Application of factor analysis in psychodiagnostics. Factor analysis for psychologists - Mitina

“Multivariate experimenters” emphasize the importance of mass surveys and a variety of experimental samples, followed by in-depth mathematical processing of the data obtained, calculation of correlation coefficients and application factor analysis. Actually, factor analytical research begins with the systematic selection of those samples that are usually used to measure certain qualities, for example, intellectual abilities or personality traits. This set of tests is applied to all subjects in the recruited group, obtaining a score for each characteristic for each person. Next, the relationship between each pair of features is determined. If people who score high on one attribute also score high on another, the correlation coefficient will show that they are closely related and will be close to +1.00 (which means a perfect match). Knowing a person's score on one of these characteristics, one can effectively predict his score on the other. A high negative relationship (approaching -1.00) means an inverse relationship between attributes (a high score on one predicts a low score on the other); correlation coefficients close to 0.00 indicate the absence of a relationship. All correlation coefficients are presented in the form of a correlation matrix, which is then subjected to special procedures of factor analysis in order to identify certain common factors that combine characteristics that have the closest connections within one factor, while different factors must be completely or relatively independent of each other (orthogonal ones are distinguished accordingly). and environmental factors).

After this, samples and tests are selected that “maximally load” each identified factor, or special additional studies are carried out in order to improve or create samples that make it possible to create the “best battery” for measuring each factor, considered as a kind of functional unity, in in this case- a certain psychological essence or property that can be interpreted hypothetically, based on the content and nature of the tests that “load” one or another factor, as well as from their relationship with other external data. The principles of factor analysis in psychology in relation to the study of the structure of intelligence were first developed by Professor of the University of London Charles Spearman.

Spearman came to the conclusion that intelligence is a certain general ability, which is primarily the ability to imagine relationships between phenomena. It is reflected by the so-called “general intelligence factor” (g). In addition to the g-factor, the existence of specific factors (s-factors characteristic of different types intellectual activity). The main measure of intelligence according to Spearman is the g-factor and those tests where it is expressed to the greatest extent.

It is curious that the American psychologist L. Thurstone, also based on factor analysis, made a slightly different conclusion. Having begun the study of mental abilities with more specific methods, he then emphasized the importance in measuring intelligence of specific “primary” abilities, such as numerical, spatial, verbal, etc., which were presented to him as “branches” of the secondary general factor of intelligence underlying them.

Factor analysis in psychology was first used to clarify the structure of human abilities, mainly mental or intellectual, and only then it began to be applied to the analysis of personality structure, and this was done most consistently by Cattell and his colleagues. To obtain initial data, Cattell turned to traditional methods of observation in psychology, questionnaires, and various types laboratory experiment, which were specially prepared and standardized for this study, since quantification and the reliability of the data obtained served a necessary condition their use in further mathematical processing.

Numerous studies and the results of their complex mathematical processing allowed Cattell and his collaborators to identify about 30 factors, of which 18 were later used most often, since they were included (in slightly different combinations) in various shapes Cattell's personality questionnaire, intended for subjects of different ages. Each personality factor is considered as a continuum of a certain quality or “primary trait” (in questionnaires it is measured in walls - scale units with a minimum value of 0 points, a maximum of 10 and an average of 5.5 points) and is characterized bipolarly at the extreme values ​​of this continuum. Accordingly, these bipolar contents are indicated by the + or - sign, standing nearby with letters of the alphabet representing factors. In addition to letter designations, factors also have “special” (or “technical”) and “popular” names. Cattell great importance gives the characterization of factors as special, obtained independently of voluntary operations of the mind, but objectively established categories, representing “natural unitary structures of personality” or a set of certain psychological qualities, each of which is considered as a “primary” personality trait, therefore the “popular” names of ordinary language are only approximately convey their essence. For "special" titles, Cattell sometimes makes up words or borrows obscure terms from Latin or Greek.

So, for example, the first and one of the most important factors- Factor A, which is sometimes not entirely correctly called “schizothymia-cyclothymia,” Cattell, pointing out its connection with these terms of Kretschmer, as well as referring to the corresponding concepts in Bleuler and Kraepelin, at the same time, in order to dissociate himself from the well-known psychopathological meaning of these concepts (although Kretschmer emphasizes their applicability to the norm), will give a “special” name - “sisothymia - affectothymia”. He writes: “Since the public associates schizothymic with abnormality, it is better to use the term sizothymia (Sise - means flatness, lethargy, dullness, monotony in painting, the same in relation to feelings) - the absence of living vibrating emotions. This coldness and aloofness characterizes the normal A-factor of a sizothymic individual. Normally, cyclothymia is affectothymia, since the initial characteristic is affect, emotion, and not fluctuations, cyclothymic changes, which are characteristic of abnormal ones.”

