All phenomena and processes of economic activity of enterprises are interconnected and interdependent. Some of them are directly related to each other, others indirectly. Hence, an important methodological issue in economic analysis is the study and measurement of the influence of factors on the value of the economic indicators under study.

Under economic factor analysis is understood as a gradual transition from the initial factor system to the final factor system, the disclosure of a full set of direct, quantitatively measurable factors that influence the change in the performance indicator.

Based on the nature of the relationship between indicators, methods of deterministic and stochastic factor analysis are distinguished.

Deterministic factor analysis is a methodology for studying the influence of factors whose connection with the performance indicator is functional in nature.

The main properties of the deterministic approach to analysis:

building a deterministic model through logical analysis;

the presence of a complete (hard) connection between indicators;

the impossibility of separating the results of the influence of simultaneously acting factors that cannot be combined in one model;

studying relationships in the short term.

There are four types of deterministic models:

Additive Models represent an algebraic sum of indicators and have the form

Such models, for example, include cost indicators in relation to elements of production costs and cost items; an indicator of the volume of production in its relationship with the volume of output of individual products or the volume of output in individual departments.

Multiplicative models in generalized form can be represented by the formula

An example of a multiplicative model is a two-factor model of sales volume

Where H- average number of employees;

C.B.- average output per employee.

Multiple models:

An example of a multiple model is the indicator of the turnover period of goods (in days). T O.T. :

Where Z T- average stock of goods; ABOUT R- one-day sales volume.

Mixed models are a combination of the above models and can be described using special expressions:

Examples of such models are cost indicators per 1 ruble. commercial products, profitability indicators, etc.

To study the relationship between indicators and quantitatively measure the many factors that influenced the performance indicator, we present general model transformation rules in order to include new factor indicators.

To detail the generalizing factor indicator into its components, which are of interest for analytical calculations, the technique of lengthening the factor system is used.

If the initial factor model is , a , then the model will take the form .

To identify a certain number of new factors and construct the factor indicators necessary for calculations, the technique of expanding factor models is used. In this case, the numerator and denominator are multiplied by the same number:

To construct new factor indicators, the technique of reducing factor models is used. When using this technique, the numerator and denominator are divided by the same number.

The detail of factor analysis is largely determined by the number of factors whose influence can be quantitatively assessed, therefore multifactorial multiplicative models are of great importance in the analysis. Their construction is based on the following principles:

the place of each factor in the model must correspond to its role in the formation of the effective indicator;

the model should be built from a two-factor full model by sequentially dividing factors, usually qualitative, into components;

When writing a formula for a multifactor model, factors should be arranged from left to right in the order in which they are replaced.

Building a factor model is the first stage of deterministic analysis. Next, determine the method for assessing the influence of factors.

Chain substitution method consists in determining a number of intermediate values of the generalizing indicator by sequentially replacing the basic values of the factors with the reporting ones. This method is based on elimination. Eliminate- means to eliminate, exclude the influence of all factors on the value of the effective indicator, except one. Moreover, based on the fact that all factors change independently of each other, i.e. First, one factor changes, and all the others remain unchanged. then two change while the others remain unchanged, etc.

In general, the application of the chain production method can be described as follows:

where a 0, b 0, c 0 are the basic values of factors influencing the general indicator y;

a 1 , b 1 , c 1 - actual values of factors;

y a, y b, are intermediate changes in the resulting indicator associated with changes in factors a, b, respectively.

The total change ∆ y=y 1 –y 0 consists of the sum of changes in the resulting indicator due to changes in each factor with fixed values of the remaining factors:

Let's look at an example:

Table 2 – Initial data for factor analysis

Indicators	Legend	Basic values	Actual values	Change
Indicators	Legend	Basic values	Actual values	Absolute (+,-)	Relative (%)
Volume of commercial products, thousand rubles.
Number of employees, people
Output per worker thousand roubles.

We will analyze the impact of the number of workers and their output on the volume of marketable output using the method described above based on the data in Table 2. The dependence of the volume of commercial products on these factors can be described using a multiplicative model:

Then the effect of a change in the number of employees on the general indicator can be calculated using the formula:

Thus, the change in the volume of marketable products was positively influenced by a change in the number of employees by 5 people, which caused an increase in production volume by 730 thousand rubles. and a negative impact was had by a decrease in output by 10 thousand rubles, which caused a decrease in volume by 250 thousand rubles. The combined influence of two factors led to an increase in production volume by 480 thousand rubles.

The advantages of this method: versatility of application, ease of calculations.

The disadvantage of the method is that, depending on the chosen order of factor replacement, the results of factor decomposition have different meanings. This is due to the fact that as a result of applying this method, a certain indecomposable residue is formed, which is added to the magnitude of the influence of the last factor. In practice, the accuracy of factor assessment is neglected, highlighting the relative importance of the influence of one or another factor. However, there are certain rules that determine the substitution sequence:

if there are quantitative and qualitative indicators in the factor model, the change in quantitative factors is considered first;

if the model is represented by several quantitative and qualitative indicators, the substitution sequence is determined by logical analysis.

Under quantitative factors in analysis they understand those that express the quantitative certainty of phenomena and can be obtained by direct accounting (number of workers, machines, raw materials, etc.).

Qualitative factors determine the internal qualities, signs and characteristics of the phenomena being studied (labor productivity, product quality, average working hours, etc.).

Absolute difference method is a modification of the chain substitution method. The change in the effective indicator due to each factor using the method of differences is defined as the product of the deviation of the factor being studied by the basic or reporting value of another factor, depending on the selected substitution sequence:

Relative difference method used to measure the influence of factors on the growth of a performance indicator in multiplicative and mixed models of the form y = (a – c) . With. It is used in cases where the source data contains previously determined relative deviations of factor indicators in percentages.

For multiplicative models like y = a . V . The analysis technique is as follows:

find the relative deviation of each factor indicator:

determine the deviation of the performance indicator at due to each factor

Example. Using the data in table. 2, we will analyze using the method of relative differences. The relative deviations of the factors under consideration will be:

Let's calculate the impact of each factor on the volume of commercial output:

The calculation results are the same as when using the previous method.

Integral method allows you to avoid the disadvantages inherent in the chain substitution method and does not require the use of techniques for distributing the indecomposable remainder among factors, because it has a logarithmic law of redistribution of factor loads. The integral method makes it possible to achieve a complete decomposition of the effective indicator into factors and is universal in nature, i.e. applicable to multiplicative, multiple and mixed models. The operation of calculating a definite integral is solved using a PC and is reduced to constructing integrand expressions that depend on the type of function or model of the factor system.

You can also use already formed working formulas given in the specialized literature ∆ 4∆:

1. Model view:

2. View model :

3. View model:

4. View model:

Let's consider the possibility of using the main methods of deterministic analysis, summarizing the above in the form of a matrix (Table 3).

