[ARA] CH11 변수 선택

Author

김보람

Published

December 5, 2022

해당 강의노트는 전북대학교 이영미교수님 2022-2 고급회귀분석론 자료임

dt <- 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)
)
pairs(dt, pch=16)

cor(dt)
A matrix: 5 × 5 of type dbl
x1 x2 x3 x4 y
x1 1.0000000 0.2285795 -0.8241338 -0.2454451 0.7307175
x2 0.2285795 1.0000000 -0.1392424 -0.9729550 0.8162526
x3 -0.8241338 -0.1392424 1.0000000 0.0295370 -0.5346707
x4 -0.2454451 -0.9729550 0.0295370 1.0000000 -0.8213050
y 0.7307175 0.8162526 -0.5346707 -0.8213050 1.0000000 
m <- lm(y~., dt)  ##FM
summary(m)

Call:
lm(formula = y ~ ., data = dt)

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
anova(m)
A anova: 5 × 5
Df Sum Sq Mean Sq F value Pr(>F)
<int> <dbl> <dbl> <dbl> <dbl>
x1 1 1450.0763281 1450.0763281 242.36791816 2.887559e-07
x2 1 1207.7822656 1207.7822656 201.87052753 5.863323e-07
x3 1 9.7938691 9.7938691 1.63696188 2.366003e-01
x4 1 0.2469747 0.2469747 0.04127972 8.440715e-01
Residuals 8 47.8636394 5.9829549 NA NA 
##### 후진제거법
summary(m) #x3 제거
# drop1(m, test = "F")

Call:
lm(formula = y ~ ., data = dt)

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
m1 <- update(m, ~ . -x3)
summary(m1)  #x4 제거
# drop1(m1, test = "F")

Call:
lm(formula = y ~ x1 + x2 + x4, data = dt)

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

m2 <- update(m1, ~ . -x4)
summary(m2)  #x4 제거
# drop1(m2, test = "F")

Call:
lm(formula = y ~ x1 + x2, data = dt)

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

##### 전진선택법 
m0 = lm(y ~ 1, data = dt)
add1(m0, 
     scope = y ~  x1 + x2 + x3+ x4,
     test = "F")  ## x4추가
A anova: 5 × 6
Df Sum of Sq RSS AIC F value Pr(>F)
<dbl> <dbl> <dbl> <dbl> <dbl> <dbl>
<none> NA NA 2715.7631 71.44443 NA NA
x1 1 1450.0763 1265.6867 63.51947 12.602518 0.0045520446
x2 1 1809.4267 906.3363 59.17799 21.960605 0.0006648249
x3 1 776.3626 1939.4005 69.06740 4.403417 0.0597623242
x4 1 1831.8962 883.8669 58.85164 22.798520 0.0005762318 
m1 <- update(m0, ~ . +x4)
summary(m1)

Call:
lm(formula = y ~ x4, data = dt)

Residuals:
    Min      1Q  Median      3Q     Max 
-12.589  -8.228   1.495   4.726  17.524 

Coefficients:
            Estimate Std. Error t value Pr(>|t|)    
(Intercept) 117.5679     5.2622  22.342 1.62e-10 ***
x4           -0.7382     0.1546  -4.775 0.000576 ***
---
Signif. codes:  0 ‘***’ 0.001 ‘**’ 0.01 ‘*’ 0.05 ‘.’ 0.1 ‘ ’ 1

Residual standard error: 8.964 on 11 degrees of freedom
Multiple R-squared:  0.6745,    Adjusted R-squared:  0.645 
F-statistic:  22.8 on 1 and 11 DF,  p-value: 0.0005762
add1(m1, 
     scope = y ~  x1 + x2 + x3+ x4,
     test = "F")  ## x1추가
A anova: 4 × 6
Df Sum of Sq RSS AIC F value Pr(>F)
<dbl> <dbl> <dbl> <dbl> <dbl> <dbl>
<none> NA NA 883.86692 58.85164 NA NA
x1 1 809.10480 74.76211 28.74170 108.2239093 1.105281e-06
x2 1 14.98679 868.88013 60.62933 0.1724839 6.866842e-01
x3 1 708.12891 175.73800 39.85258 40.2945802 8.375467e-05 
m2 <- update(m1, ~ . +x1)
summary(m2)

Call:
lm(formula = y ~ x4 + x1, data = dt)