Factor analysis is a statistical method that is used when processing large amounts of experimental data. The objectives of factor analysis are: reducing the number of variables (data reduction) and determining the structure of relationships between variables, i.e.

Classification of variables, so factor analysis is used as a data reduction method or as a structural classification method.

An important difference between factor analysis and all the methods described above is that it cannot be used to process primary, or, as they say, “raw” experimental data, i.e. obtained directly from the examination of subjects. The material for factor analysis is correlations, or more precisely, Pearson correlation coefficients, which are calculated between the variables (i.e. psychological characteristics) included in the survey.

Factor analysis has three main applications in psychology. First, it can be used to construct tests. For example, you could write 50 items to measure some ability, personality trait, or attitude (such as conservatism). The items will then be presented to a representative sample of several hundred individuals and manipulated (in the case of aptitude tests) so that a correct response is coded "1" and an incorrect response coded "O". The responses obtained when using rating scales (as in most personality and attitude questionnaires) are simply entered in their raw form: one point if answer option (a) is selected, two points if answer option (b) is selected, etc. d. Responses to these 50 items are then correlated and factor analyzed. Items that have high loadings on each factor measure the same underlying psychological construct and thus form a scale. This allows you to determine how to handle future questionnaires simply by looking at the factor matrix: if items 1, 2, 10, and 12 are the only ones that have significant loadings on one factor, then one test scale will consist of only these four items.

In addition, each of the scales needs to be validated, for example, by calculating each person's score on each factor and assessing the construct and/or predictive validity of these scales. For example, scores obtained from factors can be correlated with scores obtained from other questionnaires used to predict learning success, etc. The second task that factor analysis can solve is data reduction, or “conceptual cleaning.” A huge number of personality tests have been developed, based on different theoretical positions, and it is not always clear to what extent they overlap.

Third, factor analysis is used to test the psychometric properties of questionnaires, especially when they are used in new cultures or populations. For example, suppose that, according to the manual for the Australian Personality Test, it should be processed by adding the scores obtained on all odd-numbered items, which form one scale, while the sum of the scores obtained on all even-numbered items forms another scale .

The main concept of factor analysis is factor. This is an artificial statistical indicator that arises as a result of special transformations of the table of correlation coefficients between the psychological characteristics being studied, or the intercorrelation matrix. The procedure for extracting factors from an intercorrelation matrix is ​​called matrix factorization. As a result of factorization from the correlation matrix, m.b. a different number of factors were extracted, up to a number equal to the number of original variables. However, the factors identified as a result of factorization, as a rule, are unequal in importance. The formal criterion for the quality of the factor analysis procedure is the percentage of combined variance of the original characteristics.

Currently, factor analysis is widely used both to solve research problems and to design psychodiagnostic techniques.

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More on topic 31. Application of factor analysis in psychology:

1. 1. Methods of factor analysis, their types, features of application.
2. Factor analysis, complete and fractional factorial experiment and mathematical model.
3. 62. Planning experiments. Factor analysis, complete and fractional factorial experiment and mathematical model.

Factor analysis is one of those methods that, having been developed within the framework of the needs of one science, subsequently acquired broader interdisciplinary significance. The merit of psychology can be considered the development of just such a method.

The basic ideas of factor analysis were laid down in the works of the famous English psychologist F. Galton (1822-1911), the founder of eugenics, who made a great contribution to the study of individual differences. The further development and implementation of factor analysis (FA) in psychology is associated with the names of C. Spearman, R. Cattell, L. Thurstone.

The need to use FA in psychology as one of the methods of multidimensional quantitative description of observed variables primarily follows from the multidimensionality of the objects studied by this science. A multidimensional representation of an object is understood as the result of its evaluation according to several different and essential characteristics for its description - dimensions, i.e. assigning several numerical values ​​to it at once.

