Assumptions of Simple Linear Regression

atmruns<-read.table(header=TRUE, text="atms minutes
200 5.8
200 35.9
200 20.9
200 35.4
200 10.2
200 24.5
200 16.6
400 86.1
400 59.6
400 64.9
400 61.2
400 54.3
400 96.8
400 55.8
600 114
600 82.9
600 74
600 88.4
600 75.3
600 113.3
600 93.9
800 119.1
800 131.3
800 122
800 120
800 127
800 139.4
800 149.9
1000    175.1
1000    165.8
1000    214.4
1000    166.6
1000    145.4
1000    185
1000    160.2
")

4 main assumptions to check:

\(\textbf{Normality}\)- distribution of errors is approximately normal

\(\textbf{Independence}\)- Each \(Y_i\) is independent of each other.

\(\textbf{Linearity}\)-The relationship of \(Y\) and \(X\) is linear

\(\textbf{Homoscedasticity}\)- the variance of the errors is constant for all values in the dependent variable

\(+\) Check for influence of any outliers

fit<-lm(minutes~atms,data=atmruns)

All four assumptions can be examined by plotting the fit

plot(fit)

Checking Normality in the errors In addition to examining the QQ Plot can 1)calculate Shapiro Wilks Test

resids<-residuals(fit)
qqnorm(resids,pch=20,cex=1.5)
qqline(resids,col=2,lwd=3)

shapiro.test(residuals(fit))

    Shapiro-Wilk normality test

data:  residuals(fit)
W = 0.94281, p-value = 0.06816

2)Examine the histogram of the residiuals

#Customized function:
residplot<-function(fit,nbreaks=10){
  z<-rstudent(fit)
  hist(z,breaks=nbreaks,freq=FALSE,
       xlab="Studentized Residual",
       main="Distribution of Errors")
  rug(jitter(z),col="brown")
  curve(dnorm(x,mean=mean(z),sd=sd(z)),
        add=TRUE, col="blue",lwd=2)
  lines(density(z)$x, density(z)$y,
        col="red",lwd=2,lty=2)
  legend("topright",
         legend=c("Normal Curve","Kernel Density Curve"),
         lty=1:2,col=c("blue","red"), cex=.7)
}
residplot(fit)

Check for independence:

library(car)
library(ggplot2)

durbinWatsonTest(fit)

 lag Autocorrelation D-W Statistic p-value
   1      -0.1616609      2.274309   0.534
 Alternative hypothesis: rho != 0

acf(fit$residuals)

Checking Linearity:

Checking Homoscedasticity:

Outlier-Influence: