# Doing normal Q-Q plots in R:

# generating some normal data:

norm.data <- rnorm(300)

qqnorm(norm.data)

# generating some heavy-tailed data:

t.data <- rt(300, df=2)

qqnorm(t.data)


# generating some light-tailed data:

unif.data <- runif(300, min=-1, max=1)

qqnorm(unif.data)


# generating some right-skewed data:

expon.data <- rexp(300)

qqnorm(expon.data)

# generating some left-skewed data:

neg.expon.data <- -rexp(300)

qqnorm(neg.expon.data)

### Note the characteristic shape of the Q-Q plot for each data set.

#### Wilcoxon Signed Rank Test:


## Data from Example 1, Page 355:


first<-c(86,71,77,68,91,72,77,91,70,71,88,87)
second<-c(88,77,76,64,96,72,65,90,65,80,81,72)

wilcox.test(second,first,paired=T, alternative="less", exact=T)

# Note that it gives the T+ value of 24.5 (see the last paragraph of the example) 
# and says it can't give the exact p-value because of the ties. 
# The p-value it does give is very close to the 0.238 that the example reports using equation 7. 


# The same answer could have been gotten using the differences instead of the paired option:
diffs <- second-first
wilcox.test(diffs,alternative="less")

## Data from Problem 1, page 365:

before <- c(.68,.64,.68,.82,.58,.80,.72,.65,.84,.73,.65,.59,.78,.67,.65,.76,.61,.86,.74,.88)
after <- c(.73,.62,.66,.92,.68,.87,.77,.70,.88,.79,.72,.60,.78,.66,.68,.77,.72,.86,.72,.97)

qqnorm(after-before)

wilcox.test(after,before, paired=TRUE, alternative="two.sided", exact=T)

#### Example 3:

# Data on pollution around a chemical plant, from Rao (p. 137).
# Two independent samples of river water were taken, 
# one upstream and one downstream.
# Pollution level was measured in ppm.


downstream <- c(32.8,30.4,32.3,26.4,27.8,26.9,29.0,31.5,31.2,26.7,25.6,25.1,32.8,34.3,35.4)

wilcox.test(downstream,mu=34,alternative="less",exact=T)


### A function to calculate the CI as described on p. 360-361:

WSRci<-function(d,conf.level=.95){
  d<-sort(d)
  n<-length(d)
  alpha<-1-conf.level
  thematrix<-matrix(d,ncol=n,nrow=n,byrow=T)+matrix(d,ncol=n,nrow=n,byrow=F)
  thematrix<-thematrix[!upper.tri(thematrix)]/2
  tablevalue<-qsignrank(alpha/2,n)
  estimator<-median(thematrix)
  clow<-sort(thematrix)[tablevalue]
  chi<-sort(thematrix)[n*(n+1)/2+1-tablevalue]
  list(matrix.values=thematrix,table.value=tablevalue,estimator=estimator,interval=c(clow,chi))}


## Example 2 data:

WSRci(after-before, conf.level=0.95)

## Example 3 data:

WSRci(downstream, conf.level=0.95)