#Simple Linear Regression – R code

# Car Weight (in 1000 pounds)
 weight <-c(3.4,3.8,4.1,2.2,2.6,2.9,2.0,2.7,1.9,3.4)    

# Fuel Consumption (gallons/100miles)
 consum <- c(5.5,5.9,6.5,3.3,3.6,4.6,2.9,3.6,3.1,4.9)

 n <- length(weight)
 obsno <- 1:n     

# sequence command (equally-spaced)
 seq1 <- seq(0, 4, 0.5)                       
 seq1 

#  replicate command
 rep0 <- rep(2:4, 2)                           
 rep0

 rep1 <- rep(2:4, 1:3)                 
 rep1

## combine vectors by columns
 car<-cbind(obsno, weight, consum)          

# first row (observation) in car data
 car[1,]

# second column in the car data
 car[,2]

# linear model fit lm(y ~ x), model  y=a+b*x  
 lm(consum ~ weight ) 

# linear model y=b*x, no intercept
 lm(consum ~ weight -1)                       
              
                                     
 M <- matrix(1:4, ncol=2, byrow=T)   # create a matrix  
 t(M)                        # transpose of a vector   
 solve(M)                      # inverse matrix of M

 X <- cbind(rep(1,n),car[,2])       # design matrix
 
 beta <- c(-0.363,1.639)  

# Calculate yhat, '%*%' matrix multiplication                 
 fitted.values <- X %*% beta  
 

# res = y - yhat
 residuals <- consum - fitted.values         
 sum(residuals)                     # sum(e_i)=0
  
 sum(t(residuals)%*%weight)       # sum(x_i * e_i)=0
  

## linear regression model

 lm(consum ~ weight )                         # model  y=a+b*x 

 
 summary(lm(consum ~ weight ) )

## ANOVA table for linear regression

 anova(lm(consum~weight))