Chương 6 - Phân tích hồi quy tương quan (phần 1)

Packages

library(tidyverse)
library(knitr)
library(broom) # install.packages("broom")

1. Phân tích tương quan

cov(mtcars$mpg, mtcars$hp)

[1] -320.7321

Tương quan âm, khá chặt chẽ

2. Phân tích hồi quy - Hồi quy đơn

m1 <- lm(
  mpg ~ hp,
  mtcars
)

summary(m1)

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

Residuals:
    Min      1Q  Median      3Q     Max 
-5.7121 -2.1122 -0.8854  1.5819  8.2360 

Coefficients:
            Estimate Std. Error t value Pr(>|t|)    
(Intercept) 30.09886    1.63392  18.421  < 2e-16 ***
hp          -0.06823    0.01012  -6.742 1.79e-07 ***
---
Signif. codes:  0 ‘***’ 0.001 ‘**’ 0.01 ‘*’ 0.05 ‘.’ 0.1 ‘ ’ 1

Residual standard error: 3.863 on 30 degrees of freedom
Multiple R-squared:  0.6024,    Adjusted R-squared:  0.5892 
F-statistic: 45.46 on 1 and 30 DF,  p-value: 1.788e-07

\[ \hat{mpg}_i = 30.09 -0.06 \times hp_i \\ ~~ \\ mpg_i = 30.09 - 0.06 \times hp_i + e_i \]

confint(m1)

                  2.5 %     97.5 %
(Intercept) 26.76194879 33.4357723
hp          -0.08889465 -0.0475619

anova(m1)

Analysis of Variance Table

Response: mpg
          Df Sum Sq Mean Sq F value    Pr(>F)    
hp         1 678.37  678.37   45.46 1.788e-07 ***
Residuals 30 447.67   14.92                      
---
Signif. codes:  0 ‘***’ 0.001 ‘**’ 0.01 ‘*’ 0.05 ‘.’ 0.1 ‘ ’ 1

m2 <- lm(
  hp ~ mpg - 1,
  mtcars
)

summary(m2)

Call:
lm(formula = hp ~ mpg - 1, data = mtcars)

Residuals:
    Min      1Q  Median      3Q     Max 
-138.66  -42.48   11.86   83.28  244.88 

Coefficients:
    Estimate Std. Error t value Pr(>|t|)    
mpg   6.0078     0.8673   6.927 9.06e-08 ***
---
Signif. codes:  0 ‘***’ 0.001 ‘**’ 0.01 ‘*’ 0.05 ‘.’ 0.1 ‘ ’ 1

Residual standard error: 102.8 on 31 degrees of freedom
Multiple R-squared:  0.6075,    Adjusted R-squared:  0.5948 
F-statistic: 47.98 on 1 and 31 DF,  p-value: 9.062e-08

confint(m2, level = 0.99)

                0.5 %     99.5 %
(Intercept) 248.64087 399.523758
mpg         -12.43108  -5.228378

m4 <- lm(
  hp ~ .,
  mtcars
)
summary(m4)

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

Residuals:
    Min      1Q  Median      3Q     Max 
-38.681 -15.558   0.799  18.106  34.718 

Coefficients:
            Estimate Std. Error t value Pr(>|t|)   
(Intercept)  79.0484   184.5041   0.428  0.67270   
mpg          -2.0631     2.0906  -0.987  0.33496   
cyl           8.2037    10.0861   0.813  0.42513   
disp          0.4390     0.1492   2.942  0.00778 **
drat         -4.6185    16.0829  -0.287  0.77680   
wt          -27.6600    19.2704  -1.435  0.16591   
qsec         -1.7844     7.3639  -0.242  0.81089   
vs           25.8129    19.8512   1.300  0.20758   
am            9.4863    20.7599   0.457  0.65240   
gear          7.2164    14.6160   0.494  0.62662   
carb         18.7487     7.0288   2.667  0.01441 * 
---
Signif. codes:  0 ‘***’ 0.001 ‘**’ 0.01 ‘*’ 0.05 ‘.’ 0.1 ‘ ’ 1

Residual standard error: 25.97 on 21 degrees of freedom
Multiple R-squared:  0.9028,    Adjusted R-squared:  0.8565 
F-statistic:  19.5 on 10 and 21 DF,  p-value: 1.898e-08

ggplot(mtcars, aes(hp, mpg)) +
  geom_point() +
  geom_smooth(method = "lm", se = F) +
  geom_smooth(method = "lm", formula = y ~ x + I(x^2), se = F, col = "red") +
  geom_smooth(method = "lm", formula = y ~ I(1/x), se = F, col = "yellow") +
  geom_smooth(method = "lm", formula = y ~ I(1/x), se = F, col = "yellow")

m5 <- lm(
  mpg ~ hp + I(hp^2),
  mtcars
)
summary(m5)

Call:
lm(formula = mpg ~ hp + I(hp^2), data = mtcars)

Residuals:
    Min      1Q  Median      3Q     Max 
-4.5512 -1.6027 -0.6977  1.5509  8.7213 

Coefficients:
              Estimate Std. Error t value Pr(>|t|)    
(Intercept)  4.041e+01  2.741e+00  14.744 5.23e-15 ***
hp          -2.133e-01  3.488e-02  -6.115 1.16e-06 ***
I(hp^2)      4.208e-04  9.844e-05   4.275 0.000189 ***
---
Signif. codes:  0 ‘***’ 0.001 ‘**’ 0.01 ‘*’ 0.05 ‘.’ 0.1 ‘ ’ 1

Residual standard error: 3.077 on 29 degrees of freedom
Multiple R-squared:  0.7561,    Adjusted R-squared:  0.7393 
F-statistic: 44.95 on 2 and 29 DF,  p-value: 1.301e-09

summary(m1)

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

Residuals:
    Min      1Q  Median      3Q     Max 
-5.7121 -2.1122 -0.8854  1.5819  8.2360 

Coefficients:
            Estimate Std. Error t value Pr(>|t|)    
(Intercept) 30.09886    1.63392  18.421  < 2e-16 ***
hp          -0.06823    0.01012  -6.742 1.79e-07 ***
---
Signif. codes:  0 ‘***’ 0.001 ‘**’ 0.01 ‘*’ 0.05 ‘.’ 0.1 ‘ ’ 1

Residual standard error: 3.863 on 30 degrees of freedom
Multiple R-squared:  0.6024,    Adjusted R-squared:  0.5892 
F-statistic: 45.46 on 1 and 30 DF,  p-value: 1.788e-07

anova(m5)

Analysis of Variance Table

Response: mpg
          Df Sum Sq Mean Sq F value    Pr(>F)    
hp         1 678.37  678.37  71.633 2.514e-09 ***
I(hp^2)    1 173.04  173.04  18.273 0.0001889 ***
Residuals 29 274.63    9.47                      
---
Signif. codes:  0 ‘***’ 0.001 ‘**’ 0.01 ‘*’ 0.05 ‘.’ 0.1 ‘ ’ 1

anova(m1)

Analysis of Variance Table

Response: mpg
          Df Sum Sq Mean Sq F value    Pr(>F)    
hp         1 678.37  678.37   45.46 1.788e-07 ***
Residuals 30 447.67   14.92                      
---
Signif. codes:  0 ‘***’ 0.001 ‘**’ 0.01 ‘*’ 0.05 ‘.’ 0.1 ‘ ’ 1