The goal of an enterprise's economic activity is always a certain result, which depends on numerous and varied factors. Obviously, the more detailed the influence of factors on the magnitude of the result is studied, the more accurate and reliable the forecast about the possibility of achieving it will be. Without a deep and comprehensive study of factors, it is impossible to draw informed conclusions about the results of operations, identify production reserves, justify a business plan and make management decisions. Factor analysis, by definition, is a methodology that includes unified methods for measuring (constant and systemic) factor indicators, a comprehensive study of their impact on the value of performance indicators, and the theoretical principles underlying forecasting.

The following are distinguished: types factor analysis:

– analysis of functional dependencies and correlation analysis (probabilistic dependencies);

– direct and reverse;

– single-stage and multi-stage;

– static and dynamic;

– retrospective and prospective.

Factor analysis of functional dependencies is a technique for studying the influence of factors in the case when the resulting indicator can be presented in the form of a product, quotient or algebraic sum of factors.

Correlation analysis is a technique for studying factors whose connection with an effective indicator is probabilistic (correlation). For example, labor productivity at different enterprises at the same level of capital-labor ratio may also depend on other factors, the impact of which on this indicator is difficult to predict.

In direct factor analysis, the research is conducted from the general to the specific (deductively). Reverse factor analysis carries out research from particular, individual factors to general ones (using the induction method).

Single-stage factor analysis is used to study factors of only one level (one level) of subordination without detailing them into their component parts. For example, y = А·В. In multi-stage factor analysis, factors are detailed A And IN: dividing them into constituent elements in order to study interdependencies.

Static factor analysis is used to study the influence of factors on performance indicators as of the corresponding date. Dynamic - is a technique for studying the relationships between factor indicators in dynamics.

Retrospective factor analysis studies the reasons for changes in performance indicators over past periods, while prospective factor analysis predicts the behavior of factors and performance indicators in the future.

The main objectives of factor analysis are the following:

– selection, classification and systematization of factors that influence the studied performance indicators;

– determination of the form of dependence between factors and the performance indicator;

– development (application) of a mathematical model of the relationship between the result and factor indicators;

– calculation of the influence of various factors on the change in the value of the effective indicator and comparison of this influence;

– making a forecast based on a factor model.

From the point of view of impact on the results of financial and economic activities of the enterprise, factors are divided into basic and secondary, internal and external, objective and subjective, general and specific, constant and variable, extensive and intensive.

The main factors include those that have the most significant impact on the result. Others call them minor. It should be noted that, depending on the circumstances, the same factor can be both primary and secondary.

Internal factors are factors that an enterprise can influence. They should be given the most attention. However, external factors (market conditions, inflation processes, conditions of supply of raw materials, materials, their quality, cost, etc.) certainly affect the results of the enterprise. Their study makes it possible to more accurately determine the degree of influence of internal factors and provide a more reliable forecast of production development.

Objective factors do not depend on the will and desires of people (in contracts, the term force majeure is used to refer to these factors; this could be a natural disaster, an unexpected change of political regime, etc.). Unlike objective reasons, subjective reasons depend on the activities of individuals and organizations.

Common factors are characteristic of all sectors of the economy. Specific are those that operate in a particular industry or enterprise. This division of factors allows us to more fully take into account the characteristics of individual enterprises and make a more accurate assessment of their activities.

Constant and variable factors are distinguished by the duration of their impact on production results . Constant factors influence the phenomenon under study continuously throughout the entire period under study (reporting period, production cycle, product life, etc.). The impact of variable factors is one-time, irregular.

Extensive factors include factors that are associated with a quantitative rather than a qualitative increase in the performance indicator, for example, an increase in the volume of production by expanding the sown area, increasing the number of livestock, the number of workers, etc. Intensive factors characterize qualitative changes in the production process, for example, an increase in crop yields as a result of the use of new types of fertilizers.

Factors are also divided into quantitative and qualitative, complex and simple, direct and indirect. Quantitative factors, by definition, can be measured (number of workers, equipment, raw materials, labor productivity, etc.). But often the process of measuring or searching for information is difficult, and then the influence of individual factors is characterized qualitatively (more - less, better - worse).

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

Factors that have a direct impact on the performance indicator are called direct (factors of direct action). Indirect ones influence through the mediation of other factors. Depending on the degree of indirect influence, factors of the first, second, third and subsequent levels of subordination are distinguished. Thus, direct action factors — first level factors. Factors that determine the performance indicator indirectly, using first-level factors, are called second level factors etc.

Any factor analysis of indicators begins with modeling a multifactor model. The essence of building a model is to create a specific mathematical relationship between factors.

When modeling functional factor systems, a number of requirements must be met.

1. Factors included in the model must actually exist and have a specific physical meaning.

2. Factors that are included in the system of factor analysis of indicators must have a cause-and-effect relationship with the indicator being studied.

3. The factor model must provide measurement of the influence of a specific factor on the overall result.

In factor analysis of indicators, the following types of the most common models are used.

1. When the resultant indicator is obtained as an algebraic sum or the difference of the resulting factors, apply additive models, for example:

where is the profit from product sales,

- revenues from sales,

– production cost of goods sold,

– business expenses,

– administrative expenses.

Multiplicative models are used when the resulting indicator is obtained as a product of several resulting factors:

where is return on assets,

– profitability of sales,

– return on assets,

– the average value of the organization’s assets for the reporting year.

3. When the effective indicator is obtained by dividing one factor by another, apply multiples models:

Various combinations of the above models give mixed or combined models:

;

etc.

In the practice of economic analysis, there are several ways to model multifactor models: lengthening, formal decomposition, expansion, reduction and dismemberment of one or several factor indicators into component elements.

For example, using the expansion method, you can build a three-factor model of the organization’s return on assets as follows:

;

where is the organization’s equity capital turnover,

– independence coefficient or the share of equity capital in the total assets of the organization,

– the average cost of the organization’s equity capital for the reporting period.

Thus, we have obtained a three-factor multiplicative model of the organization's return on assets. This model is widely known in the economic literature as the Dupont model. Considering this model, we can say that the profitability of an organization’s assets is influenced by the return on sales, equity turnover and the share of equity in the total assets of the organization.

Now consider the following return on assets model:

where is the share of revenue per 1 rub. full production costs,

– share current assets in the formation of assets,

– share of inventories in the formation of current assets,

– inventory turnover.

The first factor of this model speaks about the pricing policy of the organization; it shows the basic markup that is included directly in the price of the products sold.

The second and third factors show the structure of assets and current assets, the optimal value of which makes it possible to save working capital.

The fourth factor is determined by the volume of production and sales of products and speaks of the efficiency of use of inventories; physically it expresses the number of revolutions that inventories make during the reporting year.

