ANOVA

• 使用時機
• 公式
• 圖形
• 表格

使用時機

$$\begin{array}{cccccc} \text{treatment 1} & \text{treatment 2} & \cdots & \text{treatment k} \\ x_{11} & x_{12} & \cdots & x_{1k} \\ x_{21} & x_{22} & \cdots & x_{2k} \\ \vdots & \vdots & \ddots & \vdots \\ x_{n_1 1} & x_{n_2 2} & \cdots & x_{n_k k} \\ \downarrow & \downarrow & \cdots & \downarrow & \searrow \\ \overline{x}_1 & \overline{x}_2 & \cdots & \overline{x}_k & & \bar{\bar{x}} \end{array}$$

公式

$$\begin{array}{rcl} \text{SSE} &=& \sum_{j=1}^{k}\sum_{i=1}^{n_j}(x_{ij}-\overline{x}_j)^2=\sum_{j=1}^{k} (n_j-1)s_j^2 \\ \text{SSTR} &=& \sum_{j=1}^{k} n_j(\overline{x}_j-\bar{\bar{x}})^2 \\ \text{SST} &=& \sum_{j=1}^{k}\sum_{i=1}^{n_j}(x_{ij}-\bar{\bar{x}})^2 \end{array}$$

圖形

$$\begin{array}{cccc} & & & \text{SST}\\ \text{SSE} & \downarrow & \searrow & \\ & & \longrightarrow \\ & & \text{SSTR} \end{array}$$

表格

source of variation	sum of squares	degrees of freedom	mean square	$F$
Treatments	$\text{SSTR}$	$k-1$	$\text{MSTR}=\frac{\text{SSTR}}{k-1}$	$F=\frac{\text{MSTR}}{\text{MSE}}$
Error	$\text{SSE}$	$n-k$	$\text{MSE}=\frac{\text{SSE}}{n-k}$
Total	$\text{SST}$	$n-1$

與Simple Linear Regression $\text{E}(y)=\beta_0+\beta_1 x$比較一下

source of variation	sum of squares	degrees of freedom	mean square	$F$
Regression	$\text{SSR}$	$1$	$\text{MSR}=\frac{\text{SSR}}{1}$	$F=\frac{\text{MSR}}{\text{MSE}}$
Error	$\text{SSE}$	$n-2$	$\text{MSE}=\frac{\text{SSE}}{n-2}$
Total	$\text{SST}$	$n-1$

Multiple Regression $\text{E}(y)=\beta_0+\beta_1 x_1+\beta_2 x_2+\cdots+\beta_p x_p.$

source of variation	sum of squares	degrees of freedom	mean square	$F$
Regression	$\text{SSR}$	$p$	$\text{MSR}=\frac{\text{SSR}}{p}$	$F=\frac{\text{MSR}}{\text{MSE}}$
Error	$\text{SSE}$	$n-p-1$	$\text{MSE}=\frac{\text{SSE}}{n-p-1}$
Total	$\text{SST}$	$n-1$