Index: The Book of Statistical ProofsStatistical Models ▷ Univariate normal data ▷ Analysis of variance ▷ Two-way ANOVA

Definition: Let there be two factors $A$ and $B$ with levels $i = 1, \ldots, a$ and $j = 1, \ldots, b$ that are used to group measurements $y_{ijk} \in \mathbb{R}$ from distinct objects $k = 1, \ldots, n_{ij}$ into $a \cdot b$ categories $(i,j) \in \left\lbrace 1, \ldots, a \right\rbrace \times \left\lbrace 1, \ldots, b \right\rbrace$.

Then, in two-way analysis of variance (ANOVA), these measurements are assumed to come from normal distributions

$\label{eq:anova2-p1} y_{ijk} \sim \mathcal{N}(\mu_{ij}, \sigma^2) \quad \text{for all} \quad i = 1, \ldots, a, \quad j = 1, \ldots, b, \quad \text{and} \quad k = 1, \dots, n_{ij}$

with

$\label{eq:anova2-p2} \mu_{ij} = \mu + \alpha_i + \beta_j + \gamma_{ij}$

where

• $\mu$ is called the “grand mean”;

• $\alpha_i$ is the additive “main effect” of the $i$-th level of factor $A$;

• $\beta_j$ is the additive “main effect” of the $j$-th level of factor $B$;

• $\gamma_{ij}$ is the non-additive “interaction effect” of category $(i,j)$;

• $\mu_{ij}$ is the expected value in category $(i,j)$; and

• $\sigma^2$ is common variance across all categories.

Alternatively, the model may be written as

$\label{eq:anova2-alt} \begin{split} y_{ijk} &= \mu + \alpha_i + \beta_j + \gamma_{ij} + \varepsilon_{ijk} \\ \varepsilon_{ijk} &\sim \mathcal{N}(0, \sigma^2) \end{split}$

where $\varepsilon_{ijk}$ is the error term corresponding to observation $k$ belonging to the $i$-th level of $A$ and the $j$-th level of $B$.

As the two-way ANOVA model is underdetermined, the parameters of the model are additionally subject to the constraints

$\label{eq:anova2-cons} \begin{split} \sum_{i=1}^{a} w_{ij} \alpha_i &= 0 \quad \text{for all} \quad j = 1, \ldots, b \\ \sum_{j=1}^{b} w_{ij} \beta_j &= 0 \quad \text{for all} \quad i = 1, \ldots, a \\ \sum_{i=1}^{a} w_{ij} \gamma_{ij} &= 0 \quad \text{for all} \quad j = 1, \ldots, b \\ \sum_{j=1}^{b} w_{ij} \gamma_{ij} &= 0 \quad \text{for all} \quad i = 1, \ldots, a \end{split}$

where the weights are $w_{ij} = n_{ij}/n$ and the total sample size is $n = \sum_{i=1}^{a} \sum_{j=1}^{b} n_{ij}$.

Sources:

Metadata: ID: D182 | shortcut: anova2 | author: JoramSoch | date: 2022-11-06, 13:41.