#The data give the speed of cars and the distances taken to stop. 
 data(cars)

#scatterplot
 plot(cars$speed, cars$dist)  #plot(x,y) syntax
 plot(dist~speed, data=cars)  #same plot using model(~) syntax [PREFERRED SYNTAX]
 plot(speed, dist, data=cars) #NB: this syntax does not work

#lm() is a function for linear models
 lm(cars$dist~cars$speed)
 
#attributes of the model 
 mod <- lm(dist~speed, data=cars)
 mod
 attributes(mod) #lists properties of the model
 mod$fitted    # predicted values
 mod$resid     # residuals
 mod$coeff     # estimated coefficients

#plotting the regression line
 abline(-17.579, 3.932, col=2) #regression line  in red
 abline(mod$coeff,      col=3) #alternate syntax in green
 abline(mod,            col=4) #alternate syntax in blue

#Assessing the residuals
 cars$speed[4] # x.4
 cars$dist[4]  # y.4
 mod$fitted[4] # y_hat.4
 mod$resid[4]  # resid.4 = y_hat.4 - y.4
 cars$dist[4] - mod$fitted[4]
 

 hist(mod$resid)   #Histogram of the residuals:
 qqnorm(mod$resid) #Normal QQ plot of the residuals
 qqline(mod$resid) #add the reference line
 
 plot(mod$res~mod$fit) #Residuals versus fitted (predicted)
 abline(0,0)           #Add a zero line to a residual plot
 
#inference for the regression parameters 
 summary(mod)
 anova(mod)
 
#CI for Beta_1 (95% --> alpha=.05)
 df <- 50-2
 t.crit <- qt(.975, df)  # .975 = 1-alpha/2
 b1.hat <- 3.932
 se.b1 <-  0.4155
 lwr.95 <- b1.hat - t.crit * se.b1 #lower endpoint of CI
 upr.95 <- b1.hat + t.crit * se.b1 #upper endpoint of CI
 c(lwr.95, upr.95) #95% CI for Beta_1
 

#prediction at new x-values & CI vs. PI
 x.new.df <- data.frame(speed=12.5) #NOTE: the var.name must match (speed) 
 x.new.df
 predict(mod, x.new.df, interval="confidence") 
 predict(mod, x.new.df, interval="prediction")

 predict(mod, x.new.df, se.fit=TRUE) 
 










 
 
#Extrapolation 
#Note: The 'cars' data was recorded in the 1920's & includes speeds up to 25 mph.
#      Problem "ch05q07.txt" has recent data with speeds over 60 mph.
 mod <- lm(dist~speed, data=cars)
 plot(dist~speed, data=cars)   
#new data from chapter 5-7
 source("http://www.uvm.edu/~rsingle/stat221/data/scripts-221.R")
 dat2 <- bookdata("ch05q07.txt")
 plot(dat2$MPH, dat2$DIST)
 mod2 <- lm(DIST~MPH,data=dat2)
 mod     # older 'cars' data
 mod2    # newer data from "ch05q07.txt"
 
 abline(mod2,col=2) #red line for the SLR from new data
 abline(mod, col=3) #green line for SLR from 'cars' data
 
 plot(dat2$MPH,sqrt(dat2$DIST))
 mod3 <- lm(sqrt(DIST)~MPH,data=dat2)
 mod3
 abline(mod3)
 multifig(2,2)
 plot(dist~speed, data=cars); abline(mod, col=3)
 plot(dat2$MPH, dat2$DIST); abline(mod2,col=2)
 plot(dat2$MPH,sqrt(dat2$DIST)); abline(mod3)
 plot(0,0)

 plot(mod$resid, mod$pred)  ; abline(h=0)
 plot(mod2$resid, mod2$pred); abline(h=0)
 plot(mod3$resid, mod3$pred); abline(h=0)
 plot(0,0)