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	\(\text{SST}\)	\(n-1\)