An example of the results of factor analysis. Factor analysis for psychologists - Mitina

Factor analysis- 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 answers 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 guidelines for using 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.

The second multivariate procedure is factor analysis. During factor analysis, the values ​​are determined large quantity variables, the correlation between them is found, and then groups of variables that form “factors” are identified. Let's explain this idea with a simple example. Let's say you give the students the following assignments:

vocabulary test (VT);

reading comprehension test (RT);

analogy test (for example, the doctor is related to the patient as a lawyer is to_) (AN);

geometry test (GEOM);

puzzle solving test (RG);

figure rotation test (RF).

For all possible pairs of tests, one can calculate Pearson r, resulting in the so-called correlation matrix:

Notice how some of the correlation values ​​form groups (I circled two groups). The correlations between vocabulary, reading comprehension, and analogies are all quite high. This is true for geometry, puzzles, and rotating shapes. Correlations between tests belonging to different groups, are practically equal to zero. This suggests that these tests are aimed at examining two significantly different mental abilities, or “factors.” We can label these as “verbal fluency” and “spatial skills.”

Factor analysis is a complex statistical method that extracts individual factors from a set of cross-correlations. When analyzing this matrix, the same two factors will undoubtedly be highlighted. The analysis also determines “factor loadings,” which are correlations between each of the tests and each of the identified factors. In the example above, the first three tests would have a “high loading” on factor 1 (verbal fluency), and the second three would have a “high loading” on factor 2 (spatial skills). Of course, in a real study the correlations never cluster as clearly as in in this example, and the results often lead researchers to heated discussions about whether the various factors were actually discovered. There are also discrepancies in how to correctly name factors, because factor analysis itself only identifies factors, and what to call them is decided by the researchers themselves.

Factor analysis was used in one of the longest debates in psychology - whether intelligence is a single property of a person. Charles Spearman, the founder of factor analysis (early 20th century), believed that all intelligence tests have the same load on one factor, which he called the general intelligence factor, or g (from the English general). Moreover, in his opinion, each test should load heavily on a second factor involving the skill being tested by the test (for example, math ability). He designated these second-order, or “special” factors as s (from the English special). According to his "two-factor" theory, performance on intelligence tests is directly related to a person's general intelligence (g) and his specific skills (k). Spearman believed that g is inherited and the various 5-factors are acquired through learning (Fruchter, 1954).

Other researchers, including Lewis Thurstone, believed that intelligence was composed of many factors and rejected the existence of a general g factor. From the results of factor analysis, Thurstone concluded that there were seven different factors, which he called "primary mental abilities": speech comprehension, verbal fluency, numeracy, spatial skills, memory, perceptual speed, and reasoning ability.

The question of whether intelligence is a single entity continues to baffle scientists who measure it, and its discussion is beyond the scope of this chapter. It is important for us that factor analysis can lead to different results. This is due to the fact that a) there are several varieties of factor analysis, which differently assess how high the correlation should be to identify individual factors, and b) c various studies Various intelligence tests are used for this problem. Therefore, researchers using different approaches and tests, get very different results. In short, just like the rest statistical methods, factor analysis is only a tool, and it cannot by itself solve theoretical questions such as the nature of intelligence.

As has become clear from this short introduction, correlation procedures play a prominent role in modern psychological research. Very often they are necessary if experimental procedures cannot be used. In addition, the development of complex multivariate procedures has made addressing the issue of cause and effect easier than in the past, when most correlation procedures were bivariate in nature.

Many correlational studies take place outside of laboratories. In the next chapter we will explore in more detail the

Results: main effect and interaction
Factorial studies produce two types of results: main effect and interaction. The main effect shows the common influence of independent variables, and the interaction reflects the joint effect of...

Correlation and Regression: Basics
Variables are considered to be correlated if there is any relationship between them. This is implied by the term “correlation” itself: “co” means mutual action, and “relation” (from the English relation ...

Interpersonal communication
Interpersonal communication is informal interaction that occurs one-on-one or in small groups. Are we talking with our neighbors in the student dormitory, are we chatting on the phone with...

Factor analysis is widely used in psychology in various directions related to solving both theoretical and practical problems.

