7.1 One-way ANOVA

One-way ANOVA compares means across multiple groups, extending the t-test concept. It determines if there are statistically significant differences between group means, revealing the overall effect of an independent variable on a dependent variable.

The F-statistic quantifies the ratio of between-group to within-group variance, indicating group mean differences. Post-hoc tests like Tukey's HSD identify specific group differences when the overall ANOVA result is significant, controlling for Type I error inflation.

Understanding One-Way ANOVA

Purpose of one-way ANOVA

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• Compares means of three or more independent groups (college majors, treatment types, age categories)
• Determines statistically significant differences between group means revealing overall effect of independent variable
• Extends t-test concept to multiple groups simultaneously reducing Type I error risk

Interpretation of ANOVA results

• F-statistic quantifies ratio of between-group variance to within-group variance indicating group mean differences
• Larger F-values suggest greater differences between group means (F = 10 vs F = 2)
• p-value represents probability of obtaining observed F-statistic under null hypothesis of equal means
• Typically compared to significance level α = 0.05 for decision-making
• Reject null hypothesis if p < α, indicating at least one group mean differs significantly
• Fail to reject null hypothesis if p ≥ α, suggesting insufficient evidence for group differences

Post-hoc tests for group differences

• Identifies specific groups that differ when overall ANOVA result is significant
• Controls Type I error inflation from multiple comparisons
• Common methods: Tukey's HSD, Bonferroni correction, Scheffé's method
• Pairwise comparisons assess each group mean against others with adjusted p-values
• Results highlight statistically significant differences between specific group pairs
• Reveals magnitude and direction of mean differences (Group A > Group B by 5 units)

Homogeneity of variances assumption

• Levene's test assesses equality of group variances
• Null hypothesis: All group variances are equal
• Alternative hypothesis: At least one group variance differs
• Reject null if p < α, indicating heterogeneous variances
• Violation can increase Type I error rate and reduce ANOVA power
• Alternatives for heteroscedasticity: Welch's ANOVA, Kruskal-Wallis test
• Other important assumptions: normality of residuals, independence of observations