Calculate the empirical value of the correlation coefficient by the formula:
We determine the critical values for the obtained correlation coefficient according to the table. 19 appendix 6.
Note that in table. 19 Annex 6 the magnitudes of the critical values of the linear Pearson correlation coefficients are given in absolute value. Consequently, when both positive and negative correlation coefficients are obtained by the formula, the significance level of this coefficient is estimated using the same application table without taking the sign into account, and the sign is added to further interpret the nature of the relationship between X and Y variables.
When finding the critical values for the calculated Pearson correlation coefficient, the number of degrees of freedom is calculated as.
In our case, k = 20, so n – 2 = 20 – 2 = 18. In the first column of the table. 19 of annex 6 in the line indicated by the number 18, we find:
Build the appropriate “ significance axis ”:
Due to the fact that the value of the calculated correlation coefficient fell into the zone of significance, the hypothesis is rejected and accepted. In other words, the relationship between the time for solving visual-figurative and verbal tasks is statistically significant at the 1% level and is positive. The obtained directly proportional dependence suggests that the higher the average time for solving visual-figurative tasks, the higher the average time for solving verbal and vice versa.
To apply the Pearson correlation coefficient, the following conditions must be met:
The variables to be compared must be obtained on an interval scale or a relationship scale.
The distributions of the variables X and Y should be close to normal.
The number of varying signs in the compared variables X and Y should be the same.
The tables of significance levels for the Pearson correlation coefficient (table 19 of annex 6) are calculated from n = 5 to n = 1000. The significance level is estimated from the tables with the number of degrees of freedom k = n – 2.
The correlation matrix is a symmetric square matrix of size M * M, where M is the number of factors studied, the main diagonal of which is filled with ones (or zeros for convenience of further analysis), and non-diagonal elements are a measure of the closeness of the relationship between a pair of factors (correlation coefficient, correlation ratio modified Fechner index and
In practice, there are often cases of gross misses of paired samples, which are very difficult to identify, as well as noticeable deviations of factors from the normal distribution law. The use of a classical correlation analysis with a measure of closeness in the form of a correlation coefficient in these conditions requires some caution, since it is not easy to decide whether a particular pair of numbers of the studied two-dimensional population belongs to the background or is a gross miss. In doubtful cases (both random variables are not distributed according to the normal distribution law; there is a suspicion that the pair sampling may contain gross blunders) it is recommended to use the modified Fechner index as a measure of link closeness. The modified Fechner index is, of course, less accurate than the correlation coefficient, but only in the absence of gross blunders and distortion of the distribution law. Even with one blunder, the correlation coefficient changes significantly, giving an incorrect result, while the modified Fechner index based on one of the most robust (stable to baseline conditions) estimates of mathematical statistics — on arithmetic average — yields a result much closer to the true . Of course, if one or both random values are discrete or vary over a large number of levels, then as a measure of the tightness of the correlation link, choose one that is suitable from the rich arsenal of measures partially described in section
Direct analysis of the correlation matrix presents a significant difficulty, since the correlations between the factors form trees, chains, cycles, and other graph figures. To isolate the main dependencies, he will resort to resorting to one of the methods for analyzing such matrices, the simplest of which is the correlation pleiades method.
The method consists in the fact that in the correlation matrix there is a non-diagonal element with the maximum modulo value | rij | = max. Columns with numbers i and j are deleted from the matrix, and the next maximum modulo element is selected from rows with numbers i and j, for example, | ril |. The column with the number l is deleted, and the next maximum modulo element is selected from the lines with the numbers i, j and l, and so on until the data is exhausted.
The result of this work is conveniently represented in the figure as a graph, the vertices of which are factors, the edges are the maximum connections, and the length of the edges is inversely proportional to the magnitude of the corresponding correlation coefficients. Selecting a certain threshold value of the correlation coefficient, for example, | rthr | =
Within each pleiad, the connection between the factors is recognized as close, and between the constellations it is recognized as weak. This means that if we select one representative from each galaxy, then the new total number of factors, reduced to the number of galaxies, will carry practically the same information about the object under study as before. At the same time, the factors of the new data table will be weakly correlated with each other, which is one of the main conditions for the transition to mathematical modeling.
Correlation matrix – a table in which the correlation coefficients between all the studied variables are presented.
2. Structuralogram – a form of graphical representation of the results of the correlation analysis of the relationship of variables that are part of the psychological structure or system (components of the psychological structure of the intellect, components of the psychological system of activity, components of the structure of professionally important qualities and
3. A correlogram (correlation pleiad) is a form of graphical representation of the results of correlation analysis of the relationship between variables that are not part of the psychological structure or system at the level of qualitative analysis or theoretical ideas of a scientist.
1. File SPSS 42 DZ.
2. File SPSS 43 ДЗ.
3. File SPSS 44 DZ.
4. SPSS 45 remote sensing file.
The result of the assignments: the ability to interpret and present the results of the correlation analysis in the form of a correlation pleiad.
The factors are interpreted according to the table of factor loads after rotation (table of matrix of rotated components) in the following order:
1. For each variable (line), the largest in absolute value is allocated – as the dominant one. If the second largest load in the line differs from the one already allocated by less than