#The Wilcoxon Rank Sum Test, also known as the Mann-Whitney Test, is the non-parametric analogue to the two sample t-test
#This test answers the same type of question in which you wish to compare two different samples to determine if they are significantly different from eachother
#As this is a non-parametric test the assumption that the data follows a normal distribution is relaxed
#At times the question of normality can be ambiguous when the data follows a nearly normal distribution so at times it is useful to compare this test to a two-sample t-test
#Assumptions:
#The observed values represent a random sampling of the population
#The two variables are independent
#The underlying distributions are continuous
#The measurements have an ordinal scale, in other words the data can be ranked from lowest to highest

#Hypotheses:
#Ho: No ordinal difference in median for the population
#H1: There is an ordinal difference in the medians for the populations


#For this example we will use a dataset of plant growth under different treatments that comes pre-loaded with R
PlantGrowth
#There are three different groups, a control and two different treatment groups: ctrl, trt1, trt2
#As such we would like to compare the control group to each of the treatment groups
attach(PlantGrowth)
control=weight[group=="ctrl"]
treatment1=weight[group=="trt1"]
treatment2=weight[group=="trt2"]

#For a Wilcoxon Rank Sum test the decision rule is very similar to most tests of significance
#You must set an alpha value and compare that to your p-value
#For this test we will use a=.05, if p < a then the results are significant

#Running a Wilcoxon Rank Sum test in R is very simple using the wilcox.test function.
wilcox.test(control,treatment1,exact=T,alternative="two.sided",paried=F)

#Since the default settings are for a two sided test for unpaired data it is not necessary to include these
#exact=T specifies that  you want the test to compute exact p-values

#We obtained a p-value greater than .05 so we cannot reject the null that treatment 1 is significantly different from the control
wilcox.test(control,treatment2,exact=T)

#The p-value is greater than alpha, however it is so close that it is worth further investigation
wilcox.test(treatment1,treatment2,exact=T)

#The p-value is less than alpha, treatment 1 is significantly different from treatment 2

#The results for the wilcox.test can be very similar to that for the two-sample t.test, but different in very significant ways.
#To compare lets run two sample t.tests for the same data, using the same value of a=.05 and a very similar null and alternative hypothesis.
t.test(control,treatment1,alternative="two.sided",conf.level=.95)
#The p-value is greater than alpha so we reject the null for this test as well
t.test(control,treatment2,alternative="two.sided",conf.level=.95)
#The p-value is less than alpha so we cannot reject the null. Treatment 2 is significantly different from the control.
#This conflicts directly with the results of the wilcox.test. Notice that both are very close the the alpha value; surrounding it on both above and below.
#At times like this the question of normality becomes the principal concern. To determine which test we should use it is necessary to investigate how closely the data follows a normal distribution.
t.test(treatment1,treatment2,alternative="two.sided",conf.level=.95)
#The p-value is less than alpha so we reject the null. This agrees with the wilcox.test.

qqnorm(control,main="Control Group")

qqnorm(treatment2,main="Treatment Group Two")