### Regression, mean or median estimation?

# regress systolic blood presure (SBP) as a function of age

Age = c(39,47,45,47,65,46,67,42,67,56,64,56,59,34,42)
SBP = c(144,220,138,145,162,142,170,124,158,154,162,150,140,110,128)

X11()
plot(Age, SBP)

# using standard ols, minimuze sum of square errors (SSE)

Pred = lm(SBP ~ Age)
Pred

X11()
plot(Age, SBP)
lines(Age, fitted(Pred))

plot(Age, residuals(Pred))

# using quantile regression, minimized least absolute deviation (LAD)

library(quantreg)

Pred = rq(formula = SBP ~ Age)
Pred

b = coef(Pred)
pred = b[1] + b[2]*Age
X11()
plot(Age, SBP)
lines(Age,pred)

# Is quantile regression is more robust to outliers?

# Consider estimating the tau precentile in the scatterplot.
# Consider estimating the 10% and 90% in the data by quantile regression.

X11()
plot(Age, SBP)

taus = c(0.1,0.5,0.9)
xx = seq(min(Age),max(Age),0.5)
for(tau in taus){
	Pred = rq(SBP ~ Age, tau=tau)
	b = coef(Pred)
	yy = b[1] + b[2]*xx
	lines(xx,yy)
}

###########################################################################

data(engel)	# a dataset in the quantreg library
attach(engel)

X11()
plot(x,y,xlab="household income",ylab="food expenditure")

# remove an outlier

x1 = engel$x[-138]
y1 = engel$y[-138]

# using ols

Pred = lm(y1 ~ x1)
Pred
X11()
plot(x1,y1,xlab="household income",ylab="food expenditure")
lines(x1, fitted(Pred))

# bootstrap the linear model

library(simpleboot)

lboot <- lm.boot(Pred, R = 1000)
summary(lboot)
X11()
plot(lboot)

# using quantile regression

X11()
plot(x1,y1,xlab="household income",ylab="food expenditure")
taus = c(0.1,0.5,0.9)
xx = seq(min(x1),max(x1),100)
for(tau in taus){
	Pred = rq(y1~x1,tau=tau)
	print(Pred)
	b = coef(Pred)
	yy = b[1] + b[2]*xx
	lines(xx,yy)
}