4.2 and 4.3

Overview of One-Way ANOVA

• One-Way ANOVA (Analysis of Variance):
    - An extension of the T-test that is utilized when comparing three or more groups.
    - The focus of this module is to understand its use, design requirements, assumptions, and interpretation of results.

Key Learning Objectives

• Understand the distinction between One-Way ANOVA and T-tests.

• Explain why One-Way ANOVA is preferable to using multiple T-tests.

• Identify the research design requirements for One-Way ANOVA:
    - A categorical independent variable with 3 or more groups.
    - A continuous dependent variable.

• Evaluate assumptions regarding One-Way ANOVA:
    - Understand what the assumptions are.
    - Assess whether they have been met or violated.
    - Identify alternative approaches if assumptions are violated.

• Interpret the results of an ANOVA including:
    - F-ratio (F statistic or F value).
    - Degrees of freedom.
    - P-values.
    - Effect sizes.

Comparison of ANOVA and T-test

• T-tests are suitable for comparing two groups but become problematic with multiple groups due to increased Type I error.

• ANOVA controls the Type I error rate, which is the risk of incorrectly rejecting the null hypothesis (false positive).

• Family-wise error rate: The risk of a Type I error across multiple tests is higher when multiple T-tests are performed.

• ANOVA is considered an omnibus test, testing all comparisons simultaneously to determine if there are any differences among group means.

One-Way ANOVA Design

• Independent Variable:
    - Must have only one independent variable with three or more levels.

• Dependent Variable:
    - Needs to be continuous (e.g., driving errors).

Example Scenario

• Independent Variable: Alcohol consumption (categorical variable with three conditions):
    - Condition 1: No Alcohol (placebo).
    - Condition 2: Moderate Alcohol (0.05 Blood Alcohol Concentration).
    - Condition 3: High Alcohol (0.08 Blood Alcohol Concentration).

• Dependent Variable: Number of errors made in a driving simulator after 12 hours.

Hypothesis in One-Way ANOVA

• Null Hypothesis ($H_0$): No differences among the group means.
    - $H_0: ext{Mean}{ ext{placebo}} = ext{Mean}{ ext{moderate}} = ext{Mean}_{ ext{high}}$

• Alternative Hypothesis ($H_a$): At least one group mean is different from another.
    - $H_a: ext{Not all means are equal}$.

• The hypothesis is non-directional.

Assumptions of One-Way ANOVA

1. Independence:
      - Scores must be independent across groups (no overlap of participants).

2. Continuous Dependent Variable:
      - The dependent variable must be on an interval or ratio scale.

3. Normality:
      - Dependent variable should be normally distributed within each group.
      - Testing normality in Jamovi can be done using:
        - Shapiro-Wilk Test.
        - QQ Plot.
        - Visual inspection through histograms.

4. Homogeneity of Variance:
      - Variability within each group should be approximately equal.
      - Typically assessed with Levene's test.
      - Non-significant outcomes indicate the assumption is satisfied.

Evaluating Assumptions in Context

• Analyze the hangover example for assumptions:
    1. Independence: Each participant is only in one experimental condition.
    2. Continuous variable: Number of driving errors is indeed continuous.
    3. Normal distribution: Assessed using tests and graphical plots.
    4. Homogeneity of variances: Levene's test checked for equal variances among groups.

Running the One-Way ANOVA

• Calculate test statistic including:
    - Degrees of Freedom (df):
      - Between groups: $(k - 1) = 2$ (where k = number of groups).
      - Within groups: $(N - k) = 117$ (where N = total number of participants).
    - F statistic formula:
      - $F = rac{ ext{Variance Between Groups}}{ ext{Variance Within Groups}}$.

• Example output table will include:
    - Values for variance between groups and within groups.
    - Calculated F-value (critical for significance testing).

Interpretation of Results

• Determine if the F statistic is statistically significant (P-value < 0.05).

• Assess the magnitude of the effect size, utilizing eta squared ($ ext{η²}$) which indicates the proportion of variance explained by the independent variable:
    - Small effect: 0 < ext{η²} < 0.01 Medium effect: 0.01 < ext{η²} < 0.06.   - Large effect: ext{η²} > 0.14.

• Properly structure a written analysis incorporating:
    - Degrees of freedom.
    - F statistic.
    - P-value.
    - Effect size.

Follow-Up Analysis with Post Hoc Tests

• Necessary only when a significant ANOVA result is found.

• Post hoc tests help determine which specific group means are different without inflating Type I error rate.

• Understand the difference between general post hoc tests and planned contrasts:
    - Planned contrasts utilize pre-specified hypotheses about the groups.

• Common post hoc tests include:
    - Tukey's Honestly Significant Difference (HSD): Ideal for multiple comparisons.
    - Bonferroni correction: Strict control for Type I error, best suited for fewer comparisons.

Conducting and Interpreting Post Hoc Tests

1. Run multiple pairwise comparisons after ANOVA:
      - For three groups, comparisons include:
        - Group A vs B
        - Group A vs C
        - Group B vs C

2. Report key statistics for each comparison:
      - Test statistic, degrees of freedom, P-value, and effect size.
      - Display results with means, standard deviations, and confidence intervals.

3. Assess clinical relevance, addressing implications of findings on driving ability after alcohol consumption.

Summary

• Utilize One-Way ANOVA for studies with multiple conditions (three or more groups).

• Maintain strict adherence to assumptions related to independence, distribution, and variance homogeneity.

• Follow up on significant ANOVA findings with appropriate post hoc tests to ascertain specific group differences while controlling for error rates.

• Clearly present, interpret, and communicate findings in reports or presentations to ensure clarity and comprehensiveness to the audience.