Analysis of variance (1) (one-way analysis of variance)

Analysis of variance is a mathematical statistical method based on experimental data to infer whether one or more factors have a significant impact on experimental indicators when their status changes. According to the number of factors that affect the test indicators, variance analysis can be divided into single-factor ANOVA and two-factor ANOVA and Multi-factor ANOVA.

In mathematical statistics, the results of the test (such as product performance, output, etc.) are called test indicators, The conditions that affect the test indicators are called factors or factors, where the factors The different states are calledlevels. Usually capital letters $A,B,...$ etc. represent different factors, using subscripts $A_1,A_2,...$ isoexpressive factor $A$ Non-uniform horizontal.

ANOVA

If in an experiment, only the level of one factor is changed and the levels of the other factors remain unchanged, then this kind of experiment is called Single factor experiment . Analysis of variance performed under a single factor test is called one-way analysis of variance.

mathematical model

factor $A$ Yes $r$ Discrete horizontal $A_1,A_2,...,A_r$, is horizontal $A_i$ Down, forward $n_i$Repeat the experiment independently and obtain the results in the following table:

level	sample	sample mean
$A_1$	$X_{11}{X}_{12}\cdots X_{1n_1}$	${\bar{X}}_1$
$A_2$	$X_{21}{X}_{22}\cdots X_{2n_2}$	${\bar{X}}_2$
$⋮$	$\cdots \cdots \cdots$	$⋮$
$A_r$	$X_{r1}{X}_{r2}\cdots X_{r{n}_r}$	${\bar{X}}_r$

Determine each horizontal $A_i$ Corresponding population $X_i$ Clothes correct distribution $N({\mu }_i,p^2)$, also self-dissimilar horizontal $A_i$The samples of are independent of each other. In short, normal population, homoscedasticity, a> are the three basic assumptions for conducting analysis of variance. Independent samples

由于 $X_{ij}\sim N({\mu }_i,p^2),j=1,2,...,n_i$，因而 $X_{ij}-m_i\sim N(0,p^2)$，记 ${ϵ}_{ij}=X_{ij}-m_i$，then
$$\begin{array}{rl}& X_{ij}=m_i+e_{ij},j=1,\cdots ,n_i;i=1,\cdots ,r\\ & e_{ij}\sim N(0,p^2),j=1,\cdots ,n_i;i=1,\cdots ,r\\ & e_{11},\cdots ,e_{r{n}_i}\text{Mutually independent}.\end{array}$$ constitutes the mathematical model of one-factor analysis of variance, where $m_i$ 和 $p^2$ are unknown parameters to be determined in the model.

The basic task of variance analysis is to test the hypothesis for the above model
$$H_0:m_1=m_2=\cdots =m_r\leftrightarrow H_1:m_1,m_2,...,m_{r}Not\; all\; equal$$ That is, through the analysis of experimental data, it is tested whether the means of the normal populations with homoscedasticity are equal, and thus inferred whether the factors have a significant impact on the experimental indicators. .

Advanced analysis, introduction below, order
$$n=\sum _{i=1}^r{n}_i,m=\frac{1}{n}\sum _{i=1}^r{n}_i{m}_i,d_i=m_i-\mu$$ inside,$\mu$ namereasoning average,$d_i$ Vertical horizontal $A_i$The effect of , which reflects the factor $i$ horizontal $A_i$The effect of on the test indicators. $d_1,...,d_r$ Full foot formula ${\sum }_{i=1}^r{n}_i{d}_i=0$.

由电影记号，previously submitted model can be changed
$$\begin{array}{l}{X}_{ij}=m+d_i+e_{ij},j=1,2,\cdots ,n_i,i=1,2,\cdots ,r\\ {\sum }_{i=1}^r{n}_i{d}_i=0,\\ e_{ij}\sim N(0,p^2),j=1,2,\cdots ,n_i;i=1,2,\cdots ,r\\ e_{11},\cdots ,e_{r{n}_i}\text{ Mutually independent}.\end{array}\}$$
For the above model, the hypothesis to be tested is
$$H_0:d_1=d_2=\cdots =d_r=0\leftrightarrow H_1:d_1,d_2,...,d_{r}Not\; all\; zeros$$
In variance analysis, the sum of squares decomposition method is used to decompose the total sum of squares of the deviations of the entire batch of data into several parts, some of which reflect the effects of factors , is called the sum of squares of the effects of factors, and some reflect errors caused by random fluctuations, which is called the sum of squares of errors. By analyzing the size of their ratios, the hypothesis testing work can be completed at once.

