8 One-Way ANOVA

(PSY206) Data Management and Analysis

Author

Md Rasel Biswas

8.1 Introduction

  • One–way Analysis of Variance (ANOVA) is one of the most widely used statistical techniques for comparing more than two group means.
  • While the independent samples t-test is limited to two groups, ANOVA allows us to test whether three or more groups differ significantly on a quantitative outcome variable.

This lecture focuses on:

  • Conceptual foundation of one–way ANOVA
  • Between–subjects and within–subjects designs
  • Assumptions of ANOVA
  • Interpretation of the F–ratio
  • Planned and post-hoc comparisons
  • Effect size reporting

8.2 When Do We Use One–Way ANOVA?

One–way ANOVA is appropriate when:

  1. There is one independent variable (factor)
  2. The factor has more than two levels or groups
  3. The dependent variable is measured on an interval or ratio scale

Example research questions:

  • Does teaching method (lecture / discussion / online) affect exam scores?
  • Does drug dose (0 mg / 10 mg / 20 mg) influence blood pressure?
  • Do three different diets lead to different weight loss?

8.3 Assumptions of One–Way ANOVA

As a parametric test, ANOVA relies on several assumptions:

  1. Scale of measurement
    • Dependent variable must be interval or ratio
  2. Normality
    • Data in each group should be approximately normally distributed
  3. Homogeneity of variance
    • Population variances should be equal across groups
  4. Independence of observations
    • Particularly important in between–subjects designs

Violation of these assumptions may require:

  • Data transformation
  • Robust ANOVA
  • Non-parametric alternative (e.g., Kruskal–Wallis)

8.4 Logic of ANOVA

  • Human performance always shows variability:
    • Variation within individuals
    • Variation between individuals
  • ANOVA asks a fundamental question:

Is the variability between group means larger than the natural variability within groups?

Two Sources of Variability

  1. Between-group variability
    • Caused by experimental manipulation
    • Reflects effect of the independent variable
  2. Within-group variability (error variance)
    • Individual differences
    • Measurement error
    • Uncontrolled factors

The F–Ratio

  • ANOVA summarizes this comparison using: \[ F = \frac{\text{Variance due to IV}}{\text{Error Variance}} \]

  • Interpretation:

    • F > 1 → possible effect of IV
    • F ≤ 1 → no evidence of effect
    • Significance determined by p value (< .05)

8.5 Types of One–Way ANOVA Designs

1. Between–Subjects ANOVA

  • Different participants in each group
  • Example: three separate teaching methods applied to different students

Sources of variance:

  • Between groups
  • Within groups
  • Total variance

2. Within–Subjects ANOVA

  • Same participants in all conditions
  • Also called repeated measures design

Advantages:

  • Controls individual differences
  • More statistical power

But:

  • Requires assumption of sphericity
  • More complex error structure:

8.6 ANOVA Terminology

Factors and Levels

  • Factor = independent variable
  • Levels = categories of the factor

Example:
Factor: Drug dose
Levels: 0 mg, 10 mg, 20 mg

Main Effect

  • The term ‘main effect’ is used to describe the effect a single independent variable has on a dependent variable.
  • If our One-way ANOVA is significant, we can say that there was a significant main effect of our IV on our DV. But it does not tell which groups differ.

8.7 Example Data From Book

  • To illustrate one–way ANOVA, the book uses a simple experiment on learning time (in seconds) for three different word lists: short, medium, and long words.
  • The same numerical values are used to demonstrate both a between–subjects and a within–subjects design so that students can clearly see the structural difference between the two approaches.

1. Between–Subjects Design

In this version, different participants were assigned to each word list. Each group contains eight independent observations. Learning time is in seconds.

  • List A: 30, 40, 35, 45, 38, 42, 36, 25
  • List B: 54, 58, 45, 60, 52, 56, 65, 52
  • List C: 68, 75, 80, 75, 85, 90, 75, 88

Here, the independent variable is Type of Word List with three levels (A, B, C), and the dependent variable is time taken to learn. Because the participants are different across lists, the observations are independent.

2. Within–Subjects Design

In the repeated–measures version, the same eight participants completed all three lists. Each person therefore contributes three scores.

Participant List A List B List C
1 35 42 64
2 48 60 90
3 36 65 75
4 40 55 70
5 38 52 85
6 25 42 58
7 30 42 60
8 42 60 90

This structure allows us to separate individual differences from the treatment effect, which increases statistical power but requires additional assumptions such as sphericity.

8.8 Entering the Data in SPSS

Data Structure for Between–Subjects ANOVA

  • For SPSS, the between–subjects data must be entered in long format, where each row represents one observation.
  • Create two variables:
    • group → identifies the list
      • 1 = List A
      • 2 = List B
      • 3 = List C
    • time → learning time in seconds

Example layout in Data View:

group time
1 30
1 40
1 35
… …
3 88

Download data in long format

Variable View Settings

  • Set group as Numeric with value labels
  • Set time as Scale measure
  • No missing values should be coded unless necessary

This structure is essential because SPSS One–Way ANOVA requires one column for the dependent variable and one column defining group membership.

8.9 Effect Size in ANOVA

  • Statistical significance is not enough. We must report:
    • Magnitude of effect
    • Practical importance
  • Common measures:
    • Eta squared (η²)
    • Partial eta squared
    • Omega squared (ω²)
  • Interpretation (rough guideline):
    • .01 → small
    • .06 → medium
    • .14 → large

8.10 SPSS Steps

Follow these steps to run a one–way between–subjects ANOVA in SPSS.

  1. Enter the data
    • Data should be in long format
    • Variables:
      • time → dependent variable (Scale)
      • group → factor (Nominal, with value labels 1 = A, 2 = B, 3 = C)
  2. Open the ANOVA dialog
    • Click Analyze → Compare Means → One-Way ANOVA
  3. Specify variables
    • Move time to Dependent List
    • Move group to Factor
  4. Request additional statistics
    • Click Options
      • ✔ Descriptive statistics
      • ✔ Homogeneity of variance test
    • Click Continue
  5. Post-hoc tests (if F is significant)
    • Click Post Hoc
      • Choose Tukey (equal variances assumed)
      • Or Games-Howell (if variances unequal)
    • Click Continue
  6. Run the analysis
    • Click OK
  • SPSS will produce:
    • Descriptive statistics table
    • Levene’s test
    • ANOVA table (F, df, p)
    • Post-hoc comparisons

8.11 Reporting ANOVA Results

  • A standard report includes:
    • F value
    • Degrees of freedom
    • p value
    • Effect size

Example:

There was a significant effect of teaching method on exam scores, F(2, 57) = 6.42, p = .003, η² = .18.

Then report post-hoc results.

8.12 Summary

Key ideas:

  • One–way ANOVA compares more than two means
  • Based on ratio of variances
  • Requires several assumptions
  • Significant F → need further comparisons
  • Always report effect size

Exercises for Students

  1. Explain why multiple t-tests are inappropriate instead of ANOVA.
  2. Describe the difference between:
    • between-subjects
    • within-subjects ANOVA
  3. Interpret:
    • F(3, 36) = 2.10, p = .11
    • F(3, 36) = 8.45, p < .001