One-way analysis of variance

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

Definition: Consider measurements $y_{ij} \in \mathbb{R}$ from distinct objects $j = 1, \ldots, n_i$ in separate groups $i = 1, \ldots, k$.

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

\[\label{eq:anova1} y_{ij} \sim \mathcal{N}(\mu_i, \sigma^2) \quad \text{for all} \quad i = 1, \ldots, k \quad \text{and} \quad j = 1, \dots, n_i\]

where

$\mu_i$ is the expected value in group $i$ and
$\sigma^2$ is the common variance across groups.

Alternatively, the model may be written as

\[\label{eq:anova1-alt} \begin{split} y_{ij} &= \mu_i + \varepsilon_{ij} \\ \varepsilon_{ij} &\overset{\mathrm{i.i.d.}}{\sim} \mathcal{N}(0, \sigma^2) \end{split}\]

where $\varepsilon_{ij}$ is the error term belonging to observation $j$ in category $i$ and $\varepsilon_{ij}$ are the independent and identically distributed.

Sources:

Bortz, Jürgen (1977): "Einfaktorielle Varianzanalyse"; in: Lehrbuch der Statistik. Für Sozialwissenschaftler, ch. 12.1, pp. 528ff.; URL: https://books.google.de/books?id=lNCyBgAAQBAJ.
Denziloe (2018): "Derive the distribution of the ANOVA F-statistic under the alternative hypothesis"; in: StackExchange CrossValidated, retrieved on 2022-11-06; URL: https://stats.stackexchange.com/questions/355594/derive-the-distribution-of-the-anova-f-statistic-under-the-alternative-hypothesi.

Metadata: ID: D181 | shortcut: anova1 | author: JoramSoch | date: 2022-11-06, 10:23.