Table 3 – Matrix of application of deterministic factor analysis methods

	Multiplicative	Additive	Mixed

Chain substitution
Absolute differences
Relative differences
Integral

Questions for self-control

What management problems are solved through economic analysis?

Describe the subject of economic analysis.

What distinctive features characterize the method of economic analysis?

What principles underlie the classification of techniques and methods of analysis?

What role does the method of comparison play in economic analysis?

Explain how to construct deterministic factor models.

Describe the algorithm for using the simplest methods of deterministic factor analysis: the method of chain substitutions, the method of differences.

Describe the advantages and describe the algorithm for using the integral method.

Give examples of problems and factor models to which each of the methods of deterministic factor analysis is applied.

Coursework in the discipline

"Industrial and Regional Aspects in Entrepreneurship"

On the topic “Methodology of factor analysis”

Completed by: Syrcina U.O.

5th year, SEF, Full-time

Checked by: Charyev R. M.

Associate Professor, Department of Economics

and management

Moscow 2008

Introduction

In today's economic conditions, an enterprise is forced to independently determine the prospects for its development. The successful solution of pressing economic problems, of course, depends on the development of the theory of activity analysis, which makes it possible to determine the effectiveness of the economic activity of an enterprise and to identify patterns of changes in the main results of its activities.

One of the most important tasks of financial analysis of any economic phenomenon is “... identifying factors, the level and changes of which have a decisive influence on the formation and change in the level of the phenomenon, considered as effective in relation to these factors.”

The value of the general indicator of the activity of structural divisions and the entire production formation depends on a large number of factors acting either in a certain sequence or simultaneously, in different directions and with varying strength. This dependence can be of different nature: probabilistic, in which the influence of one quantity on a change in another may have a possible (probabilistic) nature;
or deterministic, meaning the dependence of the effective indicator on factors: each factor value corresponds to one single value of the effective indicator. Each performance indicator depends on numerous factors. The more detailed the influence of factors on the value of the indicator is considered, the more accurate the results of the analysis and assessment of the quality of the decision made. In some situations, without a deep and comprehensive study of the direct influence of factors, it is impossible to draw reasonable conclusions about the company’s performance.

The purpose of my course work is a detailed examination of the types, tasks and stages of factor analysis, its purpose and relevance of use.

Before we start talking about one of the types of financial analysis - factor analysis, let me remind you what financial analysis is and what its goals are. The financial analysis is a method for assessing the financial condition and performance of an economic entity based on studying the dependence and dynamics of financial reporting indicators.

Financial analysis has several goals: assessing the financial situation; identifying changes in financial condition in space and time; identification of the main factors that caused changes in financial condition; forecast of main trends in financial condition.

As you know, there are the following main types of financial analysis:

· horizontal analysis;

· vertical analysis;

· trend analysis;

· method of financial ratios;

· comparative analysis;

· factor analysis.

Factor analysis- a section of multivariate statistical analysis that combines methods for estimating the dimension of a set of observed variables by examining the structure of covariance or correlation matrices. In other words, the task of the method is to move from a real large number of signs or causes that determine the observed variability to a small number of the most important variables (factors) with minimal loss of information. The method arose and was initially developed in problems of psychology and anthropology (at the turn of the 19th and 20th centuries), but now the scope of its application is much wider.

Basic models of financial analysis

Each type of financial analysis is based on the use of a model that makes it possible to evaluate and analyze the dynamics of the main indicators of the enterprise. There are three main types of models: descriptive, predicative and normative.

Descriptive models also known as descriptive models. They are fundamental for assessing the financial condition of an enterprise. These include: construction of a system of reporting balance sheets, presentation of financial statements in various analytical sections, vertical and horizontal analysis of reporting, a system of analytical coefficients, analytical notes for reporting. All of these models are based on the use of accounting information.

At the core vertical analysis lies a different presentation of financial statements - in the form of relative values that characterize the structure of the generalizing total indicators. An obligatory element of the analysis is the dynamic series of these quantities, which makes it possible to track and predict structural changes in the composition of economic assets and the sources of their coverage.

Horizontal analysis allows you to identify trends in changes in individual items or their groups included in the financial statements. This analysis is based on the calculation of the basic growth rates of balance sheet and income statement items.

System of analytical coefficients– the main element of financial analysis, used by various groups of users: managers, analysts, shareholders, investors, creditors, etc. There are dozens of such indicators, divided into several groups according to the main areas of financial analysis:

· liquidity indicators;

· indicators of financial stability;

· indicators of business activity;

· profitability indicators.

Predicative models These are predictive models. They are used to forecast a company's income and its future financial condition. The most common of them are: calculating the point of critical sales volume, constructing forecast financial reports, dynamic analysis models (strictly determined factor models and regression models), situation analysis models.

Normative models. Models of this type allow you to compare the actual results of enterprises with the expected ones calculated according to the budget. These models are used primarily in internal financial analysis. Their essence comes down to the establishment of standards for each cost item for technological processes, types of products, responsibility centers, etc. and to the analysis of deviations of actual data from these standards. The analysis is largely based on the use of strictly deterministic factor models.

As we see, modeling and analysis of factor models occupy an important place in the methodology of financial analysis. Let's consider this aspect in more detail.

Factor analysis, its types and tasks.

The functioning of any socio-economic system (which includes an operating enterprise) occurs in conditions of complex interaction of a complex of internal and external factors. Factor- this is the cause, the driving force of a process or phenomenon, determining its character or one of its main features.

Factor analysis- a methodology for a comprehensive and systematic study and measurement of the impact of factors on the value of performance indicators, a section of multivariate statistical analysis that combines methods for assessing the dimension of many observed variables. In other words, the task of the method is to move from a real large number of signs or causes that determine the observed variability to a small number of the most important variables (factors) with minimal loss of information (methods that are similar in essence, but not in mathematical terms - component analysis, canonical analysis, etc. ). The method arose and was initially developed in problems of psychology and anthropology (at the turn of the 19th and 20th centuries), but now the scope of its application is much wider. The assessment procedure consists of two stages: assessment of the factor structure - the number of factors necessary to explain the correlation between values, and factor loading, and then assessment of the factors themselves based on the results of observation. In short, under factor analysis understands the methodology for a comprehensive and systematic study and measurement of the impact of factors on the value of performance indicators.

Purpose of factor analysis

Factor analysis - definition influence of factors on the result - is one of the strongest methodological solutions in the analysis of the economic activities of companies for decision making. For managers- additional argument, additional "vision angle".

