Mann-Whitney

The non-parametric equivalent to the independent samples t-test is the Mann-Whitney test. The null hypothesis for the test is H0: The population medians are equal. The non-directional alternative hypothesis is H1: The population medians are not equal.

 

Worked Example

A study of the effect of caffeine on muscle metabolism used eighteen male volunteers who each underwent arm exercise tests. Nine of the men were randomly selected to take a capsule containing pure caffeine one hour before the test. The other men received a placebo capsule. During each exercise the subject's respiratory exchange ratio (RER) was measured. (RER is the ratio of CO2 produced to O2 consumed and is an indicator of whether energy is being obtained from carbohydrates or fats).

The question of interest to the experimenter was whether, on average, caffeine affects RER.

The two populations being compared are “men who have not taken caffeine” and “men who have taken caffeine”. If caffeine has no effect on RER the two sets of data can be regarded as having come from the same population.

The results were as follows:

RER (%)
 
Placebo
Caffeine
 
105
96
 
119
99
 
100
94
 
97
89
 
96
96
 
101
93
 
94
88
 
95
105
 
98
88
 
 
 
Median
98
94
SD
7.70
5.61

The medians show that, on average, caffeine appears to have reduced RER from about 98% to 94%, a reduction of 6%. However, there is a great deal of variation between the data values in both samples and considerable overlap between them. So is the difference between the two medians simply due to sampling variation or does the data provide evidence that caffeine does, on average, reduce RER?

The null hypothesis is that caffeine intake by men does not affect their median RER. The alternative hypothesis is that caffeine intake by men does affect their median RER. In Minitab the estimated medians for the two samples are calculated, together with an approximate 95% confidence interval for the difference between the medians.

The conclusion that we "cannot reject at alpha = 0.05" in Minitab means we "cannot reject the null hypothesis at the 5% level of significance", although in this case with a p-value of 0.0521, there is some evidence of a difference between the medians. Indeed, the SPSS output from a Mann-Whitney test gives p=0.046, which would give evidence to reject the null hypothesis!

 

The Mann-Whitney Test using Minitab

To perform this test use Stat > Nonparametrics > Mann-Whitney... and the following output is obtained:

Mann-Whitney Confidence Interval and Test

Caffeine N = 9 Median = 94.00
Placebo N = 9 Median = 98.00
Point estimate for ETA1-ETA2 is -6.00
95.8 Percent CI for ETA1-ETA2 is (-12.00,-0.00)
W = 63.0
Test of ETA1 = ETA2 vs ETA1 not = ETA2 is significant at 0.0521
The test is significant at 0.0512 (adjusted for ties)
Cannot reject at alpha = 0.05

N.B. ETA1 represents Median of group 1'!

The Mann-Whitney Test using SPSS

Use Analyze > Nonparametric Tests > 2 Independent Samples... to obtain the following output:

Mann-Whitney Test

table showing ranks

table showing test stats

NB: Data needs to be entered in 2 columns in Minitab or SPSS; one column representing the group (eg placebo=0 and caffeine=1), the other contains the RER percentage values.