# =============================
# FIRST READ THE DATA INTO R
# =============================

data <- data.frame(SAL = c(10455, 9680, 7300, 9388, 12496, 11812, 9224,11725,
11320, 12000, 12500, 13310, 12105, 6200, 11522, 8000, 12548, 7700,
10028, 13176, 13255, 13004, 8000, 8224, 10750, 11669, 12322, 11002,
10666, 10839), CGPA = c(2.58, 2.31, 2.47, 2.52, 3.22, 3.37, 2.43, 3.08,
2.78, 2.98, 3.55, 3.64, 3.72, 2.24, 2.7, 2.3, 2.83, 2.37, 2.52, 3.22,
3.55, 3.55, 2.47, 2.47, 2.78, 2.78, 2.98, 2.58, 2.58, 2.58))


# ALTERNATIVE WAYS: YOU MAY READ DATA EITHER FROM A LOCAL FILE OR FROM THE WEB.  
data <- data.frame(read.table(file="http://www-stat.stanford.edu/~olshen/hrp262spring01/spring01Datasets/EX0508.DAT", 
    header=T, col.names=c("SAL", "CGPA")))
data; summary(data)


# ==============================
# EXPLORTARY DATA ANALYSIS
# ==============================

# The above commands line contains options for graph page.
# 1x1 array of figures
# on the page, with 4 lines of margin around each
par(mfrow=c(1,1),mar=rep(4,4))
plot(data$CGPA, data$SAL, ylab="SAL", xlab="CGPA", main="Scatter Plot of the Data" )

# PEARSON'S CORRELATION COEFFICIENT
r <- cor(data$CGPA, data$SAL); r


# =============================
# FIT THE SIMPLE LINEAR MODEL
# =============================

fit <- lm(SAL ~ CGPA, data=data); 
# TABLE OF PARAMETER ESTIMATES
summary(fit)
# ANOVA TABLE
anova(fit)

# ADD THE FITTED LINE TO THE SCATTER PLOT
abline(a=435.9, b=3630.6)
# ALTERNATIVELY
abline(fit)


# =================
# PREDICTION
# =================

# 95% CI FOR ESTIMATING MEAN SAL OF ALL INDIVIDUALS WITH CGPA=2.5
est.ci <- predict( fit, newdata=data.frame(CGPA=2.5),
    se.fit=TRUE, interval="confidence", level=0.95)
est.ci

# 95% PI FOR ESTIMATING THE SAL VALUE OF AN INDIVIDUALS WITH CGPA=2.5
pred.pi <- predict( fit, newdata=data.frame(CGPA=2.5),
se.fit=TRUE, interval="prediction", level=0.95)
pred.pi


# ==================================
# CONFIDENCE AND PREDICTION BANDS
# ==================================

# FIRST GENERATE A SERIES OF X VALUES 
range(data$CGPA) 
new <- data.frame(CGPA= seq(from=2.24, to=3.72, length=100)); new

# COMPUTE THE CONFIDENCE BAND
CI95 <- predict( fit, newdata=new, se.fit=TRUE,
    interval="confidence", level=0.95)
names(CI95); CI95$fit

# COMPUTE THE PREDICTION BAND
PI95 <- predict( fit, newdata=new, se.fit=TRUE,
    interval="prediction", level=0.95)
PI95$fit; range(PI95$fit) # 6157.853 16430.056


# PLOT
par(mar=rep(4,4))
# First create a frame for the plot, which is big enough to contain the confidence bands
plot(c(2, 4.5), c(5500, 16500), type="n", ylab="SAL",
    xlab="CGPA", main="Linear Fit with Confidence/Prediction Bands" )
# Add the scatter plot of the data
points(data$CGPA, data$SAL, pty=2)
# Add the fitted line
abline(fit)
lines(new$CGPA, CI95$fit[,2], lty=2, col="blue") # The lower CI bond
lines(new$CGPA, CI95$fit[,3], lty=2, col="blue") # The upper CI bond
lines(new$CGPA, PI95$fit[,2], lty=3, col="red") # The lower PI bond
lines(new$CGPA, PI95$fit[,3], lty=3, col="red") # The upper PI bond
legend(3.2,8000,c("fitted line", "confidence bands", "prediction bands"), lty=1:3)