Residuals:
    Min      1Q  Median      3Q     Max 
-5.0234 -1.4737  0.1371  1.7305  3.7701 

Coefficients:
             Estimate Std. Error t value Pr(>|t|)    
(Intercept) 103.09738    2.12398   48.54 3.32e-13 ***
x4           -0.61395    0.04864  -12.62 1.81e-07 ***
x1            1.43996    0.13842   10.40 1.11e-06 ***
---
Signif. codes:  0 ‘***’ 0.001 ‘**’ 0.01 ‘*’ 0.05 ‘.’ 0.1 ‘ ’ 1

Residual standard error: 2.734 on 10 degrees of freedom
Multiple R-squared:  0.9725,    Adjusted R-squared:  0.967 
F-statistic: 176.6 on 2 and 10 DF,  p-value: 1.581e-08
add1(m2, 
     scope = y ~  x1 + x2 + x3+ x4,
     test = "F")  ## stop
A anova: 3 × 6
Df Sum of Sq RSS AIC F value Pr(>F)
<dbl> <dbl> <dbl> <dbl> <dbl> <dbl>
<none> NA NA 74.76211 28.74170 NA NA
x2 1 26.78938 47.97273 24.97388 5.025865 0.05168735
x3 1 23.92599 50.83612 25.72755 4.235846 0.06969226 
##### 단계적선택법
m0 = lm(y ~ 1, data = dt)
add1(m0, 
     scope = y ~  x1 + x2 + x3+ x4,
     test = "F")  ## x4추가
m1 <- update(m0, ~ . +x4)
A anova: 5 × 6
Df Sum of Sq RSS AIC F value Pr(>F)
<dbl> <dbl> <dbl> <dbl> <dbl> <dbl>
<none> NA NA 2715.7631 71.44443 NA NA
x1 1 1450.0763 1265.6867 63.51947 12.602518 0.0045520446
x2 1 1809.4267 906.3363 59.17799 21.960605 0.0006648249
x3 1 776.3626 1939.4005 69.06740 4.403417 0.0597623242
x4 1 1831.8962 883.8669 58.85164 22.798520 0.0005762318 
m1 <- update(m0, ~ . +x4)
summary(m1)

Call:
lm(formula = y ~ x4, data = dt)

Residuals:
    Min      1Q  Median      3Q     Max 
-12.589  -8.228   1.495   4.726  17.524 

Coefficients:
            Estimate Std. Error t value Pr(>|t|)    
(Intercept) 117.5679     5.2622  22.342 1.62e-10 ***
x4           -0.7382     0.1546  -4.775 0.000576 ***
---
Signif. codes:  0 ‘***’ 0.001 ‘**’ 0.01 ‘*’ 0.05 ‘.’ 0.1 ‘ ’ 1

Residual standard error: 8.964 on 11 degrees of freedom
Multiple R-squared:  0.6745,    Adjusted R-squared:  0.645 
F-statistic:  22.8 on 1 and 11 DF,  p-value: 0.0005762
add1(m1, 
     scope = y ~  x1 + x2 + x3+ x4,
     test = "F")  ## x1추가
m2 <- update(m1, ~ . +x1)
summary(m2)  #제거 없음
A anova: 4 × 6
Df Sum of Sq RSS AIC F value Pr(>F)
<dbl> <dbl> <dbl> <dbl> <dbl> <dbl>
<none> NA NA 883.86692 58.85164 NA NA
x1 1 809.10480 74.76211 28.74170 108.2239093 1.105281e-06
x2 1 14.98679 868.88013 60.62933 0.1724839 6.866842e-01
x3 1 708.12891 175.73800 39.85258 40.2945802 8.375467e-05 

Call:
lm(formula = y ~ x4 + x1, data = dt)

Residuals:
    Min      1Q  Median      3Q     Max 
-5.0234 -1.4737  0.1371  1.7305  3.7701 

Coefficients:
             Estimate Std. Error t value Pr(>|t|)    
(Intercept) 103.09738    2.12398   48.54 3.32e-13 ***
x4           -0.61395    0.04864  -12.62 1.81e-07 ***
x1            1.43996    0.13842   10.40 1.11e-06 ***
---
Signif. codes:  0 ‘***’ 0.001 ‘**’ 0.01 ‘*’ 0.05 ‘.’ 0.1 ‘ ’ 1

