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 ( $ n,m \ge 30$). 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.

$ H_A:$ The distribution of Group 1 that of Group 2

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 < $ \alpha$, 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  $ (X_i - Y_j)$ between $ X_1,\ldots,X_n$ in Group 1 and $ Y_1,\ldots,Y_m$ in Group 2. It is called the Hodges-Lehmann estimator.