ANOVA (Analysis of Variance) is a test that checks whether the means of three or more groups are equal, by comparing the variation between groups with the variation within groups using the F-ratio.
Why not many t-tests? Comparing 4 groups pairwise needs 6 t-tests; each has a 5% chance of a false alarm, so the overall error rate grows. ANOVA tests all means in one go.
Hypotheses. \(H_0: \mu_1 = \mu_2 = \dots = \mu_k\); \(H_1\): at least one mean is different.
Idea. Total variation is split into two parts:
\[SST = SSB + SSW\]
- SSB (between groups): how far group means are from the grand mean, \(\sum n_i(\bar x_i - \bar x)^2\), df \(= k-1\).
- SSW (within groups, error): spread inside each group, df \(= N-k\).
\[F = \frac{MSB}{MSW} = \frac{SSB/(k-1)}{SSW/(N-k)}\]
If \(F\) is larger than the table value \(F_{\alpha}(k-1, N-k)\), reject \(H_0\). A large F means the group means differ more than chance alone would explain.
Types. One-way ANOVA (one factor, e.g. fertiliser type); two-way ANOVA (two factors, e.g. fertiliser and soil type, and their interaction).
Assumptions. Independent observations, roughly normal data in each group, equal variances across groups.
Applications
- Agriculture: comparing crop yields under different fertilisers.
- Medicine: comparing effects of three or more drugs or doses.
- Education: comparing marks of students taught by different methods.
- Industry/quality control: comparing output of different machines or shifts.
- Marketing and A/B/n testing: comparing sales under several advertisement designs.
- Machine learning: feature selection (ANOVA F-test between a numeric feature and class labels) and comparing models.