## October 16, Q-Q plots for assesment of marginal normality
rm(list = ls())

###################################################################################
## example from the book
y <- c(-1,-0.1,0.16, 0.41, 0.62, 0.80, 1.26, 1.54, 1.71, 2.30)
n <- length(y)

## eventhough they are already sorted
sort.y <- sort(y)

## determine probabilities based upon sample data
prob <- (1:n - 0.5)/n

## determine quatiles based on the standard normal distribution
quant.norm <- qnorm(prob, mean=0, sd=1)

## make the Q-Q plot
plot(quant.norm, sort.y, pch=16, xlab="Theoretical Quantiles", ylab="Sample Quantiles", main="Normal Q-Q plot")

## let's add a line which passes throuugh the first and third quartile
q.25 <- qnorm(0.25, mean=0, sd=1)
q.75 <- qnorm(0.75, mean=0, sd=1)
y.25 <- quantile(y, prob=0.25)
y.75 <- quantile(y, prob=0.75)

b <- (y.75-y.25)/(q.75-q.25)
a <- y.25 - b*q.25
abline(a,b,lwd=2)

## R has a function that does this in one shot
qqnorm(y)
qqline(y)


## Radiation data
y <- c(0.15,0.09,0.18,0.10,0.05,0.12,0.08,0.05,0.08,0.10,0.07,
0.02,0.01,0.10,0.10,0.10,0.02,0.10,0.01,0.01,0.40,0.10,0.05,0.03,
0.05,0.15,0.10,0.15,0.09,0.08,0.18,0.10,0.20,0.11,0.30,0.02,0.20,
0.20,0.30,0.30,0.40,0.30,0.05)

qqnorm(unique(y), pch=16)
qqline(unique(y))



## let's look at some random variation in Q-Q plots
## All data are genereated from a standard normal distribution
par(mfrow=c(2,5))

for(i in 1:10){
y <- rnorm(30, mean=0, sd=1)
qqnorm(y, pch=16)
qqline(y)
}


## let's look at some random variation in Q-Q plots
## now n=500

par(mfrow=c(2,5))

for(i in 1:10){
y <- rnorm(500, mean=0, sd=1)
qqnorm(y, pch=16)
qqline(y)
}