Factor analysis what to do with cross loads. Factor analysis of profit. Example of Factor Analysis of Sales Profit

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

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

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

Financial analysis has several goals:

assessment of the financial situation;
identification of changes in the financial condition in the spatio-temporal context;
identification of the main factors that caused changes in the financial condition;
forecast of the main trends in the 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 application of a model that makes it possible to evaluate and analyze the dynamics of the main indicators of the enterprise. There are three main types of models: descriptive, predicative and normative.

Descriptive Models also known as descriptive models. They are the main ones for assessing the financial condition of the enterprise. These include: building a system of reporting balances, presentation of financial statements in various analytical sections, vertical and horizontal analysis of reporting, a system of analytical ratios, analytical notes to reporting. All these models are based on the use of accounting information.

At the core vertical analysis there is a different presentation of financial statements - in the form of relative values characterizing the structure of generalizing final indicators. A mandatory element of the analysis is the dynamic series of these values, which allows you to track and predict structural shifts in the composition of economic assets and sources of their coverage.

Horizontal Analysis allows you to identify trends 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 and income statement items.

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

liquidity indicators;
indicators of financial stability;
business activity indicators;
profitability indicators.

Predicative Models are predictive models. They are used to predict the income of the enterprise and its future financial condition. The most common of them are: calculation of the point of critical sales volume, construction of predictive financial reports, dynamic analysis models (rigidly determined factor models and regression models), situational analysis models.

normative models. Models of this type make it possible to compare the actual performance of enterprises with the expected ones calculated according to the budget. These models are mainly used in internal financial analysis. Their essence is reduced to the establishment of standards for each item of expenditure by 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 determined 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 the operating enterprise) occurs in 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 you to better understand the causes of changes 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 studied in the analysis can be classified according to different criteria.

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

Natural factors have a great influence on the results of activities in agriculture, forestry and other industries. Accounting for 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, organization health work at enterprises with hazardous production, the general level of personnel 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 the use of the enterprise's production resources and the final results of its activities.

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

The 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 company can influence.

The factors are divided into objective independent of the will and desires of people, and subjective affected by the activities of legal entities and individuals.

According to the degree of prevalence, factors are divided into general and specific. General 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 is manifested periodically are called variables(this is, for example, the introduction of new technology, new types of products).

Of great importance for assessing the activities of enterprises is the division of factors according to the nature of their action into intense and extensive. Extensive factors include those 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 the volume of production due to the increase in the number of workers. Intensive factors characterize the qualitative 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, consisting of several elements. However, there are also those that are not 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 one is the number of working days in reporting period.

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

It is clear that when studying the impact on the work of an enterprise of any group of factors, it is necessary to streamline them, that is, to analyze 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.

Creation factor systems is one of the ways of such systematization of factors. Consider the concept of a factor system.

Factor systems

All phenomena and processes of economic activity of enterprises are interdependent. Communication of economic phenomena is the joint change of two or more phenomena. Among the many forms of regular connections important role plays cause-and-effect (deterministic), in which one phenomenon gives rise to another.

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

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

The quantitative characterization of interrelated phenomena is carried out with the help of indicators. Indicators characterizing the cause are called factorial (independent); indicators characterizing the consequence are called effective (dependent). The totality of factor and resultant signs connected by a causal 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.

The relationship is called functional, or rigidly determined, if each value of the factor attribute corresponds to a well-defined non-random value of the resultant attribute.

The connection is called stochastic (probabilistic) if each value of the factor attribute corresponds to a set of values of the effective attribute, i.e., a certain statistical distribution.

Model factorial system - a mathematical formula that expresses the real relationship between the analyzed phenomena. In general, it can be represented as follows:

where is the effective sign;

Factor signs.

Thus, each performance indicator depends on numerous and varied factors. At the heart of economic analysis and its section - factor analysis- identifying, evaluating and predicting the influence of factors on the change in the effective indicator. The more detailed the dependence of the effective indicator on certain factors, the more accurate the results of the analysis and assessment of the quality of the work of enterprises. 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 refers to the methodology of complex and systematic study and measurement of the impact of factors on the magnitude of performance indicators.

V general case the following can be distinguished main stages of factor analysis:

Setting the goal of the analysis.
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 impact on the results of economic activity.
Determination of the form of dependence between factors and the performance indicator.
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 factor model (its practical use for managing economic processes).

Selection of factors for analysis one or another indicator is carried out on the basis of theoretical and practical knowledge in a particular industry. In this case, they usually proceed from the principle: the larger the complex of factors studied, the more accurate the results of the analysis will be. At the same time, it must be borne in mind that if this complex of factors is considered as a mechanical sum, without taking into account their interaction, without highlighting the main determining ones, then the conclusions may be erroneous. In the analysis of economic activity (AHA), 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, rectilinear or curvilinear. Here the theoretical and practical experience, as well as methods for comparing parallel and dynamic series, analytical groupings of initial information, graphic, etc.

Modeling economic indicators is also 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 stage of factor analysis is practical use of the factor model to calculate the reserves for the growth of the effective indicator, to plan and predict 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 whose relationship with the performance indicator is functional, i.e. when the performance indicator of the factor model is presented as a product, private or algebraic sum of factors.

This type of factor analysis is the most common, because, being quite simple to use (compared to 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 it is possible and expedient to change to increase production efficiency. Deterministic factor analysis will be discussed in detail in a separate chapter.

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

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

it is necessary to study the influence of factors on which it is impossible to build a rigidly determined factorial 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 in one quantitative indicator (for example, the level of scientific and technological progress).

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

the presence of a population;
sufficient volume of observations;
randomness and independence of observations;
homogeneity;
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 signs, 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 laws of distribution of the studied indicators);
construction of a stochastic (regression) model (refinement of the list of factors, calculation of estimates of the parameters of the regression equation, enumeration of competing models);
assessing 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 objectives);
economic interpretation and practical use of the model (determination of the spatio-temporal 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:

direct and reverse;
single-stage and multi-stage;
static and dynamic;
retrospective and prospective (forecast).

At direct factor analysis research is conducted in a deductive way - from the general to the particular. Inverse factor analysis carries out the study of cause-and-effect relationships by the method of logical induction - from private, individual factors to general ones.

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

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

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

Deterministic factor analysis.

