ANOVA: Analysis of Variance

Why ANOVA?

When comparing means of three or more groups, we use ANOVA (Analysis of Variance) instead of multiple t-tests.

ANOVA Questions

Do test scores differ among three teaching methods?
Is there a difference in crop yield across four fertilizers?
Do customer satisfaction scores vary by store location?

The Logic of ANOVA

ANOVA compares variation between groups to variation within groups.

ANOVA Logic

If group means are truly different:

Between-group variation will be large (groups are spread apart)
Within-group variation reflects natural variability

If group means are equal:

Between-group variation should be similar to within-group variation

The F-statistic measures this ratio:

$F = \frac{\text{Between-group variance}}{\text{Within-group variance}} = \frac{MS_{between}}{MS_{within}}$

Hypotheses in One-Way ANOVA

ANOVA Hypotheses

H₀: μ₁ = μ₂ = μ₃ = … = μₖ (all means equal)

H₁: At least one mean is different

ANOVA Table

Source	SS	df	MS	F
Between	SS_B	k - 1	MS_B = SS_B/(k-1)	F = MS_B/MS_W
Within	SS_W	N - k	MS_W = SS_W/(N-k)
Total	SS_T	N - 1

Where:

k = number of groups
N = total sample size
SS = Sum of Squares
MS = Mean Square

Sum of Squares

Total SS: $SS_T = \sum(x_{ij} - \bar{x}_{grand})^2$

Between SS: $SS_B = \sum n_j(\bar{x}_j - \bar{x}_{grand})^2$

Within SS: $SS_W = \sum\sum(x_{ij} - \bar{x}_j)^2$

Note: $SS_T = SS_B + SS_W$

Complete Example

One-Way ANOVA

Three fertilizers tested on plant growth (in cm):

Fertilizer A	Fertilizer B	Fertilizer C
20	24	28
22	26	30
21	25	29
23	27	31
19	23	27

Group means: A = 21, B = 25, C = 29 Grand mean: (21 + 25 + 29)/3 = 25

Calculate SS_Between: $SS_B = 5(21-25)^2 + 5(25-25)^2 + 5(29-25)^2$ $= 5(16) + 5(0) + 5(16) = 160$

Calculate SS_Within: Each group has variance = 2.5, so: $SS_W = (5-1)(2.5) + (5-1)(2.5) + (5-1)(2.5) = 4(2.5) \times 3 = 30$

ANOVA Table:

Source	SS	df	MS	F
Between	160	2	80	32
Within	30	12	2.5
Total	190	14

F = 80/2.5 = 32

Critical value: F₀.₀₅(2,12) = 3.89

P-value: < 0.0001

Decision: Since F = 32 > 3.89, reject H₀.

Conclusion: There is significant evidence that at least one fertilizer produces different plant growth.

Assumptions of ANOVA

Independence: Observations are independent within and between groups
Normality: Each group’s population is normally distributed
Homogeneity of variances: All groups have equal population variances

If Assumptions Fail

Violation	Solution
Non-normality	Kruskal-Wallis test (non-parametric)
Unequal variances	Welch’s ANOVA
Both	Kruskal-Wallis test

Effect Size: Eta-Squared

Eta-Squared

$\eta^2 = \frac{SS_{between}}{SS_{total}}$

Proportion of variance explained by group membership

η²	Interpretation
0.01	Small
0.06	Medium
0.14	Large

Effect Size Calculation

From the fertilizer example: $\eta^2 = \frac{160}{190} = 0.84$

84% of the variation in plant growth is explained by fertilizer type. This is a very large effect.

Post-Hoc Tests

If ANOVA is significant, which groups differ? Use post-hoc (multiple comparison) tests.

