One-Way ANOVA

Definition

Core Statement

One-Way ANOVA (Analysis of Variance) tests whether the means of three or more independent groups are equal. It is an extension of the t-test for multiple groups, using the F-statistic to compare between-group variance to within-group variance.

Purpose

Test for an overall difference among group means without inflating Type I error from multiple t-tests.
Partition total variance into explained (between-group) and unexplained (within-group) components.

When to Use

Use One-Way ANOVA When...

Outcome is continuous.
There are three or more independent groups.
Data is approximately normally distributed within groups.
Variances are approximately equal across groups (Levene's Test).

Alternatives

Unequal variances: Welch's ANOVA.
Non-normal data: Kruskal-Wallis Test.
Two groups: Student's T-Test or Welch's T-Test.

Theoretical Background

Hypotheses

$H_{0}$ : $μ_{1} = μ_{2} = \dots = μ_{k}$ (All group means are equal).
$H_{1}$ : At least one $μ_{j}$ is different.

ANOVA Tells You That Not Where

A significant ANOVA result only indicates that at least one group differs. To find which groups differ, you must run Post-Hoc Tests (e.g., Tukey's HSD).

The F-Ratio

F = \frac{M S_{b e t w e e n}}{M S_{w i t h i n}} = \frac{Variance Explained by Groups}{Variance Within Groups}

Term	Formula	Meaning
$S S_{b e t w e e n}$	$\sum n_{j} ({\bar{X}}_{j} - {\bar{X}}_{g r a n d})^{2}$	Variability due to group differences.
$S S_{w i t h i n}$	$\sum \sum (X_{i j} - {\bar{X}}_{j})^{2}$	Variability within groups (noise).
$M S$	$S S / d f$	Mean Square.

Large F $\to$ Group means are distinct $\to$ Reject $H_{0}$ .

Assumptions

Independence: Observations are independent.
Normality: Data within each group is normally distributed. (Check per-group: Shapiro-Wilk Test).
Homogeneity of Variance: Variances are equal across groups. (Check: Levene's Test). If violated, use Welch's ANOVA.

Limitations

Pitfalls

Sensitive to Variance Inequality: ANOVA assumes equal variances. If violated, Type I error increases.
Does Not Identify Which Groups Differ: Significant F-test requires post-hoc analysis.
Assumes Normality: For small samples, non-normality inflates error.

Python Implementation

import scipy.stats as stats
import statsmodels.stats.multicomp as mc

# Data: Three groups
group1 = [10, 12, 14, 16, 18]
group2 = [20, 22, 24, 26, 28]
group3 = [15, 17, 19, 21, 23]

# 1. One-Way ANOVA
f_stat, p_val = stats.f_oneway(group1, group2, group3)
print(f"F-statistic: {f_stat:.3f}, p-value: {p_val:.4f}")

# 2. Post-Hoc (Tukey HSD) if p < 0.05
if p_val < 0.05:
    data = group1 + group2 + group3
    labels = ['G1']*5 + ['G2']*5 + ['G3']*5
    tukey = mc.MultiComparison(data, labels).tukeyhsd()
    print(tukey)

R Implementation

# 1. Fit ANOVA Model
model <- aov(Score ~ Group, data = df)

# 2. Summary (F-test)
summary(model)

# 3. Check Assumptions
plot(model, 1)  # Residuals vs Fitted (Homoscedasticity)
plot(model, 2)  # Q-Q Plot (Normality)

# 4. Post-Hoc Test (Tukey HSD)
TukeyHSD(model)

# 5. Visualize Tukey Results
plot(TukeyHSD(model))

Worked Numerical Example

Diet Plan Comparison

Scenario: 3 Groups of 10 people each. Weight loss (lbs) after 1 month.

Means: Diet A = 5 lbs, Diet B = 7 lbs, Diet C = 12 lbs.
Grand Mean: 8 lbs.

ANOVA Table:

Between-Group Variance (MS_B): Large (driven by Diet C's 12 lbs).
Within-Group Variance (MS_W): 4.0 (Variation within diets).
F-Statistic: $M S_{B} / M S_{W} = 15.3$ .
Critical F: ~3.35 (for $α = 0.05$ ).

Result: $15.3 > 3.35$ ( $p < 0.001$ ). Reject $H_{0}$ .
Conclusion: At least one diet is effectively different. (Likely C vs A and C vs B).

Interpretation Guide

Output	Interpretation	Edge Case Notes
F = 15.3, p < 0.001	At least one group mean is significantly different.	Does not say which one (A≠B? B≠C?). Run Tukey.
F < 1	Within-group variability > Between-group.	Means are very similar, or noise is huge. $H_{0}$ holds.
Tukey: G1-G2 p < 0.05	G1 and G2 are significantly different.	Confidence interval for Diff excludes 0.
Tukey: G1-G3 p = 0.45	G1 and G3 are statistically indistinguishable.	CI includes 0.
F significant, but no Tukey significant?	Rare Paradox. Variance is partitioned broadly.	The overall pattern is non-random, but no single pair beats the strict pairwise threshold.

Common Pitfall Example

The Multiple T-Test Trap

Scenario: You have 4 groups (A, B, C, D).
Bad Approach: Run 6 separate t-tests (A-B, A-C, A-D, B-C, ...).

The Problem (Alpha Inflation):

Each t-test has 5% error chance.
With 6 tests, probability of at least one false positive is $\approx 1 - (0.95)^{6} \approx 26 %$ .
You are very likely to find a "significant" difference that doesn't exist.

Correct Approach:

Run One-Way ANOVA first.
Only if F is significant, run Post-Hoc tests (Tukey) which control error rate mathematically.

Welch's ANOVA - Robust to unequal variances.
Tukey's HSD - Post-hoc pairwise comparisons.
Bonferroni Correction - Alternative multiple comparison adjustment.
Kruskal-Wallis Test - Non-parametric alternative.
MANOVA - Multiple dependent variables.