# W3：相關分析－ANOVA與線性回歸

###### tags: `統計學系列課程`

## 變數、相關、相關分析模型與相關係數

### 卡方適合度分析（類別Y－類別X）

X=c(rep('F',10),rep('M',10))
Y=vector(length=20)
for(i in 1:20){
  if(X[i]=='F'){
    Y[i]=sample(c('G','NG'),size=1,replace=T,prob=c(0.25,0.75))
  }
  if(X[i]=='M'){
    Y[i]=sample(c('G','NG'),size=1,replace=T,prob=c(0.45,0.55))
  }
}
cbind(Y,X)
A=list(Glasses=Y,Gender=X)
table(A)

### 邏輯回歸（類別Y－數值X）

X=50:75
Y=vector(length=26)
for(i in 1:26){
  Y[i]=rbinom(1,1,prob = (exp(-55+X[i])/(1+exp(-55+X[i]))))
}
cbind(Y,X)
plot(X,Y)

### ANOVA（數值Y－類別X）

A=iris
cbind(A$Sepal.Length,A$Species)
boxplot(A$Sepal.Length~A$Species)

### 回歸（數值Y－數值X）

X=1:20
Y=5+0.1*X+rnorm(20,0,0.2)
cbind(Y,X)
plot(X,Y)

### 混合模型（數值Y－數值X＋類別X）

A=CO2
A
plot(A)

## ANOVA

X1=c(rep('apple',10),rep('grape',10),rep('orange',10))
A1=vector(length=30)
X2=rep(c(rep('PET',5),rep('Al',5)),3)
A2=vector(length=30)
X3=rep(c(rep('Jay',5),rep('Jolin',5)),3)
A3=vector(length=30)
B=vector(length=30)
for(i in 1:30){
  A1[i]=switch(X1[i],'apple'=4,'grape'=-2,'orange'=-2)
  A2[i]=switch(X2[i],'PET'=-1,'Al'=1)
  A3[i]=switch(X3[i],'Jay'=-2,'Jolin'=2)
  if(X1[i]=='orange'&&X3[i]=='Jolin'){
    B[i]=+5
  }else{
    B[i]=-1
  }
}

mu=20

Y1=mu+A1+rnorm(30,0,1)
Y2=mu+A1+A2+rnorm(30,0,1)
Y3=mu+A1+A3+B+rnorm(30,0,1)

boxplot(Y1~X1)
boxplot(Y2~X1+X2)
boxplot(Y3~X1+X3)

### Case 1：銷量與口味的關係（1-Way CRD）

### Case 2：銷量與口味、包裝的關係（1-Way RBD）

### Case 3：銷量與口味、代言人的關係（2-Way CRD）

L1=lm(Y1~X1)
L2=lm(Y2~X1+X2)
L3=lm(Y3~X1+X3)
L3t=lm(Y3~X1*X3)

anova(L1)
anova(L2)
anova(L3)
anova(L3t)

summary(L1)
summary(L2)
summary(L3)
summary(L3t)

AIC(L1)
AIC(L2)
AIC(L3)
AIC(L3t)

BIC(L1)
BIC(L2)
BIC(L3)
BIC(L3t)

> anova(L1)
Analysis of Variance Table

Response: Y1
          Df  Sum Sq Mean Sq F value    Pr(>F)    
X1         2 229.071 114.536  167.29 6.133e-16 ***
Residuals 27  18.486   0.685                      
---
Signif. codes:  
0 ‘***’ 0.001 ‘**’ 0.01 ‘*’ 0.05 ‘.’ 0.1 ‘ ’ 1
> anova(L2)
Analysis of Variance Table

Response: Y2
          Df  Sum Sq Mean Sq F value    Pr(>F)    
X1         2 238.957 119.478 168.865 1.272e-15 ***
X2         1  27.979  27.979  39.544 1.175e-06 ***
Residuals 26  18.396   0.708                      
---
Signif. codes:  
0 ‘***’ 0.001 ‘**’ 0.01 ‘*’ 0.05 ‘.’ 0.1 ‘ ’ 1
> anova(L3)
Analysis of Variance Table

Response: Y3
          Df Sum Sq Mean Sq F value    Pr(>F)    
X1         2 214.78 107.388  25.846 6.604e-07 ***
X3         1 255.94 255.938  61.598 2.527e-08 ***
Residuals 26 108.03   4.155                      
---
Signif. codes:  
0 ‘***’ 0.001 ‘**’ 0.01 ‘*’ 0.05 ‘.’ 0.1 ‘ ’ 1
> anova(L3t)
Analysis of Variance Table

