第6講各種データ解析法

交互作用

水準	$A_{1}$	$A_{2}$	$\cdots$	$A_{a}$	平均値
$B_{1}$	$x_{111} \\ \vdots \\ x_{11r}$	$x_{211} \\ \vdots \\ x_{21r}$	$\cdots$	$x_{a11}\\ \vdots \\ x_{a1r}$	$\bar{x}_{\cdot 1 \cdot}$
$B_{2}$	$x_{121} \\ \vdots \\ x_{12r}$	$x_{221} \\ \vdots \\ x_{22r}$	$\cdots$	$x_{a21}\\ \vdots \\ x_{a2r}$	$\bar{x}_{\cdot 1 \cdot}$
$\vdots$	$\vdots$	$\vdots$	$\cdots$	$\vdots$	$\vdots$
$B_{b}$	$x_{1b1}\\ \vdots \\ x_{1br}$	$x_{2b1}\\ \vdots \\ x_{2br}$	$\cdots$	$x_{ab1}\\ \vdots \\ x_{abr}$	$\bar{x}_{\cdot b \cdot}$
平均値	$\bar{x}_{1 \cdot \cdot}$	$\bar{x}_{2 \cdot \cdot}$	$\cdots$	$\bar{x}_{a \cdot \cdot}$	総平均 $\bar{\bar{x}}$

$H_{0}$ :

$\alpha_{1} = \alpha_{2} = \cdots = \alpha_{a} = 0$

$\beta_{1} = \beta_{2} = \cdots = \beta_{b} = 0$

$(\alpha\beta)_{11} = (\alpha\beta)_{12} = \cdots = (\alpha\beta)_{ab} = 0$

モデル :

x_{i j k} = μ + α_{i} + β_{j} + (α β)_{i j} + ϵ_{i j k}

$\begin{equation} x_{ijk} = \mu + \alpha_{i} + \beta_{j} + (\alpha\beta)_{ij}+ \epsilon_{ijk} \end{equation}$

ただし，

\sum_{i = 1}^{a} α_{i} = 0, \sum_{j = 1}^{b} β_{j} = 0, \sum_{i = 1}^{a} (α β)_{i j} = 0, \sum_{j = 1}^{b} (α β)_{i j} = 0

$\begin{equation} \sum_{i=1}^{a}\alpha_{i} = 0,~~~~\sum_{j=1}^{b}\beta_{j} = 0,~~~~\sum_{i=1}^{a}(\alpha\beta)_{ij} = 0,~~~~\sum_{j=1}^{b}(\alpha\beta)_{ij} = 0 \end{equation}$

ϵ_{i j} \sim N (0, σ)

$\begin{equation} \epsilon_{ij} \sim N(0,~\sigma) \end{equation}$

分散分解 :

\begin{aligned} \underset{総平方和 S_{T}}{\underset{⏟}{\sum_{i = 1}^{a} \sum_{j = 1}^{b} \sum_{k = 1}^{r} (x_{i j k} - \bar{\bar{x}})^{2}}} \\ = \underset{A間平方和 S_{A}}{\underset{⏟}{b r \sum_{i = 1}^{a} ({\bar{x}}_{i \cdot \cdot} - \bar{\bar{x}})^{2}}} + \underset{B間平方和 S_{B}}{\underset{⏟}{a r \sum_{j = 1}^{b} ({\bar{x}}_{\cdot j \cdot} - \bar{\bar{x}})^{2}}} \\ + \underset{A \times B平方和 S_{A \times B}}{\underset{⏟}{r \sum_{i = 1}^{a} \sum_{j = 1}^{b} ({\bar{x}}_{i j \cdot} - {\bar{x}}_{i \cdot \cdot} - {\bar{x}}_{\cdot j \cdot} + \bar{\bar{x}})^{2}}} + \underset{残差平方和 S_{e}}{\underset{⏟}{\sum_{i = 1}^{a} \sum_{j = 1}^{b} \sum_{k = 1}^{r} (x_{i j k} - {\bar{x}}_{i j \cdot})^{2}}} \end{aligned}

$\begin{align} &\underbrace{\sum_{i=1}^{a}\sum_{j=1}^{b}\sum_{k=1}^{r}(x_{ijk}-\bar{\bar{x}})^{2}}_{\text{総平方和$S_{T}$}} \\ &= \underbrace{br\sum_{i=1}^{a}(\bar{x}_{i \cdot\cdot} - \bar{\bar{x}})^{2}}_{\text{A間平方和$S_{A}$}} + \underbrace{ar\sum_{j=1}^{b}(\bar{x}_{\cdot j\cdot} - \bar{\bar{x}})^{2}}_{\text{B間平方和$S_{B}$}} \\ &~~~~+ \underbrace{r\sum_{i=1}^{a}\sum_{j=1}^{b}(\bar{x}_{ij\cdot}-\bar{x}_{i\cdot\cdot}-\bar{x}_{\cdot j\cdot}+\bar{\bar{x}})^{2}}_{\text{A$\times$B平方和$S_{A \times B}$}} +\underbrace{\sum_{i=1}^{a}\sum_{j=1}^{b}\sum_{k=1}^{r}(x_{ijk}-\bar{x}_{ij\cdot})^{2}}_{\text{残差平方和$S_{e}$}} \end{align}$

要因	平方和	自由度	平均変動	分散比
$A$	$S_{A}$	$\phi_{A} = a-1$	$\ds V_{A}=\frac{S_{A}}{\phi_{A}}$	$\ds F_{A}=\frac{V_{A}}{V_{e}}$
$B$	$S_{B}$	$\phi_{B} = b-1$	$\ds V_{B}=\frac{S_{B}}{\phi_{B}}$	$\ds F_{B}=\frac{V_{B}}{V_{e}}$
$A\times B$	$S_{A\times B}$	$\phi_{A\times B} = (a-1)(b-1)$	$\ds V_{A\times B}=\frac{S_{A\times B}}{\phi_{A\times B}}$	$\ds F_{A\times B}=\frac{V_{A\times B}}{V_{e}}$
$e$	$S_{e}$	$\phi_{e} = abr-ab$	$\ds V_{e}=\frac{S_{e}}{\phi_{e}}$
合計	$S_{T}$	$\phi_{T} = abr-1$

第6講 各種データ解析法

交互作用

第6講各種データ解析法