaningful social group over others, but they also
show preference for members of
trivial ingroups including groups of people who share the same
birthday, received the same
flip of a coin, or prefer the same artist (Miller et al., 1998;
Brewer and Silver, 1978; Billig and
Tajfel, 1973). Because membership in a group engenders
ingroup biases such that ‘we’ are
seen as better than ‘they’, there is reason to test the logical
inference that ‘our’ goals will also
be perceived as more important than ‘their’ goals.
The justification bias
People have an astounding aptitude for self-justification (Tavris
and Aronson, 2007), and
one such method of absolving ourselves from responsibility may
originate in perceptions of
our groups’ goals. Unlike personal goals, group goals are
commonly thought to have special
moral weight. After all, a large part of ethics education is
getting people to think less about
their own interests and more about the interests of the
collective. Because group goals are
often consistent with this social aspect of morality, it would not
be surprising to find that
people readily use group goals to ground moral justifications of

their behavior. So we predict
that, in addition to perceiving that their goals are more
important than average, people will
also think that they have a special justification to engage in
unethical behavior in the service
of overvalued goals. This justification bias is just what we
should expect from the self-
enhancement literature: people are highly motivated to justify
their morally questionable
behaviors, and there are relatively few objective limitations on
their ability to appeal to value
judgments to do so.
Leadership is an important component of group life: leaders
provide the vision, direction,
and goals, and they use social influence processes to transform
the individual action of
group members into the collective action necessary to achieve
those goals (Chemers, 2000;
Messick, 2005). Because of their role, leaders have
disproportionately greater power than
do non-leading group members – both to set collective goals and
to mobilize collective
action toward those goals (Hogg, 2001). Consistent with the
ample social psychological
literature demonstrating that people’s self-concept, or identity,
strongly influences
their beliefs, attitudes, and behaviors (Leary and Tangney,
2003), self-identification
as group leader can guide the processing of information
regarding their group. Hence,
the proposed self-enhancement biases regarding the importance
of group goals and
the related justification bias will likely be amplified for those
who self-identity as group
leader. After all, the leader’s identity is strongly associated

with the attainment of collective
goals.
The centrality of both setting and attaining group goals to the
leader’s identity is further
evidenced through people’s implicit leadership theories.
Implicit leadership theories are
people’s tacit beliefs regarding the traits, qualities, and
characteristics of leaders
(Eden and Leviatan, 1975; Forsyth and Nye, 2008). The content
of these implicit theories is
vast, but many of the assumptions focus on establishing
objectives, structuring neces-
sary tasks, and ultimately accomplishing group goals. Thus, not
only do we predict
actual leaders of organizations will show an enhanced MITA
effect over their followers,
but we predict that to the extent people rely on these implicit
theories when simply
394 Leadership 6(4)
perceiving themselves as leaders, this enhanced MITA effect
should be evidenced even when
people are randomly assigned to leadership positions.
Furthermore, in keeping with the
prediction that overvaluing group goals may be accompanied by
a greater justification to
engage in unethical behavior, leaders should also demonstrate a
greater justification bias
than non-leaders.
Research overview

We employed a multiple study, multi-method approach to
testing the following predictions:
(1) People’s beliefs about their goals will exhibit a MITA effect
– they will hold that their
group’s goals are more important than other groups’ goals; (2)
people will demonstrate a
justification bias – they will deem themselves more justified
than others to engage in what is
normally considered to be unethical behavior to attain their
group’s goals; (3) both the
MITA effect for group goals and the justification bias will be
greater in leaders than
in non-leading group members; (4) finally, we predict that the
justification bias will be
related to beliefs about group goal importance. These
hypotheses were tested across
three studies. In the first study we tested the MITA effect for
group goals with leaders
and non-leaders of university campus groups. In Study 2 we
sought to experimentally
demonstrate the MITA effect for the goals of illusory groups,
and in the final study we
replicated and extended the second study by experimentally
examining the justification bias
prediction.
Study 1
With a particular emphasis on leaders, this study explored the
extension of self-enhancement
biases to group goals. In this study we contacted leaders and
non-leading members of stu-
dent groups and asked them to rate the importance of their
group’s goals as well as other
groups’ goals to test the following hypotheses:

Hypothesis 1: Participants will demonstrate a more-important-
than-average effect with respect
to group goals.
Hypothesis 2: This effect will be stronger for leaders than for
non-leaders.
Method
Participants. One hundred and fifty-six undergraduate students
at the University of
Richmond participated in this study (17% first-years, 21%
sophomores, 29% juniors, and
33% seniors). Participants included 58 male and 98 female
leaders (n¼112) and non-leading
members (n¼44) of university organizations. The organizations
targeted were student gov-
ernments (for both of the male and female student coordinate
colleges as well as the school
of leadership studies), Greek organizations, political interest
groups, and religious interest
groups.
Procedures. Respondents were recruited during their
organizations’ meetings and informed
that they would be entered in a raffle to win one of a few prizes.
We employed two methods
to gauge participants’ ratings of group goals: they ranked their
goals in comparison to
others, and they assessed their and others’ goals on a 1 to 5
scale.
Hoyt et al. 395
Goal importance rankings for fund distribution. Participants