Response: Y3
          Df  Sum Sq Mean Sq F value    Pr(>F)    
X1         2 214.775 107.388  84.892 1.302e-11 ***
X3         1 255.938 255.938 202.323 3.431e-13 ***
X1:X3      2  77.669  38.835  30.700 2.427e-07 ***
Residuals 24  30.360   1.265                      
---
Signif. codes:  
0 ‘***’ 0.001 ‘**’ 0.01 ‘*’ 0.05 ‘.’ 0.1 ‘ ’ 1
> 
> summary(L1)

Call:
lm(formula = Y1 ~ X1)

Residuals:
     Min       1Q   Median       3Q      Max 
-1.71060 -0.50629 -0.02071  0.41901  1.99355 

Coefficients:
            Estimate Std. Error t value Pr(>|t|)    
(Intercept)  23.9926     0.2617   91.69  < 2e-16 ***
X1grape      -6.1636     0.3700  -16.66 9.93e-16 ***
X1orange     -5.5042     0.3700  -14.88 1.58e-14 ***
---
Signif. codes:  
0 ‘***’ 0.001 ‘**’ 0.01 ‘*’ 0.05 ‘.’ 0.1 ‘ ’ 1

Residual standard error: 0.8274 on 27 degrees of freedom
Multiple R-squared:  0.9253,	Adjusted R-squared:  0.9198 
F-statistic: 167.3 on 2 and 27 DF,  p-value: 6.133e-16

> summary(L2)

Call:
lm(formula = Y2 ~ X1 + X2)

Residuals:
     Min       1Q   Median       3Q      Max 
-1.72875 -0.41385 -0.06592  0.54663  1.48703 

Coefficients:
            Estimate Std. Error t value Pr(>|t|)    
(Intercept)  24.7133     0.3071  80.461  < 2e-16 ***
X1grape      -5.7358     0.3762 -15.248 1.76e-14 ***
X1orange     -6.2099     0.3762 -16.508 2.68e-15 ***
X2PET        -1.9315     0.3071  -6.288 1.17e-06 ***
---
Signif. codes:  
0 ‘***’ 0.001 ‘**’ 0.01 ‘*’ 0.05 ‘.’ 0.1 ‘ ’ 1

Residual standard error: 0.8412 on 26 degrees of freedom
Multiple R-squared:  0.9355,	Adjusted R-squared:  0.9281 
F-statistic: 125.8 on 3 and 26 DF,  p-value: 1.349e-15

> summary(L3)

Call:
lm(formula = Y3 ~ X1 + X3)

Residuals:
    Min      1Q  Median      3Q     Max 
-3.1310 -1.7087  0.0917  1.5084  3.8895 

Coefficients:
            Estimate Std. Error t value Pr(>|t|)    
(Intercept)  20.2679     0.7443  27.230  < 2e-16 ***
X1grape      -6.5538     0.9116  -7.189 1.23e-07 ***
X1orange     -3.2304     0.9116  -3.544  0.00152 ** 
X3Jolin       5.8417     0.7443   7.848 2.53e-08 ***
---
Signif. codes:  
0 ‘***’ 0.001 ‘**’ 0.01 ‘*’ 0.05 ‘.’ 0.1 ‘ ’ 1

Residual standard error: 2.038 on 26 degrees of freedom
Multiple R-squared:  0.8133,	Adjusted R-squared:  0.7918 
F-statistic: 37.76 on 3 and 26 DF,  p-value: 1.271e-09

> summary(L3t)

Call:
lm(formula = Y3 ~ X1 * X3)

Residuals:
    Min      1Q  Median      3Q     Max 
-2.0256 -0.8606 -0.1026  0.7768  1.7753 

Coefficients:
                 Estimate Std. Error t value Pr(>|t|)    
(Intercept)       21.5324     0.5030  42.809  < 2e-16 ***
X1grape           -6.8121     0.7113  -9.577 1.14e-09 ***
X1orange          -6.7655     0.7113  -9.511 1.30e-09 ***
X3Jolin            3.3127     0.7113   4.657 9.93e-05 ***
X1grape:X3Jolin    0.5167     1.0060   0.514    0.612    
X1orange:X3Jolin   7.0702     1.0060   7.028 2.88e-07 ***
---
Signif. codes:  
0 ‘***’ 0.001 ‘**’ 0.01 ‘*’ 0.05 ‘.’ 0.1 ‘ ’ 1

Residual standard error: 1.125 on 24 degrees of freedom
Multiple R-squared:  0.9475,	Adjusted R-squared:  0.9366 
F-statistic:  86.7 on 5 and 24 DF,  p-value: 1.462e-14

> 
> AIC(L1)
[1] 78.61046
> AIC(L2)
[1] 80.46443
> AIC(L3)
[1] 133.5724
> AIC(L3t)
[1] 99.49406
> 
> BIC(L1)
[1] 84.21525
> BIC(L2)
[1] 87.47042
> BIC(L3)
[1] 140.5784
> BIC(L3t)
[1] 109.3024