In theoretical terms, the use of factor analysis is associated with the development of the so-called factor-analytical approach to the study of personality structure, temperament and abilities. The use of factor analysis in these areas is based on the widely accepted assumption that observable and directly measurable indicators are only indirect and/or partial external manifestations of more general characteristics. These characteristics, unlike the first ones, are hidden, so-called latent variables, since they represent concepts or constructs that are not available for direct measurement. However, they can be established by factorizing the correlations between observed traits and isolating factors that (provided they are well structured) can be interpreted as a statistical expression of the latent variable of interest.

Although the factors are purely mathematical in nature, they are assumed to represent latent variables (theoretically postulated constructs or concepts), so

the names of the factors often reflect the essence of the hypothetical construct being studied. Thus, factor analysis, which was developed at the beginning of the 20th century by Charles Spearman to study the structure of abilities, made it possible to introduce into psychology the concept of a general ability factor - the factor g. Subsequently, L. Thurstone put forward and experimentally tested a model that included 12 ability factors. Factor-analytical studies of temperament and personality in foreign psychology cover whole line theories of the past and present, including the theories of G. Allport, R. Cattell, G. Eysenck and others.

In Russian psychology, factor analysis was most widely used in differential psychology and psychophysiology in the study of properties nervous system person in the works of B.M. Teplov and his school. Teplov gave great importance this type of statistical data processing, emphasizing that factor analysis is a valuable tool in any field where it is possible, at least in the form of a preliminary hypothesis, to assume the presence of some basic parameters, functions, properties that form the “structure” of a given field of phenomena.

Currently, factor analysis is widely used in differential psychology and psychodiagnostics. With its help, you can develop tests, establish a structure of connections between individual psychological characteristics measured by a battery of tests or test items (see Appendix 2).

Another aspect of the use of factor analysis is the so-called “reduction” of data or “conceptual purification” of a large number of tests developed from various theoretical positions to measure personality traits. As a result of factorization of the correlation matrix obtained on a large sample of subjects using various personality tests, it is possible to more accurately identify the structure of personality traits determined by the tests used.

Factor analysis is also used to standardize test methods, which is carried out on a representative sample of subjects.

For a more detailed introduction to the various options for using factor analysis in psychology, we recommend the following literature (4, 12, 15, 25, 39).

gumentation of the content actually guessed in this or that factor is the most difficult and controversial task. For example, if with large positive weights one of the identified factors includes such variables as high growth, rough voice, big muscle mass, risk-taking, broad shoulders, aggressive behavior, then most likely such a combination will be interpreted by an anthropologist as a male factor, an endocrinologist will see the influence of some hormone, and a psychologist will try to find some analogues in personality typology.

Factor analysis techniques are particularly widely used in psychology in attempts to organize (combine into scales) numerous items in long-form personality questionnaires.

Most factor analysis programs are structured in such a way that the first identified factor has the greatest influence on the dispersion of indicators in the group (explained variance), and the value of the remaining factors consistently decreases.

There are several basic forms of factor analysis, which ultimately yield different results. The choice of the required option is dictated by the specific objectives of the thesis research.

❖ Cluster analysis If you need to split a lot of your variables (objects) into a given or unknown number of classes, then it is advisable to use cluster analysis

(cluster - a bunch, a bunch, an accumulation, a group of elements characterized by some common property). This is a form of mathematical processing of empirical materials that is not very often used in theses, but is of interest in cases where there are a sufficiently large number of variables.

Rice. 3. An example of one of the options for graphical presentation of the results of a cluster analysis of six variables.

Therefore, I would like to clearly see their ordering - in what hierarchical relationships are variables of a higher level of generality to more specific, particular ones (Fig. 3).

Very interesting results, gravitating towards the field of psycholinguistics, can be obtained using cluster analysis when applied to items of psychological tests, questionnaires and questionnaires.

The results of cluster analysis must be used with caution, since it can impose on the experimenter a hypothesis about the relationships of variables based on external, formal criteria and not take into account their qualitative specificity. In order to avoid such a mistake, it is preferable to use several different calculation algorithms (there are many of them, the grouping techniques differ) and select from the results the one that is best explained from a common sense perspective. It should be understood that cluster analysis determines the “most likely significant solution.”

