9.4.18  The Wilcoxon or Mann-Whitney distribution

The Wilcoxon test polynomial: wilcoxonp

The wilcoxonp command computes the polynomial for the Wilcoxon or Mann-Whitney test.

• wilcoxonp takes one mandatory argument and one optional argument:
  • n, an integer.
  • Optionally, k an integer.
• wilcoxonp(n [,k] returns the polynomial for the Wilcoxon test.

Examples.

• Input:
  wilcoxonp(4)
  Output:

  ▯

  1 16

  ,

  1 16

  ,

  1 16

  ,

  1 8

  ,

  1 8

  ,

  1 8

  ,

  1 8

  ,

  1 8

  ,

  1 16

  ,

  1 16

  ,

  1 16

  ▯

• Input:
  wilcoxonp(4,3)
  Output:

  ▯

  1 35

  ,

  1 35

  ,

  2 35

  ,

  3 35

  ,

  4 35

  ,

  4 35

  ,

  1 7

  ,

  4 35

  ,

  4 35

  ,

  3 35

  ,

  2 35

  ,

  1 35

  ,

  1 35

  ▯

The Wilcoxon/Mann-Whitney statistic: wilcoxons

The wilcoxons command computes the Wilcoxon or Mann-Whitney statistic.

• wilcoxons takes two arguments:
  • L, a list.
  • M, either a list or a real number (a median).
• wilcoxons(L,M) returns the Wilcoxon statistic.

Examples.

• Input:
  wilcoxons([1,3,4,5,7,8,8,12,15,17],10)
  Output:

  18

• wilcoxons([1,3,4,5,7,8,8,12,15,17],[2,6,10,11,13,14,15,18,19,20])
  Output:

  128.5

The Wilcoxon or Mann-Whitney test: wilcoxont

The wilcoxont command will perform the Wilcoxon or Mann-Whitney test.

• wilcoxont takes two mandatory arguments and two optional arguments:
  • L, a sample (list).
  • M, either another sample or a number (a median).
  • Optionally, f, a function.
  • Optionally, x, a real number.
• wilcoxont(L,M  ⟨ ,f,x⟩) returns the results of the Wilcoxon test.

Examples.

• Input:
  wilcoxont([1,2,3,4,5,7,8,8,12,15,17],[2,6,10,11,13,14,15,18,19,20])
  Output:

  Mann-Whitney 2-sample test, H0 same Median, H1 <>

  ranksum 93.0, shifted ranksum 27.0

  u1=83 ,u2=27, u=min(u1,u2)=27

  Limit value to reject H0 26

  P-value 9055/176358 (0.0513444244094), alpha=0.05 H0 not rejected

  1

• Input:
  wilcoxont([1,3,4,5,7,8,8,12,15,17],[2,6,10,11,13,14,15,18,19,20],0.3)
  Output:

  Mann-Whitney 2-sample test, H0 same Median, H1 <>

  ranksum 81.5, shifted ranksum 26.5

  u1=73.5 ,u2=26.5, u=min(u1,u2)=26.5

  Limit value to reject H0 35

  P-value 316/4199 (0.0752560133365), alpha=0.3 H0 rejected

  0

• Input:
  wilcoxont([1,3,4,5,7,8,8,12,15,17] ,10,‘>‘,0.05)
  Output:

  Wilcoxon 1-sample test, H0 Median=10, H1 M<>10

  Wilcoxon statistic: 18, p-value: 0.375, confidence level: 0.05

  1