CARDS Group Differences ANOVA

Introduction to ANOVA

• ANOVA (Analysis of Variance)

  • A statistical method used to test for differences between means across two or more groups.

  • Allows for assessing the presence of statistically significant differences among categorical groups.

Key Terminology

• Factor: The independent variable (IV).

  • Types: one-way vs. two-way ANOVA.

• Levels: Different categories within a factor.

• Response Variable (DV): The dependent variable or outcome being measured.

Assumptions of ANOVA

• Independence: No relationship between subjects in each sample; groups must consist of independent samples.

• Equal Sample Sizes: Different groups/levels must have equal sample sizes for optimal results.

• Normality: The response variable should follow a normal distribution (middle scores should be most frequent).

• Homoscedasticity: Population variance must be equal across groups (homogeneity of variances).

Types of ANOVA

One-Way ANOVA

• Involves one categorical independent variable and normally distributed, continuous dependent variable.

• Assesses differences across two or more groups.

• Can only compare means among three or more groups.

Two-Way ANOVA

• Involves two or more categorical independent variables.

• Also assumes normally distributed, continuous dependent variables.

Repeated Measures ANOVA

• Uses the same subjects measured multiple times under different conditions.

• Commonly used in longitudinal studies.

Mixed-Design ANOVA

• Analyzes the effects of two or more independent variables on one or more dependent variables.

• Useful when comparing multiple dependent variables.

Analysis of Covariance (ANCOVA)

• Combines ANOVA and regression techniques.

• Tests for significant differences among group means while controlling for other variables (covariates).

ANOVA F-value

• Denotes the test statistic in ANOVA (F).

• Formula: F = Variance between groups ÷ Variance within groups.

• High F-value suggests significant differences exist between levels of the IV when p < 0.05.

Between vs. Within Groups Variance

• Between-Groups Variance: Variability among the sample means of different groups.

• Within-Groups Variance: Variability within each of the sample distributions.

• ANOVA aims to determine if the between-group variance is greater than the within-group variance.

The Logic of ANOVA

• ANOVA compares variance between data samples to variance within each sample.

• High between-group variance and low within-group variance indicate significant differences among group means.

Post-Hoc Testing

• Conducted after rejecting the null hypothesis in an ANOVA.

• Allows for multiple comparisons among group means.

• Tukey HSD: A common post-hoc test used to identify specifically which group means are significantly different.

Effect Size

• Eta squared (η²): Quantifies the magnitude of the association between IV and DV.

• R-squared (R²): Indicates the proportion of variance in the DV accounted for by the IV.

• Provides context for the practical implications of differences observed.

Reporting Results of ANOVA in APA Format

• Components to include:

  • Purpose of the study, sample size, descriptive statistics (mean, standard deviation), F-statistic, degrees of freedom, p-value, and effect size, along with interpretations and post-hoc results.

Comparison with Other Statistical Tests

• ANOVA vs. t-Test: ANOVA assesses more than two groups while t-Test compares only two samples.

• ANOVA vs. Chi-Square: ANOVA tests means of continuous data, whereas Chi-Square assesses associations between categorical variables.

• Handling Skewed Data: ANOVA is robust against small deviations; for severely skewed data, alternative tests like the Kruskal-Wallis test may be used.

When to Use ANOVA

• To compare groups: Ideal for performance comparisons among more than two groups.

• In experimental designs where subjects are assigned to different conditions.

• For evaluating interactions in two-way or factorial designs, testing how one factor's effect depends on another.

Advantages of ANOVA

• Controls Type I error across multiple comparisons.

• Facilitates the assessment of interactions between factors.

• Handles complex designs including repeated measures and MANOVA.

Disadvantages of ANOVA

• Lack of specificity: Does not identify which groups are different without post-hoc tests.

• Requires larger sample sizes than t-tests, especially for optimality with equal group sizes.

Examples

Term: ANOVA
Examples: A researcher compares the average test scores of students from different schools to see if there is a significant difference.

Term: One-Way ANOVA
Examples: Real-life example: Testing whether different teaching methods produce different average test scores among students.

Term: Two-Way ANOVA
Example: Assessing the impact of both gender and study method on student performance.

Term: Post-Hoc Testing
Examples: After finding significant differences in student performances, using Tukey HSD to determine which specific classes performed differently.

Term: Effect Size (Eta squared)
Examples: Evaluating the strength of the difference in test scores caused by a new curriculum compared to the old one.

Term: ANCOVA
Examples: Studying the effect of a teaching intervention on student performance while controlling for prior academic achievement.

Term: Homogeneity of Variance
Examples: Ensuring the variance in test scores is similar for both boys and girls before conducting an ANOVA.

Term: Tukey HSD
Examples: After finding a significant difference in average salaries across different industries, using Tukey HSD to find out which specific industries differ.