# The theory and logic of analysis of variance (ANOVA)

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

◦ 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.

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

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.

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