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

## 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.

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