❖ Discriminant analysis

Another statistical processing method that may be useful in your thesis is called discriminant analysis. Its essence is that it allows you to divide objects or states that have certain characteristics, assigning them to a class or assess the proximity of a particular state to one of the classes. The research procedure of discriminant analysis itself consists of several steps:

groups are determined that need to be distinguished in the future (for example, patients with hysterical neurosis from patients with obsessive-compulsive neurosis) - this is the so-called training sample;

these groups, each member of which already has an accurate (verified) diagnosis, are studied according to the maximum number of signs (current symptoms, personal predisposition, specificity family education, the nature of traumatic situations, etc.);

for each of the studied characteristics, the entire training sample (of both patients) is discriminated against and monitored - how accurately this characteristic divided the group into diagnoses in comparison with the actual state of affairs;

from all the examined features, the most informative ones are selected (those that most accurately divide the training sample) and in the future they begin to be used to improve the accuracy of the diagnosis for those who have not yet been diagnosed;

Along the way, if necessary, you can track how close or far each of the examined individuals is to one or another condition.

As a result of the discriminant analysis for each variable, you will receive a standardized coefficient (T - Wilks' lambda), interpreted as follows: the larger it is, the smaller the contribution of the corresponding variable to the discrimination of populations.

In other words, the basic idea of ​​discriminant analysis is to determine whether populations differ on the mean of some variable (or combination of them), and then use that variable to predict for new members their membership in a particular group (this is the task forecast). A simpler example: a height indicator can serve as a discriminatory sign for classifying a person unknown to us as male or female, since it is already known for sure that the average height of a man is higher than the average height of a woman.

One such feature, as can be guessed from the example presented, does not guarantee the reliability of the forecast, but a combination of characteristics can make it quite confident.

Below is an illustration of a graphical representation of discriminant analysis (Fig. 4).

Root 1 vs. Root2

Rice. 4. Graphic example of dividing trait carriers into three groups, obtained as a result of discriminant analysis.

❖ Nonparametric methods

Once again, I would like to emphasize that all the statistical analysis procedures discussed can be used correctly only if your experimental data obeys the so-called. normal distribution law or at least approach it. This means that in the distribution you have, extreme values ​​of a feature - both the smallest and the largest - appear rarely, and the closer the value of a feature is to the arithmetic mean, the more often it occurs (see Fig. 1).

If there is no such correspondence, which, as a rule, is explained either by small sample sizes (less than 20-30), or by measurements on ordinal scales (such as “high”, “medium”, “low”), or by the fact that the variables are objectively distributed " abnormal”, then to process the empirical materials of the thesis it is necessary to use the so-called non-parametric criteria, although they have less power and have less flexibility (for their calculation, the values ​​of the mean and standard deviation are not considered or taken into account). But they also have a number of advantages. They are insensitive to inaccurate measurements and these methods can be used to process data of a semi-quantitative nature (ranks, scores, etc.). In addition, they can be used to obtain answers to questions that are unsolvable using methods based on the normal distribution. Consequently, they are sometimes appropriate for processing normally distributed research results.

Without going into details, we will only point out the names of nonparametric procedures that make it possible to obtain indicators similar to normally distributed ones.

To determine the significance of differences between two independent samples (for example, when comparing boys and girls), nonparametric alternatives to the t test are serial criterion B ald a-Volfovich a, UMann-Whitney test And two-choice full-time test of the Kolmogorov-Smirnov type.

If the thesis reveals differences between dependent samples (for example, the indicators of one group before and after correctional work), then you need to use Wilcoxon T-test for differences of pairs, which can also be applied to ranked data. Compared to the Student's t-test, it requires significantly less computation and tests normally distributed samples almost as strictly. Its effectiveness for large and small samples is about 95%.

If the two variables under consideration have an alternative distribution (include only two gradations, such as the test scores in the group below or above some selected value before and after training, or the number of boys and girls who coped with a math test), then suitable nonparametric tests for the significance of differences will be % 2 (chi-square is not recommended for use if the number of experiments in each of the distributions being compared is less than 10) and Fisher's exact test for a four-field table. Attention: do not confuse the algorithm for calculating the mentioned non-parametric criterion % 2 With has a lot in common with the algorithm for calculating the Pearson x 2 goodness-of-fit test, which is useful when comparing empirical and theoretical distributions, usually used to establish whether the actual distribution corresponds to the normal law.

