Wilkokson test

Distribution free rang criterion for testing hypothesis about treatment effect
absence. Suppose that we have N = m + n observations in two samles:

Let us denote:

where x_i and y_j — appreciable values, e_m+1,...,e_m+n — unappreciable
random values; Parameter Δ is of unterest to us, it is unknown shift
resulting from some kind of treatment.
All N random values e are mutually independent.
All e are derived from common population.

Method

Will test hypothesis H₀: Δ=0.
Join our two samples and put their values in ascending order, i.e. form
pooled variation sequence. For every value in this set we put in correspondence
its rang (R_j — rang of j-th element in pooled array), which is equal to its
sequence number in variation sequence.
As a critical statistic in this creterion it is used:

Suppose that we test our hypothesis facing hypothesis H₁: Δ≠0.
Evaluate all possible variants of rang grouping wherein statistic W is lesser
or equal to obervated one (denote it K), after which we calculate amount of all
possible distributions of rangs obtained by two samples, which is equal to C^m_N.
We estimate temporary P-value as:

If our temporary pvalue ≤ 0.5 we take as resulting P-value 2*(temporary P-value).
If temporary pvalue > 0.5, then we evaluate all possible variants of rang grouping
wherein statistic W is greater or equal to obervated one (denote it L), and take
P-value: