## Nonparametric Test

The sample data from ``Group 1'' and ``Group 2'' must be shown separately in two different columns:Group 1:

Group 2:

Here it is assumed that the distribution of Group 1 and that of Group 2 share the same shape with possible shift, but not necessarily normally distributed. The Wilcoxon rank sum test is based on ranks--the rank 1 is assigned to the smallest measurement in both groups, and 2 to the second smallest, and so on. Recall that the validity of t-test is based on the ``normality of population distributions,'' and that it requires that either sample distributions are approximately normal, or the sample sizes are appropriately large ( ). Since a normal distribution is ``parametric,'' the rank sum test procedure is referred as ``nonparametric,'' free from the assumption of normal distribution.

The null hypothesis is stated as ``the distributions from two groups are identical,'' and the test will determine whether it is rejected in favor of the following alternative hypothesis.

It produces the estimate ```estimate.shift`'' and
the confidence interval
```(lower.bound,upper.bound)`''
for the shift (the difference of locations)
with
confidence interval.
The value `p.value`
indicates the significance of the test:
If p-value < , it suggest evidence against the null hypothesis
in favor of the alternative hypothesis.

Here the estimate
```estimate.shift`'' is calculated from the sample median of
the difference
between
in Group 1
and
in Group 2.
It is called the Hodges-Lehmann estimator.