Common Post-Hoc Tests

Test	When to Use
Tukey’s HSD	Equal sample sizes, all pairwise comparisons
Bonferroni	Conservative, any comparison
Scheffé	Complex contrasts, very conservative
Games-Howell	Unequal variances
Dunnett	Compare each treatment to control

Tukey's HSD

Groups i and j differ if: $|\bar{x}_i - \bar{x}_j| > q_{\alpha,k,df_W} \times \sqrt{\frac{MS_W}{n}}$

Where q is from the Studentized Range distribution

Tukey's HSD

From the fertilizer example:

For α = 0.05, k = 3, df = 12: q = 3.77

Critical difference = 3.77 × √(2.5/5) = 3.77 × 0.707 = 2.67

Pairwise differences:

|A - B| = |21 - 25| = 4 > 2.67 ✓ Significant
|A - C| = |21 - 29| = 8 > 2.67 ✓ Significant
|B - C| = |25 - 29| = 4 > 2.67 ✓ Significant

All three fertilizers produce significantly different growth!

Two-Way ANOVA (Preview)

Two-way ANOVA examines:

Main effect of Factor A
Main effect of Factor B
Interaction between A and B

Two-Way ANOVA Setup

Study how fertilizer type (A, B, C) and watering schedule (daily, weekly) affect plant growth.

Questions answered:

Is there an effect of fertilizer type?
Is there an effect of watering schedule?
Does the effect of fertilizer depend on watering schedule? (interaction)

ANOVA vs Multiple T-Tests

ANOVA	Multiple T-Tests
Controls Type I error rate	Inflates Type I error
One test for overall difference	Multiple tests needed
Needs post-hoc for specifics	Directly compares pairs
More efficient	Less efficient

Summary

In this lesson, you learned:

ANOVA compares means of 3+ groups while controlling Type I error
F-statistic = Between-group variance / Within-group variance
H₀: All group means are equal; H₁: At least one differs
Large F leads to rejecting H₀
Assumptions: Independence, normality, equal variances
Eta-squared (η²) measures effect size
Post-hoc tests (Tukey, Bonferroni) identify which groups differ

Practice Problems

1. Four teaching methods were compared on final exam scores:

Method 1: n=10, mean=75, SD=8
Method 2: n=10, mean=80, SD=9
Method 3: n=10, mean=72, SD=7
Method 4: n=10, mean=85, SD=10

The ANOVA yields F = 4.25. With df = (3, 36) and F-critical = 2.87, what’s the conclusion?

2. An ANOVA has SS_Between = 450 and SS_Total = 1500. a) What is SS_Within? b) Calculate η²

3. If an ANOVA with 5 groups (n=20 each) yields F = 6.2 and is significant, how many post-hoc pairwise comparisons are possible?

4. Why is it inappropriate to use multiple t-tests to compare 5 groups?

Click to see answers

1. F = 4.25 > F-critical = 2.87 Reject H₀. There is significant evidence that at least one teaching method produces different results.

2. a) SS_Within = SS_Total - SS_Between = 1500 - 450 = 1050 b) η² = SS_B/SS_T = 450/1500 = 0.30 (large effect)

3. Number of pairwise comparisons = k(k-1)/2 = 5(4)/2 = 10 comparisons

4. With 5 groups, there would be 10 pairwise t-tests. At α = 0.05 each:

P(at least one Type I error) = 1 - (0.95)¹⁰ ≈ 0.40
40% chance of false positive!
ANOVA maintains the overall α at 0.05

Next Steps

Continue with advanced analysis:

Chi-Square Tests - Categorical data analysis
Correlation - Relationships between variables
Linear Regression - Predicting outcomes

Why ANOVA?

The Logic of ANOVA

Hypotheses in One-Way ANOVA

ANOVA Table

Complete Example

Assumptions of ANOVA

If Assumptions Fail

Effect Size: Eta-Squared

Post-Hoc Tests

Common Post-Hoc Tests

Two-Way ANOVA (Preview)

ANOVA vs Multiple T-Tests

Summary

Practice Problems

Next Steps

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