oneway anova

One-Way Analysis of Variance

Introduction

  • Course: PSYCHOLOGY 272, Spring 2025

  • Dedicated to the memory of Stephanie S. Deese (1991-2012)

Objectives

  • Understanding the reasons for using more than two levels of an Independent Variable (IV).

  • Introduction to Analysis of Variance (ANOVA).

    • Definition and purpose.

    • Comparison with t-tests: Why use ANOVA instead of multiple t-tests?

  • Exploring:

    • Advantages and assumptions of ANOVA.

    • One-Way Randomized ANOVA.

    • Types of variance: within-groups, between-groups, and error.

    • Understanding the F-ratio: calculation, interpretation, and effect size.

    • Discussion of Tukey’s HSD (Honestly Significant Difference).

Why Use ANOVA?

  • Prior knowledge of tests:

    • z-test and one-sample t-test: single level of IV.

    • Example: Comparing IQ scores of a sample against the general population.

    • z-test: population parameter (σ) is known.

    • one-sample t-test: uses sample standard deviation (s).

  • Independent-samples t-test and correlated means t-test:

    • Comparison of two levels of IV.

    • Example 1: Comparing IQ scores of male versus female students.

    • Example 2: Comparing pretest and posttest scores after intervention.

Functionality of ANOVA

  • ANOVA is utilized for evaluating differences in the Dependent Variable (DV) across three or more levels of an IV.

  • ANOVA can handle both independent and correlated samples.

  • Examples of scenarios using ANOVA:

    • Comparing IQ scores of different classes: freshmen, sophomores, juniors, and seniors.

    • Medical drug trials with various treatment groups: control, placebo, and different dosages (50 mg, 100 mg, 150 mg).

    • Quarterly exam scores in an academic setting.

    • Measuring stress management across different timelines using the same assessment tool.

Comparing ANOVA vs t-tests

  • t-tests are straightforward, but multiple pairwise t-tests can lead to issues:

    • Example: In a medical drug trial with a control group, placebo, and three treatment groups, 10 t-tests would be necessary for all pairwise comparisons:

    • Control vs Placebo

    • Control vs 50 mg

    • Control vs 100 mg

    • Control vs 150 mg

    • Placebo vs 50 mg

    • Placebo vs 100 mg

    • Placebo vs 150 mg

    • 50 mg vs 100 mg

    • 50 mg vs 150 mg

    • 100 mg vs 150 mg

  • Type I error (α): Probability of incorrectly rejecting the null hypothesis (H0).

    • The α level for a single t-test is commonly set at 0.05.

    • As more t-tests are conducted, the chance of Type I error increases.

    • Highlighted with a formula to estimate elevated α as a function of number of t-tests (k).

    • In the drug trial example with 10 tests, the Type I error probability can peak as high as 0.401 (40.1%), highlighting impractical reliance on multiple t-tests.

    • To keep overall α at 0.05 while running multiple tests, the Bonferroni adjustment is used.

Bonferroni Adjustment

  • Formula: Set individual t-test α = desired overall α / number of tests (

    • In our medical example with 10 tests: individual α becomes 0.005.

  • Risk: Setting α too low leads to increased Type II errors (β).

  • Conclusion: Running multiple t-tests is counterproductive due to conflicting risks of Type I and Type II errors – recommend using ANOVA instead.

Advantages of ANOVA

  • ANOVA facilitates the examination of more than two treatment types simultaneously.

  • Reduces the number of participants needed compared to individual t-tests.

  • Allows for complex study designs that yield richer insights into variable interrelations.

  • Conducting a single ANOVA is quicker, cost-effective, and more efficient than multiple t-tests.

  • Avoids missing significant relationships due to the limitations of simpler t-tests.

  • Example: Investigating aerobic exercise's effect on anxiety shows that complex relationships can emerge, which simple pairwise comparisons may overlook: there exists a threshold where more exercise may inversely affect anxiety levels.

  • Use of placebo groups in experiments is possible with ANOVA.

    • Case: A study examining the effects of therapy on anxiety revealed that while both therapy and placebo demonstrated notable benefits, a correct assessment would only be possible via ANOVA.

Assumptions of ANOVA

  • Many t-test assumptions hold for ANOVA as well:

    • Null hypothesis (H0): all group means are equal (H0: μ1 = μ2 = μ3 = … = μk).

    • Alternative hypothesis (HA): at least one group mean is different (HA: μ1 ≠ μ2 ≠ μ3 ≠ … ≠ μk).