The feasibility of using factor analysis

As you know, you can analyze everything ad infinitum. It is advisable at the first stage to implement an analysis of deviations, and where necessary and justified, to apply the factor analysis method. In many cases, a simple analysis of deviations is enough to understand that the deviation is “critical”, and when it is not at all necessary to know the degree of its influence.

MAIN TASKS OF FACTOR ANALYSIS.

1. Selection of factors determining the performance indicators under study.

2. Classification and systematization of factors in order to provide an integrated and systematic approach to the study of their influence on the results of economic activity.

3. Determination of the form of dependence between factors and performance indicators.

4. Modeling the relationships between factors and performance indicators.

5. Calculation of the influence of factors and assessment of the role of each of them in changing the performance indicator.

6. Working with the factor model. Methodology of factor analysis.

However, in practice, factor analysis is rarely used for several reasons:
1) implementation of this method requires some effort and a specific tool (software product);
2) companies have other “eternal” priorities.
It is even better if the factor analysis method is “built-in” into the financial model, and is not abstract application.

In general, the following can be distinguished: main stages of factor analysis :

1. Setting the purpose of the analysis.

2. Selection of factors that determine the performance indicators under study.

3. Classification and systematization of factors in order to provide an integrated and systematic approach to the study of their influence on the results of economic activity.

4. Determination of the form of dependence between factors and the performance indicator.

5. Modeling the relationships between performance and factor indicators.

6. Calculation of the influence of factors and assessment of the role of each of them in changing the value of the effective indicator.

7. Working with the factor model (its practical use for managing economic processes).

Selection of factors for analysis of a particular indicator is carried out on the basis of theoretical and practical knowledge in a particular industry. In this case, they usually proceed from the principle: the larger the complex of factors studied, the more accurate the results of the analysis will be. At the same time, it must be borne in mind that if this complex of factors is considered as a mechanical sum, without taking into account their interaction, without identifying the main, determining ones, then the conclusions may be erroneous. In business activity analysis (ABA), an interconnected study of the influence of factors on the value of performance indicators is achieved through their systematization, which is one of the main methodological issues of this science.

An important methodological issue in factor analysis is determining the form of dependence between factors and performance indicators: functional or stochastic, direct or inverse, linear or curvilinear. It uses theoretical and practical experience, as well as methods for comparing parallel and dynamic series, analytical groupings of source information, graphical, etc.

Modeling of economic indicators also represents a complex problem in factor analysis, the solution of which requires special knowledge and skills.

Calculation of the influence of factors- the main methodological aspect in ACD. To determine the influence of factors on the final indicators, many methods are used, which will be discussed in more detail below.

The last stage of factor analysis is practical use of the factor model to calculate reserves for the growth of the effective indicator, to plan and predict its value when the situation changes.

Classification and systematization of factors in the analysis of economic activity.

In economic analysis, a factor is an active force that causes positive or negative changes in the condition of an object and in the indicators that reflect it. The concept of “factor” is used in economic analysis in 2 meanings:

Condition for carrying out a business transaction;

The reason for the change in the state of the object.

FACTORS are the reasons that shape the results of economic and financial activities. Identification and quantitative measurement of the degree of identification of individual factors on changes in the performance indicators of the economic and financial activities of an enterprise is one of the most important tasks of economic analysis. The influence of factors is reflected in different ways on changes in the performance indicators of economic activity. The classification of factors will allow us to understand the reasons for changes in the phenomena under study and more accurately assess the place and role of each factor in the formation of the value of effective indicators. The factors studied in the analysis can be classified according to different criteria.

The classification of factors is their distribution into groups depending on common characteristics. It allows you to gain a deeper understanding of the reasons for changes in the phenomena under study, and to more accurately assess the place and role of each factor in the formation of the value of effective indicators.

Classification of factors in economic analysis

1. extensive and intensive

2. permanent and temporary

3. major and minor (Barnholtz). It is customary to use the concept of rank (order) of a factor.

By their nature, factors are divided into natural, socio-economic and production-economic.

Natural factors have a great influence on the results of activities in agriculture, forestry and other industries. Taking into account their influence makes it possible to more accurately assess the results of the work of business entities.

Socio-economic factors include the living conditions of workers, the organization of health-improving work at enterprises with hazardous production, the general level of personnel training, etc. They contribute to a more complete use of the enterprise’s production resources and increase the efficiency of its work.

Production and economic factors determine the completeness and efficiency of use of the enterprise's production resources and the final results of its activities.

Based on the degree of impact on the results of economic activity, factors are divided into major and minor. The main ones include factors that have a decisive impact on the performance indicator. Those that do not have a decisive impact on the results of economic activity in the current conditions are considered secondary. It should be noted that, depending on the circumstances, the same factor can be both primary and secondary. The ability to identify the main ones from the entire set of factors ensures the correctness of the conclusions based on the results of the analysis.

Factors are divided into internal And external, depending on whether the activities of a given enterprise affect them or not. The analysis focuses on internal factors that the enterprise can influence.

Factors are divided into objective, independent of the will and desires of people, and subjective subject to the influence of the activities of legal entities and individuals.

According to the degree of prevalence, factors are divided into general and specific. Common factors operate in all sectors of the economy. Specific factors operate within a particular industry or a specific enterprise.

In the process of an organization's work, some factors influence the indicator under study continuously throughout the entire time. Such factors are called permanent. Factors whose influence appears periodically are called variables(this is, for example, the introduction of new technology, new types of products).

Of great importance for assessing the activities of enterprises is the division of factors according to the nature of their action into intensive And extensive. Extensive factors include factors that are associated with changes in quantitative, rather than qualitative, characteristics of the functioning of an enterprise. An example is an increase in the volume of production due to an increase in the number of workers. Intensive factors characterize the qualitative side of the production process. An example would be an increase in production volume by increasing the level of labor productivity.

Most of the factors studied are complex in composition and consist of several elements. However, there are also those that cannot be broken down into their component parts. In this regard, factors are divided into complex (complex) And simple (elemental). An example of a complex factor is labor productivity, and a simple one is the number of working days in the reporting period.

Based on the level of subordination (hierarchy), factors of the first, second, third and subsequent levels of subordination are distinguished. TO first level factors These include those that directly affect the performance indicator. Factors that influence the performance indicator indirectly, with the help of first-level factors, are called second level factors etc.

It is clear that when studying the influence of any group of factors on the work of an enterprise, it is necessary to organize them, that is, to carry out an analysis taking into account their internal and external connections, interaction and subordination. This is achieved through systematization. Systematization is the placement of the phenomena or objects being studied in a certain order, identifying their relationship and subordination.

Systematization of factors in the analysis of economic activity is due to a systematic approach in the analysis of economic activity, and means placing the studied factors in a certain order, identifying their relationship and subordination. One of the ways to systematize factors is to create deterministic factor systems, which means presenting the phenomenon under study in the form of an algebraic sum of a particular or a product of several factors that determine its value and are in functional dependence with it.

