UNDERSTANDING
ANOVA
A HOW-TO GUIDE
University of Arizona
Military REACH
U N D E R S T A N D I N G A N O V A
Understanding ANOVA: A How-To Guide
Do children of dierent ethnicies aend youth programs at dierent rates? Do these dierences
depend on the gender of the youth? Could an intervenon aimed at increasing youth parcipaon in
programs be eecve? Does the eecveness of the intervenon depend on the gender of the youth?
These quesons and more can be answered (at least in part) by using Analysis of Variance (ANOVA). In
this How-To Guide we will discuss the uses of ANOVA to answer such quesons where dierences are
explored between three or more groups of individuals. This guide is intended for individuals who are
unfamiliar with ANOVA and serves as a basic guide for the contexts in which ANOVA is typically used.
Analysis of Variance, more commonly referred to as ANOVA, is similar to t-tests (see the Understanding
t-Tests: A How-To Guide for more informaon on t-tests). t-tests are used when you want to compare
two means, ANOVA is used when you want to compare more than two means. We begin with asking the
queson: “Why can’t we just do mulple t-tests between three or four groups?”
Let’s say we have three groups and we want to compare the mean of each group to each other.
Group 1 compared to Group 2
Group 2 compared to Group 3
Group 1 compared to Group 3
One opon might be to conduct mulple t-tests, in this case three separate tests. The problem with
conducng these t-tests has to do with the rate of error. For each t-test conducted there would be a
certain degree of error associated with each test. In other words, each test has a probability of being
wrong. The probability of being wrong is typically small, but when mulple tests are conducted the
chances that at least one of the tests is wrong increases. To solve this problem of increasing error, we
use ANOVA to make comparisons of means between mulple groups instead of lots of t-tests.
ANOVA, like a t-test, tells us if the means of dierent groups are the same or dierent, and the p-value
associated with ANOVA tells us if in fact the groups are dierent, if that dierence is stascally
signicant (see the How-To Guide on Stascal Language for more informaon on p-values and
signicance).
There are a few types of ANOVA including:
One-Way ANOVA
Factorial ANOVA
Repeated Measures ANOVA
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Below we discuss each of these dierent types of ANOVA and the contexts in which they might be used.
One-Way ANOVA
Use
The One-Way ANOVA is used when comparing the means from three or more groups. Specically, a One
-Way ANOVA is used when there is a single independent variable that has three or more categories.
The One-Way ANOVA tells us if the three groups dier from one another on a dependent variable.
Example
Imagine a study in which researchers implemented two separate intervenons, one designed to
improve skill building among youth (Intervenon A) and another designed to increase supporve rela-
onships between instructors and youth (Intervenon B). In addion, the researchers also employed a
control group¹ in their study.
The researchers want to know if either of these two intervenons improved instrucon quality. In this
study there are three groups, parcipants who received Intervenon A, those that received Intervenon
B, and the control group. Below is a graph of what the means for each group might look like. Research-
ers interested in dierences between these three groups in terms of instrucon quality would imple-
ment a One-Way ANOVA. The One-Way ANOVA provides informaon about if there were stascally
signicant instrucon quality dierences between these three groups
Interpretaon
The result of a One-Way ANOVA indicates that there are dierences between the three means.
However, ANOVA on its own does not provide informaon about where these dierences actually are.
In this example there could be a dierence in instruconal quality between the control and Intervenon
A, between the control and Intervenon B, and/or between Intervenon A and Intervenon B. To get at
these dierences addional analyses must be conducted. See the secon below on Post-Hoc Analyses
and Planned Contrasts for more informaon.
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¹ A control group is used in experimental researcher designs. In such designs, typically one group of individuals is given some sort of treatment
or intervention while the other group is not (or is given something similar but innocuous). The control group is the group that is not given the
treatment or intervention. In well-designed experiments, difference between these groups would reflect the effect of the treatment or intervention.
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Factorial ANOVA
Use
Unlike a One-Way ANOVA, a Factorial ANOVA is used when there is more than one independent
variable. In the previous example there was only one independent variable with three levels
(Intervenon A, Intervenon B, control group). Now suppose that a researcher also wanted to know if
there were addional group dierences between boys and girls in the youth program. When this second
independent variable is added to the analysis, a Factorial ANOVA must be used.
Example
Factorial ANOVA’s typically use a mathemacal notaon to indicate the kind of Factorial ANOVA being
conducted. In the present example 3 x 2 Factorial ANOVA is being conducted. Below is an example of
how to break down this notaon:
In this example there are two independent variables, one with three levels (Intervenon A, Intervenon
B, control group) and one with two levels (boy, girl). Thus, the analysis would be a 3 x 2 Factorial
ANOVA.
The results of a Factorial ANOVA consist of several parts. First are called main eects. Main eects tell
you if there is a dierence between groups for each of the independent variables. For example, a main
eect of intervenon type (intervenon A, intervenon B, control group) would indicate that there is a
signicant dierence between these three groups. That is, somewhere among these three means, at
least two of them are signicantly dierent from one another. Like the One-Way ANOVA, a Factorial
ANOVA does not provide informaon on where these dierences are and addional analyses are
required, these addional analyses are discussed below (Post-Hoc Analyses and Planned Contrasts)
A main eect of gender would indicate that there is a dierence between the means of boys and girls.
In addion to the main eects, Factorial ANOVA also oen provides informaon about interacon
eects as well. Interacon eects provide informaon about whether an observed group dierence in
one independent variable varies as a funcon of another independent variable.
An example will clarify:
Suppose that a researcher found a main eect for intervenon type. This tells us that there is a signi-
cant dierence in instrucon quality between these three groups. If there were a signicant interacon
between intervenon type and gender, this would mean that the dierences found between interven-
on types were only there for one gender and not the other. Below is an example of such an interac-
on.
