R_reg2.Rmd

---
title: "R的回归分析之二"
author: "李峰"
date: '`r strftime(Sys.time(), format = "%B %d, %Y")`'
output:
  html_document: default
---




---


####  1. 回归建模策略

```{r}
CarData<-read.table(file="CarData.txt",header=TRUE)
```

$R^2$和$Adjusted$ $R^2$的公式


\begin{equation}\label{eq:1}
  R^2=\frac{SSR}{SST},
\end{equation}

\begin{equation}\label{eq:2}
  {\bar{R}}^2=\frac{SSR/p}{SST/(n-1)},
\end{equation}

```{r}
Fit1<-lm(log(MPG)~weight+horsepower,data=CarData)
Fit2<-lm(log(MPG)~weight+horsepower+displacement,data=CarData)
summary(Fit1)
summary(Fit2)
anova(Fit1,Fit2)
```



####  2. AIC和BIC信息准则

对回归分析而言，
\begin{equation}\label{eq:3}
  AIC=n+nln(2\pi)+nln(RSS/n)+2(p+1),
\end{equation}


\begin{equation}\label{eq:4}
  BIC=n+nln(2\pi)+nln(RSS/n)+ln(n)(p+1),
\end{equation}

p是参数个数。


```{r}
Fit1<-lm(log(MPG)~weight+horsepower,data=CarData)
Fit2<-lm(log(MPG)~weight+horsepower+displacement,data=CarData)
AIC(Fit1,Fit2)
```

```{r}
(392+392*log(2*pi)+392*log(9.587/392)+2*4)
```


```{r}
BIC(Fit1,Fit2)
```



####  3. 变量筛选策略



```{r}
Fit<-lm(log(MPG)~weight+horsepower+displacement+cylinders+acceleration,data=CarData)
library("MASS")
stepAIC(Fit,direction="backward")
```

```{r}
stepAIC(Fit,direction="both")
```




####  4. 全子集回归

```{r}
library("leaps")
leapsFit<-regsubsets(log(MPG)~weight+horsepower+displacement+cylinders+acceleration,data=CarData,nbest=2)
summary(leapsFit)
coef(leapsFit, c(3,5))
```



```{r}
plot(leapsFit,scale="bic")
```


```{r}
plot(leapsFit,scale="adjr2")
```

```{r}
library("car")
subsets(leapsFit,statistic="cp",main="全子集回归模型评价图",legend=TRUE)
abline(1,1,lty=2,col="red")
```



####  5. 虚拟变量的回归分析

```{r}
CarData$ModelYear<-as.factor(CarData$ModelYear)
Fit<-lm(log(MPG)~ModelYear,data=CarData)
summary(Fit)
```

```{r}
(table(CarData$ModelYear))
```


```{r}
tmp<-subset(CarData,CarData$ModelYear=="70"|CarData$ModelYear=="71")
tresult<-t.test(log(MPG)~ModelYear,data=tmp,paired=FALSE,var.equal=TRUE)
```


```{r}
(tresult$estimate[1]-tresult$estimate[2])
```



```{r}
Fit<-lm(log(MPG)~weight+horsepower+ModelYear,data=CarData)
summary(Fit)
```