A 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 should depend on the final results of the experiment, that is, on empirically found values. This function will be the tool for assessing the discrepancy between the samples.
Statistically significant value. General information
Statistical significance is a quantity whose probability of accidental occurrence is very small. Its more extreme indicators are also insignificant. The difference is called statistically significant if there is data whose probability of occurrence is insignificant if it is argued that these differences do not exist. But this does not mean at all that this difference must necessarily be great and significant.
The 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 false positive. In most cases, the process relies on a p-value ("pi-value"). This is the accumulated probability when observing the level of a statistical criterion. It, in turn, is sampled at the time of adoption of the null hypothesis. The assumption will be rejected if this p-value is less than the level declared by the analyst. The significance of the test quantity directly depends on this indicator: the smaller it is, the, respectively, and there is more reason to reject the hypothesis.
The significance level is usually indicated by the letter b (alpha). Popular indicators among specialists: 0.1%, 1%, 5% and 10%. If, say, it is said that the chances of coincidence are 1 to 1000, then definitely we are talking about the level of 0.1% of the statistical significance of a random variable. Different in value b-levels have their pros and cons. If the indicator is less, it is more likely that the alternative hypothesis is significant. Although there is a risk that the false null assumption will not be rejected. We can conclude that the choice of the optimal b-level depends on the balance of significance-power or, accordingly, on a compromise of the probabilities of false-positive and false-negative decisions. A synonym for "statistical significance" in the domestic literature is the term "reliability".
Hypothesis Definition
In mathematical statistics, this is an assumption that is checked for consistency with empirical data already in stock. In most cases, the hypothesis that there is no correlation between the studied variables or that there are no differences in homogeneity in the studied distributions is taken as the null hypothesis. In standard research, a mathematician tries to refute the null hypothesis, that is, to prove that it is not consistent with experimentally obtained data. Moreover, there should be an alternative assumption, which is accepted instead of zero.
Key definition
The U criterion (Mann-Whitney) in mathematical statistics allows us to evaluate 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 assessing differences in small samples. This simple criterion was proposed by Frank Wilcoxon in 1945. And already in 1947, the method was revised and supplemented by scientists H. B. Mann and D. R. Whitney, whose names it 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 theoretical research results.
Description
The Mann-Whitney test is a relatively simple method without parameters. Its power is significant. It is significantly higher than the power of the Rosenbaum Q-test. The method estimates how small the cross-sectional area between the samples, namely between the ranked series of values of the first and second selection. The smaller the value of the criterion, the greater the likelihood that the discrepancy between the parameter values is significant. To correctly apply the U criterion (Mann-Whitney), do not forget about some restrictions. Each sample must have at least 3 characteristic values. A situation is possible when in one case there are two values, but in the second, then there must be at least five of them. In the studied samples, there should be a minimum number of matching indicators. All numbers should be different in the ideal case.
Using
How to use the Mann-Whitney test? The table compiled by this method contains certain critical values. First you need to create a single series of both matched samples, which is then ranked. That is, the elements are arranged according to the degree of growth of the attribute, and a lower rank is assigned to a lower value. As a result, we obtain 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. Next, a single ranked series of values is divided into two categories. Units, respectively, from the first and second samples. Now the sum of ranks of 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 find out the critical value of this criterion for specific N1 and N2 from the table for the selected level of significance.
The resulting indicator may be less than or equal to the value from the table. In this case, a significant difference in the levels of the trait in the studied samples is noted. If the obtained value is larger than the tabulated one, then the null hypothesis is accepted. When calculating the Mann-Whitney criterion, it should be noted that if the null hypothesis is true, the criterion will have a
mathematical expectation, as well as variance. Note that for sufficiently large volumes of these samples, the method is considered to be practically normally distributed. The significance of differences is higher, the smaller the Mann-Whitney criterion takes.