7  Model liniowy

Martyna Kosińska

7.1 📘 Model liniowy z 1 zmienną objaśniającą

7.1.1 Zmienna objaśniana: spalanie paliwa, zmienna objaśniająca: waga samochodu.

model1<-lm(mpg~wt,data=mtcars)

#Podstawowe informacje.
model1

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

Coefficients:
(Intercept)           wt  
     37.285       -5.344  

#Rozszerzony widok informacji o modelu.
summary(model1)

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

Residuals:
    Min      1Q  Median      3Q     Max 
-4.5432 -2.3647 -0.1252  1.4096  6.8727 

Coefficients:
            Estimate Std. Error t value Pr(>|t|)    
(Intercept)  37.2851     1.8776  19.858  < 2e-16 ***
wt           -5.3445     0.5591  -9.559 1.29e-10 ***
---
Signif. codes:  0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1

Residual standard error: 3.046 on 30 degrees of freedom
Multiple R-squared:  0.7528,    Adjusted R-squared:  0.7446 
F-statistic: 91.38 on 1 and 30 DF,  p-value: 1.294e-10

7.1.2 Wyświetlenie wyników

7.1.2.1 Współczynniki

model1$coefficients

(Intercept)          wt 
  37.285126   -5.344472 

7.1.2.2 Współczynniki i inne informacje (dotyczące testowania istotności parametrów)

summary(model1)$coefficients

             Estimate Std. Error   t value     Pr(>|t|)
(Intercept) 37.285126   1.877627 19.857575 8.241799e-19
wt          -5.344472   0.559101 -9.559044 1.293959e-10

7.1.2.3 Wartość współczynnika kierunkowego

summary(model1)$coefficients[2,1]

[1] -5.344472

summary(model1)$coef[2,1]

[1] -5.344472

7.1.2.4 Wartość wyrazu wolnego

summary(model1)$coefficients[1,1]

[1] 37.28513

7.1.2.5 Współczynnik R-kwadrat

summary(model1)$r.squared

[1] 0.7528328

7.1.2.5.1 Reszty

model1$residuals

          Mazda RX4       Mazda RX4 Wag          Datsun 710      Hornet 4 Drive 
         -2.2826106          -0.9197704          -2.0859521           1.2973499 
  Hornet Sportabout             Valiant          Duster 360           Merc 240D 
         -0.2001440          -0.6932545          -3.9053627           4.1637381 
           Merc 230            Merc 280           Merc 280C          Merc 450SE 
          2.3499593           0.2998560          -1.1001440           0.8668731 
         Merc 450SL         Merc 450SLC  Cadillac Fleetwood Lincoln Continental 
         -0.0502472          -1.8830236           1.1733496           2.1032876 
  Chrysler Imperial            Fiat 128         Honda Civic      Toyota Corolla 
          5.9810744           6.8727113           1.7461954           6.4219792 
      Toyota Corona    Dodge Challenger         AMC Javelin          Camaro Z28 
         -2.6110037          -2.9725862          -3.7268663          -3.4623553 
   Pontiac Firebird           Fiat X1-9       Porsche 914-2        Lotus Europa 
          2.4643670           0.3564263           0.1520430           1.2010593 
     Ford Pantera L        Ferrari Dino       Maserati Bora          Volvo 142E 
         -4.5431513          -2.7809399          -3.2053627          -1.0274952 

summary(model1)$residuals

          Mazda RX4       Mazda RX4 Wag          Datsun 710      Hornet 4 Drive 
         -2.2826106          -0.9197704          -2.0859521           1.2973499 
  Hornet Sportabout             Valiant          Duster 360           Merc 240D 
         -0.2001440          -0.6932545          -3.9053627           4.1637381 
           Merc 230            Merc 280           Merc 280C          Merc 450SE 
          2.3499593           0.2998560          -1.1001440           0.8668731 
         Merc 450SL         Merc 450SLC  Cadillac Fleetwood Lincoln Continental 
         -0.0502472          -1.8830236           1.1733496           2.1032876 
  Chrysler Imperial            Fiat 128         Honda Civic      Toyota Corolla 
          5.9810744           6.8727113           1.7461954           6.4219792 
      Toyota Corona    Dodge Challenger         AMC Javelin          Camaro Z28 
         -2.6110037          -2.9725862          -3.7268663          -3.4623553 
   Pontiac Firebird           Fiat X1-9       Porsche 914-2        Lotus Europa 
          2.4643670           0.3564263           0.1520430           1.2010593 
     Ford Pantera L        Ferrari Dino       Maserati Bora          Volvo 142E 
         -4.5431513          -2.7809399          -3.2053627          -1.0274952 

7.1.2.5.2 Wartości teoretyczne

model1$fitted.values

          Mazda RX4       Mazda RX4 Wag          Datsun 710      Hornet 4 Drive 
          23.282611           21.919770           24.885952           20.102650 
  Hornet Sportabout             Valiant          Duster 360           Merc 240D 
          18.900144           18.793255           18.205363           20.236262 
           Merc 230            Merc 280           Merc 280C          Merc 450SE 
          20.450041           18.900144           18.900144           15.533127 
         Merc 450SL         Merc 450SLC  Cadillac Fleetwood Lincoln Continental 
          17.350247           17.083024            9.226650            8.296712 
  Chrysler Imperial            Fiat 128         Honda Civic      Toyota Corolla 
           8.718926           25.527289           28.653805           27.478021 
      Toyota Corona    Dodge Challenger         AMC Javelin          Camaro Z28 
          24.111004           18.472586           18.926866           16.762355 
   Pontiac Firebird           Fiat X1-9       Porsche 914-2        Lotus Europa 
          16.735633           26.943574           25.847957           29.198941 
     Ford Pantera L        Ferrari Dino       Maserati Bora          Volvo 142E 
          20.343151           22.480940           18.205363           22.427495 

# Uwaga! Funkcja poniżej nie zwróci wyników.
# summary(model1)$fitted.values

plot(mtcars$mpg,mtcars$wt)

7.2 📘 Model liniowy z 2 zmiennymi objaśniającymi

7.2.1 Zmienna zależna: spalanie, dwie zmienne niezależne: waga samochodu i przespieszenie

model2<-lm(mpg~wt+qsec,data=mtcars)
model2

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

Coefficients:
(Intercept)           wt         qsec  
    19.7462      -5.0480       0.9292  

summary(model2)

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

Residuals:
    Min      1Q  Median      3Q     Max 
-4.3962 -2.1431 -0.2129  1.4915  5.7486 

Coefficients:
            Estimate Std. Error t value Pr(>|t|)    
(Intercept)  19.7462     5.2521   3.760 0.000765 ***
wt           -5.0480     0.4840 -10.430 2.52e-11 ***
qsec          0.9292     0.2650   3.506 0.001500 ** 
---
Signif. codes:  0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1

Residual standard error: 2.596 on 29 degrees of freedom
Multiple R-squared:  0.8264,    Adjusted R-squared:  0.8144 
F-statistic: 69.03 on 2 and 29 DF,  p-value: 9.395e-12