ECON20003 · Quantitative Methods 2
One-Way ANOVA & Nonparametric Alternatives
One-Way ANOVA & Nonparametric Alternatives extends the two-sample comparison to three or more groups. The idea is a partition of total variation: total sum of squares splits into a between-group part (SST) and a within-group part (SSE). Dividing each by its degrees of freedom gives mean squares, and their ratio F = MST/MSE tests whether all group means are equal. The same assumption gate applies — independent random samples, normal populations, equal variances — and when normality fails you switch to the Kruskal-Wallis test.
What this chapter covers
- 01Sum-of-squares partition: SS_total = SST (between) + SSE (within)
- 02Mean squares: MST = SST/(k−1), MSE = SSE/(n−k)
- 03F = MST/MSE with df = (k−1, n−k); H₀: μ₁ = … = μ_k
- 04ANOVA assumptions: independence, normality, equal variances
- 05Kruskal-Wallis as the nonparametric alternative (H ~ χ²_{k−1})
- 06Randomised-block ANOVA: SS = SST + SSB + SSE
One-way ANOVA F-test with an assumption gate
- 1 markState the hypotheses: H₀: μ₁ = μ₂ = μ₃ versus H₁: at least one mean differs.
- 1 markMean square between (treatment): MST = SST/(k−1) = 48/(3−1) = 48/2 = 24.
- 1 markMean square within (error): MSE = SSE/(n−k) = 162/(30−3) = 162/27 = 6.
- 2 marksTest statistic: F = MST/MSE = 24/6 = 4.0, with df = (k−1, n−k) = (2, 27).
- 1 markDecision rule: reject if F > F₀.₀₅,₂,₂₇ ≈ 3.35. Since 4.0 > 3.35, reject H₀.
- 2 marksConclude in context: at least one production line has a different mean monthly output. Assumption gate: if Shapiro-Wilk on the residuals returned a small p (non-normal), report the Kruskal-Wallis result instead.
Key terms
- Sum-of-squares partition
- ANOVA splits the total variation into a between-group component SST (differences among group means) and a within-group component SSE (variation inside each group): SS_total = SST + SSE.
- Mean square
- A sum of squares divided by its degrees of freedom: MST = SST/(k−1) measures between-group spread and MSE = SSE/(n−k) estimates the common within-group variance.
- ANOVA F-statistic
- F = MST/MSE compares between-group to within-group variation under H₀ of equal means, with df = (k−1, n−k). A large F (small p) signals that at least one group mean differs.
- Kruskal-Wallis test
- The nonparametric alternative to one-way ANOVA. It ranks all observations and tests for differences in location across groups, with H ~ χ²_{k−1}; used when the normality assumption fails.
One-Way ANOVA & Nonparametric Alternatives FAQ
If ANOVA rejects, which group is different?
ANOVA's F-test only says at least one mean differs — it does not identify which. You then run a post-hoc multiple-comparison procedure to find the specific pairs that differ, controlling the overall error rate.
When should I use Kruskal-Wallis instead of ANOVA?
When the ANOVA normality assumption fails (a small Shapiro-Wilk p on the residuals, or visibly skewed groups). Kruskal-Wallis works on ranks and only needs similarly shaped distributions, so it is the safe nonparametric substitute.
Exam move
Learn to fill in an ANOVA table from partial information — given any two of SST, SSE, df and the mean squares, you can recover the rest and the F. Rehearse the assumption gate aloud so you automatically mention Kruskal-Wallis whenever normality is in doubt.
