逐步回归就是从自变量x中挑选出对y有显著影响的变量，已达到最优

http://www.cda.cn/uploadfile/image/20180318/20180318121348_87669.png

用step()函数

导入数据集

cement<-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)
)

> lm.sol<-lm(Y ~ X1+X2+X3+X4, data=cement)
> summary(lm.sol)
Call:
lm(formula = Y ~ X1 + X2 + X3 + X4, data = cement)
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

可以看出效果不明显

所以要进行逐步回归进行变量的筛选有forward：向前，backward:向后，both：2边，默认情况both

lm.step<-step(lm.sol)
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
> lm.step$anova
  Step Df Deviance Resid. Df Resid. Dev      AIC
1      NA       NA         8   47.86364 26.94429
2 - X3  1  0.10909         9   47.97273 24.97388

显然去掉X3会降低AIC

此时step()函数会帮助我们自动去掉X3

summary(lm.step)
Call:
lm(formula = Y ~ X1 + X2 + X4, data = cement)
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

很显然X2和X4效果不好

可以用add1()和drop1()函数进行增减删除函数

> drop1(lm.step)
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

我们知道除了AIC标准外，残差和也是重要标准，除去x4后残差和变为9.93

更新式子

> lm.opt<-lm(Y ~ X1+X2, data=cement)
> summary(lm.opt)


Call:
lm(formula = Y ~ X1 + X2, data = cement)
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

显然效果很好