ANOVA.md
Analysis of Variance (ANOVA) is a statistical technique used to compare the means of three or more groups. It tests whether there are any statistically significant differences between the means of independent (unrelated) groups.
1. When to Use ANOVA
Comparing mean blood pressure across three treatment groups
Testing whether multiple teaching methods affect exam scores differently
Analyzing differences in plant growth under different fertilizer types
2. Types of ANOVA
a. One-way ANOVA
Used when comparing means across groups based on one independent variable (factor).
Example: Comparing average test scores across four different schools
b. Two-way ANOVA
Used when examining the effect of two independent variables and their interaction.
Example: Testing the effects of diet (A/B) and exercise (yes/no) on weight loss
3. ANOVA Terminology
Between-group variance: Variability due to differences between group means
Within-group variance: Variability due to differences within each group
F-statistic: Ratio of between-group variance to within-group variance
4. Assumptions of ANOVA
Independence of observations
Normally distributed residuals within each group
Homogeneity of variances (equal variances across groups)
5. Interpreting ANOVA
Null hypothesis (H0): All group means are equal
Alternative hypothesis (Ha): At least one group mean is different
If the p-value from the F-test < 0.05, reject H0
6. Post-hoc Tests
If ANOVA is significant, use post-hoc tests to identify which group means differ:
Tukey’s HSD
Bonferroni correction
Scheffé’s method
7. Example (R Code)
# One-way ANOVA
result <- aov(score ~ group, data = mydata)
summary(result)
# Post-hoc test
TukeyHSD(result)8. Summary
ANOVA is a powerful method for testing differences among group means. It relies on specific assumptions and, when significant, should be followed by post-hoc comparisons to identify which groups differ.
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