To clarify the connections between characteristics (correlation), you can calculate the already mentioned tetrachoric index(G), Spearman rank correlation coefficients(R or p) and may(T) Kendall. The last two can be used to determine the closeness of connections between both quantitative and qualitative characteristics, provided that their values ​​are ordered or ranked according to the degree of decrease or increase of the characteristic.

❖ Computer processing and graphic illustrations

Don’t be confused by some of the overload of statistical procedures recommended for use in your thesis. In most cases, you do not need (although it is desirable) to be familiar with their mathematical apparatus. To date, numerous computer programs have been developed for the needs of science, allowing even a person not versed in mathematics to quite easily calculate most of the desired indicators. The most famous and popular of them are the Statistica packages (tabular and graphical examples using it are given above) and SPSS. Both programs are equipped with reference material in the form of Help and special information support with an overview of the main calculation algorithms. When deriving difference indicators, in correlation matrices and other tables, numerical values ​​that are of particular interest to the researcher (in terms of reliability, importance, priority, etc.) are automatically highlighted in color and boldness.

These same packages can significantly improve the appearance thesis by introducing greater clarity into it. This is achieved by replacing some hard-to-read tables and digital data with graphs, histograms, and other forms of illustrations that fit well into the semantic outline of the presented results (but nothing superfluous!).

The choice of graph form should not be accidental. For example, changes over time are better perceived in a linear representation, comparisons of indicators of two groups - in a columnar representation, proportions - in circular histograms, and dispersion - in a dotted one (Fig. 5-8).

42. Factor analysis

Factor analysis– a set of analytical methods that make it possible to identify hidden signs, as well as the reasons for their occurrence and the internal patterns of their relationship.

Factor analysis is aimed at transforming the original set of characteristics into a simpler and more meaningful form. The central task of the method is the transition from a set of directly measured characteristics of the phenomenon being studied to complex generalized factors, behind which there are combinations of initial characteristics, identified on the basis of their internal patterns, reflecting the structure of the studied area of ​​phenomena.

According to the point of view of one of the creators of factor analysis - L. Thurston, this method is used to “condense” test scores, reduce them to a relatively small number of independent variables, and to isolate the factors needed to describe individual differences in test scores. Factor analysis is not only a method of statistical processing of initial data for generalizations, but also a broad scientific method confirmation of hypotheses regarding the nature of processes.

The initial information for carrying out factor analysis is the correlation matrix, or the matrix of intercorrelations of test indicators. In some factor analysis models, the matrix may include other characteristics of connections and contingencies between the studied features (for example, cluster relationships, distances in semantic space). Isolated through the analysis of intercorrelations or other characteristics of the relationship, generalized first-order factors can be presented in the form of a new matrix reflecting the correlations between factors. Based on such matrices, higher order factors can be determined.

In the history of psychology, factor analysis is associated with solving a number of theoretical problems. At first it was perceived as one of the main approaches to revealing the content of psychological properties. So, C. Spearman(1931) based on an analysis of correlations between the results various tests the idea of ​​a single general factor (G factor) was put forward, which underlies the success of any tests related to the measurement of intellectual properties. L. Thurston(1931) developed a multifactor analysis of the assessment of many correlated (oblique) and relatively independent (orthogonal) factors, explaining the multifactor concept of intelligence. Currently, factor analysis methods constitute a complex special field of mathematical statistics.

Analysis Thesis: “Sisyphus” works hard and almost succeeds. But at one point he relaxes, stops his efforts and loses everything he has achieved. He must start again, repeating the whole cycle. Diagnosis: depressive reaction. Myth: Sisyphus. Roles: Orphaned Child,

Analysis Thesis: Little Betky sits on a stool, withdrawn into herself and waiting for “Spider”, since she has nothing else to hope for. He appears and tries to scare her, but she decides that this is the most beautiful “Spider” in the world and chooses to stay with him. He periodically