Equity method used when it is difficult to establish the dependence of the analyzed indicator on private indicators. The method is that the deviation according to the general indicator is proportionally distributed among the individual factors under the influence of which it occurred. For example, you can calculate the impact of changes in book profit on the level of profitability using the formula:

Ri = R·(  i / b) ,

where  Ri- change in the level of profitability due to an increase in profit under the influence of a factor i, %;

R-change in the level of profitability due to changes in balance sheet profit,%;

 b - change in balance sheet profit, rub.;

 i- change in balance sheet profit due to the factor i.

Chain substitution method allows you to measure the influence of individual factors on the result of their interaction - generalizing ( target) indicator, calculate deviations of actual indicators from standard (planned) indicators.

Substitution is the replacement of a basic or standard value of a particular indicator with an actual one. Chain substitutions are sequential replacements of the basic values of particular indicators included in the calculation formula with the actual values of these indicators. Then these influences (the influence of the replacement made on the change in the value of the general indicator being studied) are compared with each other. The number of substitutions is equal to the number of partial indicators included in the calculation formula.

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

IN general view The application of the chain production method can be described as follows:

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

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

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

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

The algorithm of the chain substitution method can be demonstrated by the example of calculating the impact of changes in the values of partial indicators on the value of the indicator, presented in the form of the following calculation formula: F = a· b· c· d.

Then the base value F will be equal F 0 = a 0 · b 0 · c 0 · d 0 ,

and the actual one: F 1 = a 1 · b 1 · c 1 · d 1 .

Total deviation of the actual indicator from the basic indicator  F (F=F 1 –F 0) is obviously equal to the sum of deviations obtained under the influence of changes in particular indicators:

F = F 1 +F 2 +F 3 +F 4 .

And changes in private indicators are calculated by successive substitutions into the formula for calculating the indicator F actual parameter values a, b, c, d instead of basic ones:

The calculation is checked by comparing the balance of deviations, i.e. the total deviation of the actual indicator from the basic indicator should be equal to the sum of deviations under the influence of changes in private indicators:

F 1 –F 0 = F 1 +F 2 +F 3 +F 4 .

Advantages this method: versatility of application, ease of calculations.

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

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

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

In analysis, quantitative factors are understood as those that express the quantitative certainty of phenomena and can be obtained by direct accounting (number of workers, machines, raw materials, etc.).

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

A variation of the method of chain substitutions is the method of calculation using absolute differences. The target function in this case, as in the previous example, is presented in the form of a multiplicative model. The change in the value of each factor is determined in comparison with the base value, for example, the planned one. Then these differences are multiplied by the remaining partial indicators - the factors of the multiplicative model. But, note, when moving from one factor to another, a different value of the multiplier is taken into account. The factors that appear after the factor (on the right) by which the difference is calculated remain in the value of the base period, and all those remaining before it (on the left) are taken in the values of the reporting period.

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

We will show this using the example of the influence of individual factors on the amount of materials costs TS m, which are formed under the influence of three factors: volume of production in physical terms Q, norms of material consumption per accounting unit of production m and prices for materials Pm.

TS m = Q· m· Pm.

First, the change in each factor compared to the plan is calculated:

change in production volume  Q= Q 0 – Q 1 ;

change in material consumption rates per accounting unit  m = m 0 – m 1 ;

change in price per unit of material  Pm = Pm 1 – Pm 0 .

Next, the influence of individual factors on the general indicator is determined, i.e. the amount of costs for materials. In this case, the partial indicators that stand before the indicator by which the difference is calculated are left in their actual value, and all those following it are left in the basic value.

In this case, the impact of changes in production volume  Q the amount of materials costs will be:

TS mQ = Q· m 0 · Pm 0 ;

influence of changes in material consumption rates  TS mm:

TS mm = Q 1 · m· Pm 0 ;

impact of changes in prices for materials  TS mp:

TS mp = Q 1 · m 1 · Pm.

The total deviation of the amount of costs for materials will be equal to the sum of the deviations of the influence of individual factors, i.e.

TS m = TS mQ + TS mm + TS mp.

However, in practice there are more often situations when one can only assume the existence of a functional dependence (for example, the dependence of revenue ( TR) from the number of products produced and sold ( Q): TR = TR(Q)). To test this assumption, use regression analysis, with the help of which a function of a certain type is selected ( F r(Q)). Then, on the set of function definition (on the set of values of the factor indicator), the set of function values is calculated.

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

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

find the relative deviation of each factor indicator:

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

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

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

1. Model view:

2. View model :

3. View model:

4. View model:

A comprehensive analysis of financial condition involves a broad and full research all factors that influence or may influence the final financial results of the organization, which, ultimately, are the main goal of the organization.

The results of the analysis must be used to make the correct management decisions administration of the organization and informed investment decisions by shareholders-owners.

TASK 2

It is known that during the reporting period the average number of workers on the payroll increased from 500 to 520 people, the average number of hours worked per working day - from 7.4 to 7.5 hours; the average number of days worked by workers per year decreased from 290 to 280 days; the average hourly output of a worker decreased from 26.5 rubles to 23 rubles. The volume of production decreased from 28434.5 tr. up to 25116 tr. Using the method of relative differences, evaluate the influence of factors on changes in production volume. Draw reasoned conclusions.

SOLUTION

Relative difference method used to measure the influence of factors on the growth of a performance indicator only in multiplicative and additive-multiplicative models.

Table 1

Initial data for calculation

Index	Designation	Base year	Reporting year	Deviations (+;-)
Average number of workers, people.
Average number of hours worked by one worker per day, hours.
Average number of days worked by a worker per year, days
Average hourly output, rub.		26,5
Product output volume, t.r.	VP	28434,5	25116	3318,5

We have a model of the form

VP = H*t*N*F,

IN in this case the change in the performance indicator is determined as follows

According to this rule, to calculate the influence of the first factor, it is necessary to multiply the basic (planned) value of the effective indicator by the relative increase of the first factor, expressed as a decimal fraction.

To calculate the influence of the second factor, you need to add the change in it due to the first factor to the planned (basic) value of the effective indicator and then multiply the resulting amount by the relative increase in the Second factor.

The influence of the third factor is determined similarly: to planned value of the effective indicator, it is necessary to add its growth due to the first and second factors and multiply the resulting amount by the relative growth of the third factor.

The influence of the quadruple factor is similar

Let’s summarize the factors that influenced the formation of revenue in the reporting year:

increase in the number of workers 1137.38 thousand rubles.

increasing the number of hours worked by one worker

per day 399.62 tr.

changes in the number of working days -1033.5 tr.

Changes in average hourly output -3821.95 tr.

Total -3318.45 t.r.

Thus, based on the method of relative differences, it was found that overall impact of all factors amounted to -3318.45 tr, which coincides with the absolute dynamics of the volume of production according to the conditions of the problem. A small discrepancy is determined by the degree of rounding in the calculations. The growth in average payroll workers by 20 people in the amount of 1137.8 tr, a slight increase in the working day of one worker by 0.1 hour led to an increase in output by 399.62 tr. A negative impact was caused by a decrease in the average hourly work per worker by 3.5 rubles. per hour, which resulted in a decrease in production volume by -3821.5 tr. A decrease in the average number of days worked by one worker per year by 10 days led to a decrease in production volumes by -1033.5 tr.

