added 2 definitions

Author: JoramSoch

D/anova1.md

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+---
+layout: definition
+mathjax: true
+
+author: "Joram Soch"
+affiliation: "BCCN Berlin"
+e_mail: "joram.soch@bccn-berlin.de"
+date: 2022-11-06 10:23:00
+
+title: "One-way analysis of variance"
+chapter: "Statistical Models"
+section: "Univariate normal data"
+topic: "Analysis of variance"
+definition: "One-way ANOVA"
+
+sources:
+  - authors: "Bortz, Jürgen"
+    year: 1977
+    title: "Einfaktorielle Varianzanalyse"
+    in: "Lehrbuch der Statistik. Für Sozialwissenschaftler"
+    pages: "ch. 12.1, pp. 528ff."
+    url: "https://books.google.de/books?id=lNCyBgAAQBAJ"
+  - authors: "Denziloe"
+    year: 2018
+    title: "Derive the distribution of the ANOVA F-statistic under the alternative hypothesis"
+    in: "StackExchange CrossValidated"
+    pages: "retrieved on 2022-11-06"
+    url: "https://stats.stackexchange.com/questions/355594/derive-the-distribution-of-the-anova-f-statistic-under-the-alternative-hypothesi"
+
+def_id: "D181"
+shortcut: "anova1"
+username: "JoramSoch"
+---
+
+
+**Definition:** Consider measurements $y_{ij} \in \mathbb{R}$ from disctinct 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](/D/norm)
+
+$$ \label{eq:anova1}
+y_{ij} \sim \mathcal{N}(\mu_i, \sigma^2) \quad \text{for all} \quad i = 1, \ldots, k \quad \text{and} \quad j = 1, \dots, n_i
+$$
+
+where
+
+* $\mu_i$ is the [expected value](/D/mean) in group $i$ and
+
+* $\sigma^2$ is the common [variance](/D/var) 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](/D/slr) belonging to observation $j$ in category $i$ and $\varepsilon_{ij}$ are the [independent and identically distributed](/D/iid).

D/anova2.md

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+---
+layout: definition
+mathjax: true
+
+author: "Joram Soch"
+affiliation: "BCCN Berlin"
+e_mail: "joram.soch@bccn-berlin.de"
+date: 2022-11-06 13:41:00
+
+title: "Two-way analysis of variance"
+chapter: "Statistical Models"
+section: "Univariate normal data"
+topic: "Analysis of variance"
+definition: "Two-way ANOVA"
+
+sources:
+  - authors: "Bortz, Jürgen"
+    year: 1977
+    title: "Zwei- und mehrfaktorielle Varianzanalyse"
+    in: "Lehrbuch der Statistik. Für Sozialwissenschaftler"
+    pages: "ch. 12.2, pp. 538ff."
+    url: "https://books.google.de/books?id=lNCyBgAAQBAJ"
+  - authors: "ttd"
+    year: 2021
+    title: "Proof on SSAB/s2~chi2(I-1)(J-1) under the null hypothesis HAB: dij=0 for i=1,...,I and j=1,...,J"
+    in: "StackExchange CrossValidated"
+    pages: "retrieved on 2022-11-06"
+    url: "https://stats.stackexchange.com/questions/545807/proof-on-ss-ab-sigma2-sim-chi2-i-1j-1-under-the-null-hypothesis"
+
+def_id: "D182"
+shortcut: "anova2"
+username: "JoramSoch"
+---
+
+
+**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](/D/norm)
+
+$$ \label{eq:anova2-p1}
+y_{ij} \sim \mathcal{N}(\mu_{ij}, \sigma^2) \quad \text{for all} \quad i = 1, \ldots, a, \quad j = 1, \ldots, b, \quad \text{and} \quad k = 1, \dots, n_{ij}
+$$
+
+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](/D/mean) in category $(i,j)$; and
+
+* $\sigma^2$ is common [variance](/D/var) 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](/D/slr) 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 $n = \sum_{i=1}^{a} \sum_{j=1}^{b} n_{ij}$.