Deterministic factor analysis has a fairly rigid sequence of procedures performed:

building an economically sound deterministic factor model;
choice of method of factor analysis and preparation of conditions for its implementation;
implementation of computational 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 built 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-and-effect relationship. As an example, consider a model that links the volume of sales (P), headcount (H) and labor productivity (PT). Theoretically, three models can 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 one makes sense, since in it the indicators on the right side of the formula are factors, i.e. 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. A brief comparative description of these methods will be discussed below.

Types of deterministic factor models.

There are the following models of deterministic analysis:

additive model, i.e., a model in which factors are included in the form of an algebraic sum, as an example, we can cite the commodity balance model:

where R- implementation;

Stocks at the beginning of the period;

P- receipt of goods;

Stocks at the end of the period;

V- other disposal of goods;

multiplicative model, i.e., a model in which the factors are included in the form of a product; An example is the simplest two-factor model:

where R- implementation;

H- number;

Fri- labor productivity;

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

where - 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- cost of fixed assets;
About- price working capital.

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

Typical problems of deterministic factor analysis.

There are four typical tasks in deterministic factor analysis:

Evaluation of the influence of the relative change in factors on the relative change in the performance indicator.
Assessment of the influence of the absolute change of the i-th factor on the absolute change of 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.
Determining the share of the absolute change in the performance indicator caused by the change in the i-th factor in the total change in the performance indicator.

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

Example.

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

The data for calculations are given in Table 1.

Table 1. Data for factor analysis of the volume of gross output.

Task 1.

The problem makes sense for multiplicative and multiple models. 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 gross output index is equal to the product of the indices of the number of employees and the average annual output, i.e.

Obviously, if we directly calculate the gross output index, 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 factor and performance indicators are related by the same dependence as the indicators in the original model. This task is solved by answering questions like: "What will happen if i-th indicator will change by n%, and the j-th indicator will change by k%?".

Task 2.

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

Let - a rigidly determined model that characterizes the change in the effective indicator y from n factors; all indicators received an increment (for example, in dynamics, in comparison with the plan, in comparison with the standard):

It is required to determine which part of the increment of the effective indicator y is due to the increment of the i-th factor, i.e., write down the following dependence:

where is the overall change in the performance indicator, which is formed under the simultaneous influence of all factor characteristics;

The change in the effective indicator under the influence of only the factor .

Depending on which method of model analysis is chosen, factorial expansions may differ. Therefore, in the context of this task, we will consider the main methods for analyzing factorial models.

Basic methods of deterministic factor analysis.

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

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

Let's give brief description the most common ways.

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 with the actual value in the reporting period. For this purpose, a number of conditional values of 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 a change in the level of a particular factor allows you to determine the impact specific factor on the growth of the effective indicator, excluding the influence of other factors. When using this method, complete decomposition is achieved.

Recall that when using this method, the order of changing the values of the factors is of great importance, since it depends on quantification influence of each factor.

First of all, it should be noted that there is not and cannot be 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 of great importance, since in a retrospective analysis, trends and the relative importance of a particular factor are important, and not accurate estimates of their influence.

Nevertheless, in order to comply with a more or less unified approach to determining the order of replacement of factors in the model, general principles can be formulated. Let us introduce some definitions.

A sign 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 summarized in space and time. As an example, we can cite the volume of sales, number, cost of working capital, etc.

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

A rigidly determined factor model is called complete if the effective indicator is quantitative, and incomplete if the effective indicator is qualitative. In a complete two-factor model, one factor is always quantitative, the second is qualitative. In this case, the replacement of factors is recommended to start with a quantitative indicator. If there are several quantitative and several qualitative indicators, then first you should change the value of the factors of the first level of subordination, 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 look at our example, the procedure for applying the method of chain substitutions.

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 output differs from the first one in that when calculating it, the actual number of workers was taken instead of the planned one. The average annual output by one worker in both cases is planned. This means that due to the increase in the number of workers, output increased by 32,000 million rubles. (192,000 - 160,000).

The third indicator differs from the second one 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 necessarily be equal to the total increase in the effective indicator:

The absence of such equality indicates errors in the calculations.

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

Task 3.

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

Economic interpretation: the coefficient shows how many percent the performance indicator has changed compared to the baseline under the influence of the i-th factor.

Calculate the coefficients α for our example, using the factorial expansion 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 output amounted to 50%.

Task 4.

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

Economic interpretation: the coefficient shows the share of the increase in the effective indicator due to the change in the i-th factor. There is no question here if all factor signs change in the same direction (either increase or decrease). If this condition is not met, the solution of the problem can 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 considered that 100% of the increase in the effective indicator is due to a change in the dominant factor sign, i.e., a sign that changes unidirectionally with the effective indicator.

Calculate the coefficients γ for our example, using the factorial expansion 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%. Hence, the increase in production in this situation is the determining factor.

are called factor analysis. The main varieties of factor analysis are deterministic analysis and stochastic analysis.

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

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 of 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. In reverse factor analysis the influence of factors is studied by the inductive method - in the direction from private factors to generalizing economic indicators.

Classification of factors affecting the effectiveness of the organization

The factors whose influence is studied during the conduct are classified according to various criteria. First of all, they can be divided into two main types: internal factors, depending on the activity of this , and external factors independent of this organization.

Internal factors depending on the magnitude of their impact on, can be divided into main and secondary. The main ones include factors related to the use and materials, as well as factors due to the supply and marketing activities and some other aspects of the functioning of the organization. The main factors have a fundamental impact on the general economic indicators. External factors, which do not depend on this organization, are determined by natural and climatic (geographical), socio-economic, as well as external economic conditions.

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

Factors can be divided 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 the change in the number of workers will be an extensive factor, and the change in the labor productivity of one worker will be an intensive factor.

Factors affecting economic performance, 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 may include weather conditions, natural disasters, which do not depend on human activity. Subjective factors are entirely dependent on people. The vast majority of factors should be classified as subjective.

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

According to their structure, the factors are divided into simple and complex. The overwhelming majority of factors are complex, including several components. However, there are also factors that cannot be divided. For example, capital productivity can serve as an example of a complex factor. The number of days the equipment has worked in a given period is a simple factor.

By the nature of the impact on generalizing economic indicators, there are direct and indirect factors. Thus, the change in products sold, although it has an inverse effect on the amount of profit, should be considered direct factors, that is, a factor of the first order. The change in magnitude 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 activity of organizations, or, in other words, reserves for improving the analyzed economic indicators.

Factor economic analysis

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

The combination of factor and resultant signs that are in the same causal relationship is called factor system. There is also the concept of a factor system model. It characterizes the relationship between the resultant feature, denoted as y, and factor features, denoted as . In other words, the factor system model expresses the relationship between general economic indicators and individual factors that affect this indicator. At the same time, 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 dependencies between generalizing (effective) and influencing factors is called economic and mathematical modeling.

Two types of relationships between generalizing indicators and factors influencing them are studied:

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

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

Stochastic connection- this is such a relationship in which each value of a factor (factorial attribute) corresponds to a set of values of a generalizing indicator (effective attribute). Under these conditions, for each value of the factor x, the values of 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 general indicator y.

In accordance with the two considered types of relationships, there are methods of deterministic factor analysis and methods of stochastic factor analysis. 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 results of the analysis is ensured by the use of economic and mathematical research methods.

These methods have a number of advantages over traditional and statistical methods of analysis.

Thus, they provide a more accurate and detailed calculation of the influence of individual factors on the change in the values of 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 otch * (U cr otch. - U cr. base.) / 100
At kr.otch. and bases - columns 6 and 7.

5. Calculation of the factor "management costs"

Pupr. =Watch. *(Uuro -U urb)/100
Where Uuro and U ur are the levels of management expenses in the reporting and base periods, respectively

6. Calculation of the totality of the influence of all factors on sales profit

The amount of "Total" 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 erroneous and further analysis is meaningless.

Factor analysis can be continued up to the net profit. The methodology for its implementation is as follows:

1. According to the above scheme, the profit from sales is analyzed.
2. The influence of all other factors (operating income, expenses, 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, others indirectly. Hence, an important methodological issue in economic analysis is the study and measurement of the influence of factors on the magnitude of the studied economic indicators.

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

Factor analysis begins its history in psychometrics and is currently 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 down by the English psychologist and anthropologist F. Galton. The development and implementation of factor analysis in psychology was carried out by such scientists as: Ch. Spearman, L. Terstone and R. Kettel. Mathematical factor analysis was developed by Hotelling, Harman, Kaiser, Terstone, Tucker and other scientists.

This type of analysis allows the researcher to solve two main tasks: to describe the subject of measurement compactly 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 scores obtained on several scales, the researcher notes that they are similar to each other and have a high correlation coefficient, in which case he can assume that there is some latent variable that can explain the observed similarity of the scores obtained. Such a latent variable is called a factor that affects numerous indicators of other variables, which leads to the possibility and need to mark it as the most general, higher order.

Thus, we can distinguish two goals of factor analysis:

Determination of relationships between variables, their classification, i.e. "objective R-classification";
reduction in the number of variables.

To identify the most significant factors and, as a result, 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 restrict the most informative principal 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 justified 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 as a product, private or algebraic sum of factors.
2. Stochastic (correlation) - the relationship between the performance and factor indicators is incomplete or probabilistic.
3. Direct (deductive) - from the general to the particular.
4. Reverse (inductive) - from the particular to the general.
5. Single stage and multi stage.
6. Static and dynamic.
7. Retrospective and prospective.

Also, factor analysis can be exploratory - it is carried out in the study of a hidden factor structure without an assumption about 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.

Mandatory conditions for factor analysis:

All signs must be quantitative;
The number of features should be twice the number of variables;
The sample must be homogeneous;
The source variables must be distributed symmetrically;
Factor analysis is carried out on correlating variables.

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

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

Stages of factor analysis:

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 indicators, methods of deterministic and stochastic factor analysis are distinguished

Deterministic factor analysis is the influence of factors whose relationship with the performance indicator is functional, i.e. when the performance indicator of the factor model is presented as a product, quotient or algebraic sum of factors.

Methods of deterministic factor analysis: Method of chain substitutions; Method of absolute differences; Relative difference method; Integral method; Logarithm method.

This type of factor analysis is the most common, because, being quite simple to use (compared to 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, it is possible and expedient to change for boost .

Stochastic analysis is a technique for studying factors whose relationship with a performance indicator, in contrast to a functional one, is incomplete, probabilistic (correlative). If with a functional (full) dependence, a corresponding change in the function always occurs with a change in the argument, then with a correlation, a change in the argument can give several values of the increase in the function, depending on the combination of other factors that determine this indicator.

Methods of stochastic factor analysis: - Method of pair correlation;
- 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 for the corresponding date. Another type is a methodology for studying cause-and-effect relationships in dynamics.

And, finally, factor analysis can be retrospective, which studies the reasons for the increase in performance indicators for past periods, and prospective, 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 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 factor environment for the formation of profits and income of companies.

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

To analyze profitability, use the following factorial model:

R = P/N, or
R = (N - S)/N * 100
where R - profit; N - revenue; S - cost.

In this case, the influence of the factor of price changes on products is determined by the formula:

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

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

Deterministic factor analysis

Deterministic modeling of factor systems is limited by the length of the factor 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 a 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 performance of individual farms or one farm in separate periods, the only question that can arise is the comparability of the quantitative analytical results identified on the basis of the model.

Deterministic factor analysis is a technique for studying the influence of factors whose relationship with the performance indicator is functional in nature, i.e. can be expressed mathematically.

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

Factor analysis of the enterprise

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 indicators, can be divided into main and secondary. The main ones include factors related to the use and materials, as well as factors due to the supply and marketing activities and some other aspects of the functioning of the organization. The main factors have a fundamental impact on the general 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 performance, fixed and variable factors can be distinguished. The first type of factors has an impact on economic performance, which is not limited in time. Variable factors affect economic performance only for a certain period of time.

Factors can be divided 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 the change in the number of workers will be an extensive factor, and the change in one worker will be an intensive factor.

Factors influencing 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 may include weather conditions, natural disasters, which do not depend on human activity. Subjective factors are entirely dependent on people. The vast majority of factors should be classified as subjective.

