In Unit 9, we will study the theory and logic of analysis of variance (ANOVA). Recall that a t test requires a predictor variable that is dichotomous (it has only two levels or groups). The advantage of ANOVA over a t test
is that the categorical predictor variable can have two or more groups. Just like a t test, the outcome variable in
ANOVA is continuous and requires the calculation of group means.
Logic of a “One-Way” ANOVA
The ANOVA, or F test, relies on predictor variables referred to as factors. A factor is a categorical (nominal)
predictor variable. The term “one-way” is applied to an ANOVA with only one factor that is defined by two or
more mutually exclusive groups. Technically, an ANOVA can be calculated with only two groups, but the t test is
usually used instead. Instead, the one-way ANOVA is usually calculated with three or more groups, which are
often referred to as levels of the factor.
If the ANOVA includes multiple factors, it is referred to as a factorial ANOVA. An ANOVA with two factors is
referred to as a “two-way” ANOVA; an ANOVA with three factors is referred to as a “three-way” ANOVA, and
so on. Factorial ANOVA is studied in advanced inferential statistics. In this course, we will focus on the theory
and logic of the one-way ANOVA.
ANOVA is one of the most popular statistics used in social sciences research. In non-experimental designs, the
one-way ANOVA compares group means between naturally existing groups, such as political affiliation
(Democrat, Independent, Republican). In experimental designs, the one-way ANOVA compares group means
for participants randomly assigned to different treatment conditions (for example, high caffeine dose; low
caffeine dose; control group).
Avoiding Inflated Type I Error
You may wonder why a one-way ANOVA is necessary. For example, if a factor has four groups ( k = 4), why not
just run independent sample t tests for all pairwise comparisons (for example, Group A versus Group B, Group
A versus Group C, Group B versus Group C, et cetera)? Warner (2013) points out that a factor with four groups
involves six pairwise comparisons. The issue is that conducting multiple pairwise comparisons with the same
data leads to inflated risk of a Type I error (incorrectly rejecting a true null hypothesis—getting a false positive).
The ANOVA protects the researcher from inflated Type I error by calculating a single omnibus test that
assumes all k population means are equal.
Although the advantage of the omnibus test is that it helps protect researchers from inflated Type I error, the
limitation is that a significant omnibus test does not specify exactly which group means differ, just that there is a
difference “somewhere” among the group means. A researcher therefore relies on either (a) planned contrasts
of specific pairwise comparisons determined prior to running the F test or (b) follow-up tests of pairwise
comparisons, also referred to as post-hoc tests, to determine exactly which pairwise comparisons are
significant.
Hypothesis Testing in a One-Way ANOVA
The null hypothesis of the omnibus test is that all k (group) population means are equal, or H0: μ1 = μ2 = … μk.
By contrast, the alternative hypothesis is usually articulated by stipulating that “at least one” pairwise
Unit 9 – One-Way ANOVA: Theory and Logic
INTRODUCTION
comparison of population means is unequal. Keep in mind that this prediction does not imply that all groups
must significantly differ from one another on the outcome variable.
Assumptions of a One-Way ANOVA
The assumptions of ANOVA reflect assumptions of the t test. ANOVA assumes independence of observations.
ANOVA assumes that outcome variable Y is normally distributed. ANOVA assumes that the variance of Y scores
is equal across all levels (or groups) of the factor. These ANOVA assumptions are checked in the same process
used to check assumptions for the t test discussed earlier in the course—using the Shapiro-Wilk test and the
Levene test).
Effect Size for a One-Way ANOVA
The effect size for a one-way ANOVA is eta squared (η2). It represents the amount of variance in Y that is
attributable to group differences. Recall the concept of sum of squares ( SS ) from Unit 2. Eta squared for the
one-way ANOVA is calculated by dividing the sum of squares of between-group differences ( SS between) by the
total sums of squares in the model ( SS total), which is reported in SPSS output for the F test. Eta squared for the
one-way ANOVA is interpreted by referring to Table 5.2 in the Warner text (p. 208).
Journal Article Assignment
By the conclusion of Unit 9, you will have studied three fundamental statistics used in research, including
correlation, t tests, and one-way analysis of variance (ANOVA). You are now prepared to analyze a journal article
in your career specialization that reports one of these statistical tests. In the journal article assignment, you will
follow the general process used in completing DAA assignments:
1. Provide a brief summary of the research study and, in this assignment, why it is relevant to your career.
2. Identify the predictor variables and outcome variables including scales of measurement.
3. Articulate the research question, null hypothesis, and alternative hypothesis.
4. Report the test statistic and interpret it.
5. Provide conclusions as well as the strengths and limitations of the study.
Reference
Warner, R. M. (2013). Applied statistics: From bivariate through multivariate techniques (2nd ed.). Thousand
Oaks, CA: Sage.
OBJECTIVES
To successfully complete this learning unit, you will be expected to:
1. Analyze the use of one-way ANOVA in research situations.
2. Describe how ANOVA protects against the risk of inflated Type I error.
3. Analyze the assumptions, calculation, and effect size of one-way ANOVA.
[u09s1] Unit 9 Study 1- Readings
Use your Warner text, Applied Statistics: From Bivariate Through Multivariate Techniques , to complete
the following:
• Read Chapter 6, “One-Way Between-Subjects Analysis of Variance,” pages 219–260. This reading
addresses the following topics:
◦ Research situations using one-way ANOVA.
◦ Assumptions of one-way ANOVA.
◦ Calculation of one-way ANOVA.
◦ Effect size.
◦ Planned contrasts and post-hoc tests.
◦ Reporting and interpreting SPSS output.
PSY Learners – Additional Required Readings
In addition to the other required study activities for this unit, PSY learners are required to read the following:
Wang, P., Rau, P. P., & Salvendy, G. (2015). Effect of information sharing and communication on driver’s risk
taking. Safety Science , 77, 123–132. doi:10.1016/j.ssci.2015.03.013
SOE Learners – Suggested Readings
Wabed, A., & Tang, X. (2010). Analysis of variance (ANOVA). In N. J. Salkind (Ed.), Encyclopedia of research
design (pp. 27–29). Thousand Oaks, CA: Sage. doi:10.4135/9781412961288.n11
Resources
• DAA Template.
• SPSS Data Analysis Report Guidelines.
• IBM SPSS Step-by-Step Guide: One-Way ANOVA.
[u09d1] Unit 9 Discussion 1 – Application ofF Tests
For this discussion:
• Identify a research question from your professional life or career specialization that can be addressed
by a one-way ANOVA.
• Indicate why a one-way ANOVA would be the appropriate analysis for this research question.
• Describe the variables and their scale of measurement.
• Discuss the expected outcome.