TASK 3

Using the economic information of your enterprise, assess its financial stability based on the calculation of relative indicators.

SOLUTION

Joint-stock company "KRAITEKHSNAB", registered by the Registration Chamber of the Krasnodar City Hall No. 10952 dated May 14, 1999, OGRN 1022301987278, hereinafter referred to as the "Company", is a closed joint-stock company.

Society is legal entity and operates on the basis of the Charter and legislation of the Russian Federation. Society has round stamp, containing its full corporate name in Russian and an indication of its location, stamps and forms with its name, its own emblem, as well as a trademark registered in the prescribed manner and other means of visual identification.

Full corporate name of the Company in Russian:
Closed Joint-Stock Company"KRAITECHSNAB". The abbreviated corporate name of the Company in Russian is ZAO KRAITECHSNAB.

Location (mailing address) of the Company: 350021, Russian Federation, Krasnodar region, Krasnodar, Karasunsky administrative District, st. Tramway, 25.

Closed joint stock company "KRAITECHSNAB" was created without any limitation on the period of activity.

The main subject of the Company's activity is trade and purchasing activities, intermediary, brokerage.

Let us analyze the financial stability indicators of the organization under study (Table 2).

table 2

Analysis of financial stability indicators of Kraytekhsnab CJSC in absolute terms

Indicators	2003	2004	2005	2005 to 2003
Indicators	2003	2004	2005	(+,-)	Growth rate, %
1. Sources of own funds	7371212,4	6508475,4	7713483,3	342 270,9	1004,6
2. Non-current assets	1339265,0	1320240,0	1301215,0	38 050,0	97,2
3. Sources of your own working capital for the formation of reserves and costs	6031947,4	5188235,4	6412268,4	380 321,0	1006,3
4. Long-term loans and borrowings
5. Sources of own funds, adjusted for the amount of long-term borrowed funds	6031947,4	5188235,4	6412268,4	380 321,0	106,3
6. Short-term credit and borrowed funds	1500000,0	2000000,0	1500000,0
7. The total amount of sources of funds, taking into account long-term and short-term borrowed funds	7531947,4	7188235,4	7912268,4	380 321,0	105,0
8. The amount of inventories and costs circulating in the balance sheet asset	9784805,7	10289636,4	11152558,8	1367753,1	114,0

End of table 2

Indicators	2003	2004	2005	2005 to 2003
Indicators	2003	2004	2005	(+,-)	Growth rate, %
9. Excess sources of own working capital	3752858,3	5101401,1	4740290,4	987432,2	126,3
10. Excess of sources of own funds and long-term borrowed sources	3752858,3	5101401,1	4740290,4	987432,2	126,3
11. Surplus of the total value of all sources for the formation of reserves and costs	2252858,3	3101401,1	3240290,4	987 432,2	143,8
12. Three complex indicator (S) of the financial situation	(0,0,0)	(0,0,0)	(0,0,0)

Analyzing the type of financial stability of an enterprise over time, a noticeable decrease in the financial stability of the enterprise is observed.

As can be seen from Table 2, both in 2003, and in 2004, and in 2005, the financial stability of Kraytekhsnab CJSC according to the 3-complex indicator of financial stability can be characterized as “Crisis-unstable state of the enterprise”, since the enterprise does not have enough funds to form reserves and costs to carry out current activities.

Let's calculate the financial stability coefficients of Kraytekhsnab CJSC (Table 3).

Table 3

Financial stability ratios of Kraytekhsnab CJSC

Indicators	2003	2004	2005	(+,-)
Indicators	2003	2004	2005	2004 2003	2005 to 2004
Autonomy coefficient	0,44	0,37	0,30	0,06	0,08
Debt to equity ratio (financial leverage)	1,28	1,67	2,34	0,39	0,67
Ratio of mobile and immobilized assets	11,56	13,32	18,79	1,76	5,47
Debt to equity ratio	0,78	0,60	0,43	0,18	0,17
Maneuverability coefficient	0,82	0,80	0,83	0,02	0,03
Inventory and cost coverage ratio with own funds	0,62	0,50	0,57	0,11	0,07
Industrial property ratio	0,66	0,61	0,48	0,05	0,13
Short-term debt ratio, %	15,9	18,4			10,1
Accounts payable ratio, %	84,1	81,6	91,7		10,1

Analysis of financial stability by relative indicators presented in Table 3 suggests that, according to the indicators presented in the table, compared with the base period (2003), the situation at Kraytekhsnab CJSC generally worsened in 2004 and improved slightly in the reporting year 2005 G.

The indicator “Autonomy coefficient” for the period from 2003 to 2004 decreased by -0.06 and in 2004 amounted to 0.37. This is below the standard value (0.5) at which borrowed capital can be compensated by the property of the enterprise. The indicator “Autonomy coefficient” for the period from 2004 to 2005 decreased by -0.08 and in 2005 amounted to 0.30. This is also below the standard value (0.5) at which borrowed capital can be compensated by the property of the enterprise.

The indicator “Ratio of debt and equity” (financial leverage) increased by 0.39 from 2003 to 2004 and amounted to 1.67 in 2004. The indicator for 2004 to 2005 increased by 0.67 and in 2005 amounted to 2.34. The more this ratio exceeds 1, the greater the enterprise's dependence on borrowed funds. The acceptable level is often determined by the operating conditions of each enterprise, primarily by the rate of turnover of working capital. Therefore, it is additionally necessary to determine the rate of turnover of inventories and receivables for the analyzed period. If accounts receivable turn over faster than working capital, which means a fairly high intensity of cash flow to the enterprise, i.e. the result is an increase in own funds. Therefore, with a high turnover of tangible working capital and an even higher turnover of accounts receivable, the ratio of equity and borrowed funds can greatly exceed 1.

The indicator “Ratio of mobile and immobilized assets” increased by 1.76 from 2003 to 2004 and amounted to 13.32 in 2004. The indicator for 2004 to 2005 increased by 5.47 and in 2005 amounted to 18.79. The standard value is specific to each individual industry, but all other things being equal, an increase in the coefficient is a positive trend.

Indicator "Maneuverability coefficient", for the period 2003 - 2004. decreased by -0.02 and at the end of Dec. 2004 was 0.80. This is higher than the standard value (0.5). The indicator for the period 2004 to 2005 increased by 0.03 and in 2005 amounted to 0.83. This is higher than the standard value (0.5). The agility coefficient characterizes what share of sources of own funds is in mobile form. The normative value of the indicator depends on the nature of the enterprise’s activities: in capital-intensive industries its normal level should be lower than in material-intensive ones. At the end of the analyzed period, Kraytekhsnab CJSC had a light asset structure. The share of fixed assets in the balance sheet currency is less than 40.0%. Thus, the enterprise cannot be classified as a capital-intensive industry.