  • Set significance level (α) usually at 0.05 unless specified otherwise.

  • Type of data must be interval-level or ratio-level data.

  • Assumption of normal distribution; skewness and kurtosis should range between -1.0 and +1.0.

  • Functions of observed statistics and critical values remain consistent.

One-Way ANOVA Explained

  • Also referred to as one-way randomized ANOVA.

  • Utilized when three or more groups (between-subjects design) are involved with a single IV.

  • Example: Comparing memorization techniques across three groups:

    • Group 1: Rote memorization.

    • Group 2: Imagery mnemonic technique.

    • Group 3: Story mnemonic technique.

Null and Alternate Hypothesis in ANOVA

  • Null hypothesis states that there are no differences between the three groups (H0: μrote = μimagery = μstory).

  • Alternate hypothesis suggests that at least one group differs from the others (HA: at least one μ ≠ another μ).

  • The focus of ANOVA is more about the variability between groups than within groups, establishing an interest in IQ variance as influenced by the IV (type of rehearsal).

Sources of Variation in ANOVA

  • Within-groups variance: undesirable randomness (error variance).

  • Between-groups variance: systematic variance; ideally due to IV effects.

  • Goal: Enlarge between-groups variance while minimizing within-groups variance; this is assessed using the F-ratio.

    • A larger F ratio indicates a significant variance in means across treatment groups.

Steps to Calculate ANOVA

  1. Calculate Sums of Squares (SS) for three sources:

    • Between groups.

    • Within groups.

    • Total.

  2. Determine Degrees of Freedom (df) for each source.

  3. Calculate Mean Squared Deviation (Mean Squares) for groups.

  4. Calculate F-ratio using computed mean squares.

  5. Consult F-table to assess statistical significance.

Sums of Squares Calculation Steps

  • Total sum of squares: take each score, subtract the grand mean, square the value, and sum.

  • Within-groups sum of squares: measures variability within each group from their group mean (error variance).

  • Between-groups sum of squares: variance attributed to the differences between group means.

Degree of Freedom Calculations

  • dfbetween: number of groups - 1 (k - 1).

    • Example for three groups = 3 - 1 = 2.

  • dfwithin: number of subjects - number of groups (N - k).

    • Example for 24 subjects and 3 groups = 24 - 3 = 21.

  • dftotal: number of subjects - 1 (N - 1).

    • Example = 24 - 1 = 23.

Mean Squares and F-Ratio

  • Mean squares are obtained by dividing SS by df for both between and within groups.

  • F-ratio derived by dividing MSbetween by MSwithin, indicating how much larger the variability between groups is than that within groups.

Interpretation of ANOVA

  • Comparisons against critical values in F-corrected tables identify if hypotheses can be rejected.

  • Two sets of degrees of freedom (dfbetween and dfwithin) utilized to establish critical benchmarks.

  • Once a significant F value is achieved, investigate which groups differ with effect sizes calculations.

Effect Size in ANOVA

  • Conceptualized through eta-squared (η²), indicating the proportion of variance in the DV explained by the IV.

    • Values of η² interpretations:

    • η² ≈ .10: small effect.

    • η² ≈ .25: moderate effect.

    • η² ≈ .40: large effect.

  • Example demonstrates 51% of variance attributable to the condition assigned, correlating to a large effect.

Tukey's HSD Test

  • Important for determining specific significant differences among groups post-ANOVA.

  • Allows for comprehensive comparisons avoiding inflated Type I error rates.

  • Only implemented if the overall F indicates significance.

  • Formula relates Q to within-groups mean square and identifies commonly significant mean differences based on calculated differences (e.g., minimum significant difference threshold).

Application of Tukey's HSD

  • Identifying significant differences among groups:

    • Differences must be larger than the calculated HSD.

  • Example provided illustrates comparative means for calculations.

Conducting ANOVA in SPSS

  • Streamlined process via software, which also provides potential for automatic post hoc testing.

  • Important: Clearly label groups in datasets for accurate analyses.

  • Process includes moving DV and grouping variables into designated SPSS fields, followed by proper settings adjustments for descriptive outputs.

References

  • Cohen, J. (1966). Some statistical issues in psychological research. Handbook of Clinical Psychology.

  • Dunn, O.J. (1959 & 1961). Estimation of the medians for dependent variables & Multiple comparisons among means.

  • Paul, G.L. (1966 & 1967). Insight vs. desensitization in psychotherapy.

  • Tukey, J.W. (1953). The problem of multiple comparisons.