According to the nature of the impact on generalizing economic indicators, direct and indirect factors are distinguished. So, the change in the cost of goods sold, although it has an inverse 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 a given factor on a general economic indicator, there are measurable and non-measurable factors.

Factor analysis models

Suppose you are doing a (somewhat "stupid") study in which you measure the height of a hundred people in inches and centimeters. Thus, you have two variables. If you want to further investigate, for example, the effect of different nutritional supplements on height, would you continue to use both variables? height is one characteristic of a person, regardless of the units in which it is measured.

Now let's say you want to measure people's satisfaction with life, for which you compile a questionnaire with various items; among other questions, you ask the following: are people satisfied with their hobby (point 1) and how intensively they engage in it (point 2). The results are converted so that the average responses (for example, for satisfaction) correspond to a value of 100, while below and above the average responses are lower and higher values, respectively. Two variables (responses to two different items) are correlated with each other. (If you are not familiar with the concept of the correlation coefficient, we recommend that you refer to the section Basic statistics and tables - Correlations). From the high correlation of these two variables, we can conclude that the two items of the questionnaire are redundant.

Combining two variables into one factor. The relationship between variables can be found using a scatterplot. The line obtained by fitting gives a graphical representation of the relationship. If a new variable is defined based on the regression line depicted 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 a single factor shows the main idea of a factor analysis model, or more specifically principal component analysis (this distinction will be discussed later). If the two-variable example is extended to include more variables, the calculations become more complex, but the basic principle of representing two or more dependent variables by a single factor remains valid.

Selection of the main components. Basically, the procedure for extracting 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 lines up with the regression line. This type of rotation is called a variance-maximizing rotation, since the criterion (goal) of the rotation is to maximize the variance (variability) of the "new" variable (factor) and minimize the scatter around it (see Rotation Strategies).

Generalization to the case of many variables. When 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 plot a 3D scatterplot.

For the case of more than three variables, it becomes impossible to represent the points on the scatterplot, however the logic of rotating the axes 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 maximum, there is some scatter of data around it. And naturally 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 line), and so on. Thus, the factors are sequentially allocated one after another. 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 distinguished Recall that principal component analysis is a method of reducing or reducing data, i.e. method of reducing the number of variables. A natural question arises: how many factors should be distinguished. Note that in the process of sequential selection of factors, they include less and less variability. The decision as to when to stop the factor extraction 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 standard results of Principal Component Analysis. With repeated iterations, you extract factors with less and less variance. For simplicity, we assume that work usually 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 variance that can potentially be isolated is 10 times 1. Assume that in your life satisfaction survey, you include 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 extracted factor. The third column for each factor gives the percentage of the total variance (10 in this example) 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 accumulated or cumulative variance. The variances distinguished by the factors are called eigenvalues. This name comes from the calculation method used.

Eigenvalues and the problem of the number of factors. Once you have information about how much variance each factor has allocated, you can return to the question of how many factors should be left. As mentioned above, by its nature, this decision is arbitrary. However, there are some general guidelines, and in practice, following them gives the best results.

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

The scree criterion. The scree criterion is a graphical method first proposed by Cattell (1966). You can plot the eigenvalues presented in the table above as a simple graph.

Cattell suggested finding a place on the graph where the decrease in eigenvalues from left to right slows down as much as possible. It is assumed that only "factorial scree" is located to the right of this point - "scree" is a geological term for debris rocks accumulating in the lower part of the rocky slope. According to this criterion, 2 or 3 factors can be left in this example.

What criteria should be used. Both criteria have been studied in detail by Brown (Browne, 1968), Cattell and Jaspers (Cattell, Jaspers, 1967), Hakstian, Rogers, and Cattell (Hakstian, Rogers, Cattell, 1982), Linn (Linn, 1968), Tucker, Koopman and Lynn (Tucker, Koopman, Linn, 1969). Theoretically, one can calculate their characteristics by generating random data for a specific number of factors. Then it can be seen whether a sufficiently accurate number of significant factors has been detected 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 resulting solution can be meaningfully interpreted. Therefore, it is common to examine several solutions with more or less factors, and then choose the one that makes the most sense. This question will be further considered in terms of factor rotations.

Analysis of the main factors. Before continuing with the various aspects of the derivation of Principal Component Analysis, let's introduce Principal Factor Analysis. Let's go back to the example of the life satisfaction questionnaire to formulate another "thinkable model". You can imagine that subjects' responses depend on two components. First, we select some relevant general factors, such as, for example, "satisfaction with one's hobbies" discussed earlier. Each item measures some part of this overall aspect of satisfaction. In addition, each item includes a unique aspect of satisfaction not shared by any other item.

Generalities. 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 a single variable that belongs to common factors (and is shared with other variables) is called commonality. Therefore, the additional work facing the researcher when applying this model is the assessment of the commonality for each variable, i.e. the proportion of variance that is common to all items. The proportion of variance for which each item is responsible is then equal to the total variance corresponding to all variables minus the commonality. 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 about the theory of multiple regression, we will refer to the Multiple Regression section). Several authors propose various iterative "post-solution improvements" to the initial generality estimate obtained using multiple regression; 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 the residual (unexplained) sums of squares.

Principal factors versus principal components. Principal factors versus principal components. The main difference between the two factor analysis models is that Principal Component Analysis assumes that all the variability of the variables must be used, while in Principal Factor Analysis you use only the variance of the variable that is common to other variables. A detailed discussion of the pros and cons of each approach is beyond the scope of this introduction. In most cases, these two methods lead to very close results. However, Principal Component Analysis is often preferred as a method of data reduction, while Principal Factor Analysis is best used to determine the structure of data (see next section).

Factor analysis of sales

Similarly, we derive models for factorial analysis of sales profitability.

The original indicator looks like:

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

Change in sales profitability under the influence of relevant factors:

Lrpr \u003d Prp1 / RP1- PrpO / RP0 \u003d (RP1 - Srp1) / RP1 - (RP0 - Srp0) / RL0 \u003d - CpnJ / RSH + Crp0 / RP0 \u003d (Crp0 / RSH - Crp1 / RP1) + (Crp0 / RP0 - Cp0/RP1) = LrsPRS + A/V.

Here, the component Ap prS characterizes the impact of changes in the cost products sold on the dynamics of profitability of sales. A component A//PPR - the impact of changes in the volume of sales. They are determined accordingly: ArsPRs = Cp0/RP1 - Cp1/RP1; A / nPr \u003d Cp0 / RP0 - Cp0 / RP1.

Using the method of chain substitutions, the factor analysis of the profitability of sales can be continued by studying the influence on the component of Ap prS of the dynamics of such factors as:

A) the cost of selling goods, products, works, services:
ArsPrr \u003d (Cp0 - Cp1) / RP1,
where СрО, Cpl - the cost of sales of goods, products, works, services, respectively, in the base and reporting periods (line 020 of form 2), rubles;

B) management expenses:

Ар „, y = (СуО - Сu1) / RP1, where СuО, Сu1 - administrative expenses, respectively, in the base and reporting periods (line 030 of form 2), rub.,

B) business expenses

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

If the company maintains cost and revenue records certain types products, then in the process of analysis it is necessary to evaluate the influence of the sales structure on the change in the profitability of products. However, such a study is possible only according to operational data, that is, it is carried out in the process of intra-company analysis. Let's demonstrate it with the following example.

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

Products Share of the j-th Product Profitability of the j-th product in the volume of products, Pj of sales, %, dj

Profitability of sold products:

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

This change in profitability is the result of the influence of two factors:

Change in the profitability of individual products:
pwp1 =ip>jd)-ipw =
P 1=1
= 0,1748 - (0,25*0,4 + 0,125*0,6) = 0,1748 - 0,1750 = -0,0002.
Changing the implementation structure:
Pmd. = Z P°Jd) ~ Z P°JdJ = °"1750 " °"1625 = +0"0125 "" M M

Conclusion: The increase in the level of profitability of sold products occurred due to a change in the structure of sales. 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 amounted to 1.23%.

Tasks of factor analysis

1. Selection of factors for the analysis of the studied performance indicators and their classification.
2. Determining the form of dependence between factor and performance indicators, building 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 factors according to their content: extensive (quantitative), intensive (qualitative); and the level of subordination.

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

At present, in the analysis actual cost of manufactured goods, identifying 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 effective management organization, it is of interest to assess the impact of various factors or causes on this indicator when they change in the production process, in particular, deviations from planned values, values in the base period, etc.

Economic forces most fully cover all elements of the production process - means, objects of labor and labor itself. They reflect the main directions of the work of teams of enterprises to reduce costs: increasing labor productivity, introducing advanced equipment and technology, better use of equipment, cheaper procurement and better use of labor items, reducing administrative and managerial and other costs, reducing waste and eliminating unproductive costs and losses.

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

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

For this group of factors for each event, the economic effect which translates into lower production costs. The savings from the implementation of measures is determined by comparing the cost per unit of output before and after the implementation of measures and multiplying the resulting difference by the volume of production in the planned year:

EC \u003d (Z0 - Z1) * Q, (7.8)
where EK - saving direct current costs;
Z0 - direct current costs per unit of output before the implementation of the measure;
Z1 - direct current costs per unit of output after the implementation of the measure;
Q - the volume of output of goods in physical units from the beginning of the implementation of the measure 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 specialization in production; improvement of production management and cost reduction; improved use ; improvement of material and technical supply; reduction of transport costs; other factors that increase the level of organization of production. With the simultaneous improvement of technology and the organization of production, it is necessary to establish the savings for each factor separately and include them in the appropriate groups. If it is difficult to make such a division, then the savings can be calculated based on the targeted nature of the activities or by groups of factors.

The reduction in current costs occurs as a result of improving the maintenance of the main production (for example, the development of in-line production, increasing the shift ratio, streamlining utility work, improving tool management, improving the organization of quality control of work and goods). A significant reduction in the cost of living labor can occur with an increase in norms and service areas, a reduction in losses, and a decrease in the number of workers who do not meet production standards. These savings can be calculated by multiplying the number of redundant workers by the average in the previous year (with accruals for social insurance and taking into account the costs of overalls, food, etc.). Additional savings arise from the improvement of 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 accruals on it in connection with the release of managerial personnel.

When improving the use of fixed assets, the savings are calculated as the product of the absolute reduction in costs (except depreciation) per unit of equipment (or other fixed assets) by the average operating amount of equipment (or other fixed assets).

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

3) Change in the volume and structure of goods: change 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 fixed costs (except for depreciation), a relative decrease in . Semi-fixed costs do not directly depend on the number of goods produced, with an increase in the volume of production, their number per unit of goods decreases, which leads to a decrease in its cost.

Relative savings on semi-fixed costs is determined by the formula

EKP \u003d (TV * ZUP0) / 100, (7.9)
where EKP - savings of semi-fixed costs;
ZUP0 - the amount of conditionally fixed costs in the base period;
ТV is the growth rate of output compared to the base period.

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

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

In order to avoid a repeated account, the total amount of savings is reduced (increased) by the part that is taken into account by other factors.

Changing the range and range of goods is one of the important factors affecting the level of production costs. With different profitability of individual products (in relation to the cost price), 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 is analyzed by variable costs according to the calculation items of the standard nomenclature. The calculation of the influence of the structure of goods on the cost price must be linked to indicators of increasing labor productivity.

4) Improved usage natural resources: change in the composition and quality of raw materials; change in the productivity of deposits, the volume of preparatory work during extraction, methods of extraction of natural raw materials; changing others natural conditions. These factors reflect the influence of natural (natural) conditions on the value variable costs. The analysis of their impact 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 shops and facilities.

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

EKP \u003d (З1 / Q1 - З0 / Q0) * Q1, (7.11)
where ECP is the change in the costs of preparing and mastering production;
Z0, Z1 - the sum of the costs of the base and reporting period;
Q0, Q1 - the volume of output of goods in the base and reporting period.

If changes in the value of costs in the analyzed period are 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 cost reduction factors and reserves identified as a result of the analysis must be summarized in the final conclusions, the total influence of all factors on reducing the total cost per unit of goods should be determined.

In order to conduct a factor analysis of labor productivity, i.e. determine how one or another technical and economic factor affects the changes in this indicator, calculate the relative savings (increase) in the number of employees. The 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 influence of all factors:

L = 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 the growth in production volume (extensive factor), and the growth of 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 the company'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 profitability indicator. However, a simple calculation may not be enough and will need to be supplemented by analysis. The most popular method is, perhaps, factorial analysis of profitability. equity. Let us dwell in more detail on the methodology of its implementation and the main features.

Factor analysis of return on equity is usually associated with DuPont's formulas, which allow you to quickly produce everything. 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, obviously, is determined by the ratio received to the value of this capital. The factor model is obtained from this relation by elementary transformations. Their essence is to multiply the numerator and denominator by revenue and assets. After that, it is easy to see that the efficiency of using this part of the capital, its profitability, is determined by the product of the indicator of the degree of financial dependence on the turnover of property (assets) and the level of profitability of sales. After compiling mathematical model it is analyzed directly. It can be carried out in any way suitable for deterministic models. Factor analysis of return on equity using DuPont formulas is one of the variations of the method of absolute differences. It, in turn, is also a special case of the chain substitution method. The main principle of this method lies in the successive determination of the impact of each factor in isolation, regardless of the others.

It should be noted that a 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 as the product of the turnover of the company's property by the profitability of sales. The subsequent analysis proceeds in the same way.

Necessary Special attention pay 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 income statement are cumulative in nature, as 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 value.

The above methods, that is, the method of chain substitutions and its modifications, can be used to analyze almost any deterministic factorial model. For example, factor analysis of the current liquidity ratio can be carried out extremely simply. For greater detail, it is advisable to disclose the formula of 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 pointed out 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 return on equity and other indicators is one of best practices facilitating acceptance of the faithful management decisions. Revealing the strong negative impact of this or that factor 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 the generalization of the patterns of variation in the values of economic indicators - quantitative characteristics of factors and results of economic activity. The quantitative parameters of the relationship are identified on the basis of a comparison of the values of the studied indicators in the totality 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 repeatedly 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 of the indicators should occur within the unambiguous certainty of the qualitative side of the phenomena, the characteristics of which are the modeled economic indicators (within the variation, there should not be a qualitative jump 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 population (with respect to the relationships under study).

The studied pattern of changes in economic indicators (the modeled relationship) appears in a hidden form. It is intertwined with random from the point of view of the study (not studied) components of variation and covariance of indicators. The law of large numbers says that only in a large population is a regular connection more stable than a random coincidence of the direction of variation (random variation).

From this follows the third premise of stochastic analysis - sufficient dimension (number) of the set of observations, which makes it possible to identify the studied patterns (modeled relationships) with sufficient reliability and accuracy.

The fourth premise of the stochastic approach is the availability of methods that make it possible to identify quantitative parameters of economic indicators from mass data of varying the level of indicators. The mathematical apparatus of the applied methods sometimes imposes specific requirements on the empirical material being modeled. The fulfillment of these requirements is an important prerequisite for the applicability of the methods and the reliability of the results obtained.

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

Methods of stochastic factor analysis:

pair 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, does not appear in each individual case, but in a certain dependence. With the help of pair correlation, two main tasks are solved: a model of acting factors is left (regression equation); a quantitative assessment of the closeness of connections (correlation coefficient) is given.

matrix models. Matrix models represent a schematic reflection of an economic phenomenon or process using scientific abstraction. The most widespread here is the method of analysis "cost-output", which is built according to a chess scheme and allows in the most compact form to present the relationship between costs and production results.

Mathematical programming is the main tool for solving problems of optimizing production and economic activities.

The method of operations research is aimed at studying, including the production and economic activities of enterprises, in order to determine such a combination of structural interrelated elements of systems, which to the greatest extent will allow determining the best economic indicator from a number of possible ones.

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

Integral method of factor analysis

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

Using this method allows you to get more accurate results of calculating the influence of factors compared to the methods of chain substitution, absolute and relative differences and avoid an 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 distributed equally among them.

To distribute an additional increase, it is not enough to take a part of it corresponding to the number of factors, since the factors can act in different directions. Therefore, the change in the effective indicator is measured over infinitely small periods of time, i.e., the increment of the result is summed up, defined as partial products multiplied by the increments of factors over infinitely small intervals. The operation of calculating a definite integral is solved with the help of a PC and is reduced to the construction of integrands that depend on the type of function or model of the factorial 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 recommend reading our article

Net profit is such an indicator of the company's performance, which, on the one hand, is influenced by the greatest number of factors compared to 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 profit. As the name implies, studying profit in this way involves identifying those factors that affect it the most, as well as determining the specific magnitude of this impact.

Before considering factor analysis of net profit, it is necessary to study how it is formed. Analysis of the formation of net profit is carried out according to the income statement. This is understandable, since 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 this reporting form. It involves finding the share 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 basis of comparison, 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 because this form of reporting makes it easy and simple to compile a mathematical model that will include factors that affect 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 to judge its changes under the influence of price and sales volume. These factors are extremely important, so they must be additionally taken into account in the model, dividing the impact on revenue revenue into two corresponding parts. After compiling a mathematical model, it is necessary to directly subject it to analysis according to a certain method. Most often resort to the use of 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 process of formation and dynamics, it is necessary to analyze the use of net profit. The most logical and easiest way to study this process is by conducting a vertical analysis, which has already been mentioned above. Obviously, in this case, it is necessary to take net profit as the basis. 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 figures must 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 profit, many conclusions can be drawn about pricing policy, cost management, and much more. Conclusions and recommendations are the basis for making managerial decisions that are vital to the firm's operations.

Factor analysis method of chain substitutions

The chain substitution method is the most universal 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 with the actual value in the reporting period. For this purpose, a number of conditional values of 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. Comparing the value of the effective indicator before and after changing the level of one or another factor allows you to eliminate the influence of all factors except one, and determine the impact of the latter on the growth of 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 are 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 conditional 1= a 1*b 0*C 0 ; Y a = Y arb. 1 - Y 0;
Y conditional 2= a 1*b 1*C 0; Y b= Y conv.2– Y conv.1;
Y f \u003d a 1 * b 1 * C 1; Y c \u003d Y f - Y conditional 2, etc.

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 errors 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 chain substitution method, it is very important to ensure a strict substitution sequence, since its arbitrary change can lead to incorrect results. In the practice of analysis, first of all, the influence of quantitative indicators is revealed, and then - 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 the influence of the quantitative indicator of the number of employees is established, and then the qualitative indicator of labor productivity. If the influence of quantity and price factors on the volume of industrial products sold is clarified, then the influence of quantity is first calculated, 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 substitution sequence changes the quantitative weight of a particular indicator. The more significant the deviation of the actual indicators from the planned ones, the greater the differences in the assessment of the factors calculated with different substitution sequences.

The chain substitution method has a significant drawback, the essence of which is the appearance of an indecomposable remainder, 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. The noted drawback is eliminated when a more complex integral method is used in analytical calculations.

Factor analysis of wages

It is carried out taking into account the analysis of the use labor resources at the enterprise and the level of labor productivity. It is known that with the growth of labor productivity, real prerequisites are created for increasing the level of its payment. At the same time, funds for wages 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 wage bill begins with the calculation of the absolute and relative deviations of its actual value from the planned one.

We make a sequential calculation

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

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

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

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

Initial data for the analysis of payroll

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

FZPotn \u003d FZPf - FZPsk \u003d FZPag - (FZP pl..trans * Kvp + FZP pl.. post) \u003d 21465 - (13120 * 1.026 + 7380) \u003d 21465 - 20841 \u003d + 424 million rubles
where FZPsk is the planned salary fund, adjusted for the coefficient of fulfillment of the plan for output;
FZP pl..per and FZP pl..post - variable and constant amount of the planned planned salary fund.

When calculating the FZPotn, you can use the so-called correction factor Kp, which reflects the share of the variable salary in the general fund. It shows by what fraction of a percentage the planned wage bill should be increased for each percentage of overfulfillment of the production plan (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 available for direct measurement) generalizing characteristics of the organized structure and mechanism of development of the studied phenomena or processes.

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

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. This - explicit variables. If the signs change in concert, then we can assume the existence of certain common causes this variability, i.e. the existence of some hidden (latent) factors. The task of the analysis is to find these factors.

Since the factors are a combination of certain variables, it follows that these variables are related to each other, i.e. have a correlation (covariance), moreover, greater among themselves than with other variables included in another factor. Methods for finding factors are based on the use of correlation coefficients (covariance) between variables. Factor analysis gives a non-trivial solution, i.e. the solution cannot be predicted without applying a special factor extraction technique. This solution is of great importance for characterizing the phenomenon, since at first it was characterized by a sufficiently large number of variables, and 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 be correlated X i , but also observable objects N i. Depending on what type of correlation is considered - between features or objects - R and Q data processing techniques are distinguished, respectively.

In accordance with general principles factor analysis, the result of each measurement is determined by the action of general factors, specific factors and the "factor" of measurement error. General the factors influencing 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 error it implies a set of uncountable causes that determine the results of a measurement. The variability of the obtained empirical data is usually described using their dispersion.

You are already well aware that 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 empirical data and 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 represented 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. In this topic, we will not discuss trigonometric function cosine, it is enough to know where to find the corresponding data.

Table 7.1 lists several values for the cosines of angles to give you a general idea of them.

Table 7.1

Table of cosines for a graphic image

correlations between variables.

According to this total positive correlation table ( 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 are directed in opposite directions ( cos 180 -1). (Fig. 7.3 b).

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

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

V 1

V 2

a 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 representation of the entire correlation matrix and subsequent interpretation of the data in factor analysis.

The construction of a matrix begins with the construction of a vector representing any variable. The other variables are represented by vectors of equal length, all coming 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 variable vectors would need to be at an angle to the page. This fact is not a problem for the actual mathematical procedures, but requires some imagination from the reader. Figure 7.5. it can be seen 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 of 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 to each other. High correlations between V1 and V2 are evidence 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 independent of each other, i.e. having minimal correlations between themselves, or angles corresponding to 90  (Fig. 7.5.)

V 1

Figure 7.5. Geometric interpretation of the correlation matrix
This figure shows 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, consequently, large cosine values). Similarly, the correlation between variables V 4 and V 5 is also large and positive. But between these groups of variables, the correlation is close to zero, since these groups of variables are almost orthogonal to each other, i.e. located relative to each other at right angles. The above 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 those variables and the corresponding factors can be calculated. The correlation between a variable and a common factor is usually called factor load. The geometric interpretation of this concept is given in fig. 7.6.

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

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

For example, let's hypothetically imagine a legislature at the regional level, consisting of 100 deputies. Among the various issues on the agenda for voting are: 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 production to state ownership. The contingency matrix shows the following distribution of deputies' votes:

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

It is obvious that the votes are statistically related: the overwhelming majority of deputies who support the idea of restoring the monument to Lenin also support the return to state ownership strategic enterprises. 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, the voting is completely unrelated to each other thematically.

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 hidden split in the deputies according to 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). Having identified such splits, we can describe a large number of individual votes in terms of 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 would have been asked to determine their attitude towards conservative socialist values. We detect the presence of this factor based on a meaningful analysis of quantitative relationships between variables. Moreover, if nominal variables are deliberately taken in our example - support for 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 problems that can be solved with its help. Thus, there are at least three “typical” goals of factor analysis:

dimensionality reduction (reduction) of data. Factor analysis, highlighting the nodes of interrelated features and reducing them to some generalized factors, reduces the initial basis of features of the description. The solution of 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 a semantic feature. The transition from many variables to several factors makes it possible to make the description more compact, to get rid of uninformative and duplicate variables;

Revealing the structure of objects or features (classification). This problem is close to that which is solved by the cluster analysis method. But if cluster analysis takes their values for 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, with the help of factor analysis, one can evaluate the similarity and difference of objects in the space of their correlations, or in the factor space. The resulting latent variables act as the coordinate axes of the factor space, the objects under consideration are projected onto these axes, which makes it possible to create a visual geometric representation of the studied data, convenient for meaningful interpretation;

indirect measurement. 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 factor analysis statistics on the example of data on the results of parliamentary elections in Ryazan region 1999 (general federal district). To simplify the example, let's take electoral statistics only for those parties that have overcome the 5% barrier. The data is taken in the context of territorial election commissions (by cities and districts of the region).

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

TEAK (territorial election commission)	"Apple"	"Unity"	Block Zhirinovsky	OVR	CPRF	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 scores (g-scores)
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. (total 32 cases)

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

Already a visual analysis of the matrix of pair 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 (pairs "Yabloko" - OVR, "Yabloko" - SPS and OVR - SPS). At the same time, these three variables are negatively correlated with the CPRF (support for the CPRF), to a lesser extent with Unity (support for Unity), and even less with the BZ variable (support for the Zhirinovsky Bloc). Thus, we presumably have two pronounced correlation pleiades:

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

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

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

To test our assumptions, we need to CALCULATE the eigenvalues of the factors (eigenvalues), factor scores, and factor loadings for each variable. Such calculations are quite complicated and require serious skills in working with matrices, so we will not consider the computational aspect here. We will only say that these calculations can be carried out in two ways: the method of principal components (principal components) and the method of principal factors (principal factors). Principal component method is more common, statistical programs use it "by default".

Let us dwell on the interpretation of eigenvalues, factorial values and factor loadings.

The eigenvalues of the factors for our case are as follows:

bgcolor=white>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 the factor, the greater its explanatory power (the maximum value is equal to the number of variables, in our case 6). One of key elements factor analysis statistics is the indicator "% total variation" (% total variance). It shows what proportion of the variation (variability) of variables explains the extracted factor. In our case, the weight of the first factor outweighs 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 - 12.6%, and so on. descending.

Having the eigenvalues of the factors, we can start solving the problem of data dimension reduction. 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 follow. So, factors 5 and 6 are clearly superfluous, which together explain a little more than 1% of the entire variation. But the fate of factors 3 and 4 is no longer so obvious.

As a rule, factors remain in the model, the eigenvalue of which exceeds unity (the 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 chart got its name from its resemblance to the side of a mountain. "Scree" is a geological term for rock fragments that accumulate at the bottom of a rocky slope. "Rock" is truly influential factors, "scree" 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 from left to right is greatly slowed down). 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 a 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, visually). Thus, in this case, it is better to sacrifice some accuracy in favor of compactness, leaving the first and second factors.

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

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

Reproduced correlations
	"Apple"	"Unity"	BJ	OVR	CPRF	THX
"Apple"	0,89
"Unity"	-0,53	0,80
BJ	-0,47	0,59	0,44
OVR	0,73	-0,72	-0,56	0,76
CPRF	-0,70	0,01	0,12	-0,34	0,89
THX	0,88 -0,43		-0,40	0,66	-0,77	0,88
Residual odds
	"Apple"	"Unity"	BJ	OVR	CPRF	THX
"Apple"	0,11
"Unity"	-0,02	0,20
BJ	0,00	-0,31	0,56
OVR	-0,13	-0,01	0,09	0,24
CPRF	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, being generally adequate, does not explain individual relationships well. Thus, the generality of the BZ variable is very low (only 0.56), the value of the residual coefficient of connection between BZ and “Unity” is too high (-0.31).

Now it is necessary to decide how important an adequate representation of the BJ variable is for this particular study. 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 leave a simpler model.

Factor loads can be represented as correlation coefficients of each variable with each of the identified factors 1ak, the correlation between the values of the first factor variable and the values of the "Apple" variable is -0.93. All factor loadings are given in the factor mapping matrix-

The closer the relationship of the variable with the factor under consideration, the higher the value of the factor load. The positive sign of the factor load indicates a direct, and the negative sign indicates the feedback of the variable with the factor.

Having the values of factor loads, we can construct a geometric representation of the results of factor analysis. On the X axis, we plot the loads of variables on factor 1, on the Y axis, the loads of variables on factor 2, and we get a two-dimensional factor space.

Before proceeding to a meaningful analysis of the results obtained, let's perform one more operation - rotation. The importance of this operation is dictated by the fact that there is not one, but many variants of the matrix of factor loads that equally explain the relationships of variables (the matrix of intercorrelations). It is necessary to choose a solution that is easier to interpret meaningfully. This is considered a load matrix in which the values of each variable for each factor are maximized or minimized (close to one or zero).

Consider a schematic example. There are four objects located in the factor space as follows:

Loads on both factors for all objects are significantly different from zero, and we are forced 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 unity (simple structure principle). Accordingly, for a meaningful interpretation of the position of objects, we will involve only one factor - factor 2.

There are quite a number of methods for rotating factors. Thus, the group of orthogonal rotation methods always preserves a right angle between the coordinate axes. These include vanmax (minimizes the number of variables with a high factor load), quartimax (minimizes the number of factors needed to explain the variable), equamax (a combination of the two previous methods). Oblique rotation methods do not necessarily preserve a right angle between the axes (eg 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 for most policy research tasks. In addition, as with many other methods, it is recommended to experiment with various techniques rotation.

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

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

Now we can proceed to a meaningful interpretation of the results obtained. 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. In terms of content - based on the specifics of the ideological attitudes of the named subjects of the electoral process - we can interpret this demarcation as a "left-right" split, which is "classical" for political science.

Opposition on factor 2 is formed by OVR and Unity. The “Zhirinovsky 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 relationships 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 offered 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 of 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 electoral choice of territory space. 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 build a two-dimensional diagram. The corresponding column shows factor loadings.

If you look closely at these results, you will notice that the same main split is reproduced in the republic from election to election: “the “party of power” is all the rest.” In 1995, the “party of power” was the block “Our Home is Russia "(NDR), 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 a one-dimensional political choice.

At the end of the chapter, we will give one practical advice. The success of the development of statistical methods, by and large, is possible only with intensive practical work with special programs (the already 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 the 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 problems is a fundamentally important component of the model of a political scientist.

Control questions and tasks

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

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

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

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

5. What can be learned about the relationship between variables based on the values of the chi-square and lambda statistical tests?

6. Define the concept of "error" in statistical research. How can this indicator be used to judge the quality of the constructed statistical model?

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

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

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

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

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

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

13. Explain the term "multidimensional classification method".

14. Explain the main differences between clustering using 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 task solved by discriminant analysis? Define a discriminant function.

17. Name three classes of problems solved using factor analysis. Define the term "factor".

18. Describe the three main methods for checking the quality of a model in factor analysis (Kaiser's criterion, the "scree" criterion, the matrix of reproduced correlations).

International migration of financial resources in the context of factor analysis

25. J.-B. Say entered the history of economic science as the author of the factorial theory of value. What are the main provisions of this theory?

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