Indicator “Coefficient of coverage of inventories and costs with own funds”, for 2003 – 2004. decreased by -0.11 and in 2004 amounted to 0.50. The indicator for the period 2004–2005 increased by 0.07 and in 2005 amounted to 0.57. This is lower than the standard value (0.6 - 0.8), as in 2003, 2004 and 2005. The company lacks its own funds for the formation of reserves and costs, as shown by the analysis of financial stability indicators in absolute terms.

BIBLIOGRAPHY

The procedure for monitoring the financial condition of organizations and recording their solvency. Federal Service of Russia for Insolvency and Financial Recovery: Order No. 13-r dated March 31, 1999 // Economics and Life. 1999. No. 22.
Bakanov M.I., Sheremet A.D. Theory of economic analysis. –M.: Finance and Statistics, 2006.
ASSESSMENT OF THE ECONOMIC INDICATORS OF THE ACTIVITY OF A TRADING ENTERPRISE USING THE EXAMPLE OF THE MAIN INDICATORS OF THE ENTERPRISE'S ACTIVITY SHOW THE USE OF 6 PARTIAL METHODS AND TECHNIQUES OF ECONOMIC ANALYSIS Financial condition trade organization and assessment of economic indicators
2013-11-12

All processes occurring in business are interconnected. There is both a direct and indirect connection between them. Various economic parameters change under the influence of various factors. Factor analysis (FA) allows you to identify these indicators, analyze them, and study the degree of influence.

The concept of factor analysis

Factor analysis is a multidimensional technique that allows you to study the relationships between the parameters of variables. In the process, the structure of covariance or correlation matrices is studied. Factor analysis is used in a variety of sciences: psychometrics, psychology, economics. The basics of this method were developed by psychologist F. Galton.

Objectives of the

For getting reliable results a person needs to compare indicators on several scales. In the process, the correlation of the obtained values, their similarities and differences is determined. Let's consider the basic tasks of factor analysis:

Detection of existing values.
Selection of parameters for a complete analysis of values.
Classification of indicators for system work.
Detection of relationships between resultant and factor values.
Determining the degree of influence of each factor.
Analysis of the role of each value.
Application of the factor model.

Every parameter that affects the final value must be examined.

Factor analysis techniques

FA methods can be used both in combination and separately.

Deterministic Analysis

Deterministic analysis is used most often. This is due to the fact that it is quite simple. Allows you to identify the logic of the impact of the company’s main factors and analyze their impact in quantitative terms. As a result of the DA, you can understand what factors should be changed to improve the company's performance. Advantages of the method: versatility, ease of use.

Stochastic Analysis

Stochastic analysis allows you to analyze existing indirect relationships. That is, there is a study of indirect factors. The method is used if it is impossible to find direct connections. Stochastic analysis is considered complementary. It is only used in certain cases.

What is meant by indirect connections? With a direct connection, when the argument changes, the value of the factor will also change. An indirect connection involves a change in the argument followed by a change in several indicators at once. The method is considered auxiliary. This is due to the fact that experts recommend studying direct connections first. They allow you to create a more objective picture.

Stages and features of factor analysis

Analysis for each factor gives objective results. However, it is used extremely rarely. This is due to the fact that complex calculations are performed in the process. To carry them out you will need special software.

Let's consider the stages of FA:

Establishing the purpose of the calculations.
Selection of values that directly or indirectly affect final result.
Classification of factors for complex research.
Detecting the relationship between the selected parameters and the final indicator.
Modeling of mutual relationships between the result and the factors influencing it.
Determining the degree of impact of the values and assessing the role of each parameter.
Use of the generated factor table in the activities of the enterprise.

FOR YOUR INFORMATION! Factor analysis involves very complex calculations. Therefore, it is better to entrust it to a professional.

IMPORTANT! When carrying out calculations, it is extremely important to correctly select factors that influence the results of the enterprise. The selection of factors depends on the specific area.

Factor analysis of profitability

A profitability analysis is carried out to analyze the rationality of resource allocation. As a result, it is possible to determine which factors most influence the final result. As a result, we can retain only those factors that the best way affect efficiency. Based on the data received, you can change pricing policy companies. The following factors may influence the cost of production:

fixed costs;
variable costs;
profit.

Reducing costs provokes an increase in profits. In this case, the cost does not change. We can conclude that profitability is affected by existing costs, as well as the volume of products sold. Factor analysis allows us to determine the degree of influence of these parameters. When does it make sense to do it? The main reason for this is to reduce or increase profitability.

Factor analysis is carried out using the following formula:

Rв= ((W-SB -KRB-URB)/W) - (WB-SB-KRB-URB)/WB, Where:

VT – revenue for the current period;

SB – cost price for the current period;

KRB – commercial expenses for the current period;

URB – management expenses for the previous period;

VB – revenue for the previous period;

KRB – commercial expenses for the previous period.

Other formulas

Let's consider the formula for calculating the degree of impact of cost on profitability:

Rс= ((W-SBot -KRB-URB)/W) - (W-SB-KRB-URB)/W,

CBO is the cost of production for the current period.

Formula for calculating the impact of management expenses:

RUR= ((W-SB -KRB-URot)/W) - (W-SB-KRB-URB)/W,

URot is management expenses.

The formula for calculating the impact of business costs is:

Rк= ((W-SB -KRo-URB)/W) - (W-SB-KRB-URB)/W,

CR is commercial expenses for the previous time.

The total impact of all factors is calculated using the following formula:

Rob=Rv+Rс+Rur+Rk.

IMPORTANT! When making calculations, it makes sense to calculate the influence of each factor separately. Overall PA results are of little value.

Example

Let's consider the organization's indicators for two months (for two periods, in rubles). In July, the organization's income amounted to 10 thousand, production costs - 5 thousand, administrative expenses - 2 thousand, commercial expenses - 1 thousand. In August, the company's income amounted to 12 thousand, production costs - 5.5 thousand, administrative expenses - 1.5 thousand, commercial expenses - 1 thousand. The following calculations are carried out:

R=((12 thousand-5.5 thousand-1 thousand-2 thousand)/12 thousand)-((10 thousand-5.5 thousand-1 thousand-2 thousand)/10 thousand)=0.29-0, 15=0.14

From these calculations we can conclude that the organization’s profit increased by 14%.

Factor analysis of profit

P = RR + RF + RVN, where:

P – profit or loss;

РР – profit from sales;

RF – results of financial activities;

RVN is the balance of income and expenses from non-operating activities.

Then you need to determine the result from the sale of goods:

PP = N – S1 – S2, where:

N – revenue from the sale of goods at selling prices;

S1 – cost of products sold;

S2 – commercial and administrative expenses.

The key factor in calculating profit is the sales turnover of the company.

