#
# Exercise in Class
#
# AGRESTI (2015) Foundations of Linear and Generalized Linear Models 
#
# Each year the Scottish Hill Runners Association (www.shr.uk.com)
# publishes a list of hill races in Scotland for the year. 
#
# Data on the record time for some of the races (in minutes).
# Explanatory variables listed are the distance of the race (in miles) 
# and the cumulative climb (in thousands of feet).
# 


x<-scan()
2.5 2.5 6 7.5 8 16 28


y<-scan()
16.08 17.50 33.65 45.60 62.27 204.62 192.67

cbind(x,y)
cbind(x,y,x*y,x*x)
cbind(sum(x),sum(y),sum(x*y),sum(x*x))
cbind(mean(x),mean(y))

n<-length(x)
Sxy<-sum(x*y)-sum(x)*sum(y)/n
Sxx<-sum(x*x)-sum(x)^2/n
b1<-Sxy/Sxx
b0<-mean(y)-b1*mean(x)
cat("Y=",b0,"+",b1,"*X","\n")

plot(x,y,pch=16,xlim=c(0,30),ylim=c(0,230))
abline(a=b0,b=b1,col=2,lwd=2)

summary(lm(y~x))
abline(lm(y~x),lwd=2)

predict(lm(y~x),list(x=c(2.5,20)))

yhat<-fitted(lm(y~x))
cbind(x,y,yhat,y-yhat)

join <- function(i){
lines(c(x[i],x[i]),c(y[i],yhat[i]),col="green")}
sapply(1:n,join)

lm.yx1<-lm(y~-1+x)
summary(lm.yx1)
plot(x,y,pch=16,xlim=c(0,30),ylim=c(0,230))
abline(lm.yx1,col=2,lwd=2)

#
# Full Example
#

ScotsRaces<-read.table("https://users.stat.ufl.edu/~aa/glm/data/ScotsRaces.dat",header=T)

ScotsRaces

attach(ScotsRaces)
ScotsRaces[order(distance),]

pairs(ScotsRaces[2:4])
cor(ScotsRaces[2:4])

summary(lm(time~distance,data=ScotsRaces))

attach(ScotsRaces)
plot(distance,time,pch=16,xlim=c(0,30),ylim=c(0,230))
abline(lm(time~distance))

summary(lm(time~-1+distance))
plot(distance,time,pch=16,xlim=c(0,30),ylim=c(0,230))
abline(lm(time~-1+distance))

summary(lm(time~distance+climb))
summary(lm(time~-1+distance+climb))