ANOVA Quick Reference Card
Module 10: One-Page Cheat Sheet
Purpose & Hypotheses
ANOVA: Compare means of 3+ independent groups simultaneously
H₀: μ₁ = μ₂ = μ₃ = ... (all means equal) | Hₐ: At least one mean differs
Why not multiple t-tests? Inflates Type I error (α inflation)
Essential Formulas
Sum of Squares
SST = Σ(xij - x̄)²
SSB = Σnj(x̄j - x̄)²
SSW = Σ(xij - x̄j)²
SST = SSB + SSW
SST = Σ(xij - x̄)²
SSB = Σnj(x̄j - x̄)²
SSW = Σ(xij - x̄j)²
SST = SSB + SSW
Degrees of Freedom
dfbetween = k - 1
dfwithin = N - k
dftotal = N - 1
(k=groups, N=total n)
dfbetween = k - 1
dfwithin = N - k
dftotal = N - 1
(k=groups, N=total n)
Mean Squares
MSB = SSB / (k-1)
MSW = SSW / (N-k)
MSB = SSB / (k-1)
MSW = SSW / (N-k)
F-Statistic
F = MSB / MSW
Large F → groups differ
F ≈ 1 → groups similar
F = MSB / MSW
Large F → groups differ
F ≈ 1 → groups similar
Tukey's HSD
HSD = q × √(MSW / n)
q from Studentized Range table
If |x̄i-x̄j| > HSD → significant
HSD = q × √(MSW / n)
q from Studentized Range table
If |x̄i-x̄j| > HSD → significant
Bonferroni
αadj = α / c
c = k(k-1)/2 comparisons
Use αadj for each test
αadj = α / c
c = k(k-1)/2 comparisons
Use αadj for each test
ANOVA Table Structure
| Source | SS | df | MS | F |
|---|---|---|---|---|
| Between | SSB | k-1 | SSB/(k-1) | MSB/MSW |
| Within | SSW | N-k | SSW/(N-k) | — |
| Total | SST | N-1 | — | — |
Assumptions & Post-Hoc Tests
Three Assumptions:
- Independence: Random assignment/sampling
- Normality: Check plots, OK if n≥30
- Equal Variances: max(s²)/min(s²) < 2
If Violated:
- Independence → Repeated Measures ANOVA
- Normality → Kruskal-Wallis test
- Equal Variance → Welch's ANOVA
Post-Hoc Tests:
- Tukey's HSD: Standard choice, all pairs
- Bonferroni: Unequal n, conservative
- Scheffé: Most conservative
- Dunnett's: Compare to control only
When to Use Post-Hoc:
- ANOVA is significant (p < α)
- 3 or more groups
- Don't use if ANOVA not significant
ANOVA Procedure (11 Steps)
- State H₀ and Hₐ
- Choose α (usually 0.05)
- Check assumptions
- Calculate group means (x̄j) and grand mean (x̄)
- Calculate SSB, SSW, SST
- Find dfbetween, dfwithin
- Calculate MSB, MSW
- Calculate F = MSB/MSW
- Find p-value or Fcritical
- Make decision (reject or fail to reject H₀)
- If significant: Conduct post-hoc tests
Which Test to Use?
| Situation | Test |
|---|---|
| 2 groups, independent | Two-sample t-test |
| 3+ groups, independent, assumptions met | One-way ANOVA |
| 3+ groups, severe non-normality | Kruskal-Wallis |
| 3+ groups, unequal variances | Welch's ANOVA |
| Same subjects, multiple times | Repeated Measures ANOVA |
| 2+ factors | Two-way/Factorial ANOVA |
Common Mistakes to Avoid
DON'T:
- Use multiple t-tests for 3+ groups
- Do post-hoc if ANOVA not significant
- Confuse SSB and SSW
- Forget to divide by df for MS
- Use ANOVA with repeated measures
- Ignore assumptions
DO:
- Check all assumptions first
- Report F with both df: F(df₁, df₂)
- Verify SST = SSB + SSW
- Use post-hoc only after significant F
- Interpret in context
- Report effect size
Key Takeaways
- ANOVA tests: Are all population means equal? (3+ groups)
- F-statistic: Ratio of between-group to within-group variance
- Large F: Groups differ significantly
- Significant ANOVA: Use post-hoc tests to identify which groups differ
- Three assumptions: Independence (critical!), normality, equal variances
- Robust when: Large n (≥30), equal sample sizes, moderate violations
- Report format: F(df₁, df₂) = value, p = value