One-way ANOVA is a statistical method used to compare means across three or more groups. It helps determine if there are significant differences between group means, such as test scores or blood pressure levels, for different categories like age groups or treatment conditions.

The analysis involves setting up hypotheses, where the null hypothesis assumes no difference between group means. ANOVA components include the grand mean, omnibus test, and effect size. Box plots are useful for visualizing group differences and checking assumptions.

One-Way ANOVA

Purpose of one-way ANOVA

• Compares means of three or more groups or levels of an independent variable (age groups, treatment conditions)
• Determines if there is a statistically significant difference between the means (test scores, blood pressure levels)
• Independent variable is categorical with at least three levels (low, medium, high dosage)
• Dependent variable is continuous (weight, height, income)

Hypotheses for multiple group comparisons

• Null hypothesis $H_0$
  • States that there is no significant difference between the means of the groups
  • Mathematically: $\mu_1 = \mu_2 = \mu_3 = ... = \mu_k$, where $k$ is the number of groups
• Alternative hypothesis $H_a$
  • States that at least one group mean is significantly different from the others
  • Mathematically: At least one $\mu_i \neq \mu_j$, where $i \neq j$
• Two-tailed test checks for any difference between group means, without specifying which group differs (comparing exam scores between schools)
• One-tailed test checks if a specific group mean is greater than or less than the others (testing if a new drug increases performance compared to a placebo)

Analysis of Variance (ANOVA) Components

• Grand mean: The overall mean of all observations across all groups
• Omnibus test: ANOVA serves as an omnibus test to determine if there are any significant differences among group means
• Effect size: Measures the magnitude of the difference between group means
• Multiple comparisons: Post-hoc tests conducted after a significant ANOVA result to determine which specific groups differ
  • Pairwise comparisons: Comparing means between pairs of groups to identify specific differences

Interpreting ANOVA Results

Box plots for ANOVA visualization

• Box plots display the distribution of the dependent variable for each group
  • Median represented by the line inside the box
  • Interquartile range (IQR) contains the middle 50% of the data
  • Whiskers extend to the minimum and maximum values within 1.5 times the IQR
  • Outliers shown as individual points beyond the whiskers
• Comparing box plots across groups
  • Non-overlapping boxes suggest significant differences between group means (income levels between countries)
  • Overlapping boxes do not necessarily imply non-significant differences (consider extent of overlap and sample sizes)
  • Similar box sizes across groups suggest homogeneity of variances (consistent variability in test scores across schools)
  • Outliers can affect the one-way ANOVA results and should be investigated (extremely high or low values)
• Box plots provide a visual representation of group differences
  • Complement the numerical results of the one-way ANOVA
  • Help identify potential issues with assumptions heterogeneity of variances