# Exercise 6
# a
ds <- read.csv("http://www.math.ntnu.no/~erikblys/TMA4255-V14/Data/data4.csv")
colnames(ds)
fit <- lm(Volume~.,data=ds)
summary(fit)
# look at standardized or studentized residuals
plot(ds$Height,rstudent(fit))
plot(ds$Diameter,rstudent(fit))
plot(fit$fitted.values,rstudent(fit))
qqnorm(rstudent(fit))
qqline(rstudent(fit))
# or by using
plot(rstudent(fit))
# b - adding IQ
set.seed(123) # reproducible results
iq <- rnorm(31,100,16)
fit2 <- lm(Volume~Height+Diameter+iq,data=ds)
summary(fit2)
plot(iq,rstudent(fit2))
plot(fit2$fitted.values,rstudent(fit2))
qqnorm(rstudent(fit2))
qqline(rstudent(fit2))
library(nortest)
ad.test(rstudent(fit2))
# focus on R2 and R2 adjusted we see that
# R2 has increased from 0.948 to 0.9486 to and R2adj has decreased from 0.9442 to 0.9429

# c
# new variable
x <- ds$Diameter^2*ds$Height
fitx <- lm(ds$Volume~x)
summary(fitx)
plot(fitx$fitted,rstudent(fitx))
plot(rstudent(fitx))
qqnorm(rstudent(fitx))
fitx2 <- lm(ds$Volume~x-1) #without intercept
summary(fitx2)
newobs <- data.frame("x"=15^2*80)
predict(fitx2,newdata=newobs,interval="prediction")
# d
lnD <- log(ds$Diameter)
lnV <- log(ds$Volume)
lnH <- log(ds$Height)
lnds <- data.frame(lnD,lnH,lnV)
fitln <- lm(lnV~lnD+lnH,data=lnds)
summary(fitln)
plot(fitln$fitted,rstudent(fitln))
plot(rstudent(fitln))
qqnorm(rstudent(fitln))
qqline(rstudent(fitln))
ad.test(rstudent(fitln))
newobs <- data.frame("lnD"=log(15),"lnH"=log(80))
predint <- predict(fitln,newdata=newobs,interval="predict") 
predint
# backtransform to original units
exp(predint)