FOR YOUR INFORMATION! Factor analysis is extremely difficult to perform manually. You can use special programs for it. The simplest program for calculations and automatic analysis - Microsoft Excel. It has tools for analysis.

Called factor analysis. The main types 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 general economic indicator is functional. The latter means that the generalizing indicator is either a product, 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 general economic indicator is probabilistic, otherwise - correlation.

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

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

First type of analysis carries out the study of the influence of factors by a deductive method, that is, in the direction from the general to the specific. In reverse factor analysis the influence of factors is studied inductively - in the direction from particular factors to general economic indicators.

Classification of factors influencing the efficiency of an organization

Factors, the influence of which is studied during the study, are classified according to various signs. 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 major and minor. The main ones include factors related to the use of materials and materials, as well as factors determined by supply and sales activities and some other aspects of the functioning of the organization. The main factors have a fundamental impact on general economic indicators. External factors beyond the control of a given organization are determined by natural-climatic (geographical), socio-economic, and foreign economic conditions.

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

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

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

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

According to their structure, factors are divided into simple and complex. The overwhelming majority of factors are complex, including several components. At the same time, there are also factors that cannot be separated. For example, capital productivity can serve as an example of a complex factor. The number of days the equipment was used during a given period is a simple factor.

According to the nature of the influence on general economic indicators, they are distinguished direct and indirect factors. Thus, a change in products sold, although it has an inverse effect on the amount of profit, should be considered direct factors, that is, a first-order factor. A change in the amount of material costs has an indirect effect on profit, i.e. affects profit not directly, but through cost, which is a first-order factor. Based on this, the level of material costs should be considered a second-order factor, that is, an indirect factor.

Depending on whether you can give quantification the influence of a given factor on a general economic indicator, a distinction is made between measurable and unmeasurable factors.

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

Factor economic analysis

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

The set of factor and resultant characteristics that are in the same cause-and-effect relationship is called factor system. There is also the concept of a factor system model. It characterizes the relationship between the resultant characteristic, denoted as y, and the factor characteristics, denoted as . In other words, the factor system model expresses the relationship between general economic indicators and individual factors influencing this indicator. In this case, other economic indicators act as factors, representing the reasons for changes in the general indicator.

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

Establishing dependencies between generalizing (resulting) and influencing factors is called economic-mathematical modeling.

We study two types of relationships between generalizing indicators and the factors influencing them:

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

Functional connection- this is a relationship in which each value of a factor (factorial characteristic) corresponds to a completely definite non-random value of a generalizing indicator (resultative characteristic).

Stochastic communication- this is a relationship in which each value of a factor (factor characteristic) corresponds to a set of values of a general indicator (resultative characteristic). Under these conditions, for each value of factor x, the values of the general indicator y form a conditional statistical distribution. As a result, a change in the value of factor x only on average causes a change in the general indicator y.

In accordance with the two types of relationships considered, a distinction is made between 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 analysis results is ensured by the use of economic and mathematical research methods.

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

Thus, they provide a more accurate and detailed calculation of the influence of individual factors on changes 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.

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

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

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

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

The main properties of the deterministic approach to analysis:

· construction of a deterministic model through logical analysis;
· the presence of a complete (hard) connection between indicators;
· the impossibility of separating the results of the influence of simultaneously acting factors that cannot be combined in one model;
· study of relationships in the short term.

There are four types of deterministic models:

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

Multiplicative models can be summarized by the formula

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

Multiple models:

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

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

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

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

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

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

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

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

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

· the place of each factor in the model must correspond to its role in the formation of the effective indicator;
· the model should be built from a two-factor complete model by sequentially dividing factors, usually qualitative, into components;
· when writing a formula for a multifactor model, factors should be arranged from left to right in the order of their replacement.

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

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

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

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

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

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

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

Let's look at an example:

Table 2 Initial data for factor analysis

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

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

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

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

· if there are quantitative and qualitative indicators in the factor model, the change in quantitative factors is considered first;
· if the model is represented by several quantitative and qualitative indicators, the substitution sequence is determined by logical analysis.

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

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

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

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

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

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

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

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

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

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

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

1. View model:
2. View model:
3. View model:
4. View model:

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

Table 3 Matrix of application of deterministic factor analysis methods

Analysis is a very capacious concept that underlies all practical and scientific human activity. Analytical methods are so widespread that the word “analysis” often means any research in general, both in the natural sciences and humanities, and in practical activities. Procedures and methodological principles of analysis are an integral part of any scientific and practical research, when the researcher moves from simple description phenomena to the study of its structure.

According to the classical definition, analysis is presented as only one of the logical methods of thinking. “Thinking is characterized by processes such as abstraction, generalization, analysis and synthesis, setting certain problems and finding ways to solve them.”

The characteristic of analysis as a way of thinking suggests that with its help it is possible to identify the structure of a process or phenomenon, reduce the complex to the simple, construct a classification of phenomena, and highlight the essence of an object. “Logical analysis consists of mentally dividing the object under study into its component parts and is a method of obtaining new knowledge. The purpose of analysis is to understand the parts as elements of a complex whole.” Thus, research as a concept of the process of cognition is still broader. The existence of analytical and synthetic methods of cognition allows us to formulate analysis as the first, most important, indispensable stage of any research.

Based on this, the term “economic analysis” implies analytical stage of economic research - economic systems, relationships, processes, that is, both objects and subjects of the economy. Curricula of higher economic educational institutions As a rule, a step-by-step study of economic analysis is provided. The main attention is paid to a review of various techniques that can be used when carrying out analytical calculations that justify any management decision. Any specialist related to the organization and management of a business must own certain analytical tools, know and understand the logic of conducting analytical procedures. The adoption of any decision is preceded by analytical calculations, so any representative of the enterprise management apparatus - from top managers to ordinary specialists - simply must be a good analyst. The threat of potential bankruptcy is invisibly present when making management decisions, especially when it comes to a strategic decision of a financial nature. This means that the analysis should be performed not only in retrospect, but also in the future. At the same time, there is no need to strive for absolute accuracy - it is necessary to identify trends, both already established and emerging. To do this, the analyst must have such qualities as the ability to generalize, the ability to compare and evaluate the mutual influence of a large number of factors, and the ability to notice seemingly insignificant signs of a change in the situation. In addition, conducting a qualified analysis requires knowledge of many sciences - economics, accounting, marketing, the fundamentals of industrial psychology. All analytical procedures are based on knowledge mathematical analysis, statistics and econometrics. IN modern conditions analysis is impossible without the use of computer technology, which means that economic analysis is impossible without knowledge of computer science.

Based on the nature of the relationship between indicators, deterministic and stochastic models are distinguished. Deterministic (functional) relationship is a relationship in which each value of a factor characteristic corresponds to a well-defined non-random value of the resultant characteristic. A relationship in which each value of a factor characteristic corresponds to a set of values of the resulting characteristic is called stochastic or probabilistic.

