# 方差分析

（重定向自方差分析法

## ANOVA的模式假設

1.各組樣本背後所隱含的族群分布必須為常態分布或者是逼近常態分布。

2.各組樣本必須獨立。

3.族群的變異數必須相等。

i為組別（i=1,2...,I），j為觀測值個數（j=1,2,3,...,J），${\displaystyle Y_{ij}}$ 為第i組第j個觀測值，${\displaystyle {\overline {Y}}_{total}}$ 為所有觀測值的平均數。

${\displaystyle n_{i}}$ 為i組內觀測值總數，${\displaystyle {\overline {Y}}_{i}}$ 為第i組的平均數

### 雙因子變異數分析（Two-way ANOVA）

#### 1. 固定效應

A因子的主效應（ASS）：${\displaystyle nb\sum _{i}({\overline {Y}}_{i}-{\overline {Y}}_{total})^{2}}$  其均方AMSS為：${\displaystyle {\frac {ASS}{a-1}}}$

B因子的主效應（BSS）：${\displaystyle na\sum _{j}({\overline {Y}}_{j}-{\overline {Y}}_{total})^{2}}$  其均方BMSS為：${\displaystyle {\frac {BSS}{b-1}}}$

AB因子的交互作用（ABSS）：${\displaystyle n\sum _{i}\sum _{j}({\overline {Y}}_{ij}-{\overline {Y}}_{i}-{\overline {Y}}_{j}+{\overline {Y}}_{total})^{2}}$  其均方ABMSS為：${\displaystyle {\frac {ABSS}{(a-1)(b-1)}}}$

A因子的F檢定為：${\displaystyle {\frac {AMSS}{WMSS}}}$

B因子的F檢定為：${\displaystyle {\frac {BMSS}{WMSS}}}$

## 事後檢定

1. Bonferroni T tests
2. 杜凱氏範圍檢定(Tukey's range test)
3. 丹肯新多重範圍檢定(Duncan's new multiple range test)
4. Dunnett's two-tailed test
5. Dunnett's one-tailed test
6. Gabriel's multiple-comparison procedure
7. 雷文檢定(Levene's test)
8. Waller-Duncan test
9. Ryan-Einot-Gabriel-Welsch multiple range test
10. Scheffé's multiple-comparison procedure
11. Student-Newman-Keuls multiple range test
12. Fisher's least-significant-difference test
13. Waller-Duncan K-ratio T test

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