Mann-Whitney U test

U-test Mann-Whitney

Comparison of central tendencies of two independent samples.

Requirements:

Dependent variable must be ordinal scaled (rank order scaled). Distribution free.


Idea:

The values of both samples are brought in one rank order. For H0 it is expected that 'x<y' occurrs as frequently as 'y<x'.

Measure:

U measures how often a y-value is smaller than an x-value. Definition:

if y<x, z=1,

if x<y, z=0.

U=Σ z.

Calculation of U:

formula Mann Whitney U Test

T1 = sum of ranks for sample 1.

U = n1 * n2 - U = Σ(x < y)


For large sample sizes (n1 or n2 > 10) U is approximately normal distributed with:

Expected value of U

and

Expected variance of U


Test:

test of significance


If there are tied ranks, the standard deviation of U is corrected as follows:


formula for correction when tied ranks are present

Whereas

n = n1 + n2

ti = Number of subjects sharing rank i

k = Number of tied ranks



Example of a Mann-Whitney U-test

A physician is interested in the effect of an anaesthetic on reaction times. Two groups are compared, one with (A) and one without (B) taking the anaesthetic. Subjects had to react on a simple visual stimulus. Reaction times are not normally distributed in this experiment, so data is analysed with the Mann-Whitney U-Test for ordinal scaled measurements. The table below shows the rank-ordered data:


Mean RT  Rank
Group
131
1
B
135
2
A
138
3.5
B
138
3.5
B
139
5
A
141
6
B
142
8
B
142
8
A
142
8
B
143
10
B
144
11
A
145
12
B
156
13
B
158
14
A
165
15
A
 167 16
B
171
17
A
178
18
A
191
19
B
230
20
B
244
21
A
245
22
A
256
23
A
267
24
A
268
25
A
289
26
A


Table showing Ranked Measures for each Group separately:


  Group A
Group B
  2
1
  5
3.5
  8
3.5
  11
6
  14
8
  15
8
  17
10
  18
12
  21
13
  22
16
  23
19
  24
20
  25
 
  26
 
     
Sum of Ranks
231
120
Average Ranks
16.5
10


Mann-Wihtney U



Mann-Wihtney U



Mann-Wihtney expected U



Mann-Wihtney expected variance of U



Mann-Wihtney corrected U
Mann-Wihtney corrected U


Mann-Wihtney significance test


The observed Z value is greater than the Z-value (5%, two-tailed). The anaesthetic group shows significantly slower reaction times than the non-anaesthetic group.

 

BrightStat output of Mann-Whitney U test example

 

This is a fictitious example

How to do this example on BrightStat webapp


Wiki link Mann-Whitney U test


References

Mann, H.B. & Whitney, D.R. (1947). On a Test of Whether one of Two Random Variables is Stochastically Larger than the Other.Annals of Mathematical Statistics 18 (1), 50 – 60. doi:10.1214/aoms/1177730491.

Bortz, J. (2005). Statistik für Human- und Sozialwissenschaftler (6th Edition). Heidelberg: Springer Medizin Verlag.

Conover, W.J. (1999). Practical nonparametric Statistics.(3rd edition). Wiley.




 

Gallery

 
 
map kinase