Residual standard error: 2.734 on 10 degrees of freedom
Multiple R-squared:  0.9725,    Adjusted R-squared:  0.967 
F-statistic: 176.6 on 2 and 10 DF,  p-value: 1.581e-08
add1(m2, 
     scope = y ~  x1 + x2 + x3+ x4,
     test = "F")  ## x2추가
m3 <- update(m2, ~ . +x2)
summary(m3)  #x4 제거
A anova: 3 × 6
Df Sum of Sq RSS AIC F value Pr(>F)
<dbl> <dbl> <dbl> <dbl> <dbl> <dbl>
<none> NA NA 74.76211 28.74170 NA NA
x2 1 26.78938 47.97273 24.97388 5.025865 0.05168735
x3 1 23.92599 50.83612 25.72755 4.235846 0.06969226 

Call:
lm(formula = y ~ x4 + x1 + x2, data = dt)

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 ***
x4           -0.2365     0.1733  -1.365 0.205395    
x1            1.4519     0.1170  12.410 5.78e-07 ***
x2            0.4161     0.1856   2.242 0.051687 .  
---
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
m4 <- update(m3, ~ . -x4)
add1(m4, 
     scope = y ~  x1 + x2 + x3+ x4,
     test = "F") #stop
A anova: 3 × 6
Df Sum of Sq RSS AIC F value Pr(>F)
<dbl> <dbl> <dbl> <dbl> <dbl> <dbl>
<none> NA NA 57.90448 25.41999 NA NA
x3 1 9.793869 48.11061 25.01120 1.832128 0.2088895
x4 1 9.931754 47.97273 24.97388 1.863262 0.2053954 
# install.packages("leaps")
library(leaps)
fit<-regsubsets(y~., data=dt, nbest=1,nvmax=4,
                # method=c("exhaustive","backward", 
                #          "forward", "seqrep")
                method='forward',
                )
a <- summary(fit)
str(a)
List of 8
 $ which : logi [1:4, 1:5] TRUE TRUE TRUE TRUE FALSE TRUE ...
  ..- attr(*, "dimnames")=List of 2
  .. ..$ : chr [1:4] "1" "2" "3" "4"
  .. ..$ : chr [1:5] "(Intercept)" "x1" "x2" "x3" ...
 $ rsq   : num [1:4] 0.675 0.972 0.982 0.982
 $ rss   : num [1:4] 883.9 74.8 48 47.9
 $ adjr2 : num [1:4] 0.645 0.967 0.976 0.974
 $ cp    : num [1:4] 138.73 5.5 3.02 5
 $ bic   : num [1:4] -9.46 -39.01 -42.21 -39.68
 $ outmat: chr [1:4, 1:4] " " "*" "*" "*" ...
  ..- attr(*, "dimnames")=List of 2
  .. ..$ : chr [1:4] "1  ( 1 )" "2  ( 1 )" "3  ( 1 )" "4  ( 1 )"
  .. ..$ : chr [1:4] "x1" "x2" "x3" "x4"
 $ obj   :List of 28
  ..$ np       : int 5
  ..$ nrbar    : int 10
  ..$ d        : num [1:5] 13 3362 390.2 154.7 10.5
  ..$ rbar     : num [1:10] 30 7.4615 48.1538 11.7692 -0.0863 ...
  ..$ thetab   : num [1:5] 95.423 -0.738 1.44 0.416 0.102
  ..$ first    : int 2
  ..$ last     : int 5
  ..$ vorder   : int [1:5] 1 5 2 3 4
  ..$ tol      : num [1:5] 1.80e-09 9.67e-08 2.36e-08 1.19e-07 3.72e-08
  ..$ rss      : num [1:5] 2715.8 883.9 74.8 48 47.9
  ..$ bound    : num [1:5] 2715.8 1265.7 57.9 48.1 47.9
  ..$ nvmax    : int 5
  ..$ ress     : num [1:5, 1] 2715.8 883.9 74.8 48 47.9
  ..$ ir       : int 5
  ..$ nbest    : num 1
  ..$ lopt     : int [1:15, 1] 1 1 5 1 5 2 1 5 2 3 ...
  ..$ il       : int 15
  ..$ ier      : int 0
  ..$ xnames   : chr [1:5] "(Intercept)" "x1" "x2" "x3" ...
  ..$ method   : chr "forward"
  ..$ force.in : Named logi [1:5] TRUE FALSE FALSE FALSE FALSE
  .. ..- attr(*, "names")= chr [1:5] "" "x1" "x2" "x3" ...
  ..$ force.out: Named logi [1:5] FALSE FALSE FALSE FALSE FALSE
  .. ..- attr(*, "names")= chr [1:5] "" "x1" "x2" "x3" ...
  ..$ sserr    : num 47.9
  ..$ intercept: logi TRUE
  ..$ lindep   : logi [1:5] FALSE FALSE FALSE FALSE FALSE
  ..$ nullrss  : num 2716
  ..$ nn       : int 13
  ..$ call     : language regsubsets.formula(y ~ ., data = dt, nbest = 1, nvmax = 4, method = "forward",      )
  ..- attr(*, "class")= chr "regsubsets"
 - attr(*, "class")= chr "summary.regsubsets"