The information content of a multidimensional description of the object of study increases with the number of signs or measurement scales used. However, it is very difficult to immediately select both essential and independent characteristics. As a rule, the researcher starts with a obviously redundant number of features and in the process of work is faced with the need to adequately interpret a large volume of data obtained and their compact visualization. Analyzing the data obtained, the researcher notices the fact that the assessments of the object being studied, obtained on some scales, are similar to each other. In other words, the question arises that many of the characteristics by which our object was measured are likely to duplicate each other to some extent, and all the information obtained is generally redundant. Behind the related variables, there appears to be an underlying influence of some hidden, latent variable that can explain the observed similarity between the scores. Very often this variable is called a factor.

So the method scientific knowledge- generalization - leads us to the possibility and necessity of identifying factors as variables of a more general, higher order. Generalization allows you to notice those connections between the original characteristics that were not previously obvious, and then reach a higher level of understanding of the essence of the measured object.

There are several statistical methods, which allow you to explore the relationships between variables without determining which of them are dependent and which are independent. For these methods, all variables are treated equally - none of them is more important than the other. The first method we'll look at, principal component analysis, explains the most variance in terms of the fewest linear combinations of variables. The second method, factor analysis, explains relationships between variables in terms of multiple factors that cannot be directly measured. Both methods have the same number of original variables. However, the factors determined as a result of factorization, as a rule, are not equivalent in importance.

The coefficients defining a new variable are selected in such a way that the new variables (principal components, factors) describe maximum amount data variability and did not correlate with each other. They represent the correlation coefficient between the original variable and the new variable (factor). The coefficients are called factor loadings. They are usually presented in the form of a table, where the factors are arranged in the form

To analyze the variability of a trait under the influence of controlled variables, the dispersion method is used.

To study the relationship between values ​​- the factorial method. Let's take a closer look at the analytical tools: factorial, dispersion and two-factor dispersion methods for assessing variability.

Analysis of Variance in Excel

Conventionally, the goal of the dispersion method can be formulated as follows: to isolate 3 partial variations from the general variability of the parameter:

• 1 – determined by the action of each of the studied values;
• 2 – dictated by the relationship between the studied values;
• 3 – random, dictated by all unaccounted for circumstances.

In a programme Microsoft Excel ANOVA can be performed using the Data Analysis tool (Data tab - Analysis). This is a spreadsheet add-on. If the add-in is not available, you need to open Excel Options and enable the Analysis setting.

The work begins with the design of the table. Rules:

1. Each column should contain the values ​​of one factor under study.
2. Arrange the columns in ascending/descending order of the value of the parameter being studied.

Let's look at variance analysis in Excel using an example.

The company's psychologist analyzed the behavior strategies of employees in a conflict situation using a special technique. It is assumed that behavior is influenced by the level of education (1 – secondary, 2 – specialized secondary, 3 – higher).

Let's enter the data into an Excel table:

Significant parameter is filled in yellow. Since the P-Value between groups is greater than 1, Fisher's test cannot be considered significant. Consequently, behavior in a conflict situation does not depend on the level of education.

Factor analysis in Excel: example

Factorial analysis is a multidimensional analysis of relationships between the values ​​of variables. By using this method you can solve the most important problems:

• comprehensively describe the object being measured (and succinctly, compactly);
• identify hidden variable values ​​that determine the presence of linear statistical correlations;
• classify variables (identify relationships between them);
• reduce the number of required variables.

Let's look at an example of factor analysis. Let's say we know the sales of some goods over the last 4 months. It is necessary to analyze which titles are in demand and which are not.

Now you can clearly see which product sales are generating the main growth.

Two-way ANOVA in Excel

Shows how two factors influence the change in the value of a random variable. Let's look at two-factor analysis of variance in Excel using an example.

Task. A group of men and women were presented with sounds of different volumes: 1 – 10 dB, 2 – 30 dB, 3 – 50 dB. Response times were recorded in milliseconds. It is necessary to determine whether gender influences the response; Does volume affect response?

Unlike representatives natural sciences(physicists, chemists, biologists, doctors) concerned with measuring weight (molecules, atoms, planets, living cells, humans), pressure (gas, steam or blood), temperature (in a nuclear reactor or in a patient), using instruments for this ( ruler, tonometer, thermometer) and receiving data in appropriate units (grams, millimeters or degrees), psychologists most often have to comprehend (understand, describe, measure) some more general, abstract characteristics, often invented by themselves, existing hypothetically: introversion , altruism, intelligence. How, for example, can one measure the degree of love one person has for another? Using what device? On what scale? In what units?

