FACTOR ANALYSIS — the section of mathematical statistics combining the methods and models providing a possibility of the compact (compressed) submission of empirical information on various phenomena and events. Model F. and. are based on idea that observed parameters are indirect characteristics of bodies of interest, and in practice there is rather small number (t) of the internal (hidden) factors defining values of observed parameters (their number — p). At the same time the number of t are much less than number of the item. Task F. and. expression of observed parameters in the form of linear combinations of factors, and also (if necessary) in the form of nek-ry additional sizes is. The defined factors usually well are interpreted as new essential characteristics of bodies of interest.
In most F. and. it is possible to allocate a method main a component (the component analysis) and methods actually F. and.
The task of the component analysis consists in finding of a small number of factors (component), linear combinations to-rykh well describe the observed (measured) parameters, and also in determination of coefficients of the corresponding linear combinations (so-called factor loadings). The model of the component analysis registers in a look:
Xj — cijiFi - j-• • • N - yajnfni
where Xj — j-y the parameter, aJl5 a, 2... ajn — coefficients, Fx, F2... Fn — - I a component.
Coefficients of aji it is possible to present matrixes of factor loadings in the form:
And = | | aJt | |.
All Fx components... F „— are assumed by not interconnected. Everyone next a component makes the maximum contribution to total dispersion of the parameters which remained after the account пре^ dydushchy a component (see Probabilities the theory). Usually (t «n) the first the component is the share of a small number rather big percent of total dispersion, i.e. these components are sufficient for the adequate description of bodies of interest, and the size of Hts = = anFt 4-• • • ajmFm can serve as good model of parameter X }.
For definition of matrixes of factor loadings and factor «scales» (values of factors on bodies of interest) the corresponding receptions of the component analysis are used.
Problem of methods actually F. and. consists in definition of small set of the factors explaining the correlations existing between parameters. Model F. and. registers in a look:
X/— djlFi +... + OimFm ~ \-d jU j, where all characteristic factors of U (;' = 1,2... o), connected with specifics of parameters, shall be uncorrelated among themselves, with the general factors (Ft... Fm). Search of a matrix of factor loadings is carried out by an ekstremization of the nek-ry criterion which is completely determined by this matrix and differing in various methods F. and. All criteria are under construction on comparison of matrixes selective and calculated on model F. and. correlation coefficients of parameters, but if in one methods (e.g., a method of the minimum remains) it is required to estimate number of the general factors in advance, then in others (e.g., a method of the main factors) estimates of communities shall be received in advance, and the number of the general factors is defined in the course of the analysis.
Initially methods F. and. were used in psychology where with their help the hypothesis that all range of mental abilities of the person is defined by one or several factors was checked. These methods were used also for designing of small set of the tests describing intellectual activity. Then methods F. and. found application in the field of clinical medicine, especially in cardiology. The method main a component is widely used in social and economic researches.
See also Mathematical methods.
Bibliography: Zhukovsky V. M. and
Muchnik I. B. Factor analysis in social and economic researches, M., 1976; And e r l and To. Factor analysis, the lane with it., M., 1980; Perch I. Factor analysis, the lane with polsk., M., 1974; Harman G. Modern factor analysis, the lane with English, M., 1972.
P. O. Aven.