ANOVA (one-way)

Why ANOVA

If you compare 3+ groups with many t-tests, you increase false positives.

ANOVA tests:

• H0: all group means are equal
• H1: at least one mean differs

Example

One-way ANOVA

import numpy as np
from scipy import stats
 
A = np.array([10, 12, 11, 13, 12])
B = np.array([7, 8, 9, 8, 7])
C = np.array([14, 15, 13, 16, 14])
 
f_stat, p = stats.f_oneway(A, B, C)
print("F:", f_stat)
print("p:", p)

If ANOVA is significant

ANOVA says “some difference exists” but not where.

Next steps:

• Post-hoc tests (e.g., Tukey HSD)
• Pairwise comparisons with correction

Assumptions

• Independence
• Normality within groups (approx)
• Homogeneity of variances

If variances differ a lot, consider Welch ANOVA or non-parametric alternatives.

Why not just run many t-tests?

diagram ANOVA vs. repeated t-tests mermaid

  flowchart TD
  A["3+ groups to compare"] --> B["Many pairwise t-tests"]
  A --> C["One-way ANOVA"]
  B --> D["False positive rate
compounds with each test"]
  C --> E["Single F-test:
any group differ?"]
  E --> F["If significant,
run post-hoc tests"]

Running many pairwise t-tests inflates false positives; ANOVA tests all groups at once.

From the book

This is the same idea Patsy/statsmodels formalize with formula syntax like v2 ~ key1 + key2 + key1:key2 in Chapter 12 — ANOVA is really a linear model with categorical predictors. Once you’re comfortable with f_oneway, smf.ols(...).fit() plus sm.stats.anova_lm(...) gets you the same F-test with richer diagnostics.

🧪 Try It Yourself

Exercise 1 – Run a one-way ANOVA

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Exercise 2 – Reading the result

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Exercise 3 – Group means and variances

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Next

Continue to Chi-Square Test to compare categorical variables instead of numeric means.