with(summary(fit),
     round(cbind(which,rss,rsq,adjr2, cp, bic),3))
A matrix: 4 × 10 of type dbl
(Intercept) x1 x2 x3 x4 rss rsq adjr2 cp bic
1 1 0 0 0 1 883.867 0.675 0.645 138.731 -9.463
2 1 1 0 0 1 74.762 0.972 0.967 5.496 -39.008
3 1 1 1 0 1 47.973 0.982 0.976 3.018 -42.211
4 1 1 1 1 1 47.864 0.982 0.974 5.000 -39.675 
###Backward - AIC
model_back = step(m, direction = "backward")
summary(model_back)
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
<none>              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
<none>               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 = dt)

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
###Forward - AIC
model_forward = step(
  m0, 
  scope = y ~  x1 + x2 + x3+ x4, 
  direction = "forward")
summary(model_forward)
Start:  AIC=71.44
y ~ 1

       Df Sum of Sq     RSS    AIC
+ x4    1   1831.90  883.87 58.852
+ x2    1   1809.43  906.34 59.178
+ x1    1   1450.08 1265.69 63.519
+ x3    1    776.36 1939.40 69.067
<none>              2715.76 71.444

Step:  AIC=58.85
y ~ x4

       Df Sum of Sq    RSS    AIC
+ x1    1    809.10  74.76 28.742
+ x3    1    708.13 175.74 39.853
<none>              883.87 58.852
+ x2    1     14.99 868.88 60.629

Step:  AIC=28.74
y ~ x4 + x1

       Df Sum of Sq    RSS    AIC
+ x2    1    26.789 47.973 24.974
+ x3    1    23.926 50.836 25.728
<none>              74.762 28.742

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

       Df Sum of Sq    RSS    AIC
<none>              47.973 24.974
+ x3    1   0.10909 47.864 26.944

Call:
lm(formula = y ~ x4 + x1 + x2, data = dt)

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 ***
x4           -0.2365     0.1733  -1.365 0.205395    
x1            1.4519     0.1170  12.410 5.78e-07 ***
x2            0.4161     0.1856   2.242 0.051687 .  
---
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
###Step - AIC
model_step = step(
  m0, 
  scope =  y ~  x1 + x2 + x3+ x4, 
  direction = "both")
summary(model_step)
Start:  AIC=71.44
y ~ 1

       Df Sum of Sq     RSS    AIC
+ x4    1   1831.90  883.87 58.852
+ x2    1   1809.43  906.34 59.178
+ x1    1   1450.08 1265.69 63.519
+ x3    1    776.36 1939.40 69.067
<none>              2715.76 71.444

Step:  AIC=58.85
y ~ x4

       Df Sum of Sq     RSS    AIC
+ x1    1    809.10   74.76 28.742
+ x3    1    708.13  175.74 39.853
<none>               883.87 58.852
+ x2    1     14.99  868.88 60.629
- x4    1   1831.90 2715.76 71.444

Step:  AIC=28.74
y ~ x4 + x1

       Df Sum of Sq     RSS    AIC
+ x2    1     26.79   47.97 24.974
+ x3    1     23.93   50.84 25.728
<none>                74.76 28.742
- x1    1    809.10  883.87 58.852
- x4    1   1190.92 1265.69 63.519