Creation factor systems is one of the ways of such systematization of factors. Let's consider the concept of a factor system.

Factor systems

All phenomena and processes of economic activity of enterprises are interdependent. Relationship between economic phenomena is a joint change in two or more phenomena. Among the many forms of regular relationships, an important role is played by cause-and-effect (deterministic), in which one phenomenon gives rise to another.

In the economic activity of an enterprise, some phenomena are directly related to each other, others - indirectly. For example, the amount of gross output is directly influenced by factors such as the number of workers and the level of their labor productivity. Many other factors indirectly affect this indicator.

In addition, each phenomenon can be considered as a cause and as a consequence. For example, labor productivity can be considered, on the one hand, as the reason for changes in production volume and the level of its cost, and on the other hand, as a result of changes in the degree of mechanization and automation of production, improvement in labor organization, etc.

Quantitative characterization of interrelated phenomena is carried out using indicators. Indicators characterizing the cause are called factorial (independent); indicators characterizing the consequence are called effective (dependent). The set of factor and resultant characteristics related by cause and effect is called factor system.

Modeling any phenomenon is the construction of a mathematical expression of an existing relationship. Modeling is one of the most important methods of scientific knowledge. There are two types of dependencies studied in the process of factor analysis: functional and stochastic.

A relationship is called functional, or strictly determined, if each value of a factor characteristic corresponds to a well-defined non-random value of the resultant characteristic.

A relationship is called stochastic (probabilistic) if each value of a factor characteristic corresponds to a set of values of the resulting characteristic, i.e., a certain statistical distribution.

Model factor system is a mathematical formula that expresses real connections between the analyzed phenomena. In general, it can be presented as follows:

where is the resultant sign;

Factor signs.

Thus, each performance indicator depends on numerous and varied factors. The basis of economic analysis and its section is factor analysis- identify, evaluate and predict the influence of factors on changes in the performance indicator. The more detailed the dependence of the performance indicator on certain factors is studied, the more accurate the results of the analysis and assessment of the quality of the enterprises’ work. Without a deep and comprehensive study of factors, it is impossible to draw informed conclusions about the results of operations, identify production reserves, and justify plans and management decisions.

Types of factor analysis

Depending on the type of factor model, there are two main types of factor analysis- deterministic and stochastic.

is a technique for studying the influence of factors whose connection with the effective indicator is functional in nature, that is, when the effective indicator of the factor model is presented in the form of a product, quotient or algebraic sum of factors.

This type of factor analysis is the most common, since, being quite simple to use (compared to stochastic analysis), it allows you to understand the logic of the action of the main factors of enterprise development, quantify their influence, understand which factors and in what proportion it is possible and advisable to change to increase production efficiency.

Deterministic factor analysis has a fairly strict sequence of procedures:

· construction of an economically sound deterministic factor model;

· choosing a method of factor analysis and preparing conditions for its implementation;

· implementation of counting procedures for model analysis;

Basic methods of deterministic factor analysis

· One of the most important methodological factors in ACD is determining the magnitude of the influence of individual factors on the increase in performance indicators. In deterministic factor analysis (DFA), the following methods are used for this: identifying the isolated influence of factors, chain substitution, absolute differences, relative differences, proportional division, integral, logarithm, etc.

· The first three methods are based on the elimination method. Eliminate means to eliminate, reject, exclude the influence of all factors on the value of the effective indicator, except one. This method is based on the fact that all factors change independently of each other: first one changes, and all others remain unchanged, then two change, then three, etc., while the rest remain unchanged. This allows us to determine the influence of each factor on the value of the indicator under study separately.

Stochastic analysis is a methodology for studying factors whose connection with a performance indicator, unlike a functional one, is incomplete and probabilistic (correlation). The essence of the stochastic method is to measure the influence of stochastic dependencies with uncertain and approximate factors. It is advisable to use the stochastic method for economic research with incomplete (probabilistic) correlation: for example, for marketing problems. If with a functional (complete) dependence with a change in the argument there is always a corresponding change in the function, then with a correlation connection a change in the argument can give several values of the increase in the function depending on the combination of other factors that determine this indicator. For example, labor productivity at the same level of capital-labor ratio may be different at different enterprises. This depends on the optimal combination of other factors affecting this indicator.

Stochastic modeling is, to a certain extent, a complement and deepening of deterministic factor analysis. In factor analysis, these models are used for three main reasons:

· it is necessary to study the influence of factors for which it is impossible to build a strictly determined factor model (for example, the level of financial leverage);

· it is necessary to study the influence of complex factors that cannot be combined in the same strictly determined model;

· it is necessary to study the influence of complex factors that cannot be expressed in one quantitative indicator (for example, the level of scientific and technological progress).

In contrast to the strictly deterministic approach, the stochastic approach requires a number of prerequisites for implementation:

a) the presence of a population;

b) a sufficient volume of observations;

c) randomness and independence of observations;

d) homogeneity;

e) the presence of a distribution of characteristics close to normal;

f) the presence of a special mathematical apparatus.

The construction of a stochastic model is carried out in several stages:

· qualitative analysis (setting the purpose of the analysis, defining the population, determining the effective and factor characteristics, choosing the period for which the analysis is carried out, choosing the analysis method);

· preliminary analysis of the simulated population (checking the homogeneity of the population, excluding anomalous observations, clarifying the required sample size, establishing distribution laws for the indicators being studied);

· construction of a stochastic (regression) model (clarification of the list of factors, calculation of estimates of regression equation parameters, enumeration of competing model options);

· assessment of the adequacy of the model (checking the statistical significance of the equation as a whole and its individual parameters, checking the compliance of the formal properties of the estimates with the objectives of the study);

· economic interpretation and practical use of the model (determining the spatio-temporal stability of the constructed relationship, assessing the practical properties of the model).

In addition to dividing into deterministic and stochastic, the following types of factor analysis are distinguished:

o direct and reverse;

o single-stage and multi-stage;

o static and dynamic;

o retrospective and prospective (forecast).

At direct factor analysis The research is conducted in a deductive manner - from the general to the specific. Reverse factor analysis carries out the study of cause-and-effect relationships using the method of logical induction - from particular, individual factors to general ones.

Factor analysis can be single stage And multi-stage. The first type is used to study factors of only one level (one level) of subordination without detailing them into their component parts. For example, . In multi-stage factor analysis, factors are detailed a And b into constituent elements in order to study their behavior. The detailing of factors can be continued further. In this case, the influence of factors at different levels of subordination is studied.

It is also necessary to distinguish static And dynamic factor analysis. The first type is used when studying the influence of factors on performance indicators on the corresponding date. Another type is a technique for studying cause-and-effect relationships in dynamics.

