MANN'S U TEST - DUDAS

The Mann - Whitney U test is applied for the comparison of two independent samples when they have little data or do not follow a normal distribution. In this way, it is considered to be a non-parametric test, unlike its homologous Student's t test, which is used when the sample is large enough and follows the normal distribution.

Frank Wilcoxon proposed it for the first time in 1945, for samples of identical sizes, but two years later it was extended for the case of samples of different sizes by Henry Mann and DR Whitney.

Figure 1. The Mann-Whitney U test is applied for the comparison of independent samples. Source: Pixabay.

The test is often applied to check whether there is a relationship between a qualitative and a quantitative variable.

An illustrative example is to take a set of hypertensive people and extract two groups, from whom daily blood pressure data is recorded for one month.

Treatment A is applied to one group and treatment B to another. Here blood pressure is the quantitative variable and the type of treatment is the qualitative one.

We want to know if the median, and not the mean, of the measured values is statistically the same or different, to establish if there is a difference between both treatments. To obtain the answer, the Wilcoxon statistic or Mann - Whitney U test is applied.

Statement of the problem in the Mann-Whitney U test

Another example in which the test can be applied is the following:

Suppose you want to know whether the consumption of soft drinks differs significantly in two regions of the country.

One of them is called region A and the other region B. A record is kept of the liters consumed weekly in two samples: one of 10 people for region A and another of 5 people for region B.

The data is as follows:

-Region A: 16, 11, 14, 21, 18, 34, 22, 7, 12, 12

-Region B: 12,14, 11, 30, 10

The following question arises:

Qualitative variables versus quantitative variables

-Qualitative variable X: Region

-Quantitative variable Y: consumption of soft drinks

If the amount of liters consumed is the same in both regions, the conclusion will be that there is no dependence between the two variables. The way to find out is to compare the mean or median trend for the two regions.

Normal case

If the data follow a normal distribution, two hypotheses are proposed: the null H0 and the alternative H1 through the comparison between the means:

- H0: there is no difference between the mean of the two regions.

- H1: the means of both regions are different.

Case with non-normal trend

On the contrary, if the data do not follow a normal distribution or the sample is simply too small to know it, instead of comparing the mean, the median of the two regions would be compared.

- H0: there is no difference between the median of the two regions.

- H1: the medians of both regions are different.

If the medians coincide, then the null hypothesis is fulfilled: there is no relationship between consumption of soft drinks and the region.

And if the opposite happens, the alternative hypothesis is true: there is a relationship between consumption and region.

It is for these cases where the Mann - Whitney U test is indicated.

Paired or unpaired samples

The next important question in deciding whether to apply the Mann Whitney U test is whether the number of data in both samples is identical, which is to say that they are on par.

If the two samples are paired, the original Wilcoxon version would apply. But if not, as is the case in the example, then the modified Wilcoxon test is applied, which is precisely the Mann Whitney U test.

Characteristics of the Mann Whitney U test

The Mann - Whitney U test is a non-parametric test, applicable to samples that do not follow the normal distribution or with little data. It has the following characteristics:

1.- Compare the medians

2.- It works on ordered ranges

3.- It is less powerful, meaning power is the probability of rejecting the null hypothesis when it is actually false.

Taking these characteristics into account, the Mann - Whitney U test is applied when:

-The data are independent

-They do not follow the normal distribution

-The null hypothesis H0 is accepted if the medians of the two samples coincide: Ma = Mb

-The alternative hypothesis H1 is accepted if the medians of the two samples differ: Ma ≠ Mb

Mann - Whitney formula

The variable U is the contrast statistic used in the Mann - Whitney test and is defined as follows:

This means that U is the smallest of the values between Ua and Ub, applied to each group. In our example it would be to each region: A or B.

The variables Ua and Ub are defined and calculated according to the following formula:

Ua = Na Nb + Na (Na +1) / 2 - Ra

Ub = Na Nb + Nb (Nb +1) / 2 - Rb

Here the Na and Nb values are the sizes of the samples corresponding to the regions A and B respectively, and for their part, Ra and Rb are the rank sums that we will define below.

Steps to apply the test

1.- Order the values of the two samples.

2.- Assign an order rank to each value.

3.- Correct the existing ties in the data (repeated values).

4.- Calculate Ra = Sum of the ranks of sample A.

5.- Find Rb = Sum of the ranks of sample B.