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

       Df Sum of Sq    RSS    AIC
<none>               47.97 24.974
- x4    1      9.93  57.90 25.420
+ x3    1      0.11  47.86 26.944
- x2    1     26.79  74.76 28.742
- x1    1    820.91 868.88 60.629

Call:
lm(formula = y ~ x4 + x1 + x2, data = dt)

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 ***
x4           -0.2365     0.1733  -1.365 0.205395    
x1            1.4519     0.1170  12.410 5.78e-07 ***
x2            0.4161     0.1856   2.242 0.051687 .  
---
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
str(mtcars)
'data.frame':   32 obs. of  11 variables:
 $ mpg : num  21 21 22.8 21.4 18.7 18.1 14.3 24.4 22.8 19.2 ...
 $ cyl : num  6 6 4 6 8 6 8 4 4 6 ...
 $ disp: num  160 160 108 258 360 ...
 $ hp  : num  110 110 93 110 175 105 245 62 95 123 ...
 $ drat: num  3.9 3.9 3.85 3.08 3.15 2.76 3.21 3.69 3.92 3.92 ...
 $ wt  : num  2.62 2.88 2.32 3.21 3.44 ...
 $ qsec: num  16.5 17 18.6 19.4 17 ...
 $ vs  : num  0 0 1 1 0 1 0 1 1 1 ...
 $ am  : num  1 1 1 0 0 0 0 0 0 0 ...
 $ gear: num  4 4 4 3 3 3 3 4 4 4 ...
 $ carb: num  4 4 1 1 2 1 4 2 2 4 ...
round(cor(mtcars),2)
A matrix: 11 × 11 of type dbl
mpg cyl disp hp drat wt qsec vs am gear carb
mpg 1.00 -0.85 -0.85 -0.78 0.68 -0.87 0.42 0.66 0.60 0.48 -0.55
cyl -0.85 1.00 0.90 0.83 -0.70 0.78 -0.59 -0.81 -0.52 -0.49 0.53
disp -0.85 0.90 1.00 0.79 -0.71 0.89 -0.43 -0.71 -0.59 -0.56 0.39
hp -0.78 0.83 0.79 1.00 -0.45 0.66 -0.71 -0.72 -0.24 -0.13 0.75
drat 0.68 -0.70 -0.71 -0.45 1.00 -0.71 0.09 0.44 0.71 0.70 -0.09
wt -0.87 0.78 0.89 0.66 -0.71 1.00 -0.17 -0.55 -0.69 -0.58 0.43
qsec 0.42 -0.59 -0.43 -0.71 0.09 -0.17 1.00 0.74 -0.23 -0.21 -0.66
vs 0.66 -0.81 -0.71 -0.72 0.44 -0.55 0.74 1.00 0.17 0.21 -0.57
am 0.60 -0.52 -0.59 -0.24 0.71 -0.69 -0.23 0.17 1.00 0.79 0.06
gear 0.48 -0.49 -0.56 -0.13 0.70 -0.58 -0.21 0.21 0.79 1.00 0.27
carb -0.55 0.53 0.39 0.75 -0.09 0.43 -0.66 -0.57 0.06 0.27 1.00 
m_full <- lm(mpg~., mtcars)
summary(m_full)

Call:
lm(formula = mpg ~ ., data = mtcars)

Residuals:
    Min      1Q  Median      3Q     Max 
-3.4506 -1.6044 -0.1196  1.2193  4.6271 

Coefficients:
            Estimate Std. Error t value Pr(>|t|)  
(Intercept) 12.30337   18.71788   0.657   0.5181  
cyl         -0.11144    1.04502  -0.107   0.9161  
disp         0.01334    0.01786   0.747   0.4635  
hp          -0.02148    0.02177  -0.987   0.3350  
drat         0.78711    1.63537   0.481   0.6353  
wt          -3.71530    1.89441  -1.961   0.0633 .
qsec         0.82104    0.73084   1.123   0.2739  
vs           0.31776    2.10451   0.151   0.8814  
am           2.52023    2.05665   1.225   0.2340  
gear         0.65541    1.49326   0.439   0.6652  
carb        -0.19942    0.82875  -0.241   0.8122  
---
Signif. codes:  0 ‘***’ 0.001 ‘**’ 0.01 ‘*’ 0.05 ‘.’ 0.1 ‘ ’ 1