Finally, factor analysis can be retrospective, which studies the reasons for the increase in performance indicators over past periods, and promising, which examines the behavior of factors and performance indicators in perspective.

Characteristics of the DuPont multifactor model

Developments in the field of factor analysis, which have been ongoing since the beginning of the 20th century, are of great importance for expanding the possibilities of using analytical coefficients for intra-company analysis and management.

First of all, this relates to the development in 1919 of a factor analysis scheme proposed by specialists from the DuPont company (The DuPont System of Analysis). By this time, indicators of return on sales and asset turnover had become quite widespread. However, these indicators were used on their own, without linking them with factors of production. In the DuPont model, for the first time, several indicators were linked together and presented in the form of a triangular structure, at the top of which is the return on total capital ROA as the main indicator characterizing the return received from funds invested in the company's activities, and at the base are two factor indicators - profitability sales NPM and resource efficiency TAT.

This model was based on a strictly determined dependence

where is net profit;

The amount of assets of the organization;

- (production volume) sales revenue.

The original representation of the DuPont model is shown in Figure 1:

Figure 1. Schematic of the DuPont model.

In theoretical terms, DuPont specialists were not innovators; they used the original idea of interrelated indicators, first expressed by Alfred Marshall and published by him in 1892 in the book “Elements of Industrial Economics”. Nevertheless, their merit is obvious, since these ideas have not previously been applied in practice.

Subsequently, this model was expanded into a modified factor model, presented in the form of a tree structure, at the top of which is the return on equity (ROE) indicator, and at the base are signs characterizing the factors of the production and financial activities of the enterprise. The main difference between these models is a more detailed identification of factors and a change in priorities relative to the effective indicator. It must be said that the factor analysis models proposed by DuPont specialists remained unclaimed for quite a long time, and only recently they began to pay attention.

The mathematical representation of the modified DuPont model is:

where is return on equity;

Emergency- net profit;

A - the amount of the organization's assets;

VR -(production volume) sales revenue.

SK- the organization's own capital.

From the presented model it is clear that return on equity depends on three factors: return on sales, asset turnover and the structure of advanced capital. The significance of the identified factors is explained by the fact that they, in a certain sense, generalize all aspects of the financial and economic activities of the enterprise, its statics and dynamics, in particular the financial statements: the first factor summarizes form No. 2 “Profit and Loss Statement”, the second - the balance sheet asset, the third – balance sheet liability.

Now let's characterize each of the main indicators included in the DuPont model.

Return on equity.

Return on equity is calculated using the formula:

where is the amount of equity at the beginning and end of the period.

In the practice of analysis, many indicators of enterprise performance are used. The return on equity indicator was chosen because it is the most important for the company's shareholders. It characterizes the profit that owners receive from the ruble of funds invested in the enterprise. This coefficient takes into account such important parameters as interest payments for the loan and income tax.

Asset turnover (resource productivity).

The formula for calculating the indicator is:

Where VR– sales revenue for the billing period;

A np, A kp

This indicator can be interpreted in two ways. On the one hand, asset turnover reflects how many times during a period the capital invested in the assets of the enterprise is turned over, i.e. it evaluates the intensity of use of all assets, regardless of the sources of their formation. On the other hand, resource productivity shows how many rubles of revenue an enterprise has per ruble invested in assets. The growth of this indicator indicates an increase in the efficiency of their use.

Sales profitability.

Sales profitability is also one of the most important indicators of a company's performance. It is calculated as:

where is the revenue from product sales,

net profit of the enterprise.

This ratio shows how much net profit the company receives from each ruble of products sold. In other words, how much money remains with the enterprise after covering the cost of production, paying interest on loans and paying taxes. The return on sales indicator characterizes the most important aspect of the company's activities - the sale of main products, and also allows you to estimate the share of cost in sales.

Return on assets.

The return on assets indicator is calculated using the following formula:

net profit,

A np, A kp– the value of assets at the beginning and end of the period.

Return on assets is an indicator of the efficiency of an enterprise's operational activities. It is the main production indicator and reflects the efficiency of using invested capital. From the point of view of accounting, this indicator connects the balance sheet and the profit and loss account, that is, the main and investment activities of the enterprise, therefore it is very important for financial management (we will consider the types of activities of the enterprise in detail in the next chapter).

Financial leverage (leverage).

This indicator reflects the structure of capital advanced into the activities of the enterprise. It is calculated as the ratio of the entire advanced capital of the enterprise to the equity capital.

Advance capital,

Equity.

The level of financial leverage can be interpreted, on the one hand, as a characteristic of the financial stability and riskiness of a business, and on the other hand, as an assessment of the efficiency of the enterprise’s use of borrowed funds.

Before moving on to factor analysis itself, we will make a number of important reservations regarding the scope of application of the DuPont model.

When analyzing return on equity in the spatiotemporal aspect, it is necessary to take into account three important features of this indicator, which are essential for formulating valid conclusions.

The first is related to the temporary aspect of the activities of a commercial organization. The return on sales ratio is determined by the performance of the reporting period; it does not reflect the probable and planned effect of long-term investments. For example, when a business organization makes a transition to new promising technologies or types of products that require large investments, profitability indicators may temporarily decrease. However, if the strategy was chosen correctly, the costs incurred will pay off in the future, and in this case, a decrease in profitability in the reporting period does not mean low efficiency of the enterprise.

The second feature is determined by the problem of risk. One of the indicators of the riskiness of a business is the financial dependence ratio - the higher its value, the more risky the business is from the position of shareholders, investors and creditors.

Thus, it is necessary to take into account the relationships between factors that are not directly reflected in the DuPont model. For example, based only on the mathematical formula of the model, it may seem that an infinite increase in financial leverage will lead to an equally infinite increase in return on equity. However, as the share of borrowed funds in the advanced capital increases, payments for using loans also increase. As a result, net profit decreases and return on equity does not increase. In addition, one cannot ignore the financial risk that accompanies the use of borrowed sources.

The third feature is related to the problem of evaluation. The numerator and denominator of the return on equity ratio are expressed in monetary units of different purchasing power. Profit is a dynamic indicator; it reflects the results of operations and the current level of prices for goods and services, mainly for the past period. Unlike profit, equity capital accumulates over a number of years. It is expressed in an accounting estimate, which may differ greatly from the current market value.

In addition, the accounting estimate of equity does not reflect the future earnings of the enterprise. Not everything can be reflected in the balance sheet; for example, the prestige of a company, a trademark, the latest technologies, and highly qualified personnel do not have an adequate monetary value in the reporting (unless we are talking about the sale of the business as a whole). Thus, the market price of a company's shares may greatly exceed its book value, in which case a high return on equity does not mean a high return on the capital invested in the company. Therefore, the market value of the company should be taken into account.

