#######################
# multiple regression #
#######################
# part 1
set.seed(12345)
n = 100

# w is omitted variable. By construction, x does not cause y. w does 
w = rnorm(n)
x = w + rnorm(n)
y = 4*w + rnorm(n)

# Scatterplot just shows correlation, but not causality
plot(y~x)
abline(lm(y~x))

# Simple regression is misleading: it indicates spurious causality
summary(lm(y~x))$coef
summary(lm(y~x))$r.squared
# spurious causality is gone in a multipleregression that controls for w
summary(lm(y~x+w))$coef
 
# part 2   
rm(list = ls())     # clear the workspace     
ad = "https://www.fsb.miamioh.edu/lij14/400_house.txt"
data = read.table(url(ad), header=T)
attach(data)
summary(lm(rprice~baths))$coef
summary(lm(rprice~baths+area))$coef
cor(baths, area)
mean(age)
mean(baths)
mean(baths[age>18])
mean(baths[age<18])

# part 3
m_first = lm(baths~area)
rhat = resid(m_first)
m_second = lm(rprice~rhat)
summary(m_second)$coef
summary(lm(rprice~baths+area))$coef

# part 4
m = lm(rprice~baths+area+age)
library("car")
linearHypothesis(m, c("baths=0", "area=0"))
m_u = summary(lm(rprice~baths+area+age))
m_r = summary(lm(rprice~age))
f_test = ((m_u$r.squared-m_r$r.squared)/2)/((1-m_u$r.squared)/(321-3-1))
f_test
m_u$fstatistic
summary(m)

# part 5
summary(lm(rprice~baths))$coef
set.seed(12345)
bedroom = baths + 0.1*rnorm(321)
summary(lm(rprice~baths+bedroom))$coef
summary(lm(rprice~baths+bedroom))$fstatistic
summary(lm(baths~bedroom))$r.squared

################
# mini project #
################

library(HistData)
data(GaltonFamilies)
attach(GaltonFamilies)

# multiple regression
summary(lm(childHeight~father))$coefficient
summary(lm(childHeight~father+mother))$coefficient

0.38451-0.36828 
ovb = 0.29051*cov(father, mother)/var(father)
ovb

cor(childHeight,father)
cor(childHeight,mother)