To apply the techniques of factor analysis, it is necessary to create a model, that is, present a formula for calculating the analyzed indicator. Models can be:

1. Additive. The value of the analyzed indicator is determined as the sum of indicator factors. This model looks like

Y = A + B + C.

An example of an additive model would be the gross profit of an enterprise, which consists of such components as profit from sales, results from other activities and the balance of operating and non-operating income and expenses.

2. Multiplicative. The value of the analyzed indicator is determined as the product of indicators - factors. This model looks like

Y = A * B * C.

Most models used in factor analysis are multiplicative. For example, revenue can be represented as the product quantity multiplied by the unit cost. The total material costs of an enterprise are the product of three factors - the number of products produced, the rate of material consumption per unit of production, the cost of a unit of material resources.

3. Multiples. The value of the analyzed indicator is determined as the quotient of two factors. This model looks like

An example is the capital-labor ratio, determined by dividing the cost of fixed assets by the number of employees.

4. Mixed. Such models can take different forms and represent various combinations of additive, multiplicative and multiple models:

Y = A * (B + C);

Y = A / (B + C);

Y = (A / B) * C.

An example of such a model would be the definition of the wage fund as the product of average wages and headcount. At the same time, the average wage is the sum of several components - the tariff component, additional payments of an incentive nature and additional payments of a compensating nature:

Payroll = (Salary Tar + Salary Steam + Salary Comp) * H.

When drawing up any factor analysis models, it is necessary to take into account the cause-and-effect relationships of the indicators. For example, from an arithmetic point of view, the two expressions below are true:

Revenue = Labor productivity * Headcount;

Labor productivity = Revenue / Headcount.

Both of these expressions can be used to calculate an unknown quantity. However, only one of them can be used as an analytical model - we can say that the volume of production depends on labor productivity, but it cannot be said that labor productivity depends on the enterprise's revenue.

When conducting factor analysis, the method of expanding factor models is often used. For example:

The same model can be written in the form

Y = (A/C)* (C/B).

In this case, instead of two absolute (quantitative) factors, we obtain two relative (qualitative) factors for analysis.

The most famous example is Dupont's formula:

Return on Assets = Profit / Assets;

Return on assets = (Profit/Revenue)* (Revenue/Assets).

In this case, the first factor is return on sales, the second factor is asset turnover. Indeed, the profitability (return) of assets depends on how profitable the enterprise produces products, and on how quickly the turnover of capital invested in assets occurs:

Return on Equity = Profit / Equity;

SK profitability = (Profit/Revenue)* (Revenue/Assets)* (Assets/SK).

In this case, the first factor is return on sales, the second is asset turnover, and the third is capital structure.

In conditions market relations The process of managing an enterprise, which has been granted complete economic and financial independence, becomes significantly more complicated.

The main functions of management are control and regulation. Effective management The production activities of an enterprise increasingly depend on the level of information support for managers at all levels.

But as the individual chooses the organizational form of the enterprise, type of activity, sales markets, and free pricing, the tasks facing the accounting system become more complicated.

Financial accounting as the main part information system The enterprise does not provide managers at all levels with operational information and does not provide information for planning and coordinating the future development of the enterprise in market conditions. Under these conditions, the emergence of management accounting as an independent area of accounting activity becomes inevitable.

All accounting begins to be divided into financial and management. The task of management accounting is to compile reports for the purposes of periodic planning and control, for decision-making in non-standard economic situations. These reports are compiled for internal users of accounting information and should contain information not only about the general financial position of the enterprise, but also about the state of affairs directly in the production area.

Managers (managers) need information that will help them in making decisions, monitoring and regulating management activities. These are: sales goals, production costs, demand, profitability of products produced at their enterprise, competitiveness, etc. Any information is important for a manager, regardless of whether it is an object of accounting or not. Management accounting is intended to provide such information.

This concept is not yet used in domestic practice. But it would be wrong to say that management accounting is something new for our enterprises. Many of its elements are included in our accounting (accounting for production costs and calculating production costs), operational accounting (operational reporting), economic analysis (analysis of product costs, assessment of task completion, justification of decisions, etc.).

However, this information is scattered between various services and is formed with a delay; for example, analysis of economic activity is carried out when the main financial indicators have already been formed and cannot be influenced. The efficiency of individual departments of the enterprise is practically not analyzed.

In conditions centralized system Management took measures to introduce internal economic accounting, which, in essence, is a prototype of management by responsibility centers. Management accounting methods used in the context of administrative management measures did not give the desired result. This is explained by the absence of an owner interested in reducing production costs and increasing its efficiency. Only in conditions of market relations is it possible to independently integrate cost and income accounting, regulation, planning, control and analysis in order to prepare information for operational management decisions and predict the future development of the enterprise.

2. The economic essence of management accounting in an enterprise

Management accounting is a system of accounting, planning, control, analysis of information about the costs and results of business activities necessary for management personnel to manage the activities of the organization.

Management Accounting- this is the link between the accounting process and enterprise management.

Subject Management accounting is the production activities of the organization as a whole and its individual structural divisions (responsibility centers).

Objects management accounting are the costs and results of the economic activities of the enterprise and its responsibility centers, internal pricing and internal reporting.

A wide variety of methods are used in management accounting:

elements of the financial accounting method (documentation, inventory, double entry, grouping and generalization, reporting);

index method;

techniques of economic analysis;

mathematical methods.

Consequently, the management accounting method is a systematic operational analysis of information.

The development of management accounting took place on the basis of production and cost accounting. Therefore, its main content is accounting for production costs and costing.

Modern production accounting is designed to monitor production costs, analyze the reasons for cost overruns compared to previous periods, forecasts, standards, and identify possible reserves for cost reduction. Production accounting includes accounting for costs by type, by place of their occurrence, by media.

Hence, the most important goals management accounting are:

providing information assistance to managers in making operational management decisions;

control, planning and forecasting economic efficiency activities of the enterprise;

providing a basis for pricing;

choosing the most effective ways to develop the enterprise.

2. Methodology and techniques of economic analysis

Method Economic analysis is a way of approaching the study of economic processes in their smooth development.

Characteristic features of the method economic analysis are:

· determination of a system of indicators that comprehensively characterize the economic activities of organizations;
· establishing the subordination of indicators with the identification of total effective factors and factors (major and secondary) influencing them;
· identifying the form of relationship between factors;
· selection of techniques and methods for studying the relationship;
· quantitative measurement of the influence of factors on the aggregate indicator.

The set of techniques and methods that are used in the study of economic processes is economic analysis methodology.

The methodology of economic analysis is based on the intersection of three areas of knowledge: economics, statistics and mathematics.

TO economic methods analysis includes comparison, grouping, balance sheet and graphical methods.

Statistical methods include the use of averages and relative values, the index method, correlation and regression analysis, etc.

Mathematical methods can be divided into three groups: economic (matrix methods, theory production functions, theory of intersectoral balance); methods of economic cybernetics and optimal programming (linear, nonlinear, dynamic programming); methods of operations research and decision making (graph theory, game theory, queuing theory).

