Mann-Whitney in SPSS is the most widely used non-parametric test, and it is used as an alternative to the t-test. In the Mann-Whitney in SPSS, we do not make assumptions about the parent population as in the t-test. Mann-Whitney in SPSS tests that the two sample populations are equivalent in location. The observations from both the groups are combined together and are ranked. In the case of ties in the Mann-Whitney in SPSS, the average rank is obtained. In the Mann-Whitney in SPSS, one should keep the number of ties relatively small in relation to the total number of observations.
If the populations are identical in location, the ranks should be randomly mixed between the two samples. The test calculates the number of times a score from group 1 precedes a score from group 2, and the number of times a score in group 2 precedes a score from group 1. The value of the Mann-Whitney in SPSS is the one that comes out to be the smaller of these two numbers.
While conducting the Mann-Whitney in SPSS, one needs to perform the following options: go to “Analyze menu” and click on the “non parametric tests” option, select the “Two independent sample tests” option, and select the test type (Mann-Whitney in this case).
The following are the operations which SPSS does while calculating the Mann-Whitney in SPSS:
· We rank the cases in order of increasing size, and the test statistic U, which indicates the number of times that a score from group 1 preceded a score from group 2.
· We can compute an exact level of significance if there are fewer cases. When there are more than just a few cases, we transform U into Z statistic, and a normal approximation p value is computed.
· A test statistic is then calculated for each variable.
In order to compute the Mann-Whitney in SPSS, the following actions need to be performed:
Let xi (i=1…n1) and yj (j=1…n2) be an independent sample of n1 and n2 from the population probability density f ( ) and f2 ( ) respectively. If we want to test the null hypothesis H1 : f1 ( ) = f2 ( ), let T be the sum of ranks of the y’s in the combined ordered sample. The test statistic U is defined in terms of T as follows:
U= n1 n2+ n2 (n2 + 1)/2 – T
If T is significantly larger or smaller, then the null hypothesis is rejected. The problem is finding the distribution of T under null hypothesis. Unfortunately, it is very troublesome to obtain the distribution of T under null hypothesis. However, Mann-Whitney in SPSS has obtained the distribution of T for small n1 and n2 and has shown that T is asymptotically normal. It has been established that under null hypothesis, U is asymptotically normally distributed as N (µ, σ2), where
µ=E (U) = n1 n2/2 and σ2= V (U) = n1 n2(n1 + n2+1)/12.
Asymptotic normal means that the true parameter approaches the normal distribution as the size of the sample increases.
Here, Z= U-µ/ σ which is asymptotically normal with mean 0 and variance 1.The approximation of Mann-Whitney in SPSS is fairly good if both n1 and n2 are greater than 8. This means that the size of the two independent samples should be greater than 8, and then only the approximation given by Mann-Whitney in SPSS is true.
The Asymptotic Relative Efficiency (ARE) of the Mann-Whitney in SPSS is relative to the two sample t-tests, which is greater than or equal to 0.864. For a normal population, the ARE is 0.955. Accordingly, the Mann-Whitney in SPSS is regarded as the best non parametric test for location. Asymptotic Relative Efficiency (ARE) means that it is the limit of the relative efficiency, as the size of the sample increases.
