#ccar for continuous variables
ccar<-mtcars[,c("mpg","disp","hp","drat","wt")]
head(ccar)
summary(ccar)
pairs(ccar)
# There is no colliniarity between pairs of predictors.
lm1<-lm(formula = mpg ~ hp + disp + drat + wt, data = ccar)
summary(lm1)
# Only the variables weight and horsepower have significant effect on the response.
# So we do a reduced regression by dropping other predictors.
lm2<-lm(formula = mpg ~ hp + wt, data = ccar)
summary(lm2)
par(mfrow=c(2,2))
plot(lm2)
# There is a quadratic pattern remaining in the residuals.  
hp2<-mtcars$hp^2
lm3<-lm(formula = mpg ~ hp + hp2 + wt , data = ccar)
summary(lm3)
# The new variable hp2 is significant and the adjusted R^2 has gone up. 
plot(lm3)
# There is still some quadratic type dependence.
wt2<-mtcars$wt^2
lm4<-lm(formula = mpg ~ hp + hp2 + wt +wt2, data = ccar)
summary(lm4)
plot(lm4)
# Now the quadratic effect is gone. The R^2 has gone wp and all predictor variables are significant.
# But there are four outliers.
which(rownames(ccar)=="Pontiac Firebird")
which(rownames(ccar)=="Chrysler Imperial")
which(rownames(ccar)=="Fiat 128")
which(rownames(ccar)=="Toyota Corolla")
ccarred<-ccar[-c(17,18,20,25),]
wt2<-ccarred$wt^2
hp2<-ccarred$hp^2
lm5<-lm(formula = mpg ~ hp + hp2 + wt + wt2, data = ccarred)
summary(lm5)
plot(lm5,id.n=NULL)
# The final adjusted R^2 is 0.92. All predictors are significant and all diagnostic plots are satisfactory. So this is our final model.

# The t-statistic and F statistic do not give same p-value for multiple regression. The ANOVA is computed pne-by-one, so the order matters.
lm6<-lm(formula = mpg ~ wt+hp, data = ccar)
anova(lm6)
anova(lm2)
summary(lm2)