Conclusion

The purpose of the DuPont model is to identify the factors that determine the efficiency of a business, assess the degree of their influence and emerging trends in their change and significance. This model is also used for comparative assessment of the risk of investing or lending to a given enterprise.

All factors of the model, both in terms of significance level and change trends, are characterized by industry specificity, which the analyst must take into account. Thus, the resource productivity indicator may have a relatively low value in high-tech industries that are capital intensive; on the contrary, the profitability indicator of economic activity in them will be relatively high. A high value of the financial dependence coefficient can be afforded by firms that have a stable and predictable flow of money for their products. The same applies to enterprises that have a large share of liquid assets (trade and distribution enterprises, banks). Consequently, depending on the industry specifics, as well as the specific financial and economic conditions prevailing at a given enterprise, it can rely on one or another factor to increase the return on equity.

After completing the work, I made the following conclusions.

Factor analysis is one of the strongest methodological solutions in analyzing the economic activities of companies for decision making. The main task, which is solved by various methods of factor analysis, including the method of principal components, is the compression of information, the transition from a set of values according to elementary characteristics with a volume of information to a limited set of elements of a factor mapping matrix or a matrix of latent factor values for each observed object.

Factor analysis methods also make it possible to visualize the structure of the phenomena and processes being studied, which means determining their state and predicting their development. Finally, factor analysis data provide grounds for identifying the object, i.e. solving the problem of image recognition.
Factor analysis methods have properties that are very attractive for their use as part of other statistical methods, most often in correlation and regression analysis, cluster analysis, multidimensional scaling, etc.

LITERATURE:

1. G.V. Savitskaya “Analysis of economic activity” Minsk LLC “New Knowledge”, 2002

2. V.I. Strazhev “Analysis of economic activity in industry”, Mn. Higher school, 2003

3. General and special management: Textbook/General. Ed. A.L. Gaponenko, A.P. Pankrukhin.-M.: Publishing house RAGS, 2001.

Tutoring

Need help studying a topic?

Our specialists will advise or provide tutoring services on topics that interest you.
Submit your application indicating the topic right now to find out about the possibility of obtaining a consultation.

In order to find out how profitable or unprofitable a company is, it is not enough to simply count money. To understand this for sure, and most importantly, to help increase profits, you need to regularly carry out the work of the enterprise as a whole. And for this you need to have some skills in the accounting field and certain information. It is worth considering that the company operated both during inflation and during the crisis. Prices changed constantly. Now do you understand why a banal counting of money does not make it possible to objectively assess the situation with profits or costs? After all, the price factor must also be taken into account.

So, many people find it difficult to do an example of our analysis, we hope it will help them do their own - by analogy, this type of diagnosis is compiled extremely quickly. It is presented in the form of a table. First, let's make the header of our factor analysis. Draw a table with 5 columns and 9 rows. Make the first column wider - it will contain the names of the enterprise's articles, not numbers. It will be called “Indicators”, which you should write down in the first line of the column. In it, fill out all the lines according to the sample: 1 - name, 2 - put the number 1 - column numbering, in line 3 write - “Sales revenue”, 4 - “Cost”. In the fifth line of the first column, put the item “Business expenses”. In 6, write “Process management costs.” The seventh line is called - and 8 - “Index of price changes”, and the last, 9 line - “Sales at comparable prices”.

Next, we move on to the design of column 2: in line 1 we write - “Previous period, thousand rubles.” (you can write other monetary units - euro, dollar, etc. - it depends on what currency you will carry out the calculations in), and in the second line we write the number - 2. Go to column 3 - in it 1 line has the name - “Reporting period”, thousand rubles. And the second is filled in with the number 3. Next, we draw up our factor analysis of revenue and move on to column 4. In the first line we enter “Absolute change, thousand rubles,” and the second line contains a small formula: 4 = 3-2. This means that the indicators that you will write in subsequent lines will be the result of subtracting the indicators of the second column from the indicators of the third. Let's move on to the design of the last - 5th column. In it, in line 1 you need to write: “Relative changes%”, which means that in this column all data will be recorded as a percentage. In the second line the formula is: 5=(4/2)*100%. That's it, we've designed the header, all that remains is to fill out each item in the table with the appropriate data. We carry out factor analysis, an example of which we give you. First of all, we calculate the price change index - this is perhaps the most important figure in our calculations. We write down the numbers from different periods in the appropriate columns. In columns 4 and 5 we carry out the necessary calculations. Factor analysis, an example of which you can view, requires precision in numbers. Therefore, you only need to write reliable information in 3 lines of each column. In 4 and 5 we again carry out calculations. As you understand, factorization is mainly carried out in lines 5 and 6: try to add as real, not underestimated, numbers there as possible. In the 4th and 5th columns of these lines, again carry out calculations using the formulas. Next, we carry out a factor analysis of revenue in column 7 - profit. We write down reliable numbers in columns 2 and 3, and in columns 4 and 5 we again calculate everything using the formulas. And the last column remains: we write the data, we calculate. Bottom line: factor analysis, an example of which we give you, shows what the impact of each of the factors described in the articles is on profit or production costs. Now you see the weaknesses and can correct the situation in order to get as much profit as possible.

You have done all the calculations to perform factor analysis, but they will not help you at all if you do not analyze the data obtained thoroughly.

Based on the nature of the relationship between indicators, methods of deterministic and stochastic factor analysis are distinguished.

Deterministic factor analysis is a methodology for studying the influence of factors whose connection with the performance indicator is functional in nature.

The main properties of the deterministic approach to analysis:

· construction of a deterministic model through logical analysis;

· the presence of a complete (hard) connection between indicators;

· the impossibility of separating the results of the influence of simultaneously acting factors that cannot be combined in one model;

· study of relationships in the short term.

There are four types of deterministic models:

Additive Models represent an algebraic sum of indicators and have the form

Multiplicative models in generalized form can be represented by the formula

An example of a multiplicative model is a two-factor model of sales volume

Where H- average number of employees;

C.B.- average output per employee.

Multiple models:

An example of a multiple model is the indicator of the turnover period of goods (in days). T OB.T:

Where Z T- average stock of goods; O R- one-day sales volume.

Mixed models are a combination of the above models and can be described using special expressions:

; Y = ; Y = ; Y = .

Examples of such models are cost indicators per 1 ruble. commercial products, profitability indicators, etc.

To detail the generalizing factor indicator into its components, which are of interest for analytical calculations, the technique of lengthening the factor system is used.

If the initial factor model is , a , then the model will take the form .

To construct new factor indicators, the technique of reducing factor models is used. When using this technique, the numerator and denominator are divided by the same number.