6.- Determine the value Ua and Ub, according to the formulas given in the previous section.

7.- Compare Ua and Ub, and the smaller of the two is assigned to the experimental U statistic (that is, of the data) that is compared with the theoretical or normal U statistic.

Practical application example

Now we apply the aforementioned to the problem of soft drinks raised previously:

Region A: 16, 11, 14, 21, 18, 34, 22, 7, 12, 12

Region B: 12,14, 11, 30, 10

Depending on whether the means of both samples are statistically the same or different, the null hypothesis is accepted or rejected: there is no relationship between the variables Y and X, that is, the consumption of soft drinks does not depend on the region:

H0: Ma = Mb

H1: Ma ≠ Mb

Figure 2. Soft drink consumption data in regions A and B. Source: F. Zapata.

- Step 1

We proceed to order the data jointly for the two samples, ordering the values from lowest to highest:

Notice that the value 11 appears 2 times (once in each sample). Originally it has positions or ranges 3 and 4, but in order not to overestimate or underestimate one or the other, the average value is chosen as the range, that is, 3.5.

In a similar way, we proceed with the value 12, which is repeated three times with ranges 5, 6 and 7.

Well, the value 12 is assigned the average range of 6 = (5 + 6 + 7) / 3. And the same for the value 14, which has ligature (appears in both samples) in positions 8 and 9, it is assigned the average range 8.5 = (8 + 9) / 2.

- Step 2

Next, the data for Region A and B are separated again, but now their corresponding ranges are assigned in another row:

Region A

Region B

The ranges Ra and Rb are obtained from the sums of the elements of the second row for each case or region.

Step 3

The respective Ua and Ub values are calculated:

Ua = 10 × 5 + 10 (10 + 1) / 2 - 86 = 19

Ub = 10 × 5 + 5 (5 + 1) / 2 -34 = 31

Experimental value U = min (19, 31) = 19

Step 4

It is assumed that the theoretical U follows a normal distribution N with parameters given exclusively by the size of the samples:

N ((na⋅nb) / 2, √)

In order to compare the variable U obtained experimentally, with the theoretical U it is necessary to make a change of variable. We move from the experimental variable U to its standardized value, which will be called Z, in order to be able to make the comparison with that of a standardized normal distribution.

The change of variable is as follows:

Z = (U - na.nb / 2) / √

It should be noted that for the change of variable the parameters of the theoretical distribution for U were used. Then the new variable Z, which is a hybrid between the theoretical U and the experimental U, is contrasted with a standardized normal distribution N (0,1).

Comparison criteria

If Z ≤ Zα ⇒ the null hypothesis H0 is accepted

If Z> Zα ⇒ reject the null hypothesis H0

The standardized Zα critical values depend on the required level of confidence, for example, for a confidence level α = 0.95 = 95%, which is the most usual, the critical value Zα = 1.96 is obtained.

For the data shown here:

Z = (U - na nb / 2) / √ = -0.73

Which is below the critical value 1.96.

So the final conclusion is that the null hypothesis H0 is accepted:

Online calculators for the Mann - Whitney U test

There are specific programs for statistical calculations, including SPSS and MINITAB, but these programs are paid and their use is not always easy. This is due to the fact that they provide so many options that their use is practically reserved for experts in Statistics.

Fortunately, there are a number of very accurate, free, and easy-to-use online programs that allow you to run the Mann-Whitney U test, among others.

These programs are:

-Social Science Statistics (socscistatistics.com), which has both the Mann-Whitney U test and the Wilcoxon test in the case of balanced or paired samples.

-AI Therapy Statistics (ai-therapy.com), which has several of the usual tests of descriptive statistics.

-Statistic to Use (physics.csbsju.edu/stats), one of the oldest, so its interface may look dated, although it is nevertheless a very efficient free program.

References

Dietrichson. Quantitative methods: rank test. Recovered from: bookdown.org
Marín J P. SPSS Guide: Analysis and procedures in non-parametric tests. Recovered from: halweb.uc3m.es
USAL MOOC. Nonparametric tests: Mann-Whitney U. Recovered from: youtube.com
Wikipedia. Mann-Whitney U test. Recovered from: es.wikipedia.com
XLSTAT. Help Center. Mann - Whitney test tutorial in Excel. Recovered from: help.xlsat.com

MANN'S U TEST - DUDAS - 2024