were asked to rank their
group’s goals by responding to the following:
The Director of Student Activities has decided to distribute
funds to the current officially rec-
ognized student organizations, one of which is your
organization. There are 100 such organi-
zations. If the Director wants to distribute the funds based on
the importance of each
organization’s goals, where in the ranking should your
organization be put for the distribution
of funds? 1¼most important organizational goals and 100¼ least
important organizational goals
(one organization per ranking).
Goal importance scale. Participants were asked to select the
best description of their
group’s goals: unimportant (1), somewhat important (2),
important (3), very important
(4), and extremely important (5). They were then asked to
indicate the percentage of student
groups on campus that have goals best described as being
unimportant to extremely impor-
tant. A final weighted rating of the goals of other organizations
was created by multiplying
the percentage of organizations reported in each category by the
numerical value of the
category and summing all five values.
In yet another approach to gauge goal assessments, participants
responded to the follow-
ing two items on a 7-point Likert-type scale ranging from 1
(strongly disagree) to 7 (strongly
agree): ‘The goals of my student organization are important’

and ‘the goals of the average
student organization on campus are important’. The results from
these questions directly
parallel those of the goal importance scale described in the
paragraph above, thus, for
simplicity, we have not included these results.
Results
Hypothesis 1: Participants will demonstrate a more-important-
than-average effect with respect
to group goals. First, we examined participants’ rankings of
their group goals in the fund
distribution questions.
1
They were asked to rank their group’s goals on a scale from 1
(most
important) to 100 (least important). Similar to previous research
on the above-average effect,
we assessed this effect by conducting one-sample t-tests against
the midpoint on the scale, in
this case, 50 (Alicke et al., 1995; Alicke et al., 2001).
Participants’ average ranking of their
groups’ goals was 13.67 (SD¼5.53). This ranking is
significantly better than the fiftieth
percentile (t(151)¼�17.68, p< .00). In fact, participants ranked
their group better than
the seventieth percentile (t(151)¼�9.35, p< .00).
In another approach to testing Hypothesis 1, we analyzed
participants’ responses to the
5-point goal importance scale. We conducted a factorial
ANOVA with one between-subjects
factor (Leader) and one within-subjects factor (Self/Other).

2
Analyses revealed that partic-
ipants rated the goals of their group as significantly more
important than the goals of other
student groups (F (1, 132)¼105.43, p< .00, Z2¼ .44; self:
M¼3.78 SE¼ .08, others:
M¼2.90 SE¼ .06).
Hypothesis 2: The more-important-than-average effect will be
stronger for leaders than non-
leaders. First, we conducted a one-way (Leader) between-
subjects ANOVA on the fund-dis-
tribution group-goal-importance rankings (again, rankings were
made from 1[most important]
to 100 [least important]). There was a significant main effect of
Leader such that leaders’ rank-
ings attributed greater importance to group goals than did the
rankings of non-leaders (F (1,
150)¼10.23, p< .01, Z2¼ .06; leaders: M¼11.10 SE¼ .05, non-
leaders: M¼21.66 SE¼ .12).
396 Leadership 6(4)
We also tested this hypothesis on the 5-point goal importance
scale by examining the
simple-effects tests from the mixed-factorial ANOVA (B:
Leader, W: Self/Other). These tests
revealed that leaders rated their group goals as being
significantly more important (M¼4.03,
SE¼ .09), as compared to non-leaders (M¼3.54, SE¼ .14;
simple F (1, 132)¼8.85, p< .01,
Z2¼ .06), but there was no difference in leaders’ ratings

(M¼2.96, SE¼ .06) and non-lea-
ders’ ratings (M¼2.84, SE¼ .10) of the group goals of others
(see Figure 1).
Discussion
The results from this first study established what we call the
more-important-than-average
effect: participants claimed that their group goals are more
important than other people’s
group goals. Furthermore, the more-important-than-average
effect was stronger for leaders
than for non-leading group members. These findings are quite
robust, as they were
Study 1
2
3
4
5
G
o
a
l a
ss
e
ss
m
e

n
t
Own group's goals Other groups' goals
Study 2
2
3
4
5
Own group's goals Other groups' goals
G
o
a
l a
ss
e
ss
m
e
n
t
Study 3
2
3