· the place of each factor in the model must correspond to its role in the formation of the effective indicator;

· the model should be built from a two-factor complete model by sequentially dividing factors, usually qualitative, into components;

· when writing a formula for a multifactor model, factors should be arranged from left to right in the order of their replacement.

Building a factor model is the first stage of deterministic analysis. Next, determine the method for assessing the influence of factors.

In general, the application of the chain production method can be described as follows:

y 0 = a 0 . b 0 . c 0 ;

y a = a 1 . b 0 . c 0 ;

y b = a 1 . b 1. c 0 ;

y 1 = a 1 . b 1. c 1,

where a 0, b 0, c 0 are the basic values of factors influencing the general indicator y;

a 1 , b 1 , c 1 - actual values of factors;

y a, y b, are intermediate changes in the resulting indicator associated with changes in factors a, b, respectively.

The total change Dу=у 1 –у 0 consists of the sum of changes in the resulting indicator due to changes in each factor with fixed values of the remaining factors:

Dу = SDу (а, b, с) = Dу a + Dу b + Dу c

Dу а = у а – у 0 ; Dу b = у в – у а; Dу с = у 1 – у в.

Let's look at an example:

table 2

Initial data for factor analysis

TP o = Ch o. NE o = 20. 146 = 2920 (thousand rubles).

Then the effect of a change in the number of employees on the general indicator can be calculated using the formula:

TP conv 1 = Ch 1. NE o = 25. 146 = 3650 (thousand rubles),

DTPusl 1 = TPusl 1 – TP o = 3650 – 2920 = 730 (thousand rubles).

TP 1 = Ch 1. CB 1 = 25. 136 = 3400 (thousand rubles),

DTP cond 2 = TP 1 – TPusl 1 = 3400 – 3650 = - 250 (thousand rubles).

Thus, the change in the volume of commercial output was positively influenced by a change in 5 people. number of employees, which caused an increase in production volume by 730 tons. rub. and a negative impact was exerted by a decrease in output by 10 thousand rubles, which caused a decrease in volume by 250 thousand rubles. The combined influence of two factors led to an increase in production volume by 480 thousand rubles.

The advantages of this method: versatility of application, ease of calculations.

· if there are quantitative and qualitative indicators in the factor model, the change in quantitative factors is considered first;

· if the model is represented by several quantitative and qualitative indicators, the substitution sequence is determined by logical analysis.

Qualitative factors determine the internal qualities, signs and characteristics of the phenomena being studied (labor productivity, product quality, average working hours, etc.).

y 0 = a 0 . b 0 . c 0 ;

Dу а = Da. b 0 . c 0 ;

Dу b = Db. a 1. c 0 ;

Dу с = Dс. a 1. b 1 ;

y 1 = a 1 . b 1. c 1 ;

Dу = Dу а + Dу b + Dу c.

Relative difference method used to measure the influence of factors on the growth of a performance indicator in multiplicative and mixed models of the form y = (a – b) . With. It is used in cases where the source data contains previously determined relative deviations of factor indicators in percentages.

For multiplicative models like y = a . V . The analysis technique is as follows:

· find the relative deviation of each factor indicator:

· determine the deviation of the performance indicator at due to each factor

Example. Using the data in table. 2, we will analyze using the method of relative differences. The relative deviations of the factors under consideration will be:

Let's calculate the impact of each factor on the volume of commercial output:

The calculation results are the same as when using the previous method.

Questions for self-control

1. What management problems are solved through economic analysis?

2. Describe the subject of economic analysis.

3. What distinctive features characterize the method of economic analysis?

4. What principles underlie the classification of techniques and methods of analysis?

5. What role does the method of comparison play in economic analysis?

6. Explain how to construct deterministic factor models.

7. Describe the algorithm for using the simplest methods of deterministic factor analysis: the method of chain substitutions, the method of differences.

8. Characterize the advantages and describe the algorithm for using the integral method.

9. Give examples of problems and factor models to which each of the methods of deterministic factor analysis is applied.

Galton F. (1822-1911), who also made a major contribution to the study of individual differences. But many scientists contributed to the development of Factor Analysis. The development and implementation of factor analysis in psychology was carried out by such scientists as Spearman Ch. (1904, 1927, 1946), Thurstone L. (1935, 1947, 1951) and Cattell R. (1946, 1947, 1951). It is also impossible not to mention the English mathematician and philosopher K. Pearson, who largely developed the ideas of F. Galton, and the American mathematician G. Hotelling, who developed a modern version of the principal component method. The English psychologist G. Eysenck, who widely used Factor Analysis to develop a psychological theory of personality, also deserves attention. Mathematically, factor analysis was developed by Hotelling, Harman, Kaiser, Thurstone, Tucker, etc. Today, factor analysis is included in all statistical data processing packages - SAS, SPSS, Statistica, etc.

Tasks and possibilities of factor analysis

Factor analysis allows you to solve two important problems for the researcher: to describe the object of measurement comprehensively and at the same time compact. Using factor analysis, it is possible to identify hidden variable factors responsible for the presence of linear statistical relationships of correlations between observed variables.

Thus, two goals of Factor Analysis can be distinguished:

During the analysis, variables that are highly correlated with each other are combined into one factor, as a result, the variance is redistributed between the components and the most simple and clear structure of factors is obtained. After combining, the correlation of components within each factor with each other will be higher than their correlation with components from other factors. This procedure also makes it possible to isolate latent variables, which is especially important when analyzing social ideas and values. For example, when analyzing scores obtained on several scales, a researcher notices that they are similar to each other and have a high correlation coefficient, he can assume that there is some latent variable that can be used to explain the observed similarity of the scores obtained. This latent variable is called factor. This factor influences numerous indicators of other variables, which leads us to the possibility and necessity of identifying it as the most general, of a higher order. To identify the most significant factors and, as a consequence, the factor structure, it is most justified to use the principal components method (PCA). The essence of this method is to replace correlated components with uncorrelated factors. Another important characteristic of the method is the ability to limit oneself to the most informative principal components and exclude the rest from the analysis, which simplifies the interpretation of the results. The advantage of PCA is also that it is the only mathematically based method of factor analysis.

Factor analysis can be:

exploration- it is carried out when studying the latent factor structure without assumptions about the number of factors and their loadings;
confirmation, designed to test hypotheses about the number of factors and their loadings (note 2).