From the point of view of a psychometrician, love is a latent (deep) characteristic that cannot be seen as such, but can be assessed based on the measurement of explicit (observable) variables. For example, you can highlight certain acts of behavior and interpret them as manifestations of love. If someone gives flowers to someone, is interested in his problems, reads his notes, laughs at his jokes, sacrifices something for him, etc., then we can assume that in this case the variable “love” should be “evaluated” withplus sign." Similarly, simple observable characteristics can be selected to study altruism, understanding, etc. In general, identification of deep dimensions by observable characteristics (actions) occurs in psychological research at any level: the individual (for example, the severity of neuroticism), personality (/Q), interpersonal interaction (leadership), society as a whole (ideology, moral standards).

In this case, they proceed from the hypothesis that abstract concepts can be described through simpler (observable) ones, since these abstract concepts explain the observed correlations between simple variables. For example, the postulate of the existence of something called “love” determines the correlations between actions in various situations associated with manifestations of love. It is worth paying attention to the highlighted word “connected”: by whom, where, when, how? Having decided to measure any latent variable, the researcher (experimental psychologist) compiles a list of characteristics (observable variables) that indicate the manifestations of the latent variable. This list is most often compiled on the basis of his hypothesis (for example, that the feeling of love is manifested in some obvious and generally accepted actions). Here it is necessary, of course, to take into account the sociocultural context, because the “legalized” (normative) ways of expressing certain feelings in different societies are completely different (just remember the still existing Russian proverb: “He hits, it means he loves”). Since measurements on several variable parameters are used for assessment, we speak of a latent construct - a factor.

The concept of “construct” was introduced by J. Kelly (1955), who considered personal (personal) constructs not only as a form of ordering experience, but also as a formation that mediates the perception and awareness of reality. This term also applies to public consciousness, which has absorbed personal constructs (for example, at the level of ideology, morality, social norms that determine the functioning and development of society as a whole). We can talk about group constructs inherent in representatives of a certain specialty and associated with a certain professional painting peace.

To process the data obtained during the experiment, they are widely used various methods multivariate statistics. The most common one is factor analysis- a statistical procedure used to identify a relatively small number of underlying (not explicitly observable) constructs that can be used to represent relationships among numerous observable variables.

The exact moment when the factor analysis method emerged is quite difficult to determine. If we count its history from the invention of the correlation coefficient by F. Galton, then this is the mid-1880s. Working with anthropometric data, Pearson put forward the idea of ​​“principal axes” in 1901, but the birth of factor analysis as a research method is associated with the publication in 1904 of Spearman’s article “The Objective Determination and Measurement of General Intelligence.” Based statistical analysis tests, Spearman put forward a two-factor theory of intelligence, described in terms of one general (general) factor inherent in all measures of intelligence, and a whole series of specific factors introduced by each of the tests used. However, the concept of one general factor turned out to be untenable, and further development of the theory led to the emergence of Thurstone's multifactor analysis, i.e. to what we call factor analysis today. It is now common to view scores on aptitude test batteries (observed variables) as linear combinations of factors expressing verbal skills, mathematical ability, and perceptual speed.

During the Second World War factor analysis was widely used by various US military services in connection with solving problems of qualification testing, classification and distribution of personnel. Quite soon, works appeared devoted to the use of factor analysis in the study of temperament (Guilford, Zimmerman, 1956), official morality (Roebuck, 1958), in the development of clinical therapy methods (Lorr, McNair, 1964; McNair, 1964), in identifying psychological characteristics“public relations” (Schubert, 1962; Thurstone, Began, 1951; Voiers, 1964; for more information on this, see Harman, 1972).

Factor analysis quite quickly turned into a fairly complex mathematical system, combining methods of probability theory and mathematical statistics, linear algebra and functional analysis, developed by American mathematicians and statisticians for American psychologists and mainly used by these American psychologists. Almost all books on factor analysis available to the Russian-speaking reader are translations. And the references in them illustrating the application of this method in psychology refer exclusively to English-language literature.

In our country, discussion of the fundamentals of factor analysis began back in the 1930s. However, these were mainly critical speeches that corresponded to the spirit of the era and led to the thesis about “extreme simplification of a metaphysical nature that arises when a property is decomposed into the sum of its components” (Mandryka, 1931). This position significantly slowed down the further spread and use of factor analysis in all areas of Soviet science.