Statistical Analysis

首先，引入以下记号：
$$\begin{array}{l}{\overline{ϵ}}_i=\frac{1}{n_i}{\sum }_{j=1}^{n_i}{ϵ}_{ij},i=1,\cdots ,r\\ \\ \overline{ϵ}=\frac{1}{n}{\sum }_{i=1}^r{\sum }_{j=1}^{n_i}{ϵ}_{ij}=\frac{1}{n}{\sum }_{i=1}^r{n}_i{\overline{ϵ}}_i\\ \\ {\bar{X}}_i=\frac{1}{n_i}{\sum }_{j=1}^{n_i}{X}_{ij},i=1,\cdots ,r\\ \\ \bar{X}=\frac{1}{n}{\sum }_{i=1}^r{\sum }_{j=1}^{n_i}{X}_{ij}=\frac{1}{n}{\sum }_{i=1}^r{n}_i{\bar{X}}_i\end{array}\}.$$
由电影论社示可知
$$\begin{array}{l}{\overline{ϵ}}_i\sim N(0,\frac{p^2}{n_i}),i=1,\cdots ,r\\ \overline{ϵ}\sim N(0,\frac{p^2}{n}),\\ {\bar{X}}_i=m+d_i+{\overline{ϵ}}_i\sim N(\mu +d_i,\frac{p^2}{n_i}),i=1,\cdots ,r\\ \overline{X}=m+\bar{e}\sim N(\mu ,\frac{p^2}{n})\end{array}\}$$
Homogeneous, introduce the total sum of squared deviations
$$Q_T=\sum _{i=1}^r\sum _{j=1}^{n_i}(X_{ij}-\bar{X}{)}^2$$
由于 $\bar{X}$ is the average of the entire batch of data, and $Q_T$ is the variance of the entire batch of data $n$ 倍，即 $Q_T$ reflects the degree of fluctuation of the data, so $Q_T$ is calledthe sum of squared deviations.

$Q_T$ 可电影的$Q_T=Q_A+Q_E$，其中
$$\begin{gathered} Q_A=\sum _{i=1}^r{n}_i({\bar{X}}_i-\bar{X}{)}^2 \\ {Q}_E=\sum _{i=1}^r\sum _{j=1}^{n_i}(X_{ij}-{\bar{X}}_i{)}^2 \end{gathered}$$ $Q_A$ nominal factor A A The sum of squares of the effects of Q E Q_E ), Sum of squares between groups (also known as $A$$Q_E$ is called sum of squares of errors (also called sum of squares within group< a i=4>).

H $H_0$ 成立时，$Q_A$ give $Q_E$ Individual Function,Online
$$F=\frac{\frac{Q_A/{\sigma }^2}{r-1}}{\frac{Q_E/{\sigma }^2}{n-r}}=\frac{Q_A/(r-1)}{Q_E/(n-r)}\sim F(r-1,n-r)$$
对给简的显書horizontal $\alpha$，得 $H_0$The rejection region of is W = { F ≥ F α ( r − 1 , n − r ) } W=\{F \ge F_\alpha(r-1, n-r)\} IN={ F≥Fα​(r−1,n−r)}。

The calculation results are usually listed in a variance analysis table:

Source of variance	sum of squares	degrees of freedom	sum of mean squares	$F$ 值
因素 $A$ (organization)	$Q_A={\sum }_{i=1}^r{n}_i({\bar{x}}_i-\bar{x}{)}^2$	$r-1$	${\bar{Q}}_A=\frac{Q_A}{r-1}$	$F=\frac{{\bar{Q}}_A}{{\bar{Q}}_E}$
Direction $E$ (internal)	$Q_E={\sum }_{i=1}^r{\sum }_{j=1}^{n_i}(x_{ij}-{\bar{x}}_i{)}^2$	$n-r$	${\bar{Q}}_E=\frac{Q_E}{n-r}$
sum	$Q_T={\sum }_{i=1}^r{\sum }_{j=1}^{n_i}(x_{ij}-\bar{x}{)}^2$	$n-1$

references

[1] "Applied Mathematical Statistics", Shi Yu, Xi'an Jiaotong University Press.