# R语言--逐步回归分析

### 1.载入数据 首先对数据进行多元线性回归分析

``````tdata<-data.frame(
x1=c( 7, 1,11,11, 7,11, 3, 1, 2,21, 1,11,10),
x2=c(26,29,56,31,52,55,71,31,54,47,40,66,68),
x3=c( 6,15, 8, 8, 6, 9,17,22,18, 4,23, 9, 8),
x4=c(60,52,20,47,33,22, 6,44,22,26,34,12,12),
Y =c(78.5,74.3,104.3,87.6,95.9,109.2,102.7,72.5,
93.1,115.9,83.8,113.3,109.4)
)
tlm<-lm(Y~x1+x2+x3+x4,data=tdata)
summary(tlm)
``````

``````Call:
lm(formula = Y ~ x1 + x2 + x3 + x4, data = tdata)

Residuals:
Min      1Q  Median      3Q     Max
-3.1750 -1.6709  0.2508  1.3783  3.9254

Coefficients:
Estimate Std. Error t value Pr(>|t|)
(Intercept)  62.4054    70.0710   0.891   0.3991
x1            1.5511     0.7448   2.083   0.0708 .
x2            0.5102     0.7238   0.705   0.5009
x3            0.1019     0.7547   0.135   0.8959
x4           -0.1441     0.7091  -0.203   0.8441
---
Signif. codes:  0 ‘***’ 0.001 ‘**’ 0.01 ‘*’ 0.05 ‘.’ 0.1 ‘ ’ 1

Residual standard error: 2.446 on 8 degrees of freedom
Multiple R-squared:  0.9824,    Adjusted R-squared:  0.9736
F-statistic: 111.5 on 4 and 8 DF,  p-value: 4.756e-07
``````

### 2.逐步回归分析###

``````tstep<-step(tlm)
summary(tstep)
``````
``````Start:  AIC=26.94
Y ~ x1 + x2 + x3 + x4

Df Sum of Sq    RSS    AIC
- x3    1    0.1091 47.973 24.974
- x4    1    0.2470 48.111 25.011
- x2    1    2.9725 50.836 25.728
47.864 26.944
- x1    1   25.9509 73.815 30.576

Step:  AIC=24.97
Y ~ x1 + x2 + x4

Df Sum of Sq    RSS    AIC
47.97 24.974
- x4    1      9.93  57.90 25.420
- x2    1     26.79  74.76 28.742
- x1    1    820.91 868.88 60.629
``````

……

``````Call:
lm(formula = Y ~ x1 + x2 + x4, data = tdata)

Residuals:
Min      1Q  Median      3Q     Max
-3.0919 -1.8016  0.2562  1.2818  3.8982

Coefficients:
Estimate Std. Error t value Pr(>|t|)
(Intercept)  71.6483    14.1424   5.066 0.000675 ***
x1            1.4519     0.1170  12.410 5.78e-07 ***
x2            0.4161     0.1856   2.242 0.051687 .
x4           -0.2365     0.1733  -1.365 0.205395
---
Signif. codes:  0 ‘***’ 0.001 ‘**’ 0.01 ‘*’ 0.05 ‘.’ 0.1 ‘ ’ 1

Residual standard error: 2.309 on 9 degrees of freedom
Multiple R-squared:  0.9823,    Adjusted R-squared:  0.9764
F-statistic: 166.8 on 3 and 9 DF,  p-value: 3.323e-08
``````

### 3.逐步回归分析的优化

``````drop1(tstep)
``````

``````Single term deletions

Model:
Y ~ x1 + x2 + x4
Df Sum of Sq    RSS    AIC
47.97 24.974
x1      1    820.91 868.88 60.629
x2      1     26.79  74.76 28.742
x4      1      9.93  57.90 25.420
``````

### 4.进一步进行多元回归分析

``````tlm<-lm(Y~x1+x2,data=tdata)
summary(tlm)
``````

``````Call:
lm(formula = Y ~ x1 + x2, data = tdata)

Residuals:
Min     1Q Median     3Q    Max
-2.893 -1.574 -1.302  1.363  4.048

Coefficients:
Estimate Std. Error t value Pr(>|t|)
(Intercept) 52.57735    2.28617   23.00 5.46e-10 ***
x1           1.46831    0.12130   12.11 2.69e-07 ***
x2           0.66225    0.04585   14.44 5.03e-08 ***
---
Signif. codes:  0 ‘***’ 0.001 ‘**’ 0.01 ‘*’ 0.05 ‘.’ 0.1 ‘ ’ 1

Residual standard error: 2.406 on 10 degrees of freedom
Multiple R-squared:  0.9787,    Adjusted R-squared:  0.9744
F-statistic: 229.5 on 2 and 10 DF,  p-value: 4.407e-09
``````