ANOVA in Counseling & Psychology Research

ANOVA stands for Analysis of Variance. Although the term “variance” will be explained later, for now you can think of it simply as differences. ANOVA procedures examine differences in scores among groups of people who complete a test, survey, or any measurable task (Sutton, 2021, January 5).

Cite this Post

Sutton, G. W. (2021, January 5). ANOVA in counseling & psychology research. Assessment, Statistics, and Research.  https://statistics.suttong.com/2021/01/anova-in-counseling-psychology-research.html

What's in this post?

ANOVA defined

What ANOVA does

Types of ANOVA

Understanding the F value

When to use ANOVA

A conceptual example

A worked example

A sample write-up

A One-Way ANOVA Calculator

**********

What ANOVA Does

ANOVA evaluates whether group means differ more than we would expect by chance.

For example, imagine a study examining the effects of three room temperatures on math performance:

• Independent Variable (IV): Temperature

◦ Levels: 75°, 85°, 95° Fahrenheit

• Dependent Variable (DV): Math performance

◦ Measured by: A math test score

ANOVA determines whether the mean math scores at the three temperatures differ enough to conclude that temperature truly affects performance.

*****

Types of ANOVA

The number of independent variables determines the type of ANOVA:

• One-way ANOVA: One IV

• Two-way ANOVA: Two IVs

• Three-way ANOVA: Three IVs

• Four-way ANOVA: Four IVs (rare because interactions become difficult to interpret)

Each IV and each interaction among IVs is tested with its own F value.

Understanding the F Value

ANOVA results are typically reported with an F statistic.

A larger F value suggests that the observed group differences are unlikely to be due to chance.

Each F test is accompanied by a p value, which indicates the probability that the result occurred by chance. In psychology and education, a common significance level is p < .05.

Researchers also report effect sizes, often partial eta squared, to show the magnitude of the effect. Modern reporting emphasizes effect sizes rather than relying solely on p values

*****

When to Use ANOVA

Use ANOVA when your study includes:

• Independent Variable(s): One or more

• Dependent Variable: Only one

• Dependent Measure: A quantitative score (test score, rating, performance measure)

• IV Levels: Two or more groups or conditions

*****

Examples of IV levels include temperatures, drug dosages, therapy groups, learning methods, or work teams.

ANOVA provides overall tests of whether the IV(s) produce significant differences in the DV. If the overall F test is significant, researchers may conduct post hoc tests to compare pairs of group means.

Common post hoc tests include:

• t tests

• Tukey HSD

• Bonferroni

• Newman–Keuls

These are used only after a significant overall F test.

Read more about ANOVA and data analyses in the following books.

Applied Statistics Concepts for Counselors on AMAZON   or   GOOGLE

Creating Surveys on AMAZON    or   GOOGLE  Worldwide

CONCEPTUAL EXAMPLE

Illustration of a three group design project. Volunteers in a math class are randomly assigned to three levels of room temperature (Fahrenheit) to determine if room temperature has an effect on the score. The random assignment should mean that individual characteristics of the participants would not be a factor in the score.

IV	DV
75 degrees	Math score
85 degrees	Math score
95 degrees	Math Score

Random assignment helps ensure that pre-existing math ability does not bias the results. If researchers are concerned about unequal math skills, they could administer a pretest.
In statistical notation:

• The IV (temperature) is often labeled X, with levels X₁, X₂, X₃

• The DV (math score) is labeled Y, producing Y₁, Y₂, Y₃ for the three groups

If the overall F test is significant, researchers compare the group means in pairs:

• Y₁ vs. Y₂

• Y₂ vs. Y₃

• Y₁ vs. Y₃

ANOVA analyzes differences among all group means, while post hoc tests analyze differences between specific pairs.

Other notes

The letter F in the F value comes from the surname of the British statistician, Sir Ronald Fisher (1890-1962) born in my hometown, London, England.

Daniel Fahrenheit of temperature fame was born in Poland to a German family.

**********

Post Author

Geoffrey W. Sutton, Professor Emeritus of Psychology at Evangel University, holds a master’s degree in counseling and a PhD in psychology from the University of Missouri-Columbia. His postdoctoral work encompassed education and supervision in forensic and neuropsychology and psychopharmacology. As a licensed psychologist, he conducted clinical and neuropsychological evaluations and provided psychotherapy for patients in various settings, including schools, hospitals, and private offices. During his tenure as a professor, Dr. Sutton taught courses on psychotherapy, assessment, and research. He has authored over one hundred publications, including books, book chapters, and articles in peer-reviewed psychology journals. 
His website is https://suttong.com 
You can find Dr. Sutton's books on   AMAZON    and  GOOGLE. 
Many publications are free to download at ResearchGate   and Academia  

Find chapters and essays on Substack. [ @GeoffreyWSutton ]

**********

Resource Links: 

All Measures A – Z Test Index

Spirituality Measures A - Z

Statistical Concepts A - Z

Worked Example

One‑Way ANOVA With Three Temperature Levels

Study Scenario

Researchers want to know whether room temperature affects math test performance. Students are randomly assigned to one of three temperature conditions:

75°F,  85°F,  95°F

The dependent variable is math performance measured by a test score (0–100).

*****

1. Raw Data

Group 1: 75°F

Scores: 82, 88, 91, 85, 90

Mean = 87.2

Group 2: 85°F

Scores: 78, 74, 81, 79, 77

Mean = 77.8

Group 3: 95°F

Scores: 70, 68, 65, 72, 66

Mean = 68.2

******

2. Summary of Group Means

3. ANOVA Calculations

Interpretation

• The F(2, 12) = 15.00, p < .001, indicates a significant effect of temperature on math scores.

• Students in different temperature conditions performed differently.

4. Effect Size (Partial Eta Squared)

Interpretation

• η² = .71 is a large effect.
• About 71% of the variance in math scores is explained by room temperature.

5. Post Hoc Test: Tukey HSD

Step 1: Compute HSD

Step 2: Compare Mean Differences

Interpretation

Only the 75°F vs 95°F comparison is significant.

6. Suggested write-up of results

We conducted a one-way ANOVA to examine the effect of room temperature (75°, 85°, 95° Fahrenheit) on math test performance. The results showed a significant effect of temperature on math scores, F(2, 12) = 15.00, p < .001, with a large effect size (η² = .71). Post hoc comparisons using the Tukey HSD test indicated that students in the 75°F condition scored significantly higher than those in the 95°F condition. The difference between 75°F and 85°F, and between 85°F and 95°F, was not significant. These results suggest that higher room temperatures substantially reduce math performance.

One-Way ANOVA Calculator

Instructions

Enter scores for each group below. You may use 2 to 5 groups. Separate scores using commas or spaces (example: 82 88 91 85 90). Leave unused groups blank.

Click Compute ANOVA to display:
• Group means
• SS_between, SS_within, SS_total
• df, MS, F, p-value
• Eta squared (η²)
• Tukey HSD post hoc tests (only if F is significant)

Group 1 scores:

Group 2 scores:

Group 3 scores:

Group 4 scores:

Group 5 scores: