Factor analysis of what to do with cross-loadings. Factor analysis of profit. Example of factor analysis of sales profit

The main types of models used in financial analysis and forecasting.

Before we start talking about one of the types financial analysis - factor analysis, recall what financial analysis is and what are its goals.

The financial analysis is a method for assessing the financial condition and performance of an economic entity based on the study of the dependence and dynamics of indicators financial statements.

Financial analysis serves several purposes:

assessment of the financial position;
identification of changes in the financial condition in space and time;
identification of the main factors that caused changes in the financial condition;
forecast of the 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.

Each type of financial analysis is based on the use of a model that makes it possible to assess 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 basic for assessing the financial condition of an enterprise. These include: building a balance sheet system, presenting financial statements in various analytical sections, vertical and horizontal analysis of reporting, a system of analytical ratios, analytical notes to reporting. All of these models are based on the use of accounting information.

At the heart of vertical analysis lies a different presentation of financial statements - in the form of relative values \u200b\u200bcharacterizing the structure of the summarizing final indicators. An obligatory element of the analysis is the time series of these values, which makes it possible to track and predict structural changes in the composition of household assets and the sources of their coverage.

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

Analytical coefficient system - 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;
financial stability indicators;
business activity indicators;
indicators of profitability.

Predictive models Are predictive models. They are used to predict the income of an enterprise and its future financial condition. The most common of them are: calculating the point of critical sales volume, building predictive financial reports, dynamic analysis models (rigidly deterministic factor models and regression models), situational analysis models.

Regulatory models. Models of this type allow you to compare the actual results of the activities of enterprises with the expected ones calculated according to the budget. These models are used primarily in internal financial analysis. Their essence boils down to the establishment of standards for each item of expenditure on 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 rigidly deterministic factor models.

As we can 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.

Basics of modeling.

The functioning of any socio-economic system (which includes an operating enterprise) takes place in the context of a complex interaction of a complex of internal and external factors. Factor - This is the reason, driving force any process or phenomenon that determines its nature or one of its main features.

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

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

The factors investigated in the analysis can be classified according to different criteria.

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

Natural factors have a large impact 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 wellness work at enterprises with hazardous production, the general level of training, etc. They contribute to a more complete use of the production resources of the enterprise and increase the efficiency of its work.

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

By the degree of impact on the results economic activity factors are divided into major and minor. The main factors include factors that have a decisive impact on the performance indicator. Secondary are those that do not have a decisive impact on the results of economic activities in the current environment. It should be noted that, depending on the circumstances, one and the same factor can be both primary and secondary. The ability to single out the main factors from the whole set of factors ensures the correctness of the conclusions based on the analysis results.

Factors are divided into internal and external, depending on whether the activity affects them this enterprise or not. The analysis focuses on internal factors that the enterprise can influence.

Factors are classified into objectivenot dependent on the will and desires of people, and subjectiveaffected by the activities of legal entities and individuals.

In terms 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 particular enterprise.

In the course of the organization's work, some factors affect the studied indicator continuously throughout the entire time. Such factors are called permanent... Factors whose influence manifests itself periodically are called variables (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 by the nature of their action into intense and extensive... Extensive factors include factors that are associated with a change in the quantitative rather than qualitative characteristics of the functioning of the enterprise. An example is the increase in production by increasing the number of workers. Intensive factors characterize the quality side of the production process. An example is the increase in the volume of production by increasing the level of labor productivity.

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

According to the level of subordination (hierarchy), factors of the first, second, third and subsequent levels of subordination are distinguished. TO factors of the first level include those that directly affect the performance indicator. The factors influencing the effective indicator indirectly, using the factors of the first level, are called second-tier factors etc.

It is clear that when studying the influence on the operation of an enterprise of any group of factors, it is necessary to streamline them, that is, to conduct an analysis taking into account their internal and external relations, interaction and subordination. This is achieved through systematization. Systematization is the placement of the studied phenomena or objects in a certain order with the identification of their relationship and subordination.

Creature 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. The connection of economic phenomena is a joint change of 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 are indirectly. For example, the value of gross output is directly influenced by such factors as the number of workers and the level of their labor productivity. Many other factors indirectly affect this indicator.

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

A quantitative characteristic of interrelated phenomena is carried out using indicators. Indicators characterizing the cause are called factorial (independent); the indicators characterizing the effect are called effective (dependent). The set of factorial and effective signs associated with a cause-and-effect relationship is called factor system.

Modeling any phenomenon is the construction of a mathematical expression of the existing dependence. 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 connection is called functional, or rigidly determined, if each value of a factor attribute corresponds to a well-defined non-random value of a productive attribute.

A relationship is called stochastic (probabilistic) if each value of the factor attribute corresponds to a set of values \u200b\u200bof the effective attribute, i.e., a certain statistical distribution.

Model the factor system is a mathematical formula that expresses the real connections between the analyzed phenomena. IN general view it can be represented like this:

where is the effective sign;

Factorial signs.

Thus, each performance indicator is influenced by numerous and varied factors. At the heart of economic analysis and its section - factor analysis - are the identification, assessment and forecasting of the influence of factors on the change in the effective indicator. The more detailed the dependence of the effective indicator on certain factors is investigated, the more accurate the results of the analysis and the assessment of the quality of the enterprises' work. Without a deep and comprehensive study of the factors, it is impossible to draw reasonable conclusions about the results of activities, identify production reserves, justify plans and management decisions.

Factor analysis, its types and tasks.

Under factor analysis the method of complex and systematic study and measurement of the impact of factors on the value of effective indicators is understood.

IN general case the following the main stages of factor analysis:

Analysis goal setting.
Selection of factors that determine the studied performance indicators.
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.
Determination of the form of dependence between factors and performance indicators.
Modeling the relationship between performance and factor indicators.
Calculation of the influence of factors and assessment of the role of each of them in changing the value of the effective indicator.
Working with a factorial model (its practical use for managing economic processes).

Selection of factors for analysisthis or that 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 is investigated, the more accurate the results of the analysis will be. At the same time, it should be borne in mind that if this complex of factors is considered as a mechanical sum, without taking into account their interaction, without highlighting the main, determining ones, then the conclusions may be erroneous. In the analysis of economic activity (ACA), an interconnected study of the influence of factors on the value of effective 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 determination of the form of dependence between factors and performance indicators: functional or stochastic, direct or inverse, straight or curvilinear. It uses theoretical and practical experience, as well as ways of comparing parallel and dynamic series, analytical groupings of initial information, graphical, etc.

Modeling economic indicators also presents 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 AHD. To determine the influence of factors on the final indicators, many methods are used, which will be discussed in more detail below.

The last step of factor analysis is practical use of the factor model for calculating the reserves for the growth of the effective indicator, for planning and forecasting its value when the situation changes.

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

is a methodology for studying the influence of factors, the relationship of which with the effective indicator is of a functional nature, that is, when the effective indicator of the factor model is presented as a product, a quotient or an algebraic sum of factors.

This type of factor analysis is the most common, since, being quite simple to use (in comparison with stochastic analysis), it allows you to understand the logic of the main factors of enterprise development, quantify their influence, understand which factors and in what proportion can and should be changed to increase production efficiency. We will consider in detail deterministic factor analysis in a separate chapter.

Stochastic Analysis is a technique for studying factors, the connection of which with the effective indicator, in contrast to the functional one, is incomplete, probabilistic (correlation). If, with a functional (complete) dependence with a change in the argument, a corresponding change in the function always occurs, then with a correlation connection, a change in the argument can give several values \u200b\u200bof the increase in the function, depending on a combination of other factors that determine this indicator. For example, labor productivity at the same level of capital-labor ratio may not be the same at different enterprises. It depends on the optimal combination of other factors affecting this indicator.

Stochastic modeling is, to a certain extent, an addition 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 rigidly deterministic 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 rigidly deterministic model;
it is necessary to study the influence of complex factors that cannot be expressed by one quantitative indicator (for example, the level of scientific and technological progress).

In contrast to the rigidly determined stochastic approach, implementation requires a number of prerequisites:

the presence of the aggregate;
sufficient amount of observations;
randomness and independence of observations;
uniformity;
the presence of a distribution of signs close to normal;
the presence of a special mathematical apparatus.

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

qualitative analysis (setting the goal of the analysis, determining the population, determining the effective and factor indicators, 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 the distribution laws of the studied indicators);
building a stochastic (regression) model (clarifying the list of factors, calculating estimates of the parameters of the regression equation, enumerating 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 correspondence of the formal properties of the estimates to the research tasks);
economic interpretation and practical use of the model (determination of the space-time stability of the constructed dependence, assessment of the practical properties of the model).

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

forward and backward;
single-stage and multi-stage;
static and dynamic;
retrospective and prospective (forecast).

When direct factor analysisresearch is carried out in a deductive way - from the general to the particular. Inverse factor analysiscarries out the study of cause-and-effect relationships by means of logical induction - from particular, individual factors to generalizing ones

Factor analysis can be single-stage and multistage... The first type is used to study factors of only one level (one level) of subordination without their detailing into their component parts. For instance, . In multi-stage factor analysis, factors are detailed a and b into building blocks in order to study their behavior. The detailing of the factors can be continued further. In this case, the influence of factors of various levels of subordination is studied.

It is also necessary to distinguish staticand dynamic factor analysis. The first type is used when studying the influence of factors on performance indicators at the corresponding date. Another type is a technique for studying causal relationships in dynamics.

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

Deterministic factor analysis.

Deterministic Factor Analysis has a fairly strict sequence of procedures:

construction of an economically sound deterministic factor model;
selection of the method of factor analysis and preparation of conditions for its implementation;
implementation of counting procedures for model analysis;
formulation of conclusions and recommendations based on the results of the analysis.

The first stage is especially important, since an incorrectly constructed model can lead to logically unjustified results. The meaning of this stage is as follows: any extension of a rigidly determined factor model should not contradict the logic of the “cause - effect” relationship. As an example, consider a model linking sales (P), headcount (H) and labor productivity (PT). Three models can theoretically be explored:

All three formulas are correct from the point of view of arithmetic, however, from the point of view of factor analysis, only the first makes sense, since in it the indicators on the right side of the formula are factors, that is, the cause that generates and determines the value of the indicator on the left side (consequence ).

At the second stage, one of the methods of factor analysis is selected: integral, chain substitutions, logarithmic, etc. Each of these methods has its own advantages and disadvantages. We will consider a brief comparative description of these methods below.

Types of deterministic factor models.

The following deterministic analysis models exist:

additive model, that is, a model in which factors are included in the form of an algebraic sum, as an example, we can give the model of the balance of goods:

where R - implementation;

Stocks at the beginning of the period;

P - receipt of goods;

Stocks at the end of the period;

IN - other disposal of goods;

multiplicative model, that is, a model in which factors are included in the form of a product; an example is the simplest two-factor model:

where R - implementation;

H - number;

PT - labor productivity;

multiple model, that is, a model that is a ratio of factors, for example:

where is the capital-labor ratio;

H - number;

mixed model, i.e., a model in which factors are included in various combinations, for example:

where R - implementation;

Profitability;

OS - the cost of fixed assets;
About - the cost of working capital.

A rigidly deterministic model with more than two factors is called multifactorial.

Typical tasks of deterministic factor analysis.

In deterministic factor analysis, four typical tasks can be distinguished:

Assessment of the influence of the relative change in factors on the relative change in the effective indicator.
Assessment of the influence of the absolute change in the i-th factor on the absolute change in the effective indicator.
Determination of the ratio of the magnitude of the change in the effective indicator caused by the change in the i-th factor to the base value of the effective indicator.
Determination of the share of the absolute change in the effective indicator caused by the change in the i-th factor in the overall change in the effective indicator.

Let us characterize these problems and consider the solution to each of them using a specific simple example.

Example.

The volume of gross output (GP) depends on two main factors of the first level: the number of employees (HR) and the average annual output (GW). We have a two-factor multiplicative model:. Consider a situation when both the output and the number of workers in the reporting period deviated from the planned values.

The data for calculations are shown in Table 1.

Table 1. Data for factor analysis of gross output.

Objective 1.

The problem makes sense for multiplicative and multiple models. Let's consider the simplest two-factor model. Obviously, when analyzing the dynamics of these indicators, the following relationship between the indices will be fulfilled:

where the index value is the ratio of the indicator value in the reporting period to the base one.

Let's calculate the indices of gross output, number of employees and average annual output for our example:

;

According to the above rule, the index of gross output is equal to the product of the indices of the number of employees and the average annual output, i.e.

Obviously, if we calculate directly the index of gross production, we will get the same value:

We can conclude that as a result of an increase in the number of employees by 1.2 times and an increase in average annual output by 1.25 times, the volume of gross output increased by 1.5 times.

Thus, the relative changes in the factorial and effective indicators are related by the same relationship as the indicators in the original model. This problem is solved by answering questions like: "What will happen if i-th indicator will change by n%, and j-th exponent change to k%? ".

Objective 2.

Is an main task deterministic factor analysis; its general setting is:

Let be - rigidly deterministic model characterizing the change in the effective indicator y from n factors; all indicators were incremented (for example, in dynamics, in comparison with the plan, in comparison with the standard):

It is required to determine what part of the increment in the effective indicator y is obliged to increment the i-th factor, i.e., to describe the following dependence:

where is the general change in the effective indicator, which is formed under the simultaneous influence of all factor signs;

Change in the effective indicator under the influence of only a factor.

Factor expansions may differ depending on which method of model analysis is chosen. Therefore, we will consider in the context of this problem the main methods of analysis of factor models.

Basic methods of deterministic factor analysis.

One of the most important methodological in the AHD is to determine the magnitude of the influence of individual factors on the growth of effective indicators. Deterministic factor analysis (DFA) uses the following methods 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. To eliminate means to eliminate, reject, exclude the influence of all factors on the value of the effective indicator, except for one. This method assumes that all factors change independently of each other: first one changes, and all the others remain unchanged, then two change, then three, etc., while the rest remain unchanged. This allows you to determine the influence of each factor on the value of the studied indicator separately.

Let's give a brief description of the most common methods.

The chain substitution method is a very simple and intuitive method, the most versatile of all. It is used to calculate the influence of factors in all types of deterministic factor models: additive, multiplicative, multiple and mixed. This method allows you to determine the influence of individual factors on the change in the value of the effective indicator by gradually replacing the base value of each factor indicator in the volume of the effective indicator for the actual one in the reporting period. For this purpose, a number of conditional values \u200b\u200bof the effective indicator are determined, which take into account the change in one, then two, then three, etc. factors, assuming that the rest do not change. Comparison of the value of the effective indicator before and after the change in the level of one or another factor allows you to determine the impact specific factor on the growth of the effective indicator, excluding the influence of other factors. Complete decomposition is achieved using this method.

Recall that when using this method, the sequence of changing the values \u200b\u200bof factors is of great importance, since the quantitative assessment of the influence of each factor depends on it.

First of all, it should be noted that there is not and cannot exist a single method for determining this order - there are models in which it can be determined arbitrarily. For only a small number of models, formalized approaches can be used. In practice, this problem is not very important, since in retrospective analysis, trends and the relative importance of a particular factor are important, and not accurate estimates of their impact.

Nevertheless, in order to maintain a more or less uniform approach to determining the order of replacement of factors in the model, general principles can be formulated. Let's introduce some definitions.

A feature that is directly related to the phenomenon under study and characterizes its quantitative side is called primary or quantitative... These signs are: a) absolute (volumetric); b) they can be summed up in space and time. As an example, we can cite the volume of sales, the number, the cost of working capital, etc.

Signs related to the studied phenomenon not directly, but through one or more other signs and characterizing the qualitative side of the studied phenomenon are called secondary or quality... These signs are: a) relative; b) they cannot be summed up in space and time. Examples are capital-labor ratio, profitability, etc. In the analysis, secondary factors of the 1st, 2nd, etc. orders are distinguished, obtained by successive detailing.

A rigidly deterministic factor model is called complete if the effective indicator is quantitative, and incomplete if the effective indicator is qualitative. In a full two-factor model, one factor is always quantitative, the other is qualitative. In this case, it is recommended to start replacing factors with a quantitative indicator. If there are several quantitative and several qualitative indicators, then first the value of the factors of the first level of subordination should be changed, and then the lower one. Thus, the application of the method of chain substitution requires knowledge of the relationship of factors, their subordination, the ability to correctly classify and systematize them.

Now let's consider using our example the order of applying the chain substitution method.

The algorithm for calculating by the method of chain substitution for this model is as follows:

As you can see, the second indicator of gross production differs from the first in that the actual number of workers was taken instead of the planned one. The average annual output by one worker in either case is planned. This means that due to the increase in the number of workers, production increased by 32,000 million rubles. (192,000 - 160,000).

The third indicator differs from the second in that when calculating its value, the output of workers is taken at the actual level instead of the planned one. The number of employees in both cases is actual. Hence, due to the increase in labor productivity, the volume of gross output increased by 48,000 million rubles. (240,000 - 192,000).

Thus, the overfulfillment of the plan in terms of gross output was the result of the influence of the following factors:

Algebraic sum of factors when using this method must be equal to the overall increase in the effective indicator:

The absence of such equality indicates the errors made in the calculations.

Other methods of analysis, such as integral and logarithmic, allow to achieve a higher accuracy of calculations, but these methods have a more limited scope and require a large amount of computation, which is inconvenient for real-time analysis.

Objective 3.

It is, in a certain sense, a consequence of the second typical problem, since it is based on the obtained factor expansion. The need to solve this problem is due to the fact that the elements of factorial expansion are absolute values \u200b\u200bthat are difficult to use for space-time comparisons. When solving the problem, the 3 factor decomposition is supplemented with relative indicators:

Economic interpretation: the coefficient shows by what percentage of the baseline the effective indicator has changed under the influence of the i-th factor.

Let's calculate the coefficients α for our example, using the factorization obtained earlier by the method of chain substitutions:

;

Thus, the volume of gross output increased by 20% due to an increase in the number of workers and by 30% due to an increase in output. The total increase in gross production was 50%.

Problem 4.

It is also solved on the basis of basic task 2 and is reduced to the calculation of indicators:

Economic interpretation: the coefficient shows the proportion of the increase in the effective indicator due to the change in the i-th factor. The question does not arise here if all factorial features change unidirectionally (either increase or decrease). If this condition is not met, the solution of the problem may be complicated. In particular, in the simplest two-factor model, in such a case, the calculation according to the above formula is not performed and it is believed that 100% of the increase in the effective indicator is due to a change in the dominant factor attribute, that is, the attribute changing unidirectionally with the effective indicator.

Let's calculate the coefficients γ for our example, using the factorization obtained by the method of chain substitutions:

Thus, the increase in the number of employees accounted for 40% of the total increase in gross output, and the increase in output - 60%. This means that the increase in production in this situation is the determining factor.

Called factor analysis... The main types of factor analysis are deterministic analysis and stochastic analysis.

Deterministic Factor Analysis is based on the method of studying the influence of such factors, the relationship of which with the generalizing economic indicator is functional. The latter means that the generalizing indicator is either a product, or a quotient of division, or an algebraic sum of individual factors.

Stochastic factor analysis is based on the methodology for studying the influence of such factors, the relationship of which with the generalizing economic indicator is probabilistic, otherwise - correlation.

In the presence of a functional relationship with a change in the argument, there is always a corresponding change in the function. If there is a probabilistic relationship, the change in the argument can be combined with several values \u200b\u200bof the change in the function.

Factor analysis is also subdivided into straight, otherwise deductive analysis and back (inductive) analysis.

First type of analysis carries out the study of the influence of factors by the deductive method, that is, in the direction from the general to the particular. Inverse factor analysis the influence of factors is investigated by the inductive method - in the direction from particular factors to generalizing economic indicators.

Classification of factors affecting the effectiveness of the organization

The factors, the influence of which is studied during the conduct, are classified according to various characteristics. First of all, they can be divided into two main types: internal factorsdepending on the activity of this, and external factorsnot dependent on this organization.

Internal factors depending on the magnitude of their impact on, it can be subdivided into major and minor. The main factors include factors related to the use, and materials, as well as factors due to supply and marketing activities and some other aspects of the organization. The main factors have a fundamental impact on the generalized economic indicators. External factors that do not depend on this organization are due to natural and climatic (geographical), socio-economic, as well as external economic conditions.

Depending on the duration of their impact on economic indicators, one can distinguish constant and variable factors... The first type of factors affects economic indicators, which is not limited in time. Variable factors affect economic performance only for a certain period of time.

Factors can be classified into extensive (quantitative) and intensive (qualitative) on the basis of the essence of their influence on economic indicators. So, for example, if the influence of labor factors on the volume of output is studied, then a change in the number of workers will be an extensive factor, and a change in the productivity of one worker will be an intensive factor.

Factors affecting economic indicators, according to the degree of their dependence on the will and consciousness of employees of the organization and other persons, can be divided into objective and subjective factors... Objective factors can include weather conditions, natural disasters that do not depend on human activities. Subjective factors are entirely dependent on people. The overwhelming majority of factors should be classified as subjective.

Factors can also be subdivided, depending on their scope, into factors of unlimited and factors of limited action. The first type of factors operates everywhere, in all sectors of the national economy. The second type of factors affects only within an industry or even an individual organization.

By their structure, factors are divided into simple and complex. The overwhelming majority of factors are complex, including several components. At the same time, there are also factors that do not lend themselves to dismemberment. For example, return on assets is an example of a complex factor. The number of days the equipment worked during a given period is a simple factor.

By the nature of the impact on the generalizing economic indicators, there are direct and indirect factors... So, the change in the sold products, although it has the opposite effect on the amount of profit, should be considered direct factors, that is, a factor of the first order. The change in the value material costs has an indirect effect on profit, i.e. affects profit not directly, but through the cost, which is a factor of the first order. Based on this, the level of material costs should be considered a second-order factor, that is, an indirect factor.

Depending on whether it is possible to quantify the influence of this factor on the generalizing economic indicator, distinguish between measurable and non-measurable factors.

This classification is closely interconnected with the classification of reserves for increasing the efficiency of economic activities of organizations, or, in other words, reserves for improving the analyzed economic indicators.

Factor economic analysis

The signs that characterize the cause are called factorial, independent. The same signs that characterize the effect are usually called resultant, dependent.

The set of factorial and effective features that are in one causal relationship is called factor system... There is also the concept of a factor system model. It characterizes the relationship between the effective trait, denoted as y, and factorial features, denoted as. In other words, the factor system model expresses the relationship between the generalizing economic indicators and individual factors that affect this indicator. In this case, other economic indicators act as factors, which are the reasons for the change in the generalizing indicator.

Factor system model can be mathematically expressed using the following formula:

Establishing relationships between generalizing (effective) and factors influencing them is called economic and mathematical modeling.

In examines two types of relationships between generalizing indicators and factors influencing them:

functional (otherwise - functional-deterministic, or rigidly deterministic connection.)
stochastic (probabilistic) connection.

Functional connection - this is such a relationship, in which each value of a factor (factor attribute) corresponds to a well-defined non-random value of the generalizing indicator (effective attribute).

Stochastic link - this is such a relationship, in which each value of a factor (factor attribute) corresponds to a set of values \u200b\u200bof a generalizing indicator (effective attribute). Under these conditions, for each value of the factor x, the values \u200b\u200bof the generalizing indicator y form a conditional statistical distribution. As a result, a change in the value of the factor x only on average causes a change in the generalizing indicator y.

In accordance with the two types of relationships considered, the methods of deterministic factor analysis and methods of stochastic factor analysis are distinguished. Consider the following diagram:

Methods used in factor analysis. Scheme No. 2

The greatest completeness and depth of analytical research, the greatest accuracy of the analysis results are provided by the use of economic and mathematical research methods.

These methods have several advantages over traditional and statistical analysis methods.

So, they provide a more accurate and detailed calculation of the influence of individual factors on the change in the values \u200b\u200bof economic indicators and also make it possible to solve a number of analytical problems that cannot be done without the use of economic and mathematical methods.

For the convenience of studying the material, we divide the article into topics:

P cr \u003d V och * (U cr och. -U cr. Base.) / 100
At the center of the report. and bases - columns 6 and 7.

5. Calculation of the factor "management costs"

Pupp. \u003d Watch. * (Uuro-U urb) / 100
Where Uuro and U ur are, respectively, the levels of management costs in the reporting and base periods

6. Calculation of the aggregate influence of all factors on the profit from sales

The total amount must be equal to the absolute deviation in line 050 of Form No. 2 (column 5). If this is not the case, then the calculations are wrong and further analysis is meaningless.

Factor analysis can be continued until net income. The technique is as follows:

1. According to the given scheme, the profit from sales is analyzed.
2. The influence of all other factors (operating income, expense, etc.) is assessed in column 5 in the table above.

Factor analysis methods

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

Factor analysis in educational literature is interpreted as a section of multivariate statistical analysis that combines methods for assessing the dimension of the set of observed variables by examining the structure of covariance or correlation matrices.

Factor analysis begins its history in psychometrics and is now widely used not only in psychology, but also in neurophysiology, sociology, political science, economics, statistics and other sciences. The main ideas of factor analysis were laid by the English psychologist and anthropologist F. Galton. The development and implementation of factor analysis in psychology was carried out by such scientists as: Charles Spearman, L. Thurstone and R. Kettel. Mathematical factor analysis was developed by Hotelling, Harman, Kaiser, Thurstone, Tucker and other scientists.

This type of analysis allows the researcher to solve two main tasks: to describe the subject of measurement in a compact and at the same time comprehensively. With the help of factor analysis, it is possible to identify the factors responsible for the presence of linear statistical relationships of correlations between the observed variables.

For example, when analyzing estimates obtained on several scales, the researcher notes that they are similar to each other and have a high correlation coefficient, in this case, he can assume that there is some latent variable that can explain the observed similarity of the estimates. Such a latent variable is called a factor that affects numerous indicators of other variables, which leads to the possibility and the need to mark it as the most general, higher order.

Thus, there are two goals of factor analysis:

Determination of relationships between variables, their classification, ie, "objective R-classification";
reducing the number of variables.

To identify the most significant factors and, as a consequence, the factor structure, it is most justified to use the method of principal components. The essence of this method is to replace correlated components with uncorrelated factors. Another important characteristic of the method is the ability to limit itself to the most informative main components and exclude the rest from the analysis, which simplifies the interpretation of the results. The advantage of this method is also that it is the only mathematically substantiated method of factor analysis.

Factor analysis is a method of complex and systematic study and measurement of the impact of factors on the value of the effective indicator.

There are the following types of factor analysis:

1. Deterministic (functional) - the effective indicator is presented in the form of a product, a quotient or an algebraic sum of factors.
2. Stochastic (correlation) - the relationship between the effective and factor indicators is incomplete or probabilistic.
3. Direct (deductive) - from the general to the particular.
4. Reverse (inductive) - from particular to general.
5. Single-stage and multi-stage.
6. Static and dynamic.
7. Retrospective and forward-looking.

Also, factor analysis can be exploratory - it is carried out in the study of the hidden factor structure without the assumption of the number of factors and their loads and confirmatory, designed to test hypotheses about the number of factors and their loads. The practical implementation of factor analysis begins with checking its conditions.

Prerequisites for factor analysis:

All signs must be quantitative;
The number of features must be twice the number of variables;
The sample must be uniform;
The original variables must be distributed symmetrically;
Factor analysis is carried out on correlated variables.

In the analysis, strongly correlated variables are combined into one factor, as a result, the dispersion between the components is redistributed and the most simple and clear structure of factors is obtained. After combining, the correlation of the components within each factor with each other will be higher than their correlation with the components from other factors. This procedure also makes it possible to isolate latent variables, which is especially important in the analysis of social perceptions and values.

As a rule, factor analysis is carried out in several stages.

Factor analysis stages:

Stage 1. Selection of factors.
Stage 2. Classification and systematization of factors.
Stage 3. Modeling the relationship between performance and factor indicators.
Stage 4. Calculation of the influence of factors and assessment of the role of each of them in changing the value of the effective indicator.
Stage 5. Practical use of the factor model (calculation of reserves for the growth of the effective indicator).

By the nature of the relationship between the indicators, methods of deterministic and stochastic factor analysis are distinguished

Deterministic factor analysis is the influence of factors, the connection of which with the effective indicator is of a functional nature, that is, when the effective indicator of the factor model is presented as a product, quotient, or algebraic sum of factors.

Deterministic factor analysis methods: Method of chain substitutions; Method of absolute differences; Method of relative differences; Integral method; Logarithm method.

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

Stochastic analysis is a technique for studying factors, the connection of which with the effective indicator, in contrast to the functional one, is incomplete, probabilistic (correlation). If, with a functional (complete) dependence with a change in the argument, a corresponding change in the function always occurs, then with a correlation connection, a change in the argument can give several values \u200b\u200bof the increase in the function, depending on a combination of other factors that determine this indicator.

Stochastic factor analysis methods: - Pairwise correlation method;
- Multiple correlation analysis;
- Matrix models;
- Mathematical programming;
- Operations research method;
- Game theory.

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

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

Factor analysis of profitability

The main goal of any company is to find the optimal ones aimed at maximizing profits, the relative expression of which is profitability indicators. The advantages of using these indicators in the analysis are the ability to compare the performance not only within one company, but also the use of multidimensional several companies over a number of years. In addition, profitability indicators, like any relative indicators, are important characteristics of the factorial environment for the formation of companies' profits and income.

The problematic of using analytical procedures in this area lies in the fact that the authors propose different approaches to the formation of not only the basic system of indicators, but also indicators of profitability.

To analyze profitability, the following factor model is used:

R \u003d P / N, or
R \u003d (N - S) / N * 100
where P is profit; N - revenue; S is the cost.

In this case, the influence of the factor of change in the price of products is determined by the formula:

RN \u003d (N1 - S0) / N1 - (N0 - S0) / N0
Accordingly, the influence of the factor of cost price changes will be:
RS \u003d (N1 - S1) / N1 - (N1 - S0) / N1
The sum of the factorial deviations will give the total change in profitability for the period:
R \u003d RN + RS

Using this model, we will conduct a factor analysis of the profitability indicators of the production of hardware products by a conventional enterprise. To analyze and build a factor model, data are needed: on prices for products sold, sales volumes and the cost of production or sales of one unit. product.

Deterministic Factor Analysis

Deterministic modeling of factorial systems is limited by the length of the factorial field of direct links. With an insufficient level of knowledge about the nature of direct links of one or another indicator of economic activity, a different approach to the knowledge of objective reality is often needed. The range of quantitative changes in economic indicators can only be determined by stochastic analysis of massive empirical data.

In deterministic factor analysis, the model of the phenomenon under study does not change for economic objects and periods (since the ratios of the corresponding main categories are stable). If it is necessary to compare the results of the activities of individual farms or one farm in certain periods, only the question of the comparability of the quantitative analytical results identified on the basis of the model may arise.

Deterministic factor analysis is a technique for studying the influence of factors, the connection of which with the effective indicator is of a functional nature, i.e. can be expressed by mathematical dependence.

Deterministic models can be of different types: additive, multiplicative, multiple, mixed.

Factor analysis of the enterprise

The factors, the influence of which is studied in the analysis of economic activity, are classified according to various criteria. First of all, they can be divided into two main types: internal factors that depend on the activities of a given organization, and external factors that do not depend on this organization.

Internal factors, depending on the magnitude of their impact on economic performance, can be subdivided into major and minor. The main factors include factors related to the use, and materials, as well as factors due to supply and marketing activities and some other aspects of the organization. The main factors have a fundamental impact on the generalized economic indicators. External factors that do not depend on this organization are due to natural and climatic (geographical), socio-economic, as well as external economic conditions.

Depending on the duration of their impact on economic indicators, constant and variable factors can be distinguished. The first type of factors affects economic indicators, which is not limited in time. Variable factors affect economic performance only for a certain period of time.

Factors can be subdivided into extensive (quantitative) and intensive (qualitative) based on the nature of their impact on economic indicators. So, for example, if the influence of labor factors on the volume of output is studied, then a change in the number of workers will be an extensive factor, and a change in one worker will be an intensive factor.

Factors affecting economic indicators, according to the degree of their dependence on the will and consciousness of employees of the organization and other persons, can be divided into objective and subjective factors. Objective factors can include weather conditions, natural disasters that do not depend on human activities. Subjective factors are entirely dependent on people. The overwhelming majority of factors should be classified as subjective.

By the nature of the impact on the generalizing economic indicators, direct and indirect factors are distinguished. So, the change in the cost of goods sold, although it has the opposite effect on the amount of profit, should be considered direct factors, that is, a factor of the first order. A change in the value of material costs has an indirect effect on profit, i.e. affects profit not directly, but through the cost, which is a factor of the first order. Based on this, the level of material costs should be considered a second-order factor, that is, an indirect factor.

Depending on whether it is possible to quantify the influence of this factor on the generalizing economic indicator, there are measurable and non-measurable factors.

Factor Analysis Models

Suppose you are doing a (somewhat "stupid") study that measures the height of a hundred people in inches and centimeters. Thus, you have two variables. If you want to further investigate, for example, the effects of different nutritional supplements on growth, will you continue to use both variables? Probably not, because height is one characteristic of a person, no matter what units it is measured in.

Now suppose you want to measure people's satisfaction with their lives, for which you compose a questionnaire with various items; Among other questions, ask the following: are people satisfied with their hobby (item 1) and how intensively they practice it (item 2). The results are converted so that the average answers (for example, for satisfaction) correspond to the value of 100, while the lower and higher values \u200b\u200bare located below and above the average answers, respectively. Two variables (answers to two different items) are correlated with each other. (If you are not familiar with the concept of correlation coefficient, we recommend that you refer to the section Basic statistics and tables - Correlations). From the high correlation of these two variables, it can be concluded that two items of the questionnaire are redundant.

Combining two variables into one factor. The relationship between variables can be discovered using a scatter plot. The resulting fit line gives a graphical representation of the relationship. If you define a new variable based on the regression line shown in this diagram, then such a variable will include the most significant features of both variables. So, in fact, you have reduced the number of variables and replaced two with one. Note that the new factor (variable) is actually a linear combination of the two original variables.

Principal component analysis. An example in which two correlated variables are combined into one factor shows the main idea of \u200b\u200bthe factor analysis model or, more precisely, principal component analysis (this difference will be discussed later). Extending the two-variable example to more variables makes the calculations more complicated, but the basic principle of representing two or more dependent variables as a single factor remains valid.

Selection of the main components. Basically, the procedure for identifying the principal components is similar to a rotation that maximizes the variance (varimax) of the original variable space. For example, in a scatterplot, you can view the regression line as the x-axis by rotating it so that it matches the regression line. This type of rotation is called variance-maximizing rotation, since the criterion (goal) of rotation is to maximize the variance (variability) of the "new" variable (factor) and minimize the spread around it (see Rotation Strategies).

Generalization to the case of many variables. In the case where there are more than two variables, they can be considered to define a three-dimensional "space" in the same way that two variables define a plane. If you have three variables, you can build a 3M scatterplot.

For the case of more than three variables, it becomes impossible to represent the points on the scatter diagram, however, the logic of rotating the axes in order to maximize the variance of the new factor remains the same.

Several orthogonal factors. After you have found the line for which the variance is greatest, some scatter of data remains around it. And it is natural to repeat the procedure. In principal component analysis, this is exactly what is done: after the first factor is selected, that is, after the first line is drawn, the next line is determined that maximizes the residual variation (the spread of data around the first straight line), etc. Thus, the factors are sequentially highlighted one by one. Since each subsequent factor is determined in such a way as to maximize the variability remaining from the previous ones, the factors turn out to be independent of each other. In other words, uncorrelated or orthogonal.

How Many Factors Should be Highlighted Recall that principal component analysis is a method of reducing or reducing data, i.e. by reducing the number of variables. A natural question arises: how many factors should be selected. Note that in the process of successive selection of factors, they include less and less variability. The decision as to when to stop the factoring procedure mainly depends on the point of view of what counts as small "random" variability.

Overview of Principal Component Analysis Results. Let's now look at some of the standard results of principal component analysis. With repeated iterations, you isolate factors with less and less variance. For simplicity of presentation, we assume that usually work begins with a matrix in which the variances of all variables are equal to 1.0. Therefore, the total variance is equal to the number of variables. For example, if you have 10 variables, each with a variance of 1, then the largest variability that can potentially be isolated is 10 times 1. Suppose that in your life satisfaction survey, you included 10 items to measure various aspects of home life satisfaction and work.

Eigenvalues. In the second column (Eigenvalues) of the results table, you can find the variance of the new, just selected factor. In the third column, for each factor, the percentage of the total variance (in this example, 10) is given for each factor. As you can see, the first factor (value 1) explains 61 percent of the total variance, factor 2 (value 2) explains 18 percent, and so on. The fourth column contains the cumulative or cumulative variance. The variances allocated by the factors are called eigenvalues. This name comes from the method of calculation used.

Eigenvalues \u200b\u200band the problem of the number of factors. Once you know how much variance each factor has highlighted, you can return to the question of how many factors should be kept. As stated above, this decision is arbitrary in nature. However, there are some commonly used guidelines and in practice, following them gives the best results.

Kaiser criterion. At first, you can select only factors with eigenvalues \u200b\u200bgreater than 1. Essentially, this means that if a factor does not select a variance equivalent to at least the variance of one variable, then it is omitted. This criterion was proposed by Kaiser (1960) and is probably the most widely used. In the example above, based on this criterion, you should only store 2 factors (two principal components).

Scree criterion. The scree criterion is a graphical method first proposed by Cattell (1966). You can plot the eigenvalues \u200b\u200bpresented in the table earlier as a simple graph.

Cattel suggested finding a place on the graph where the decrease in eigenvalues \u200b\u200bfrom left to right slows down as much as possible. It is assumed that only "factorial talus" is located to the right of this point - "talus" is a geological term for debris accumulating in the lower part of the rocky slope. In accordance with this criterion, 2 or 3 factors can be left in this example.

What criterion should be used. Both criteria have been studied in detail by Browne (1968), Cattell and Jaspers (1967), Hakstian, Rogers, Cattell (1982), Lynn (Linn, 1968), Tucker, Koopman and Lynn. (Tucker, Koopman, Linn, 1969). In theory, you can calculate their characteristics by generating random data for a specific number of factors. Then you can see whether a sufficiently accurate number of significant factors was found using the criterion used or not. Using this general method, the first criterion (the Kaiser criterion) sometimes retains too many factors, while the second criterion (the scree criterion) sometimes retains too few factors; however, both criteria are quite good under normal conditions when there are relatively few factors and many variables. In practice, an important additional question arises, namely: when the obtained solution can be meaningfully interpreted. Therefore, usually several solutions with more or fewer factors are investigated, and then the one most "meaningful" is selected. This issue will be further considered in terms of factor rotations.

Analysis of the main factors. Before continuing to consider the various aspects of the derivation of principal component analysis, we introduce a principal factor analysis. Let's go back to the example of a life satisfaction questionnaire to formulate another "thinkable model". You can imagine that subjects' responses depend on two components. First, select some relevant general factors, such as, for example, "satisfying your hobbies" discussed earlier. Each item measures some part of this general aspect of satisfaction. In addition, each item includes a unique aspect of satisfaction not found in any other item.

Communities. If this model is correct, then you cannot expect the factors to contain all the variance in the variables; they will contain only the part that belongs to common factors and is distributed over several variables. In the language of the factor analysis model, the proportion of the variance of an individual variable belonging to common factors (and shared with other variables) is called common. Therefore, the additional work facing the researcher when applying this model is to assess the commonality for each variable, i.e. the proportion of variance that is common to all items. The fraction of the variance for which each item is responsible is then equal to the total variance corresponding to all variables, minus the generality. From a general point of view, the multiple correlation coefficient of the selected variable with all others should be used as an estimate of generality (for information on multiple regression theory, refer to the Multiple Regression section). Several authors have proposed various iterative “post-solution improvements” to the initial multiple regression generality score; for example, the so-called MINRES method (method of minimum factorial residuals; Harman and Jones (Harman, Jones, 1966)), which tests various modifications of factor loadings in order to minimize residual (unexplained) sums of squares.

Major factors versus major components. Major factors versus major components. The main difference between the two factor analysis models is that principal component analysis assumes that all the variability of the variables should be used, whereas in the analysis of the main factors you only use the variability of the variable, common to other variables. A detailed discussion of the pros and cons of each approach is outside the scope of this introduction. In most cases, these two methods lead to very similar results. However, principal component analysis is often preferred as a data reduction technique, while principal factor analysis is best used to determine the structure of the data (see next section).

Factor analysis of sales

Similarly, we will derive models for factor analysis of sales profitability.

The initial indicator is as follows:

RPr \u003d Prp / RP \u003d SRP - Srp) / RP.

Change in profitability of sales under the influence of relevant factors:

Lrpr \u003d Prp1 / RP1- PrnO / RP0 \u003d (RP1 - Srp1) / RP1 - (RP0 - Srp0) / RL0 \u003d - CpnJ / RSh + Srp0 / RP0 \u003d (Srp0 / RSh - Srp1 / RP1) + (Srp0 / RP0 Срп0 / РП1) \u003d ЛрсПРС + А / V.

Here, the component Ar prS characterizes the effect of changes in the cost products sold on the dynamics of sales profitability. And component A // PPR - the impact of changes in the volume of sales. They are determined accordingly: ArsPRs \u003d Срп0 / РП1 - Срп1 / РП1; А / пПр \u003d Срп0 / РП0 - Срп0 / РП1.

Applying the method of chain substitutions, the factor analysis of profitability of sales can be continued by studying the influence of the dynamics of factors such as:

A) the cost of selling goods, products, works, services:
ArsPrr \u003d (Ср0 - Ср1) / РП1,
where СрО, Cpl is the cost of selling goods, products, works, services, respectively, in the base and reporting periods (line 020 of Form 2), rubles;

B) administrative expenses:

Ar „, y \u003d (SuO - Su1) / RP1, where SuO, Su1 are administrative expenses, respectively, in the base and reporting periods (line 030 of Form 2), rubles,

C) business expenses:

LrsPrk \u003d (SkO - Sk1) / RP1, where SkO, Sk1 are commercial expenses, respectively, in the base and reporting periods (line 040 of Form 2), rub.

If the company keeps records of the cost and revenues for certain types of products, then in the process of analysis it is necessary to assess the impact of the sales structure on the change in the profitability of products. However, such research is possible only on the basis of operational data, that is, it is carried out in the process of in-house analysis. Let's demonstrate it with the following example.

Example: Assess the impact of the sales structure on the change in the profitability of products sold.

Products Specific gravity of the j-th Profitability of the j-th product in the volume of the product, Pj sales,%, dj Last Reporting Last reporting year A 30 40 0.25 0.245 B 70 60 0.125 0.128

Profitability of products sold:

Last year p »t \u003d ^ podo \u003d 0.25 * 0.3 + 0.125 * 0.7 \u003d 0.1625,
reporting YEAR ^ \u003d \u003d 0.245 * 0.4 + 0.128 * 0.6 \u003d 0.1748,
LrRP \u003d p \\ p - p \\ n \u003d 0.1748 - 0.1625 \u003d 0.0123.

This change in profitability is the result of two factors:

Changes in the profitability of individual products:
pshP1 \u003d ip\u003e jd) -ipw \u003d
P 1 \u003d 1
= 0,1748 - (0,25*0,4 + 0,125*0,6) = 0,1748 - 0,1750 = -0,0002.
Changing the implementation structure:
Pmd. \u003d Z P ° Jd) ~ Z P ° JdJ \u003d ° "1750" ° "1625 \u003d +0" 0125 "" M M

Conclusion: The increase in the level of profitability of products sold was due to a change in the sales structure. The increase in the share of more profitable products (product A) from 30% to 40% in the sales volume led to an increase in the profitability of products sold by 1.25%. However, the decrease in the profitability of product A caused a decrease in the profitability of products sold by 0.02%. Therefore, the overall increase in product profitability was 1.23%.

Factor analysis tasks

1. Selection of factors for the analysis of the studied performance indicators and their classification.
2. Determination of the form of dependence between factorial and performance indicators, construction of a factor model.
3. Calculation of the influence of factors and assessment of the role of each of them in changing the value of the effective indicator.

The most important task of deterministic factor analysis is to calculate the influence of factors on the value of effective indicators, for which the analysis uses a whole arsenal of methods, essence, purpose, the scope of which is discussed below.

It is important to distinguish between factors by their content: extensive (quantitative), intensive (qualitative); and by the level of subordination.

Some factors have a direct impact on the performance indicator, others indirectly. According to the level of subordination (hierarchy), factors of the first, second, third and subsequent levels of subordination are distinguished.

Currently under analysis actual cost of manufactured goods, identification of reserves and the economic effect of its reduction, factor analysis is used.

Since the cost is a complex resulting indicator, and knowledge of the conditions for its formation is important for the effective management of an organization, it is of interest to assess the influence of various factors or reasons on this indicator when they change during the production process, in particular, deviations from planned values, values \u200b\u200bin the base period, etc. P.

Economic factors most fully embrace all elements of the production process - means, objects of labor and labor itself. They reflect the main directions of work of collectives of enterprises to reduce costs: increasing labor productivity, introducing advanced equipment and technology, better use of equipment, cheaper procurement and better use of objects of labor, reduction of administrative and managerial and others, reduction of rejects and elimination of unproductive costs and losses.

The most important groups of factors that have a significant impact on the cost price include the following:

1) Raising the technical level of production: the introduction of a new, progressive technology; mechanization and automation production processes; improving the use and application of new types of raw materials and materials; changes in the design and technical characteristics of products. They also decrease as a result of the integrated use of raw materials, the use of economical substitutes, and the complete use of waste in production. A large reserve is fraught with the improvement of products, a decrease in its material consumption and labor intensity, a decrease in the weight of machinery and equipment, a decrease in overall dimensions, etc.

For this group of factors, for each event, the economic effect, which is reflected in the reduction of production costs. Savings from the implementation of measures are determined by comparing the cost per unit of production before and after the implementation of measures and multiplying the resulting difference by the volume of production in the planned year:

EC \u003d (З0 - З1) * Q, (7.8)
where EK - saving direct current costs;
З0 - direct current costs per unit of production before the implementation of the measure;
З1 - direct current costs per unit of output after the implementation of the measure;
Q is the volume of goods output in natural units from the beginning of the implementation of the event to the end of the planning period.

2) Improving the organization of production and labor: changes in the organization of production, forms and methods of labor with the development of production specialization; improving production management and reducing costs for it; improved usage; improvement of material and technical supply; reducing transportation costs; other factors that increase the level of organization of production. With the simultaneous improvement of technology and organization of production, it is necessary to establish savings for each factor separately and include them in the appropriate groups. If such a division is difficult to make, then the savings can be calculated based on the targeted nature of the activities or by groups of factors.

A decrease in operating costs occurs as a result of improving the maintenance of the main production (for example, the development of continuous production, an increase in the shift ratio, streamlining of auxiliary technological work, improvement of the instrumental economy, improvement of the organization of quality control of work and goods). A significant reduction in the cost of living labor can occur with an increase in the norms and service areas, a reduction in losses, a decrease in the number of workers who do not fulfill the output standards. These savings can be calculated by multiplying the number of redundant workers by the average in the previous year (including social security charges and taking into account the cost of clothing, food, etc.). Additional savings arise when improving the management structure of the organization as a whole. It is expressed in the reduction of management costs and in the savings in wages and salaries due to the release of management personnel.

With the improvement in the use of fixed assets, savings are calculated as the product of the absolute cost reduction (excluding depreciation) per unit of equipment (or other fixed assets) by the average amount of equipment (or other fixed assets).

The improvement of material and technical supply and the use of material resources is reflected in a decrease in the consumption rates of raw materials and materials, a decrease in their cost price due to a decrease in procurement and storage costs. Transportation costs are reduced as a result of lower costs for the delivery of raw materials and supplies from the supplier to the organization's warehouses, from factory warehouses to places of consumption; reducing transportation costs finished products.

3) Changes in the volume and structure of goods: changes in the nomenclature and, improving the quality and volume of production of goods. Changes in this group of factors can lead to a relative decrease in conditionally fixed costs (except for depreciation), a relative decrease. Conditionally fixed costs do not directly depend on the number of goods produced; with an increase in production, their number per unit of goods decreases, which leads to a decrease in its cost.

The relative savings on conditionally fixed costs is determined by the formula

EKP \u003d (TV * ZUP0) / 100, (7.9)
where EKP is the saving of conditionally fixed costs;
ZUP0 - the amount of conditionally fixed costs in the base period;
TV - the rate of increase in the volume of production in comparison with the base period.

The relative change in depreciation charges is calculated separately. Part of the depreciation deductions (as well as other production costs) is not included in the prime cost, but is reimbursed from other sources (special funds, payment for services on the side that are not included in the marketable product, etc.), so the total amount of depreciation may decrease. The decrease is determined based on actual data for the reporting period. The total savings on depreciation deductions are calculated using the formula

ECA \u003d (AOK / QO - A1K / Q1) * Q1, (7.10)
where ECA - savings due to a relative decrease in depreciation charges;
A0, A1 - the amount of depreciation deductions in the base and reporting periods;
K is a coefficient that takes into account the amount of depreciation charges attributed to in the base period;
Q0, Q1 - the volume of goods output in natural units of the base and reporting period.

To avoid repeated counting, the total amount of savings is reduced (increased) by the part that was taken into account for other factors.

Changing the nomenclature and assortment of goods is one of the important factorsinfluencing the level of production costs. With different profitability of individual products (in relation to the cost), shifts in the composition of goods associated with improving the structure and increasing production efficiency can lead to both a decrease and an increase in production costs. The impact of changes in the structure of goods on the cost price is analyzed by variable costs according to the items of the calculation of the standard nomenclature. Calculation of the influence of the structure of goods on the cost must be linked to indicators of increasing labor productivity.

4) Improving the use of natural resources: changing the composition and quality of raw materials; changes in the productivity of deposits, the volume of preparatory work during production, methods of extracting natural raw materials; change in other natural conditions. These factors reflect the influence of natural (natural) conditions on the value variable costs... The analysis of their influence on reducing the cost of production is carried out on the basis of sectoral methods of the extractive industries.

5) Industry and other factors: commissioning and development of new shops, production units and industries, preparation and development of production; other factors.

Significant reserves are laid down in reducing the costs of preparing and mastering new types of production of goods and new technological processes, in reducing the costs of the start-up period for newly commissioned workshops and facilities.

The calculation of the amount of change in costs is carried out according to the formula:

EKP \u003d (З1 / Q1 - З0 / Q0) * Q1, (7.11)
where EKP is the change in the costs of preparing and mastering production;
З0, З1 - sums of expenses of the base and reporting period;
Q0, Q1 - the volume of production of goods of the base and reporting period.

If changes in the amount of costs in the analyzed period were not reflected in the above factors, then they are referred to others. These include, for example, a change in the size or termination of mandatory payments, a change in the amount of costs included in the cost of production, etc.

The factors of cost reduction and reserves identified as a result of the analysis must be summarized in the final conclusions, to determine the total influence of all factors on reducing the total cost per unit of goods.

In order to conduct a factor analysis of labor productivity, i.e. determine how this or that technical and economic factor affects the changes in this indicator, calculate the relative savings (increase) in the number of employees. Calculations are carried out in the following sequence.

First, the relative release of industrial and production personnel is determined in comparison with the reporting period as a result of the impact of all factors:

L \u003d L cn 0 qQ t 0.

Then, using any of the methods of factor analysis, the influence of a change in the value of the corresponding factor is determined: the output of marketable products, which can be achieved due to an increase in the volume of production (extensive factor), and an increase in the average annual output per one payroll worker, which can be achieved as a result of measures to improve the technical level of production (intensive factor).

One of the important aspects of evaluating a firm's performance is to study its effectiveness from the point of view of the owner. Efficiency in this case, as in many others, can be assessed by determining the rate of return. However, a simple calculation may not be enough and will need to be supplemented with analysis. The most popular method is probably the factor analysis of profitability. equity capital... Let us dwell in more detail on the methodology for its implementation and the main features.

The factor analysis of return on equity is usually associated with DuPont formulas that allow you to quickly make all the necessary calculations. It is important to understand how these formulas turned out, besides, there is nothing complicated about it. The profitability of the owner's capital is obviously determined by the ratio of the received capital to the value of this capital. The factorial model is obtained from this relation by elementary transformations. Their essence lies in the multiplication of the numerator and denominator by revenue and assets. After that, it is easy to see that the efficiency of using this part of capital, its profitability, is determined by the product of the indicator of the degree of financial dependence by the turnover of property (assets) and the level of profitability of sales. After drawing up a mathematical model, it is directly analyzed. It can be done in any way suitable for deterministic models. Return on equity factor analysis using DuPont formulas is one variation of the absolute difference method. It, in turn, is also a special case of the chain substitution method. The main principle of this method lies in the sequential determination of the impact of each factor in isolation, regardless of the rest.

It should be noted that factor analysis of economic profitability is carried out in a similar way. It is the ratio of profit to assets. After small transformations, this indicator can be represented by the product of the company's property turnover by the profitability of sales. Subsequent analysis proceeds in the same way.

It is necessary to pay special attention to what indicators should be used in the calculations. Obviously, it is necessary to use information for at least two periods in order to be able to observe changes. The data taken from the profit and loss account are cumulative in nature, since they represent a certain amount for a particular period. In the balance sheet, the data are presented for a specific date, so it is best to calculate their average.

The above methods, that is, the method of chain substitutions and its modifications, can be used to analyze almost any deterministic factor model. For example, factor analysis of the current liquidity ratio can be carried out very simply. For greater detail, it is advisable to disclose the formula for this coefficient, reflecting the components of current assets in the numerator, and short-term liabilities in the denominator. Then it is required to calculate the influence of each of the identified factors. It should be noted that absolute differences and the method of the same name cannot be used for this model, since it has a multiple character.

The value of any type of analysis is difficult to overestimate, and factor analysis of the return on equity and other indicators is one of best practicespromoting acceptance of the faithful management decisions... Revealing the strong negative impact one factor or another clearly indicates where the impact should be directed. On the other hand, a positive impact may indicate, for example, the presence of certain reserves for profit growth.

Stochastic factor analysis

Stochastic modeling of factor systems of interrelationships of individual aspects of economic activity is based on generalizing the patterns of variation in the values \u200b\u200bof economic indicators - the quantitative characteristics of factors and results of economic activity. The quantitative parameters of the connection are identified by comparing the values \u200b\u200bof the studied indicators in a set of economic objects or periods.

Thus, the first prerequisite for stochastic modeling is the ability to compose a set of observations, i.e., the ability to re-measure the parameters of the same phenomenon under different conditions.

In stochastic analysis, where the model itself is compiled on the basis of a set of empirical data, a prerequisite for obtaining a real model is the coincidence of the quantitative characteristics of relationships in the context of all initial observations. This means that the variation of the values \u200b\u200bof the indicators should occur within the limits of the unambiguous certainty of the qualitative side of the phenomena, the characteristics of which are the simulated economic indicators (within the limits of variation, there should not be a qualitative leap in the nature of the reflected phenomenon).

This means that the second prerequisite for the applicability of the stochastic approach to modeling relationships is the qualitative homogeneity of the set (with respect to the studied relationships).

The studied regularity of changes in economic indicators (modeled connection) appears in a latent form. It is intertwined with random (in terms of research) (not studied) components of variation and covariance of indicators. The law of large numbers states that only in a large population does a regular relationship appear more stable than a random coincidence of the direction of variation (random variation).

This implies the third prerequisite of stochastic analysis - a sufficient dimension (number) of a set of observations ”, which makes it possible to identify the studied patterns (modeled connections) with sufficient reliability and accuracy.

The fourth prerequisite of the stochastic approach is the availability of methods that make it possible to identify quantitative parameters of economic indicators from mass data on the variation in the level of indicators. The mathematical apparatus of the methods used sometimes makes specific requirements for the simulated empirical material. Fulfillment of these requirements is an important prerequisite for the applicability of methods and the reliability of the results obtained.

The main feature of stochastic factor analysis is that in stochastic analysis it is impossible to compose a model by qualitative (theoretical) analysis, a quantitative analysis of empirical data is required.

Stochastic factor analysis methods:

Pairwise correlation method. The method of correlation and regression (stochastic) analysis is widely used to determine the closeness of the relationship between indicators that are not in functional dependence, i.e. connection, manifests itself not in each individual case, but in a certain dependence. With the help of pair correlation, two main tasks are solved: the model of the acting factors is left (the regression equation); a quantitative assessment of the tightness of relations (correlation coefficient) is given.

Matrix models. Matrix models are a schematic reflection of an economic phenomenon or process using scientific abstraction. The most widespread method here is the “input-output” analysis, which is based on a checkerboard pattern and allows one to present the relationship between costs and production results in the most compact form.

Mathematical programming is the main tool for solving problems to optimize production and economic activities.

The operation research method is aimed at studying, including the production and economic activities of enterprises, in order to determine such a combination of structurally interconnected elements of systems, which will most effectively determine the best economic indicator from a number of possible ones.

Game theory as a branch of operations research is a theory of mathematical models for making optimal decisions in conditions of uncertainty or conflict of several parties with different interests.

Integral factor analysis method

Elimination as a method of deterministic factor analysis has an important disadvantage. When using it, it is assumed that the factors change independently of each other, however, in fact, they change interconnected, as a result, some indecomposable residue is formed, which is added to the magnitude of the influence of one of the factors (as a rule, the latter). In this regard, the magnitude of the influence of factors on the change in the effective indicator fluctuates depending on the place of the factor in the deterministic model. To get rid of this drawback, in deterministic factor analysis, an integral method is used, which is used to determine the influence of factors in multiplicative, multiple and mixed multiply-additive models.

Using this method allows you to obtain more accurate results of calculating the influence of factors in comparison with methods of chain substitution, absolute and relative differences and to avoid ambiguous assessment of the influence: in this case, the results do not depend on the location of the factors in the model, and an additional increase in the effective indicator arising from the interaction of factors is divided equally between them.

To distribute the additional growth, it is not enough to take its part corresponding to the number of factors, since factors can act in different directions. Therefore, the change in the effective indicator is measured at infinitely small intervals of time, i.e., the increment of the result is summed up, which is defined as partial products multiplied by the increments of factors at infinitely small intervals. The operation of calculating a definite integral is solved using a personal computer and is reduced to the construction of integrands that depend on the type of function or model of the factor system. Due to the complexity of calculating some definite integrals and additional difficulties associated with the possible action of factors in opposite directions.

Factor analysis of net profit

We advise you to read our article

Net profit is such an indicator of the efficiency of the firm, which, on the one hand, is influenced by the greatest number of factors in comparison with other types of profit, and on the other hand, is the most accurate and "honest" indicator. It is for these reasons that this value requires close attention and should be subjected to detailed study. One of the most popular and frequently used methods is the factor analysis of net income. As the name implies, the study of profit in this way involves the determination of those factors that most affect it, as well as the determination of the specific magnitude of this impact.

Before considering factor analysis of net profit, it is necessary to study how it is formed. The analysis of the formation of net profit is carried out according to the profit and loss statement. This is understandable, since exactly given form reporting reflects the order in which the formation financial result the functioning of the firm. When studying the formation of profit, it is useful to conduct a vertical analysis of the specified reporting form. It implies finding the specific weight of each of the indicators included in the report, as well as the subsequent study of its dynamics. As a rule, revenue is chosen as the comparison base, which is considered equal to one hundred percent.

Factor analysis of net profit is also advisable to carry out on the income statement. This is due to the fact that this form of reporting makes it easy and simple to draw up a mathematical model that will include factors affecting the amount of profit. The factors that have the greatest influence should be placed in the model before the factors whose influence is less significant. The profit and loss statement reflects the amount of revenue, but does not allow judging its changes under the influence of price and sales volume. These factors are extremely important, therefore, they must be additionally taken into account in the model, dividing the effect on profit of revenue into two corresponding parts. After drawing up a mathematical model, it is necessary to directly subject it to analysis according to a certain method. Most often, they resort to using the method of chain substitutions or its modifications, for example, the method of absolute differences. This choice is due to the ease of use and the accuracy of the results.

After studying the formation process and dynamics, it is necessary to analyze the use of net profit. The most logical and easiest way to study this process will be by conducting vertical analysis, which has already been mentioned above. Obviously, in this case, it is necessary to take the net profit as the base. Then you need to determine the shares of each direction of spending this profit: on, in reserve funds, on investments, and so on. Naturally, it is necessary to study the change in this structure in dynamics.

Obviously, to carry out any of the types of analysis described above, information is needed for several periods, at least for two years. This is due to the fact that on the basis of one period it is simply impossible to draw any conclusions about certain changes. However, it should be borne in mind that the indicators should be comparable, it is necessary to make adjustments in case of changes in accounting policies or any other.

Whether it is a factor analysis of net profit or some other, it must necessarily end with the formulation of certain conclusions and recommendations. Based on the study of profits, many conclusions can be drawn about pricing policy, and about cost management, and much more. Conclusions and recommendations form the basis for making management decisions that are vital to the firm's operations.

Factor analysis method of chain substitutions

The chain substitution method is the most versatile of the elimination methods. It is used to calculate the influence of factors in all types of deterministic factor models: additive, multiplicative, multiple and mixed (combined). This method allows you to determine the influence of individual factors on the change in the value of the effective indicator by gradually replacing the base value of each factor indicator in the volume of the effective indicator for the actual one in the reporting period. For this purpose, a number of conditional values \u200b\u200bof the effective indicator are determined, which take into account the change in one, then two, three, etc. factors, assuming that the rest do not change. Comparison of the value of the effective indicator before and after the change in the level of one factor or another allows one to eliminate the influence of all factors except one, and to determine the impact of the latter on the increase in the effective indicator.

The degree of influence of this or that indicator is revealed by successive subtraction: the first is subtracted from the second calculation, the second is subtracted from the third, etc. In the first calculation, all values \u200b\u200bare planned, in the last - actual.

In the case of a three-factor multiplicative model, the calculation algorithm is as follows:

Y 0 \u003d a 0 * b 0 * C 0;
Y conv. 1 \u003d a 1 * b 0 * C 0; Y a \u003d Y conv. 1 - Y 0;
Y conv. 2 \u003d a 1 * b 1 * C 0; Y L \u003d Y conv. 2 - Y conv. 1;
Y f \u003d a 1 * b 1 * C 1; Y c \u003d Y f - Y conv. 2 and so on.

The algebraic sum of the influence of factors must necessarily be equal to the total increase in the effective indicator:

Y a + Y b + Y c \u003d Y f - Y 0.

The absence of such equality indicates the errors made in the calculations.

This implies the rule that the number of calculations per unit is greater than the number of indicators of the calculation formula.

When using the chained substitution method, it is very important to maintain a strict sequence of substitutions, since arbitrary changes can lead to incorrect results. In the practice of analysis, first of all, the influence of quantitative indicators is revealed, and then of qualitative ones. So, if it is required to determine the degree of influence of the number of employees and labor productivity on the size of industrial output, then first establish the influence of the quantitative indicator of the number of employees, and then the qualitative labor productivity. If the influence of the factors of quantity and prices on the volume of industrial products sold is found out, then the influence of the quantity is calculated first, and then the influence of wholesale prices. Before proceeding with the calculations, it is necessary, firstly, to identify a clear relationship between the studied indicators, secondly, to distinguish between quantitative and qualitative indicators, and thirdly, to correctly determine the sequence of substitution in cases where there are several quantitative and qualitative indicators (main and derivatives, primary and secondary). Thus, the application of the method of chain substitution requires knowledge of the relationship of factors, their subordination, the ability to correctly classify and systematize them.

An arbitrary change in the sequence of substitution changes the quantitative weight of this or that indicator. The more significant the deviation of the actual indicators from the planned ones, the greater the differences in the assessment of factors calculated with a different sequence of substitution.

The chain substitution method has a significant drawback, the essence of which boils down to the emergence of an indecomposable residue, which is added to the numerical value of the influence of the last factor. This explains the difference in calculations when changing the substitution sequence. This disadvantage is eliminated by using a more complex integral method in analytical calculations.

Factor analysis of wages

It is conducted taking into account the analysis of use labor resources at the enterprise and the level of labor productivity. It is known that with an increase in labor productivity, real prerequisites are created for an increase in the level of labor remuneration. At the same time, funds for labor remuneration must be used in such a way that the growth rate of labor productivity outstrips the growth rate of its payment, since this creates opportunities for increasing reproduction at the enterprise.

The analysis of the use of FZP begins with the calculation of the absolute and relative deviations of its actual value from the planned one.

We make a consistent calculation

The absolute deviation of FZPabs is determined by comparing the actually used funds for wages by the planned wage fund of FZPpl as a whole for the enterprise, production units and categories of workers:

FZPabs \u003d FZPf - FZPpl. \u003d 21465-20500 \u003d +965 million rubles

However, it should be borne in mind that the absolute deviation in itself does not characterize the use of PPP, since this indicator is determined without taking into account the degree of fulfillment of the production plan.

The relative deviation of FZPf is calculated as the difference between the actually accrued wages of the FZPf and the planned fund, adjusted for the coefficient of implementation of the plan for the production of products Kvp

Initial data for the analysis of PPP

The constant part of wages does not change with an increase or decrease in the volume of production (wages of workers at tariff rates, salaries of employees in salaries, all types of additional payments, wages of workers in non-industrial production and the corresponding amount of vacation pay):

FZPrel \u003d FZPf - FZPsk \u003d FZPag - (FZP pl..per * Kvp + FZP pl..post) \u003d 21465 - (13120 * 1.026 + 7380) \u003d 21465 - 20841 \u003d +424 million rubles
where ФЗПск - the planned salary fund, adjusted for the coefficient of implementation of the plan for production;
FZP pl..per and FZP pl..post are variable and constant amounts of the planned planned salary fund.

When calculating the FZPotn, you can use the so-called correction coefficient Kp, which reflects the share of the variable salary in the general fund. It shows what percentage of the percentage should increase the planned payroll for each percentage of the overflow of the plan for production (VP,%)
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Their classification
In modern statistics, factor analysis is understood as a set of methods that, on the basis of real-life relationships of features, objects or phenomena, make it possible to identify latent (hidden and not accessible for direct measurement) generalizing characteristics of the organized structure and mechanism of development of the studied phenomena or processes.

The concept of latency is key and means the implicitness of the characteristics revealed using the methods of factor analysis.

The idea behind factor analysis is quite simple. As a result of the measurement, we are dealing with a set of elementary features X i measured on several scales. It - explicit variables.If the signs change consistently, then we can assume the existence of certain common causes. this variability, i.e. the existence of some hidden (latent) factors. The task of analysis is to find these factors.

Since factors are a union of certain variables, it follows that these variables are related to each other, i.e. have a correlation (covariance), and more with each other than with other variables included in another factor. The methods for finding factors and are based on the use of correlation coefficients (covariance) between variables. Factor analysis gives a non-trivial solution, i.e. the decision cannot be foreseen without the use of a special factor extraction technique. This decision is of great importance for characterizing the phenomenon, since at first it was characterized by a sufficiently large number of variables, but as a result of the analysis, it turned out that it can be characterized by a smaller number of other variables - factors.

Not only explicit variables can correlate X i , but also observed objects N i ... Depending on what type of correlation is considered - between features or objects - there are R and Q data processing techniques, respectively.

In accordance with the general principles of factor analysis, the result of each measurement is determined by the action of general factors, specific factors and the measurement error “factor”. Common the factors that influence the results of measurements on several measuring scales are called. Each of specific factors affects the measurement result only on one of the scales. Under measurement errormeans a set of non-accountable reasons that determine the measurement results. The variability of the obtained empirical data is usually described using their variance.

As you already know, the correlation coefficient is most often used to quantify the relationship between two variables. There are many varieties of this coefficient, and the choice of an adequate measure of connection is determined both by the specifics of the empirical data and by the measuring scale.

However, there is also a geometric possibility of describing the relationship between features. Graphically, the correlation coefficient between two variables can be depicted as two vectors - arrows, originating at one point. These vectors are located at an angle to each other, the cosine of which is equal to the correlation coefficient. The cosine of an angle is a trigonometric function, the value of which can be found in the reference book. Within the framework of this topic, we will not discuss trigonometric function cosine, it is enough to know where to find the corresponding data.

Table 7.1 provides several values \u200b\u200bfor the cosines of the angles to give a general idea of \u200b\u200bthem.

Table 7.1

Cosine table for graphic display

correlations between variables.

In accordance with this table, the complete positive correlation ( r 1) will correspond to an angle of 0 ( cos 0 1), i.e. graphically, this will correspond to the complete coincidence of both vectors (see Fig. 7.3 a).

Full negative correlation ( r  -1) means that both vectors lie on the same straight line, but directed in opposite directions ( cos 180 -1). (Fig. 7.3 b).

Mutual independence of variables ( r \u003d 0) is equivalent to the mutual perpendicularity (orthogonality) of vectors ( cos 90 ° \u003d 0). (Fig. 7.3 c).

Intermediate values \u200b\u200bof the correlation coefficient depicted as pairs of vectors forming either sharp ( r \u003e 0), or obtuse ( r   0 0, r  1  180, r  -1

V 1

V 2

and b
 90, r  0   90, r  0   90, r  0

V 2

V 1
Figure 7.3. Geometric interpretation of correlation coefficients.

Geometric approach to factor analysis

The above geometric interpretation of the correlation coefficient is the basis for the graphical presentation of the entire correlation matrix and subsequent interpretation of the data in factor analysis.

Matrix construction begins by constructing a vector representing any variable. Other variables are drawn using vectors of equal length, all of which originate from the same point. As an example, consider the geometric expression of correlations between five variables. (Figure 7.4.)

V 1

V 5 V 2

V 4
Figure 7.4. Geometric interpretation of the correlation matrix (5x5).
It is clear that it is not always possible to represent the correlation in two dimensions (on a plane). Some of the variable vectors would have to be angled to the page. This fact is not a problem for the actual mathematical procedures, but requires some imagination from the reader. Figure 7.5. you can see that the correlation between the variables V1 V2 is large and positive (because there are small angles between these vectors). Variables V2 V3 are practically independent from each other, because the angle between them is very close to 90 , i.e. the correlation is 0. Variables V3 - V5 are strongly and negatively related. The high correlations between V1 and V2 indicate that both of these variables practically measure the same property and that, in fact, one of these variables can be excluded from further consideration without significant loss of information. The most informative for us are variables that are independent of each other, i.e. having minimal correlations with each other, or angles corresponding to 90  (Fig. 7.5.)

V 1

Figure 7.5. Geometric interpretation of the correlation matrix
From this figure it can be seen that there are two groups of correlations: V 1, V 2, V 3 and V 4, V5. The correlations between the variables V 1, V 2, V 3 are very large and positive (there are small angles between these vectors, and, therefore, large values \u200b\u200bof cosines). Similarly, the correlation between variables V 4 and V 5 is also large and positive. But the correlation between these groups of variables is close to zero, since these groups of variables are practically orthogonal to each other, i.e. are located relative to each other at right angles. The given example shows that there are two groups of correlations and the information obtained from these variables can be approximated by two common factors (F 1 and F 2), which in this case are orthogonal to each other. However, this is not always the case. Varieties of factor analysis, in which correlations are calculated between factors that are not orthogonal, are called oblique solution. However, we will not consider such cases within the framework of this course, and will focus exclusively on orthogonal solutions.

By measuring the angle between each common factor and each common variable, correlations between these variables and the corresponding factors can be calculated. The correlation between a variable and a common factor is commonly called factor loading... The geometric interpretation of this concept is given in Fig. 7.6.

F 2

Factor analysis is one of the most powerful statistical data analysis tools. It is based on the procedure of combining groups of variables correlating with each other ("correlation pleiades" or "correlation nodes") into several factors.

In other words, the goal of factor analysis is to concentrate the initial information, expressing a large number of considered features through a smaller number of more capacious internal characteristics, which, however, cannot be directly measured (and in this sense are latent).

For example, let's hypothetically imagine a regional legislature with 100 deputies. Among the various issues on the agenda, the following are put to vote: a) a bill proposing to restore the monument to V.I. Lenin on the central square of the city - the administrative center of the region; b) an appeal to the President of the Russian Federation with a demand to return all strategic industries to state ownership. The contingency matrix shows the following distribution of votes of deputies:

	Monument to Lenin (for)	Monument to Lenin (against)
Appeal to the President (for)	49	4
Appeal to the President (against)	6	41

Obviously, the votes are statistically linked: the overwhelming majority of the deputies who support the idea of \u200b\u200brestoring the monument to Lenin also support the return of strategic enterprises to state ownership. Similarly, most opponents of the restoration of the monument are at the same time opponents of the return of enterprises to state ownership. At the same time, thematically, the voting is completely unrelated to each other.

It is logical to assume that the revealed statistical relationship is due to the existence of some hidden (latent) factor. Legislators, formulating their point of view on a wide variety of issues, are guided by a limited, small set of political positions. In this case, we can assume the presence of a latent split in the deputies by the criterion of support / rejection of conservative-socialist values. A group of "conservatives" stands out (according to our contingency table - 49 deputies) and their opponents (41 deputies). By identifying such splits, we will be able to describe a large number of separate votes through a small number of factors that are latent in the sense that we cannot detect them directly: in our hypothetical parliament, there has never been a vote in which MPs were asked to determine their attitude to conservative socialist values. We detect the presence of this factor based on a meaningful analysis of quantitative relationships between variables. Moreover, if in our example nominal variables are deliberately taken - support of the bill with the categories “for” (1) and “against” (0), then in reality factor analysis effectively processes interval data.

Factor analysis is very actively used both in political science and in "neighboring" sociology and psychology. One of the important reasons for the great demand for this method is the variety of tasks that can be solved with its help. Thus, there are at least three “typical” goals of factor analysis:

· Reduction of dimension (reduction) of data. Factor analysis, highlighting the nodes of interrelated features and reducing them to some generalized factors, reduces the initial basis of description features. The solution to this problem is important in a situation where objects are measured by a large number of variables and the researcher is looking for a way to group them according to semantic criteria. The transition from many variables to several factors allows you to make the description more compact, get rid of uninformative and duplicate variables;

Revealing the structure of objects or characteristics (classification). This task is close to that which is solved by the cluster analysis method. But if the cluster analysis takes the values \u200b\u200bof several variables as the "coordinates" of objects, then factor analysis determines the position of the object relative to factors (related groups of variables). In other words, using factor analysis, it is possible to assess the similarity and difference of objects in the space of their correlations, or in the factor space. The obtained latent variables are the coordinate axes of the factor space, the objects under consideration are projected onto these axes, which allows you to create a visual geometric representation of the studied data, convenient for meaningful interpretation;

Indirect measurement. The factors, being latent (empirically not observable), cannot be directly measured. However, factor analysis allows not only to identify latent variables, but also to quantify their value for each object.

Let us consider the algorithm and interpretation of the statistics of factor analysis using the example of data on the results of the parliamentary elections in the Ryazan region in 1999 (federal district). To simplify the example, let us take electoral statistics only for those parties that have overcome the 5% barrier. The data are taken in the context of territorial election commissions (by city and district of the region).

The first step is to standardize the data by converting it to standard points (the so-called L-points, calculated using the normal distribution function).

TEAK (territorial election commission)	"Yabloko"	"Unity"	Block Zhirinovsky	OVR	Communist Party	THX
Ermishinskaya	1,49	35,19	6,12	5,35	31,41	2,80
Zakharovskaya	2,74	18,33	7,41	11,41	31,59	l b 3 "
Kadomskaya	1,09	29,61	8,36	5,53	35,87	1,94
Kasimovskaya	1,30	39,56	5,92	5,28	29,96	2,37
Kasimovskaya city	3,28	39,41	5,65	6,14	24,66	4,61
The same in standardized points (g-points)
Ermishinskaya	-0,83	1,58	-0,25	-0,91	-0,17	-0,74
Zakharovskaya	-0,22	-1,16	0,97	0,44	-0,14	0,43
Kadomskaya	-1,03	0,67	1,88	-0,87	0,59	-1,10
Kasimovskaya	-0,93	2,29	-0,44	-0,92	-0,42	-0,92
Kasimovskaya city	0,04	2,26	-0,70	-0,73	-1,32	0,01
Etc. (32 cases in total)

	"Apple"	"Unity"	BZ	OVR	Communist Party	THX
"Apple"
"Unity"	-0,55
BZ	-0,47	0,27
OVR	0,60	-0,72	-0,47
Communist Party	-0,61	0,01	0,10	-0,48
THX	0,94	-0,45	-0,39	0,52	-0,67

Even a visual analysis of the matrix of pairwise correlations allows us to make assumptions about the composition and nature of the correlation pleiades. For example, positive correlations are found for the Union of Right Forces, Yabloko and the Fatherland-All Russia bloc (the Yabloko-OVR, Yabloko-SPS and OVR-SPS pairs). At the same time, these three variables are negatively correlated with the KPRF (support for the KPRF), to a lesser extent - with "Unity" (support for "Unity") and even less with the BZ variable (support for the "Zhirinovsky Bloc"). Thus, presumably, we have two pronounced correlation constellations:

("Yabloko" + OVR + SPS) - the Communist Party;

("Yabloko" + OVR + SPS) - "Unity".

