Multiple Regression

 The following dataset gives information on makes of cars taken from the
April, 1990 issue of Consumer Reports. In particular, the data frame
"fuel.frame" contains data on mileage, weight of car, engine
displacement and size of car (qualitative, 6 possibilities, saved under
"Type").

 Question: Can we use a regression model to predict mileage from some of
these variables?

 We should begin with some plots to study the relationships among the
variables.

 Size of car is qualitative, but R can figure this out under "plot":

>  plot(Mileage ~ Type, fuel.frame)

 Regression is done in R using the "lm" command, for linear model.

> fuel.fit <- lm(Mileage ~ Weight, fuel.frame) 




> fuel.fit1 <- lm(Mileage ~ Weight+ Disp., fuel.frame)





 If we wanted to use all the variables in our data frame, we'd do the
following:

>  lm(Mileage ~ ., fuel.frame)  

> fuel.fit    
Call:
lm(formula = Mileage ~ Weight, data = fuel.frame)

Coefficients:
  (Intercept)       Weight 
    48.349347    -0.008193

> summary(fuel.fit1)

Call:
lm(formula = Mileage ~ Weight + Disp., data = fuel.frame)

Residuals:
    Min      1Q  Median      3Q     Max 
-4.5726 -1.5814 -0.2569  1.8499  4.6783 

Coefficients:
             Estimate Std. Error t value Pr(>|t|)    
(Intercept) 44.379702   2.654526  16.719  < 2e-16 ***
Weight      -0.006044   0.001188  -5.088 4.23e-06 ***
Disp.       -0.016540   0.007641  -2.165   0.0346 *  
---
Signif. codes:  0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1 

Residual standard error: 2.485 on 57 degrees of freedom
Multiple R-squared: 0.7402,     Adjusted R-squared: 0.7311 
F-statistic: 81.21 on 2 and 57 DF,  p-value: < 2.2e-16 


> fuel.fit1$residuals  # Gives residuals


> residuals(fuel.fit1) # Does same thing. R prefers this method.


> coefficients(fuel.fit1) # Gives coefficient estimates


> predict(fuel.fit1)  # Gives predicted values for mileages 


> anova(fuel.fit1) # NOTE: This adds terms sequentially (first to last)

Analysis of Variance Table

Response: Mileage

Response: Mileage
          Df Sum Sq Mean Sq  F value  Pr(>F)    
Weight     1 973.75  973.75 157.7259 < 2e-16 ***
Disp.      1  28.93   28.93   4.6863 0.03461 *  
Residuals 57 351.90    6.17                     
---
Signif. codes:  0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1 


> par(mfrow=c(2,2))
> plot(fuel.fit1) # Gives some summary plots of the fitted model