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Intervenon A Intervenon B Control Group
By
Boy Girl
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x
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The graph shows that there are no dierences for boys (green line) between the control, Intervenon A,
or Intervenon B. However, there are clear dierences for girls (blue line) between these three groups.
The implicaon of an interacon is that dierences between groups are dependent on membership in
another group (in this case, being a boy or girl).
Interpretaon
In the above example, because there is an interacon between intervenon type and gender, mean
dierences in instrucon quality among intervenon types only exist for girls and not boys.
Repeated Measures ANOVA
Use
As discussed in the How-To Guide on t-tests, a dependent samples t-test is used when the scores
between two groups are somehow dependent on each other. One example of such a dependency is
when the same people are given the same measure over me to see whether there is change in that
measure. Oen this may take the form of pre- and post-test scores (see the How-To Guide on t-tests for
a refresher on stascal dependency). The Repeated Measures ANOVA takes the dependent samples
t-test one step further and allows the research to ask the queson “Does the dierence between the
pre-test and post-test means dier as a funcon of group membership?”
Example
Imagine that pre-test and post-test data were collected regarding instrucon quality among 100 youth
program instructors. Between the pre- and post-tests an intervenon was implemented aimed at
improving instrucon quality. If you were only interested in the dierence between pre- and post-test
means, a dependent samples t-test would be sucient. However, suppose that a researcher wanted to
know whether the dierence in instrucon quality between pre- and post-test was dierent between
male and female instructors; in this case a Repeated Measures ANOVA would be used.
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In this example, there would be two independent variables: (1) tesng period (pre vs. post) and gender
(male vs. female). In the language of a Repeated Measures ANOVA the tesng period independent
variable would be called a within-subjects factor² while the gender independent variable would be
called a between-subjects factor³.
Like the Factorial ANOVA, the Repeated Measures ANOVA has both main eects and an interacon.
One main eect in this example would be for tesng period. A signicant main eect for tesng period
would indicate that the means for instrucon quality between pre- and post-test were meaningfully
dierent from one another.
Interpretaon
In this example a signicant main eect of gender would indicate that there was a meaningful dierence
in instruconal quality between male and female instructors. In this case the analysis would test for an
interacon between tesng period and gender. This would provide informaon regarding if the dier-
ence observed in instrucon quality between pre- and post-test diered depending on if the instructor
were male or female. A signicant interacon would indicate that the dierence observed between
pre- and post-test instrucon quality may be present for female instructors and not male instructors. In
the gure below we can see that the means for men (green line) are the same between pre-test (1) and
post-test (2). However, there was a big dierence between pre- and post-test means for women (blue
line). From this analysis one might conclude that the intervenon was eecve for female instructors
but not male instructors.
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² A within-subjects factor is an independent variable that varies within the participants of the study. This means that the variance of the variable is
centered on the changes observed within each study participant, such as between a pre-test and a post-test
³ A between-subjects factor is an independent variable that varies only between participants but does not change within participants. Gender can
be treated as a between-subjects factor as it is static within an individual but varies between individuals.
U N D E R S T A N D I N G A N O V A
Post-Hoc Analyses and Planned Contrasts
The results of an ANOVA only tell you that there is a dierence between means of dierent groups; it
does not provide informaon about where these dierences lie. In the Factorial ANOVA described
above, an ANOVA would tell us that there is a dierence among the means between intervenon A,
intervenon B, and the control group. However, it does not tell us which means dier from each other.
Is the dierence between intervenon A and intervenon B? Is it between the control group and inter-
venon A? Is it between all three means?
In order to answer these quesons addional analyses have to be conducted.
There are two types of follow-up analyses that might be conducted with an ANOVA
post-hoc analyses
planned contrasts
Post-hoc analyses are used aer the fact, that is these analyses are used when the researcher does not
have specic predicons about where the dierences between means lie. Planned contrasts are exactly
that: planned. These analyses are used when the researcher has specic predicons about dierences
between the means.
Summary
In this How-To Guide we have gone over the basics of Analysis of Variance. We hope that we have
provided useful informaon about ANOVA, the dierent types, and informaon about how and why
ANOVA is used in research. In addion to this guide on ANOVA, our website has several other guides
that may be of interest. These include guides on t-tests, correlaon, regression, and others. We encour-
age readers to examine these guides as well to become crical consumer of youth program quality re-
search.
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The Arizona Center for Research and Outreach (AZ REACH)
Lynne M. Borden, Principal Invesgator
Extension Specialist and Professor
The Military REACH Team
Lynne M. Borden, Ph.D. – Principal Invesgator Bryna Koch, M.P.H
Leslie Bosch, M.S. Mary Koss, Ph.D.
Noel A. Card, Ph.D. Leslie Langbert, M.S.W.
Deborah M. Casper, M.S. Ann Mastergeorge, Ph.D.
Sandra Fletcher, M.S. Stephen Russell, Ph.D.
Stacy Ann Hawkins, Ph.D. Amy Schaller, M.A.
Ashley Jones, B.S.
Gabriel Schlomer, Ph.D. (Primary Author)
Chrisne Bracamonte Wiggs, M.S. (Co-Principal Invesgator)
John & Doris Norton School of Family and Consumer Sciences
The University of Arizona
Tucson, Arizona
Phone: 520.621.1075
Website: hp://ag.arizona.edu/fcs/
The Arizona Center for Research and Outreach (AZ REACH)
The University of Arizona
Tucson, Arizona
Phone: 520.621.1742
Website: hp://reachmilitaryfamilies.arizona.edu/
