Analysis of Variance

One-way ANOVA tests $H_0: \mu_1 = \mu_2 = \cdots = \mu_k$ (all $k$ group means are equal) against $H_a$: at least one pair of means differs. It replaces the multiple $t$-test approach, which would inflate the Type I error rate.

Partitioning variability: $SS_{\text{total}} = SS_{\text{between}} + SS_{\text{within}}$. $SS_{\text{between}} = \sum_j n_j(\bar{x}_j - \bar{x})^2$ measures how much group means vary around the grand mean. $SS_{\text{within}} = \sum_j \sum_i (x_{ij}-\bar{x}_j)^2$ measures variability within each group.

The $F$-statistic: $F = MS_{\text{between}}/MS_{\text{within}}$ where $MS = SS/df$. $df_{\text{between}} = k-1$, $df_{\text{within}} = N-k$ (total observations $N$, groups $k$). Under $H_0$, $F \sim F_{k-1,\,N-k}$.

Interpretation: a large $F$ means the between-group variation is large relative to within-group noise — the group means differ more than chance would predict. Reject $H_0$ if $F > F_{\alpha, k-1, N-k}$ (critical value from $F$-table).

The ANOVA table summarises the decomposition: Source, SS, df, MS, $F$, $p$-value. Each row represents one source of variability.

Independence: all observations are independent within and across groups. Violated by repeated measures on the same subjects (use repeated-measures ANOVA instead).

Normality: each group's population is approximately normally distributed. ANOVA is robust to moderate violations when group sizes are equal and $n_j \geq 5$. Check with Q-Q plots of residuals.

Homoscedasticity (equal variances): all groups share the same variance $\sigma^2$. Check with Levene's test or by comparing the largest to smallest sample standard deviation (ratio $\leq 2$ is a common rule of thumb). Welch's one-way ANOVA does not require equal variances.

If assumptions are seriously violated, consider transforming the data (e.g., log-transform for right-skewed data) or using the Kruskal-Wallis test (the non-parametric alternative to one-way ANOVA).

A significant $F$-test tells you at least one mean differs — not which pairs. Post-hoc tests identify specific differences while controlling the family-wise error rate (FWER).

Tukey's HSD (Honest Significant Difference): tests all $\binom{k}{2}$ pairwise comparisons and controls the FWER at $\alpha$ exactly. It is the most commonly used post-hoc method for balanced designs.

Bonferroni correction: for $m$ comparisons, use $\alpha/m$ for each test. Simple and widely applicable, but more conservative (lower power) than Tukey's when $m$ is large.

Scheffé's method: allows any contrast (not just pairwise) while controlling FWER. Most conservative but most flexible.

Planned contrasts: if specific comparisons are hypothesised in advance (before seeing data), they can be tested at level $\alpha$ without correction. Only pre-specified comparisons qualify.

Two-way ANOVA has two categorical factors $A$ (with $a$ levels) and $B$ (with $b$ levels). It tests three hypotheses: the main effect of $A$, the main effect of $B$, and the $A\times B$ interaction.

An interaction $A\times B$ means the effect of $A$ on the response depends on the level of $B$ (and vice versa). When an interaction is significant, interpret main effects cautiously — they may be misleading averages of varying effects.

The total SS is partitioned: $SS_{\text{total}} = SS_A + SS_B + SS_{AB} + SS_{\text{error}}$. Each has an associated $F$-ratio and $p$-value.

Two-way ANOVA with replication (more than one observation per cell) is needed to estimate $SS_{\text{error}}$ and test the interaction. Without replication, you must assume no interaction exists.

Balanced designs (equal cell sizes) simplify computation and interpretation; unbalanced designs require more care with the order of entering terms (Type I vs. Type III sums of squares).