These are two different constellations, not one, since there is no connection between Unity and the Communist Party of the Russian Federation (0.01). It is more difficult to make an assumption regarding the BZ variable, here the correlations are less pronounced.

To test our assumptions, it is necessary to CALCULATE the eigenvalues, factor scores, and factor loadings for each variable. Such calculations are quite complicated, require serious skills in working with matrices, so here we will not consider the computational aspect. Let's just say that these calculations can be done in two ways: the principal components method and the principal factors method. Principal component analysis is more common; statistical programs use it by default.

Let us dwell on the interpretation of eigenvalues, factor values \u200b\u200band factor loadings.

The eigenvalues \u200b\u200bof the factors for our case are as follows:

bgcolor \u003d white\u003e 5

Factor	Eigenvalue	% total variation
1	3,52	58,75
2	1,14	19,08
3	0,76	12,64
4	0,49	S.22
0,05	0.80
6	0,03	0,51
Total	6	100%

The greater the eigenvalue of a factor, the greater its explanatory power (the maximum value is equal to the number of variables, in our case 6). One of the key elements of factor analysis statistics is the "% total variance" indicator. It shows how much of the variation (variability) of the variables is explained by the extracted factor. In our case, the weight of the first factor exceeds the weight of all other factors combined: it explains almost 59% of the total variation. The second factor explains 19% of the variation, the third explains 12.6%, etc. decreasing.

Having the eigenvalues \u200b\u200bof the factors, we can start solving the problem of reducing the dimension of the data. The reduction will occur due to the exclusion from the model of factors that have the least explanatory power. And here the key question is how many factors to leave in the model and what criteria to be guided by. So, factors 5 and 6 are clearly superfluous, which together explain just over 1% of the total variation. But the fate of factors 3 and 4 is no longer so obvious.

As a rule, the model contains factors whose eigenvalues \u200b\u200bexceed one (Kaiser criterion). In our case, these are factors 1 and 2. However, it is useful to check the correctness of removing four factors using other criteria. One of the most widely used methods is scree plot analysis. For our case, it looks like:

The graph gets its name from its resemblance to a mountainside. "Scrap" is a geological term for debris accumulating at the bottom of a rocky slope. "Rock" is really influential factors, "talus" is statistical noise. Figuratively speaking, you need to find a place on the graph where the “rock” ends and the “scree” begins (where the decrease in eigenvalues \u200b\u200bfrom left to right slows down greatly). In our case, the choice must be made from the first and second inflections corresponding to two and four factors. Leaving four factors, we get very high accuracy of the model (more than 98% of the total variation), but make it quite complex. Leaving two factors, we will have a significant unexplained part of the variation (about 22%), but the model will become concise and easy to analyze (in particular, visual). Thus, in this case, it is better to sacrifice some degree of accuracy in favor of compactness, leaving the first and second factors.

You can check the adequacy of the resulting model using special matrices of reproduced correlations and residual correlations. The matrix of reproduced correlations contains the coefficients that were restored for the two factors left in the model. Of particular importance in it is the main diagonal, on which communities of variables are located (in the table are italicized), which show how accurately the model reproduces the correlation of a variable with the same variable, which should be one.

The residual coefficient matrix contains the difference between the original and reproduced coefficients. For example, the reproduced correlation between the ATP and Yabloko variables is 0.88, the initial one is 0.94. Remainder \u003d 0.94 - 0.88 \u003d 0.06. The lower the residual values, the higher the quality of the model.

Reproduced correlations
	"Apple"	"Unity"	BZ	OVR	Communist Party	THX
"Apple"	0,89
"Unity"	-0,53	0,80
BZ	-0,47	0,59	0,44
OVR	0,73	-0,72	-0,56	0,76
Communist Party	-0,70	0,01	0,12	-0,34	0,89
THX	0,88 -0,43		-0,40	0,66	-0,77	0,88
Residual coefficients
	"Apple"	"Unity"	BZ	OVR	Communist Party	THX
"Apple"	0,11
"Unity"	-0,02	0,20
BZ	0,00	-0,31	0,56
OVR	-0,13	-0,01	0,09	0,24
Communist Party	0,09	0,00	-0,02	-0,14	0,11
THX	0,06	-0,03	0,01	-0,14	0,10	0,12

As can be seen from the matrices, the two-factor model, while generally adequate, poorly explains individual relationships. So, the commonality of the BZ variable (only 0.56) is very low, the value of the residual coefficient of connection between BZ and "Unity" (-0.31) is too high.

It is now necessary to decide how important it is for this particular study to adequately represent the BZ variable. If the importance is high (for example, if the study is devoted to the analysis of the electorate of this particular party), it is correct to return to the four-factor model. If not, two factors can be left.
Taking into account the educational nature of our tasks, we will leave a simpler model.

Factor loadings can be represented as the correlation coefficients of each variable with each of the identified factors 1 as, the correlation between the values \u200b\u200bof the first factor variable and the values \u200b\u200bof the Yabloko variable is -0.93. All factor loadings are given in the factor display matrix -

The closer the relationship between the variable and the factor under consideration, the higher the value of the factor load. The positive sign of the factor loading indicates the straight line, and the negative sign indicates the inverse relationship of the variable with the factor.

With the values \u200b\u200bof factor loadings, we can build a geometric representation of the results of factor analysis. On the X-axis we will postpone the loads of variables by factor 1, on the Y-axis - loads of variables by factor 2 and we will get a two-dimensional factor space.

Before proceeding to a meaningful analysis of the results obtained, we will carry out one more operation - rotation. The importance of this operation is dictated by the fact that there is not one, but many variants of the factor loadings matrix, equally explaining the relationships of the variables (the intercorrelation matrix). It is necessary to choose a solution that is easier to interpret meaningfully. This is considered to be a matrix of loads, in which the values \u200b\u200bof each variable for each factor are maximized or minimized (close to one or zero).

Let's consider a schematic example. There are four objects located in factor space as follows:

The loads on both factors for all objects are significantly different from zero, and we have to use both factors to interpret the position of objects. But if we “rotate” the entire structure clockwise around the intersection of the coordinate axes, we get the following picture:

In this case, the loads on factor 1 will be close to zero, and the loads on factor 2 will be close to one (the principle of a simple structure). Accordingly, for meaningful interpretation of the position of objects, we will use only one factor - factor 2.

There are quite a few methods for rotating factors. Thus, the group of methods of orthogonal rotation always preserves the right angle between the coordinate axes. These include vanmax (minimizes the number of variables with high factor loadings), quartimax (minimizes the number of factors needed to explain a variable), equamax (a combination of the two previous methods). Oblique rotation methods do not necessarily preserve right angles between axes (e.g. direct obiimin). The promax method is a combination of orthogonal and oblique rotation methods. In most cases, the vanmax method is used, which gives good results when applied to most policy research tasks. Also, as with many other techniques, it is a good idea to experiment with different spinning techniques.

In our example, after rotation by the varimax method, we obtain the following factorial loadings matrix:

Accordingly, the geometric representation of the factor space will look like:

Now you can start meaningful interpretation of the results. The key opposition - the electoral split - according to the first factor is formed by the Communist Party of the Russian Federation, on the one hand, and Yabloko and the Union of Right Forces (to a lesser extent, OVR), on the other. Substantially - based on the specifics of the ideological attitudes of the named subjects of the electoral process - we can interpret this delimitation as a "left-right" split, which is "classic" for political science.

The opposition on factor 2 is formed by OVR and Unity. The “Zhirinovsky's block” adjoins the latter, but we cannot reliably judge its position in the factor space due to the peculiarities of the model, which poorly explains the connections of this particular variable. To explain this configuration, it is necessary to recall the political realities of the 1999 election campaign. At that time, the struggle within the political elite led to the formation of two echelons of the “party of power” - the Unity and Fatherland - All Russia blocs. The difference between them was not of an ideological nature: in fact, the population was asked to choose not from two ideological platforms, but from two elite groups, each of which had significant power resources and regional support. Thus, this split can be interpreted as “power-elite” (or, somewhat simplifying, “power-opposition”).

In general, we get a geometric representation of a certain electoral space of the Ryazan region for these elections, if we understand the electoral space as a space for electoral choice, the structure of key political alternatives (“splits”). The combination of these two splits was very typical of the 1999 parliamentary elections.

Comparing the results of factor analysis for the same region in different elections, we can judge the presence of continuity in the configuration of the space of the electoral choice of the territory. For example, a factor analysis of the federal parliamentary elections (1995, 1999 and 2003), held in Tatarstan, showed a stable configuration of the electoral space. For the 1999 elections, only one factor was left in the model with an explanatory power of 83% of the variation, which made it impossible to plot a two-dimensional diagram. Factor loadings are given in the corresponding column.

If you look closely at these results, you will notice that in the republic, from election to election, the same basic split is reproduced: “the party of power” - all the others. ”The“ party of power ”in 1995 was the bloc“ Our Home - Russia "(PDR), in 1999 - OVR, in 2003 -" United Russia. "Over time, only the" details "change - the name of the" party of power. "The new political" label "very easily fits into the static matrix of choice.

We conclude this chapter with one practical tip. By and large, the success of mastering statistical methods is possible only with intensive practical work with special programs (already repeatedly mentioned SPSS, Statistica, or at least Microsoft Excel). It is no coincidence that the presentation of statistical techniques is conducted by us in the mode of work algorithms: this allows the student to independently go through all the stages of analysis, sitting at the computer. Without attempts at practical analysis of real data, the idea of \u200b\u200bthe possibilities of statistical methods in political analysis will inevitably remain general and abstract. And today the ability to apply statistics to solve both theoretical and applied tasks - a fundamentally important component of the model of a political scientist.

Control questions and tasks

1. What measurement levels correspond to average values \u200b\u200b- mode, median, arithmetic mean? What measures of variation are characteristic for each of them?

2. For what reasons is it necessary to take into account the shape of the distribution of variables?

3. What does the statement “There is a statistical relationship between two variables” mean?

4. What useful information about relationships between variables can be obtained from the analysis of contingency tables?

5. What can you learn about the relationship between variables, based on the values \u200b\u200bof the statistical tests chi-square and lambda?

6. Give a definition of the concept of "error" in statistical research. How can the quality of the constructed statistical model be judged by this indicator?

7. What is the main purpose of correlation analysis? What characteristics of the statistical connection does this method reveal?

8. How to interpret the value of the Pearson correlation coefficient?

9. Describe the method of analysis of variance. What other statistical methods use ANOVA statistics, and why?

10. Explain the meaning of the "null hypothesis" concept.

11. What is a regression line, what method is it used to construct?

12. What does the R coefficient show in the final statistics of the regression analysis?

13. Explain the term "multivariate classification method".

14. Explain the main differences between clustering through hierarchical cluster analysis and K-means.

15. How can cluster analysis be used to study the image of political leaders?

16. What is the main problem solved by discriminant analysis? Give the definition of the discriminant function.

17. Name three classes of problems that can be solved using factor analysis. Concretize the concept of "factor".

18. Describe the three main methods of checking the quality of a model in factor analysis (Kaiser test, “scree” test, matrix of reproduced correlations).

International migration of financial resources in the context of factor analysis

25. J.-B. Say went down in the history of economics as the author of the factor theory of value. What are the main provisions of this theory?

Technical and economic analysis of a construction project and analysis of collateral for the requested construction loan

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