Conditions for using factor analysis

The practical implementation of factor analysis begins with checking its conditions. The mandatory conditions of factor analysis include:

Basic concepts of factor analysis

Factor - hidden variable
Loading - correlation between the original variable and the factor

Rotation procedure. Isolation and interpretation of factors

The essence of factor analysis is the procedure for rotating factors, that is, redistributing variance according to a certain method. The purpose of orthogonal rotations is to determine the simple structure of factor loadings, the purpose of most oblique rotations is to determine the simple structure of secondary factors, that is, oblique rotations should be used in special cases. Therefore, orthogonal rotation is preferable. According to Muljek's definition, a simple structure meets the requirements:

each row of the secondary structure matrix V must contain at least one zero element;
For each column k of the secondary structure matrix V there must be a subset of r linearly independent observed variables whose correlations with the kth secondary factor are zero. This criterion boils down to the fact that each column of the matrix must contain at least r zeros.
One of the columns of each pair of columns of the matrix V must have several zero coefficients (loadings) in those positions where they are non-zero for the other column. This assumption ensures that the secondary axes and their corresponding subspaces of dimension r-1 are distinguishable in the space of common factors.
When the number of common factors is greater than four, each pair of columns should have a number of zero loadings in the same rows. This assumption makes it possible to divide the observed variables into separate clusters.
For each pair of columns of matrix V there should be as few significant loadings corresponding to the same rows as possible. This requirement ensures that the complexity of the variables is minimized.

(In Mullake's definition, r denotes the number of common factors, and V is the secondary structure matrix formed by the coordinates (loads) of the secondary factors obtained as a result of rotation.) Rotation occurs:

orthogonal
oblique.

In the first type of rotation, each subsequent factor is determined in such a way as to maximize the variability remaining from the previous ones, so the factors turn out to be independent and uncorrelated from each other (PCA belongs to this type). The second type is a transformation in which factors are correlated with each other. The advantage of oblique rotation is the following: when it results in orthogonal factors, you can be sure that this orthogonality is really inherent in them, and not artificially introduced. There are about 13 methods of rotation in both types, five are available in the statistical program SPSS 10: three orthogonal, one oblique and one combined, but of all the most common is the orthogonal method " varimax" The varimax method maximizes the spread of squared loadings for each factor, resulting in larger and smaller factor loadings. As a result, a simple structure is obtained for each factor separately.

The main problem of factor analysis is the identification and interpretation of the main factors. When selecting components, the researcher usually faces significant difficulties, since there is no unambiguous criterion for identifying factors, and therefore subjectivity in the interpretation of the results is inevitable. There are several commonly used criteria for determining the number of factors. Some of them are alternative to others, and some of these criteria can be used together so that one complements the other:

Practice shows that if rotation does not produce significant changes in the structure of the factor space, this indicates its stability and the stability of the data. There are two more options: 1). strong redistribution of variance is the result of identifying a latent factor; 2). a very slight change (tenths, hundredths or thousandths of the load) or its absence at all, while only one factor can have strong correlations - a single-factor distribution. The latter is possible, for example, when several social groups are checked for the presence of a certain property, but only one of them has the desired property.

Factors have two characteristics: the amount of variance explained and loadings. If we consider them from the point of view of geometric analogy, then regarding the first we note that the factor lying along the OX axis can explain a maximum of 70% of the variance (the first main factor), the factor lying along the OU axis can determine no more than 30% (the second main factor). That is, in an ideal situation, all the variance can be explained by two main factors with the indicated shares. In a normal situation, two or more main factors may be observed, and there will also be a portion of uninterpretable variance (geometric distortions) that is excluded from the analysis due to insignificance. Loadings, again from the point of view of geometry, are projections from points on the OX and OU axes (with a three or more factor structure also on the OZ axis). Projections are correlation coefficients, points are observations, so factor loadings are measures of association. Since a correlation with a Pearson coefficient R ≥ 0.7 is considered strong, only strong connections should be paid attention to in the loads. Factor loadings can have the property bipolarity- the presence of positive and negative indicators in one factor. If bipolarity is present, then the indicators included in the factor are dichotomous and are in opposite coordinates.

Factor analysis methods:

Notes

Literature

Afifi A., Eisen S. Statistical analysis: Computer approach. - M.: Mir, 1982. - P. 488.
Colin Cooper. Individual differences. - M.: Aspect Press, 2000. - 527 p.
Gusev A. N., Izmailov Ch. A., Mikhalevskaya M. B. Measurement in psychology. - M.: Smysl, 1997. - 287 p.
Mitina O. V., Mikhailovskaya I. B. Factor analysis for psychologists. - M.: Educational and methodological collector Psychology, 2001. - 169 p.
Factor, discriminant and cluster analysis / collection of works, ed. Enyukova I. S.- M.: Finance and Statistics, 1989. - 215 p.
Patsiorkovsky V.V., Patsiorkovskaya V.V. SPSS for sociologists. - M.: Textbook of ISEPN RAS, 2005. - 433 p.
Büül A., Zöfel P. SPSS: The Art of Information Processing. Analysis of statistical data and recovery of hidden patterns. - St. Petersburg: DiaSoftYUP LLC, 2002. - 603 p.
Factor, discriminant and cluster analysis: Transl.

F18 from English/J.-O. Kim, C. W. Mueller, W. R. Klekka, etc.; Ed. I. S. Enyukova. - M.: Finance and Statistics, 1989.- 215 p.:

Links

Electronic textbook StatSoft. Principal components and factor analysis
Nonlinear principal component method (library website)

Wikimedia Foundation.

2010.

See what “Factor analysis” is in other dictionaries: factor analysis - - factor analysis The field of mathematical statistics (one of the sections of multivariate statistical analysis), combining computational methods that in some cases allow ...

Technical Translator's Guide Statistical method for testing hypotheses about the influence of diff. factors on the random variable being studied. A model has been developed and generally accepted in which the influence of a factor is presented in linear form. The analysis procedure is reduced to evaluation operations using...

See what “Factor analysis” is in other dictionaries: Geological encyclopedia - (from Latin factor active, producing and Greek analysis decomposition, division) a method of multidimensional mathematical statistics (see statistical methods in psychology), used in the study of statistically related characteristics with the aim of ... ...

Great psychological encyclopedia A method for studying economics and production, which is based on an analysis of the impact of various factors on the results of economic activity and its effectiveness. Raizberg B.A., Lozovsky L.Sh., Starodubtseva E.B.. Modern economic ...

Factor analysis Economic dictionary - the field of mathematical statistics (one of the sections of multivariate statistical analysis), combining computational methods, which in some cases make it possible to obtain a compact description of the phenomena under study based on... ...

Economic and mathematical dictionary FACTOR ANALYSIS, in statistics and psychometrics, a mathematical method by which a large number of measurements and studies is reduced to a small number of “factors” that fully explain the research results, as well as their... ...

Scientific and technical encyclopedic dictionary Multivariate statistical analysis section (See Multivariate statistical analysis). combining methods for estimating the dimension of a set of observed variables by examining the structure of covariance or correlation matrices.... ...