Introduction to Factor Analysis

During recent years factor analysis has found its use among a wide range of researchers mainly due to the development of high-speed computers and statistical software packages (for example, DATATEXT, BMD, OSIRIS, SAS and SPSS). This also affected a large group of users who did not have relevant mathematical training but were nevertheless interested in using the potential of factor analysis in their research (Harman, 1976; Horst, 1965; Lawley and Maxswel, 1971; Mulaik, 1972).

Factor analysis assumes that the variables under study are a linear combination of some hidden (latent) unobservable factors. In other words, there is a system of factors and a system of studied variables. A certain dependence between these two systems allows, through factor analysis, taking into account the existing dependence, to draw conclusions on the variables (factors) being studied. The logical essence of this dependence is that the causal system of factors (the system of independent and dependent variables) always has a unique correlation system of the studied variables, and not vice versa. Only under strictly limited conditions imposed on factor analysis is it possible to unambiguously interpret causal structures across factors for the presence of correlations between the variables under study. In addition, there are problems of a different nature. For example, when collecting empirical data, various kinds of errors and inaccuracies may be made, which in turn complicates the work of identifying hidden unobservable parameters and their further research.

What is factor analysis? Factor analysis refers to a variety of statistical techniques, the main task of which is to represent a set of studied characteristics in the form of a reduced system of hypothetical variables. Factor analysis is a research empirical method that primarily finds its application in social and psychological disciplines.

As an example of the use of factor analysis, we can consider the study of personality traits using psychological tests. Personality properties cannot be directly measured; they can only be judged on the basis of a person’s behavior, answers to certain questions, etc. To explain the collected empirical data, their results are subjected to factor analysis, which allows us to identify those personality traits, which influenced the behavior of the subjects in the experiments.

The first stage of factor analysis, as a rule, is the selection of new features, which are linear combinations of the previous ones and “absorb” most overall variability in the observed data, and therefore convey much of the information contained in the original observations. This is usually done using principal component method, although sometimes other techniques are used (for example, the principal factors method, the maximum likelihood method).

The principal component method is a statistical technique that allows you to transform the original variables into their linear combination (GeorgH.Dunteman). The purpose of the method is to obtain a reduced system of source data, which is much easier for understanding and further statistical processing. This approach was proposed by Pearson (1901) and independently received its further development in Hotelling (1933). The author tried to minimize the use of matrix algebra when working with this method.

The main goal of the principal component method is to isolate primary factors and determine the minimum number of common factors that satisfactorily reproduce the correlations between the studied variables. The result of this step is a matrix of factor loading coefficients, which in the orthogonal case are correlation coefficients between variables and factors. When determining the number of factors to be selected, the following criterion is used: only factors with eigenvalues greater than the specified constant (usually unity) are selected.

However, usually the factors obtained by the principal component method cannot be interpreted clearly enough. Therefore, the next step in factor analysis is to transform (rotate) the factors in such a way as to facilitate their interpretation. Rotation factors consists in finding the simplest factor structure, that is, such a variant of assessing factor loadings and residual variances, which makes it possible to meaningfully interpret common factors and loadings.

The most commonly used rotation method by researchers is the varimax method. This is a method that allows, on the one hand, by minimizing the spread of squared loadings for each factor, to obtain a simplified factor structure by increasing large and decreasing small factor loadings, on the other hand.

So, the main goals of factor analysis are:

reduction number of variables (data reduction);

structure definition relationships between variables, i.e. classification of variables.

Therefore, factor analysis is used either as a data reduction method or as a classification method.

Practical examples and advice on the use of factor analysis can be found in the book by Stevens (1986); a more detailed description is given by Cooley and Lohnes (1971); Harman (1976); Kim and Mueller (1978a, 1978b); Lawley and Maxwell (1971); Lindeman, Merenda and Gold (1980); Morrison (1967) and Mulaik (1972). An interpretation of secondary factors in hierarchical factor analysis as an alternative to traditional factor rotation is given by Wherry (1984).

Issues in preparing data for use

factor analysis

Let's look at a series of questions and short answers using factor analysis.

What level of measurement does factor analysis require or, in other words, in what measurement scales should data be presented for factor analysis?

Factor analysis requires that variables be presented on an interval scale (Stevens, 1946) and follow a normal distribution. This requirement also assumes that covariance or correlation matrices are used as input data.

Should a researcher avoid using factor analysis when the metric basis of the variables is not precisely defined, i.e. Is the data presented on an ordinal scale?

Not necessary. Many variables representing, for example, measures of subjects' opinions on a large number tests do not have a precisely established metric base. However, in general, it is assumed that many “ordinal variables” can contain numerical values that do not distort and even preserve the basic properties of the characteristic being studied. The researcher’s tasks: a) correctly determine the number of reflexively identified orders (levels); b) take into account that the sum of the admitted distortions will be included in the correlation matrix, which is the basis for the input data of the factor analysis; c) correlation coefficients are fixed as “ordinal” distortions in measurements (Labovitz, 1967, 1970; Kim, 1975).

For a long time it was believed that distortions are assigned to the numerical values of ordinal categories. However, this is unfounded, since for metric quantities distortions, even minimal ones, are possible during the experiment. In factor analysis, the results depend on the possibility of errors obtained during the measurement process, and not on their origin and correlation to data of a certain type of scale.

Can factor analysis be used for nominal (dichotomous) variables?

Many researchers argue that using factor analysis for nominal variables is very convenient. First, dichotomous values (values equal to “0” and “1”) exclude the choice of anything other than them. Secondly, as a result, the correlation coefficient is the equivalent of the Pearson correlation coefficient, which acts as the numerical value of the variable for factor analysis.

However, there is no clear positive answer to this question. Dichotomous variables are difficult to express within the framework of an analytical factor model: each variable has a weighting value of at least two main factors - general and specific (Kim, Muller). Even if these factors have two values (which is quite rare in real factor models), then the final results in the observed variables must contain at least four different values, which, in turn, justify the inconsistency of using nominal variables. Therefore, factor analysis for such variables is used to obtain a number of heuristic criteria.

How many variables should there be for each hypothetically constructed factor?

It is assumed that for each factor there should be at least three variables. But this requirement is omitted if factor analysis is used to confirm a hypothesis. In general, researchers agree that it is necessary to have at least twice as many variables as factors.

One more point regarding this issue. The larger the sample size, the more reliable the criterion value CI-square. Results are considered statistically significant if the sample contains at least 51 observations. Thus:

N-n-150,(3.33)

where N is the sample size (number of measurements),

n – number of variables (Lawley, Maxwell, 1971).

This is, of course, only a general rule.

What is the meaning of the sign of the factor loading?

The sign itself is not significant and there is no way to assess the significance of the relationship between a variable and a factor. However, the signs of the variables included in the factor have a specific meaning relative to the signs of other variables. Different signs simply mean that the variables are related to the factor in opposite directions.