Residual standard error: 2.65 on 21 degrees of freedom
Multiple R-squared:  0.869, Adjusted R-squared:  0.8066 
F-statistic: 13.93 on 10 and 21 DF,  p-value: 3.793e-07
fit<-regsubsets(mpg~., data=mtcars, nbest=1,nvmax=9,
                # method=c("exhaustive","backward", "forward", "seqrep")
                method='exhaustive',
)
summary(fit)
Subset selection object
Call: regsubsets.formula(mpg ~ ., data = mtcars, nbest = 1, nvmax = 9, 
    method = "exhaustive", )
10 Variables  (and intercept)
     Forced in Forced out
cyl      FALSE      FALSE
disp     FALSE      FALSE
hp       FALSE      FALSE
drat     FALSE      FALSE
wt       FALSE      FALSE
qsec     FALSE      FALSE
vs       FALSE      FALSE
am       FALSE      FALSE
gear     FALSE      FALSE
carb     FALSE      FALSE
1 subsets of each size up to 9
Selection Algorithm: exhaustive
         cyl disp hp  drat wt  qsec vs  am  gear carb
1  ( 1 ) " " " "  " " " "  "*" " "  " " " " " "  " " 
2  ( 1 ) "*" " "  " " " "  "*" " "  " " " " " "  " " 
3  ( 1 ) " " " "  " " " "  "*" "*"  " " "*" " "  " " 
4  ( 1 ) " " " "  "*" " "  "*" "*"  " " "*" " "  " " 
5  ( 1 ) " " "*"  "*" " "  "*" "*"  " " "*" " "  " " 
6  ( 1 ) " " "*"  "*" "*"  "*" "*"  " " "*" " "  " " 
7  ( 1 ) " " "*"  "*" "*"  "*" "*"  " " "*" "*"  " " 
8  ( 1 ) " " "*"  "*" "*"  "*" "*"  " " "*" "*"  "*" 
9  ( 1 ) " " "*"  "*" "*"  "*" "*"  "*" "*" "*"  "*" 
with(summary(fit),
     round(cbind(which,rss,rsq,adjr2, cp, bic),3))
A matrix: 9 × 16 of type dbl
(Intercept) cyl disp hp drat wt qsec vs am gear carb rss rsq adjr2 cp bic
1 1 0 0 0 0 1 0 0 0 0 0 278.322 0.753 0.745 11.627 -37.795
2 1 1 0 0 0 1 0 0 0 0 0 191.172 0.830 0.819 1.219 -46.348
3 1 0 0 0 0 1 1 0 1 0 0 169.286 0.850 0.834 0.103 -46.773
4 1 0 0 1 0 1 1 0 1 0 0 160.066 0.858 0.837 0.790 -45.099
5 1 0 1 1 0 1 1 0 1 0 0 153.438 0.864 0.838 1.846 -42.987
6 1 0 1 1 1 1 1 0 1 0 0 150.093 0.867 0.835 3.370 -40.227
7 1 0 1 1 1 1 1 0 1 1 0 148.528 0.868 0.830 5.147 -37.096
8 1 0 1 1 1 1 1 0 1 1 1 147.843 0.869 0.823 7.050 -33.779
9 1 0 1 1 1 1 1 1 1 1 1 147.574 0.869 0.815 9.011 -30.371 
fit_4 <- lm(mpg~hp+wt+qsec+am, mtcars)
summary(fit_4)

Call:
lm(formula = mpg ~ hp + wt + qsec + am, data = mtcars)

Residuals:
    Min      1Q  Median      3Q     Max 
-3.4975 -1.5902 -0.1122  1.1795  4.5404 

Coefficients:
            Estimate Std. Error t value Pr(>|t|)   
(Intercept) 17.44019    9.31887   1.871  0.07215 . 
hp          -0.01765    0.01415  -1.247  0.22309   
wt          -3.23810    0.88990  -3.639  0.00114 **
qsec         0.81060    0.43887   1.847  0.07573 . 
am           2.92550    1.39715   2.094  0.04579 * 
---
Signif. codes:  0 ‘***’ 0.001 ‘**’ 0.01 ‘*’ 0.05 ‘.’ 0.1 ‘ ’ 1

Residual standard error: 2.435 on 27 degrees of freedom
Multiple R-squared:  0.8579,    Adjusted R-squared:  0.8368 
F-statistic: 40.74 on 4 and 27 DF,  p-value: 4.589e-11