# Example of generating random permutations to see if an observed difference between the
# means of two groups is large compared to differences between randomly generated groups
library(UsingR)

x=c(6,11,15,18,29,33,34,46,49,73,102,12,13,14,23,26,33,59,59,69,75,78)
mean(x[12:22])-mean(x[1:11])

# Take 100 random samples with two groups each from x and calculate the difference between the means of the two groups.
diff=0
for(permutation in 1:100) 
{
y=sample(x,22)
diff[permutation]=mean(y[1:11])-mean(y[12:22])
}

# Print and plot the differences between the randomly generated groups
print(diff)
sort(diff)

hist(diff,
main = "Histogram of differences between the means \n for 100 randomly generated groups", 
xlab="Difference between the means")

plot(sort(diff), main = "Plot of differences between the means \n for 100 randomly generated groups", 
ylab="Difference between the means",
xlab="Permutation (sorted)")
abline(4.09,0, col="red",lty=2)
abline(-4.09,0, col="red",lty=2)

# Difference in the original data was 4.09.
# What % of random differences are > 4.09?  (Note that this is a one-tailed test.)
sort(diff)>4.09
pctGT409=100*sum(sort(diff)>4.09)/length(diff)
pctGT409 # of random differences are > 4.09
cat(pctGT409, "%  of random differences are > 4.09  \n")

# Two-tailed version
# What % of random differences are > 4.09  or < -4.09?
# abs(sort(diff))>4.09
abspctGT409=100*sum(abs(sort(diff))>4.09)/length(diff)
abspctGT409


abline(-4.09,0, col="red",lty=2)



# find the median diff
quantile(diff,0.5)


Suppose that 78 out of 100 permutations give us a difference between the means of 
A and B 
greater than the difference in the clinical data.

If the null hypothesis is true (that there is no difference between Treatment A 
and Treatment B), then
the probability that we would observe a difference between the means of A and B 
greater than the difference in the clinical data is  78/100 P=0.78.



Suppose that 1 out of 100 permutations give us a difference between 
the means of A and B 
greater than the difference in the clinical data.

If the null hypothesis is true (that there is no difference between Treatment A 
and Treatment B), then
the probability that we would observe a difference between the means of A and B 
greater than the difference in the clinical data is  1/100 P=0.01.