For example, according to the results of factor analysis, it was found that for a pair of qualities open-closed(multifactorial Catell questionnaire) there are positive and negative weight loads, respectively. Then they say that the share of quality open, in the selected factor there is more than the share of quality closed.

Principal components and factor analysis

Factor analysis as a data reduction method

Suppose a (somewhat "dumb") study is conducted in which the height of one hundred people is measured in meters and centimeters. So there are two variables. If we further investigate, for example, the effect of various nutritional supplements on growth, would it be advisable to use both variables? Probably not, because... Height is one characteristic of a person, regardless of the units in which it is measured.

Let's assume that people's satisfaction with life is being measured using a questionnaire containing various items. For example, questions are asked: are people satisfied with their hobby (item 1) and how intensively do they engage in it (item 2). The results are transformed so that average responses (for example, for satisfaction) correspond to a value of 100, while below and above the average responses are smaller and large values, respectively. Two variables (responses to two different items) are correlated. From the high correlation of these two variables, we can conclude that the two questionnaire items are redundant. This, in turn, allows the two variables to be combined into one factor.

The new variable (factor) will include the most significant features of both variables. So, in fact, the original number of variables has been reduced and two variables have been replaced by one. Note that the new factor (variable) is actually a linear combination of the two original variables.

An example in which two correlated variables are combined into a single factor shows the main idea of factor analysis or, more precisely, principal components analysis. If the example with two variables is extended to a larger number of variables, the calculations become more complex, but the basic principle of representing two or more dependent variables as one factor remains valid.

Principal component method

Principal component analysis is a method of data reduction or reduction, i.e. by reducing the number of variables. A natural question arises: how many factors should be identified? Note that in the process of sequential selection of factors, they include less and less variability. The decision about when to stop the factor selection procedure depends largely on one's view of what constitutes small "random" variability. This decision is quite arbitrary, but there are some recommendations that allow you to rationally choose the number of factors (see section Eigenvalues and number of allocated factors).

In the case where there are more than two variables, they can be considered to define a three-dimensional "space" in the same way that two variables define a plane. If there are three variables, then a three-dimensional scatterplot can be constructed (see Figure 3.10).

Rice. 3.10. 3D trait scatterplot

For the case of more than three variables, it becomes impossible to represent points on a scatterplot, but the logic of rotating the axes to maximize the variance of the new factor remains the same.

After the line for which the dispersion is maximum is found, some scatter of data remains around it and it is natural to repeat the procedure. In principal component analysis, this is exactly what is done: after the first factor highlighted, 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), etc. Thus, the factors are sequentially identified 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 (uncorrelated or orthogonal).

Eigenvalues and number of allocated factors

Let's look at some standard results from principal component analysis. With repeated calculations, factors with less and less variance are identified. For simplicity of presentation, it is believed that work usually begins with a matrix in which the variances of all variables are equal to 1.0. That's why total variance equal to the number of variables. For example, if there are 10 variables and the variance of each is 1, then the largest variance that can potentially be extracted is 10 times 1.

Suppose that a study of life satisfaction included 10 items to measure different aspects of satisfaction with home life and work. The variance explained by the sequential factors is presented in Table 3.14:

Table 3. 14

Eigenvalue table

STATISTICA FACTOR ANALYSIS	Eigenvalues (factor.sta) Highlight: Principal components
Meaning	Eigenvalues	% total variance	Cumulate. own meaning	Cumulate. %

In the second column of table 3. 14. (eigenvalues) the variance of the new, just identified factor is presented. The third column for each factor gives the percentage of the total variance (in this example it is 10) 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, etc. The fourth column contains the accumulated (cumulative) variance.

So, the variances allocated by the factors are called eigenvalues. This name comes from the calculation method used.

Once you have information about how much variance each factor contributed, you can return to the question of how many factors should be retained. As stated above, this decision is arbitrary in nature. However, there are some generally accepted recommendations, and in practice, following them gives the best results.

Criteria for selecting factors

Kaiser criterion. First, only those factors are selected eigenvalues of which there are more than 1. Essentially, this means that if a factor does not allocate variance equivalent to at least the variance of one variable, then it is omitted. This criterion was proposed by Kaiser (1960) and is the most widely used. In the above example (see Table 3.14), based on this criterion, only 2 factors (two main components) should be retained.

Scree criterion is a graphical method first proposed by Cattell (1966). It allows you to display eigenvalues in the form of a simple graph:

Rice. 3. 11. Scree criterion

Both criteria have been studied in detail by Browne (1968), Cattell and Jaspers (1967), Hakstian, Rogers and Cattell (1982), Lynn (1968), Tucker, Koopman and Lynn (Tucker, Koopman, Linn, 1969). Cattel 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 to the right of this point there is only a "factorial talus" ("talus" is a geological term for debris rocks, accumulating at the bottom of a rocky slope). In accordance with this criterion, 2 or 3 factors can be left in the example considered.

Which criterion should still be preferred in practice? Theoretically, it is possible to calculate characteristics by generating random data for a specific number of factors. Then you can see whether the criterion used has detected a sufficiently accurate number of significant factors or not. Using this general method, the first criterion ( Kaiser criterion) sometimes retains too many factors, while the second criterion ( scree criterion) sometimes retains too few factors; however, both criteria are quite good under normal conditions, when there are a relatively small number of factors and many variables.

In practice, an important additional question arises, namely: when the resulting solution can be meaningfully interpreted. Therefore, several solutions with more or less factors are usually examined, and then the one that makes the most sense is selected. This issue will be further discussed within the framework of factor rotations.

Commonalities

In the language of factor analysis, the proportion of variance in a particular variable that belongs to common factors (and is shared with other variables) is called community. That's why extra work The challenge facing the researcher when applying this model is to estimate the commonalities for each variable, i.e. the proportion of variance that is common to all items. Then share of variance, for which each item is responsible, is equal to the total variance corresponding to all variables minus the communality (Harman and Jones, 1966).

Main factors and main components

Term factor analysis includes both principal component analysis and principal factor analysis. It is assumed that, in general, it is known how many factors should be identified. One can find out (1) the significance of the factors, (2) whether they can be interpreted in a reasonable way, and (3) how to do this. To illustrate how this can be done, we work backwards, that is, start with some meaningful structure and then see how that translates into results.

The main difference between the two factor analysis models is that in principal component analysis it is assumed that all variability of variables, whereas principal factor analysis uses only the variability of a variable that is common to other variables.

In most cases, these two methods lead to very similar results. However, principal component analysis is often preferred as a data reduction method, while principal factor analysis is better used to determine the structure of the data.

Factor analysis as a method of data classification

Correlation matrix

The first stage of factor analysis involves calculating the correlation matrix (in the case of normal sampling distribution). Let's return to the satisfaction example and look at the correlation matrix for the variables related to satisfaction at work and at home.