One-Way ANOVA

Foundations of Statistics

Comparing Multiple Group Means Simultaneously

One-way ANOVA tests whether three or more group means differ significantly, controlling the family-wise error rate. The F-statistic compares between-group variance to within-group variance, with post-hoc tests identifying which specific groups differ.

Pharmaceutical Research — Compare efficacy across multiple drug dosages
Agriculture — Test crop yield differences across fertilizer types
Education — Evaluate teaching effectiveness across multiple methods

The F-test asks whether any group stands out — post-hoc tests reveal which ones.

Motivation

When comparing $k$ group means, conducting all pairwise $t$ -tests inflates the Type I error rate. With $k = 4$ groups, there are $\binom{4}{2} = 6$ pairwise comparisons. If each is tested at $\alpha = 0.05$ , the family-wise error rate can exceed $1 - (1-0.05)^6 \approx 0.26$ — far above the nominal 5%.

ANOVA (Analysis of Variance) tests all group means simultaneously with a single test, controlling the overall Type I error at $\alpha$ .

The Statistical Model

DfOne-Way ANOVA Model

Y_{ij} = \mu + \alpha_i + \varepsilon_{ij}, \quad i = 1, \ldots, k, \; j = 1, \ldots, n_i

where:

$\mu$ is the overall (grand) mean
$\alpha_i$ is the effect of group $i$ (with $\sum w_i \alpha_i = 0$ for identifiability)
$\varepsilon_{ij} \sim \mathcal{N}(0, \sigma^2)$ i.i.d. — the within-group errors
$k$ is the number of groups, $n_i$ is the size of group $i$

Hypotheses

ANOVA Hypotheses

H_0: \mu_1 = \mu_2 = \cdots = \mu_k \quad \text{vs} \quad H_1: \text{at least one } \mu_i \text{ differs}

Here,

$\mu_i$ =Population mean of group i
$k$ =Number of groups

$H_0$ states that all group means are equal (no treatment effect). $H_1$ is the composite alternative that at least one group mean differs from the others.

The ANOVA Decomposition

ThSum of Squares Decomposition

The total variability in the data decomposes into two components:

\text{SST} = \text{SSB} + \text{SSW}

where:

\text{SST} = \sum_{i=1}^k \sum_{j=1}^{n_i} (Y_{ij} - \bar{Y})^2 \quad \text{(total sum of squares)}

\text{SSB} = \sum_{i=1}^k n_i (\bar{Y}_i - \bar{Y})^2 \quad \text{(sum of squares between groups)}

\text{SSW} = \sum_{i=1}^k \sum_{j=1}^{n_i} (Y_{ij} - \bar{Y}_i)^2 \quad \text{(sum of squares within groups)}

Degrees of freedom: $\text{df}_{\text{total}} = N-1$ , $\text{df}_{\text{between}} = k-1$ , $\text{df}_{\text{within}} = N-k$ , where $N = \sum n_i$ .

The F-Statistic

F-Statistic for One-Way ANOVA

F = \frac{\text{MSB}}{\text{MSW}} = \frac{\text{SSB}/(k-1)}{\text{SSW}/(N-k)}

Here,

$MSB$ =Mean square between = SSB/(k-1)
$MSW$ =Mean square within = SSW/(N-k) — estimates s²
$F$ =F-statistic: ratio of between-group to within-group variability

ThDistribution of F Under $H_0$

Under $H_0$ (all means equal) and the normality assumption:

F = \frac{\text{MSB}}{\text{MSW}} \sim F_{k-1, \, N-k}

where $F_{k-1, N-k}$ is the $F$ -distribution with $k-1$ and $N-k$ degrees of freedom.

The intuition: if $H_0$ is true, MSB and MSW both estimate $\sigma^2$ , so $F \approx 1$ . If $H_0$ is false, MSB overestimates $\sigma^2$ (it includes the treatment effect), so $F$ is large.

Effect Size: Eta-Squared

Eta-Squared

\eta^2 = \frac{\text{SSB}}{\text{SST}} = 1 - \frac{\text{SSW}}{\text{SST}}

Here,

$\eta^2$ =Proportion of total variance explained by group membership

| $\eta^2$ | Interpretation |

|----------|---------------|

| $< 0.01$ | Negligible effect |

| $0.01 - 0.06$ | Small effect |

| $0.06 - 0.14$ | Medium effect |

| $> 0.14$ | Large effect |

Assumptions

| Assumption | What It Means | How to Check |

|-----------|---------------|-------------|

| Independence | Observations are independent within and between groups | Study design (randomization) |

| Normality | Within each group, the errors are normally distributed | Shapiro–Wilk test; Q–Q plots |

| Homogeneity of variance | $\text{Var}(\varepsilon_{ij}) = \sigma^2$ is constant across groups | Levene's test; Bartlett's test |

Robustness of ANOVA

ANOVA is robust to moderate violations of normality when group sizes are equal (or nearly equal), thanks to the CLT. However, it is sensitive to unequal variances when group sizes are unequal. If Levene's test is significant, use Welch's ANOVA instead.

Post-Hoc Tests

When $H_0$ is rejected, post-hoc tests determine which pairs of groups differ:

| Method | Controls | Best For |

|--------|----------|----------|

| Tukey HSD | Family-wise error rate | All pairwise comparisons |

| Bonferroni | Family-wise error rate | Small number of comparisons |

| Scheffé | Family-wise error rate (conservative) | All possible contrasts |

| Dunnett | Family-wise error rate | Comparisons with a control group |

| Games–Howell | Family-wise error rate | Unequal variances and group sizes |

Key Takeaways

Summary: One-Way ANOVA

ANOVA tests all means simultaneously — avoids inflated Type I error of multiple $t$ -tests
$F = \text{MSB}/\text{MSW}$ — large $F$ means between-group variation exceeds random noise
Sum of squares decomposition: $\text{SST} = \text{SSB} + \text{SSW}$
Assumptions: independence, normality within groups, equal variances
Post-hoc tests (Tukey, Bonferroni) identify which specific groups differ
$\eta^2$ tells you the proportion of total variance explained by group membership
Welch's ANOVA is an alternative when homoscedasticity is violated

One-Way ANOVA — Comparing Multiple Group Means

One-Way ANOVA

Comparing Multiple Group Means Simultaneously

Motivation

The Statistical Model

DfOne-Way ANOVA Model

Hypotheses

ANOVA Hypotheses

The ANOVA Decomposition

ThSum of Squares Decomposition

The F-Statistic

F-Statistic for One-Way ANOVA

ThDistribution of F Under $H_0$

Effect Size: Eta-Squared

Eta-Squared

Assumptions

Post-Hoc Tests

Key Takeaways

Summary: One-Way ANOVA

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