A new stage in the development of this method in the USSR began in the 1950s. in anthropology (Ignatiev, 1957). The work of V.P. Chtetsov (1960) outlined general scheme factor analysis and reviewed some of the works of foreign anthropologists. The need to use factor analysis in physical anthropology was shown in the article by P.N. Bashkirov (1960), which served as a “bridge” between anthropology and the sciences of sports, which are closely related to the science of higher nervous activity of man - the area of ​​interest of B.M. Teplova and V.D. Nebylitsyn (for more details, see: Nebylitsyn, 1960; Doktorov, 1969).

Nebylitsyn's article (1960) was quite bold at that time (let's not forget about the active struggle against bourgeois trends in Soviet biology, genetics, mathematics, etc.). Calling carefully factor analysis rather an art that provides considerable scope for subjective interpretations and conclusions, the author nevertheless invites psychologists to become acquainted with the theory, basic premises, logic and technique of this method, and also expresses the hope that it will soon be transformed into a strict logical scheme that provides a unique solution.

Teplov (1967) draws attention to two different, but not contradictory friends other tasks of factor analysis: formal-mathematical (statistical, associated with an economical description of the data obtained) and scientific-substantive (interpretive, allowing to confirm or reject hypotheses regarding the nature of the processes being studied). These two problems are closely interrelated: to solve the second (scientific-substantive) problem, you must first solve the first - mathematical one. Describing the mathematical model of factor analysis and citing examples from the research of the laboratory he leads, Teplov says that factor analysis will be a valuable tool in any field where one can assume the presence of some basic parameters, functions, properties that form a structure. Currently, all monographs on factor analysis indicate the areas of its application in psychology. It is worth noting that the method of factor analysis received its final name in Russian in this work of Teplov (previously, along with the term “factorial”, the term “factorial” was used).

If you ask any domestic psychologist to name the names of colleagues who most often use factor analysis today, the undisputed leaders of such a rating will be the “founding fathers” of the psychosemantic direction - V.F. Petrenko (1983, 1988, 1997) and A.G. Shmelev (1983). This is true. Factor analysis (along with other methods of multivariate statistics - cluster and discriminant analysis, multidimensional scaling) is included in the working arsenal of psychosemantics. And if E.Yu. Artemyeva (1980, 1999), developing a psychosemantic approach, tried to avoid data processing associated with cumbersome calculations due to certain difficulties in using large computers and the lack of personal computers (hence her semantic codes, etc.), then in Currently, the removal of these barriers allows us to “squeeze” much more out of the data obtained. more information methods of multivariate statistics. Of course, the use of factor analysis is not limited to the area of ​​psychosemantics alone, although the development of the latter significantly contributes to the development of general mathematical culture domestic psychologists. It is enough to look at psychological journals over the last two or three years to be convinced that there are practically no areas of general or applied psychology left where research has not been carried out using the method of factor analysis.

If at the first stages factor analytical procedures were carried out mainly “manually”, which required the researcher to master the theory and methods of calculation, then at present the overwhelming majority of psychologists using factor analysis to process their data, has a very vague understanding of the complex structures that justify the calculations, and perceives the corresponding computer programs (usually created by American programmers) as a “black box” into which you can enter your data matrix, and at the output you will receive a matrix of factors or some graphs. Of course, some knowledge in the field of factor analysis theory will allow the researcher to feel more free not only when processing data (choosing methods, statistical criteria, mathematical justification optimal solution), but also at the stage of planning the experiment (which variables to include, what mathematical solution to expect), as well as when interpreting the result obtained and understanding why this particular solution was obtained and whether it can be improved by choosing other methods of factor analysis. All this increases the level of research.

However, we all know that it is quite possible to drive a car without knowing it. internal structure: learned the rules traffic, got acquainted with the principles of machine movement, remembered school lessons physicists - on their way. If something breaks along the way, it is not at all necessary to go under the hood, but you can turn to a specialist for help. Probably the same thing should happen when a psychologist sits down at the computer, turns on the factor analysis program and begins to process his data. Here, the “User’s Guide” for using the factor analysis program and the general mathematical culture acquired at school and then at the institute act as assistants (it is no coincidence that a course in mathematics is considered necessary for a psychology student). the main objective of the proposed manual - as simply as possible, explain to the psychologist (or student) how to use the power of factor analysis for their own purposes. However, recalling the analogy with driving a car and taking into account the fact that a car service in currently is much better developed than the service of psychologists by specialists in the theory of factor analysis (the latter are simply very few), and also the likelihood that the psychologist himself will want (or be forced) to understand the formulas and theorems of this theory, we provide some mathematical foundations of factor analysis , and for more advanced readers we recommend additional literature.