# Correspondence of the structural model: estimation of homogeneity...

## Correspondence of the structural model: estimation of variance homogeneity in several samples

Assessing the consistency of the data obtained with the structural assumptions of a single-factor variance analysis consists in testing two hypotheses.

The first hypothesis is that all empirical results are taken from a population of data distributed according to a normal law. This hypothesis can be verified either with the help of the eye-normal method of "normal paper", either as a result of estimating the asymmetry and the kurtosis of the distribution of the values ​​obtained, or by some other method. It should be remembered, however, that if the number of measurements is not large enough, this test may not be accurate enough. It is worth noting, however, that, as a rule, such preliminary tests are not required.

The second hypothesis is the derivative of the first and consists in the fact that the intra-group variance of the empirical data does not depend on the experimental impact, or, what is the same, the experimental error is not related to the effects of the independent variable. This hypothesis can be verified in the case of a two-level plan using the variance homogeneity test described in paragraph 3.2. The application of this test at two levels of the independent variable is similar to what was described for the Student test, which is usually used when comparing the averages in two samples. For a number of independent variable levels, several variance homogeneity tests have been developed, which are also recommended when comparing two samples instead of a somewhat outdated F-test.

The most common test evaluating the dispersion homogeneity is the Livin test. The result of this test is the statistics F. Using the Livin test is preferable to a similar but somewhat outdated Bartlett test, calculating the statistics of χ2, since the Bartlett test can be too sensitive to the normal distribution of the experimental data . In addition, it should be borne in mind that the use of the test

Bartlett suggests that the number of groups can not be less than three people, while most groups should include more than five subjects.

Also, the estimate of the homogeneity of the variance of several samples can be performed using the Cochran test , which gives the C statistics:

where - the maximum value of the intra-group variance in all groups; - the sum of variances for all groups.

The statistical reliability of the result can be determined using special tables. In their absence, it can be calculated on the basis of the values ​​of the F-distribution:

where - the critical value of the test for a given level of significance a.

In addition, to evaluate the homogeneity of the variance, the Gartley test calculating the F -max statistics based on comparison of the maximum and minimum variances in the available samples:

Despite the fact that the Gartley test operates with less information than the Cochran test, the results of using these tests are almost identical. Therefore, the application of the Gartley test is most convenient in the "manual" calculations. The statistical reliability of the result obtained with this test is also assessed using special tables.

Note that not all modern statistical packages allow the use of all these tests. So, the well-known package of the statistical package IBM SPSS Statistics is limited only to Livin's test. As a rule, this is quite enough.

It should also be emphasized that the probability of obtaining a negative result when carrying out this kind of evaluation can be quite high, especially in a situation where a large enough number of groups, say, eight or ten, is compared. However, this does not mean that the method of variance analysis can be used only in those relatively rare cases when the hypotheses of the normality of the distribution and the homogeneity of the dispersion are highly probable. Theoretical calculations show that even in cases when the asymmetry and kurtosis of the distribution significantly exceed the boundary values ​​and the maximum intra-group dispersion exceeds several times the minimum, the quantiles of the distribution of only a fraction of a percent differ from the quantiles of the distribution describing the statistics constructed on the basis of the experimental data. Thus, theoretical studies have shown that even with a very small size of the groups of subjects who participated in the experiment and a threefold difference in variance in different populations, deviations from the 5% quantile for the statistics F from the theoretically assumed structural model for the three-level plan is less than 1% (Table 3.5, n = 5 and k = 3).

Table 35

The probability corresponding to a 5% quantile of the structural model of a single-factor variance analysis, with a different dispersion relation (G Box [17, p. 299])

 The collection Probability corresponding to F095 1 2 3 1 1 1 0.050 1 2 3 0.058 1 1 3 0.059

Thus, dispersion analysis for disjoint samples is usually referred to methods that are extremely resistant to underlying assumptions. That is why the researcher, as a rule, does not have a serious need to refrain from using this method in favor of other methods existing in mathematical statistics.

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