4
5
Own group's
goals
Other groups'
goals
G
o
a
l a
ss
e
ss
m
e
n
t
Leader
Non-leader
Figure 1. Leaders’ and non-leading group members’ ratings of
their own and other groups’ goals across
all three studies
Hoyt et al. 397

consistently supported across a variety of measures ranging
from the fund distribution rat-
ings to two additional approaches for assessing of goal
importance.
Study 2
Study overview and hypotheses
In Study 1 we demonstrated that the MITA effect was indeed
stronger for leaders than for
non-leading group members. Although we contend that this
effect is driven by the self-
identification as group leader, because we studied actual group
leaders and non-leading
members we cannot rule out alternative explanations
implicating factors associated with
people who become leaders. For example, we can assume that
some of these individuals
became leaders of their groups precisely because of beliefs
about the importance of the
groups’ goals. So, in this second study, we sought to test
experimentally the prediction
that the MITA effect is driven by people’s self-conception as
leaders, rather than factors
that explain why they become leaders in the first place. If
simply perceiving oneself as a
leader is sufficient to activate implicit leadership theories
(Forsyth and Nye, 2008), we should
be able to demonstrate the MITA effect in those randomly
assigned to the leader position.
To test this explanation, we conducted an experimental study in
which we randomly assigned
participants to the role of leader or non-leading member of a
group and assessed how

important they deemed their groups’ goals to test the same
hypotheses tested in Study 1.
Method
Participants and design. One hundred and seventy undergraduate
students at the
University of North Carolina, Chapel Hill participated in this
study (67 men and
103 women). Participants were recruited to participate before
their classes began. The exper-
iment employed a 2 (Leader: Leader, Non-leader) by 3 (Group:
Business, Service, Political)
between-subjects design.
Procedures. Participants were given a vignette with the
following instructions: ‘For the
purposes of this survey, imagine that you are the leader (or non-
leading member) of a
business (or service, or political) organization on campus.
Please take a minute to think
of yourself as the leader (or non-leading member) of this kind
of organization and then
complete the following items’.
Goal importance. As in Study 1, participants were asked to
select the best description of the
goals of their student organization: unimportant, somewhat
important, important, very
important, and extremely important. In addition, they were
asked to indicate which of
these best described the goals of the typical organization
(political, service, or business)
on campus.
Results

Hypotheses 1 and 2: Participants will demonstrate the more-
important-than-average effect, and
leaders will demonstrate a stronger effect than non-leaders. To
test these hypotheses, we
analyzed participants’ responses to the goal-importance
assessments by conducting a facto-
rial ANOVA with two between-subjects factors (Leader and
Organization) and one within-
398 Leadership 6(4)
subjects factor (Self/Other). As predicted, there was a main
effect of goal importance such
that all participants rated their own group’s goals as
significantly more important than other
groups’ goals (F (1, 164)¼48.12, p< .00, Z2¼ .23; own:
M¼3.76 SD¼ .84; other: M¼3.35
SD¼ .93). Additionally, in support of the second hypothesis,
there was a significant inter-
action between Self/Other and Leader (F (1, 164)¼5.72, p¼
.018, Z2¼ .03). Simple effects
tests revealed that leaders rated their group’s goals as being
significantly more important
(M¼3.90, SE¼ .09) than did non-leaders (M¼3.60, SE¼ .09;
simple F (1, 164)¼5.59,
p¼ .019, Z2¼ .03), but there was no difference in leaders’ and
non-leaders’ ratings of the
importance of the typical group’s goals (M¼3.37, SE¼ .09,
M¼3.36, SE¼ .10, respec-
tively, see Figure 1). We included organization type as a factor
to test whether the relation-
ship between leader and goal assessment differed across types
of organizations. Analyses

revealed that the three-way interaction between goal
assessment, leader, and organization
was not significant (p¼ .245).
Discussion
In Study 2, we both replicated the more-important-than-average
effect for group goals found
in Study 1 among members of real university groups, and we
demonstrated that this effect
extends to people assigned to imaginary groups. In addition,
people randomly assigned to
the role of group leader showed a stronger MITA effect than
those assigned to the role of
non-leading group member, again replicating and extending the
findings from Study 1.
Because participants were assigned to the position of
leadership, the enhanced MITA
effect for leaders appears not to be a peculiarity associated with
choosing to take on a
leadership position but, rather, the result of a more general
cognitive bias associated with
the leader role.
Study 3
In Study 3 we set out to replicate the experimental findings by
demonstrating both the MITA
effect for participants assigned to illusory groups and a greater
MITA effect for those
assigned to the leader condition. We also sought to link
perceptions of the importance of
one’s goals to the belief that one is justified in deviating from
general moral requirements.
Thus, we wanted to test three additional hypotheses:

Hypothesis 3: Participants will think they are more justified
than others in breaking basic moral
rules to achieve group goals.
Hypothesis 4: Leaders will report a greater justification bias
than non-leaders.
Hypothesis 5: Ratings of group goal importance will be
positively correlated with this justifica-
tion bias.
Method
Participants and design. Ninety-one undergraduate students at
the University of
Richmond participated in this study (28 men and 63 women).
The experiment employed a
two-group (Leader: Leader, Non-leader) between-subjects
design.
Hoyt et al. 399
Procedures. Participants were given a vignette with similar
instructions as in Study 2; how-
ever, instead of imagining they are the leader or non-leading
member of a specific type of
group (business, service, or political), they were simply asked
to imagine being a leader or
non-leading group member of an organization on campus.
Goal importance. Participants responded to the following two
questions: ‘The goals of my
organization are best described as being. . .’, and ‘The goals of

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