Mann-Whitney Criterion: example, table

The criterion in mathematical statistics is a strict rule, according to which a hypothesis with a certain level of significance is accepted or rejected. To build it, you need to find a specific function. It must depend on the final results of the experiment, that is, on empirically found values. It is this function that will be a tool for estimating the discrepancy between the samples.

Statistically significant value. General information

Statistical significance is a quantity whose probability of accidental occurrence is very small. Insignificant also more extreme its indicators. The difference is called statistically significant if there are data, the probability of occurrence of which is insignificant, if we assert that these discrepancies do not exist. But this does not mean at all that this difference must necessarily be great and meaningful.

Level of statistical reliability of the test

This term should be understood as the probability of rejection of the null hypothesis in the case of its truth. This is also called a first-kind error or a false positive solution. In most cases, the process relies on the p-value ("pi-value"). This is the accumulated probability when observing the level of the statistical criterion. It, in turn, is calculated from the sample during the adoption of the null hypothesis. The assumption will be rejected if this p-value is less than the level declared by the analyst. From this indicator directly depends the significance of the test value: the smaller it is, so, accordingly, and more grounds to reject the hypothesis. The level of significance, as a rule, is denoted by the letter b (alpha). Popular figures among specialists: 0.1%, 1%, 5% and 10%. If, say, it is said that the odds of coincidence are 1 to 1000, then it is definitely about the level of 0.1% of the statistical significance of the random variable. Different in value b-levels have their pros and cons. If the indicator is less, then the probability is greater that the alternative hypothesis is significant. Although there is a risk that a false zero assumption will not be rejected. It can be concluded that the choice of the optimal b-level depends on the "significance-power" balance or, correspondingly, on the compromise of the probabilities of false-positive and false-negative decisions. The term "reliability" is a synonym for "statistical significance" in Russian literature.

Definition of the null hypothesis

In mathematical statistics, this assumption, tested for consistency with existing empirical data in the stock. In most cases, as a null hypothesis, a hypothesis is made that there is no correlation between the variables being investigated or that there are no homogeneity differences in the studied distributions. In standard studies, the mathematician tries to disprove the null hypothesis, that is, to prove that it is not consistent with the experimentally obtained data. And there must be an alternative assumption, which is taken instead of zero.

Key Definition

The criterion U (Mann-Whitney) in mathematical statistics makes it possible to estimate the differences between the two samples. They can be given by the level of a certain attribute, which is measured quantitatively. This method is ideal for estimating the differences between small samples. This simple criterion was proposed by Frank Wilcoxon in 1945. And already in 1947 the method was revised and supplemented by the scientists H. Mann and DR Whitney, whose names he is called to this day. The Mann-Whitney criterion in psychology, mathematics, statistics and in many other sciences is one of the fundamental elements of the mathematical substantiation of the results of theoretical studies.

Description

The Mann-Whitney criterion is a relatively simple method without parameters. Its power is considerable. It is substantially higher than the power of the Rosenbaum Q-test. The method estimates how small the cross-valued area between samples, namely between the ranked series of values of the first and second collections. The smaller the value of the criterion, the greater the probability that the discrepancy between the values of the parameter is reliable. To correctly apply the U criterion (Manna-Whitney), do not forget about some limitations. Each sample must have at least 3 characteristic values. A situation is possible when there are two values in one case, but in the second case there must necessarily be at least five. The sampled samples should have a minimum number of coincident indicators. All numbers must be different in the ideal case.

Using

How to properly use the Mann-Whitney test? The table, which is compiled according to this method, contains certain critical values. First, you need to create a single row from both of the mapped samples, which is then ranked. That is, the elements are aligned according to the degree of increase in the sign, and the lower rank is assigned to a smaller value. As a result, we get the following total number of ranks:

N = N1 + N2,

Where N1 and N2 are the number of units contained in the first and second samples, respectively. Then a single ranked series of values is divided into two categories. Units, respectively, from the first and second samples. Now the sum of the ranks of the values in the first and second rows is considered in turn. The largest of them (Tx) is determined, which corresponds to a sample with nx units. To use the Wilcoxon method further, its value is calculated using the following procedure. It is necessary to determine the critical value of this criterion for specifically taken N1 and N2 according to the table for the selected significance level. The resulting indicator may be less than or equal to the value in the table. In this case, there is a significant difference in the levels of the trait in the samples studied. If the value obtained is larger than the table value, then the null hypothesis is accepted. When the Mann-Whitney criterion is calculated, it should be noted that if the null hypothesis is valid, the criterion will have a mathematical expectation, as well as a variance. Note that for sufficiently large volumes of sample data, the method is considered to be almost normally distributed. The reliability of the differences is higher the smaller the value of the Mann-Whitney criterion.