Factor analysis statistics for dummies. Factor analysis of sales profit: example of calculation of indicators. Criteria and types of analysis of variance

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

To obtain 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 the 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, only those factors that best influence efficiency can be retained. Based on the data obtained, you can change the company's pricing policy. 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 is Microsoft Excel. It has tools for analysis.

Variance analysis is a set of statistical methods designed to test hypotheses about the relationship between certain characteristics and studied factors that do not have a quantitative description, as well as to establish the degree of influence of factors and their interaction. In the specialized literature it is often called ANOVA (from the English name Analysis of Variations). This method was first developed by R. Fischer in 1925.

Types and criteria of analysis of variance

This method is used to study the relationship between qualitative (nominal) characteristics and a quantitative (continuous) variable. In essence, it tests the hypothesis about the equality of the arithmetic means of several samples. Thus, it can be considered as a parametric criterion for comparing the centers of several samples at once. If this method is used for two samples, the results of the analysis of variance will be identical to the results of the Student's t-test. However, unlike other criteria, this study allows us to study the problem in more detail.

Dispersion analysis in statistics is based on the law: the sum of squared deviations of the combined sample is equal to the sum of squared intragroup deviations and the sum of squared intergroup deviations. The study uses Fisher's test to establish the significance of the difference between intergroup variances and within-group variances. However, the necessary prerequisites for this are normality of distribution and homoscedasticity (equality of variances) of samples. There are univariate (one-factor) analysis of variance and multivariate (multifactorial). The first considers the dependence of the value under study on one characteristic, the second - on many at once, and also allows us to identify the connection between them.

Factors

Factors are controlled circumstances that influence the final result. Its level or processing method is a value that characterizes a specific manifestation of this condition. These numbers are usually presented on a nominal or ordinal measurement scale. Often output values are measured on quantitative or ordinal scales. Then the problem arises of grouping output data in a number of observations that correspond to approximately the same numerical values. If the number of groups is taken to be excessively large, then the number of observations in them may be insufficient to obtain reliable results. If you take the number too small, this can lead to the loss of significant features of the influence on the system. The specific way to group data depends on the amount and nature of variation in values. The number and size of intervals in univariate analysis are most often determined by the principle of equal intervals or the principle of equal frequencies.

Analysis of variance problems

So, there are cases when you need to compare two or more samples. It is then that it is advisable to use analysis of variance. The name of the method indicates that conclusions are drawn based on the study of variance components. The essence of the study is that the overall change in the indicator is divided into component parts that correspond to the action of each individual factor. Let's consider a number of problems that are solved by typical analysis of variance.

Example 1

The workshop has a number of automatic machines that produce a specific part. The size of each part is a random variable that depends on the setup of each machine and the random deviations that occur during the manufacturing process of the parts. It is necessary to determine, based on the measurement data of the dimensions of the parts, whether the machines are configured in the same way.

Example 2

During the manufacture of an electrical device, various types of insulating paper are used: capacitor, electrical, etc. The device can be impregnated with various substances: epoxy resin, varnish, ML-2 resin, etc. Leaks can be eliminated under vacuum at elevated pressure, with heating. Impregnation can be done by immersion in varnish, under a continuous stream of varnish, etc. The electrical apparatus as a whole is filled with a certain compound, of which there are several options. Quality indicators are the electrical strength of insulation, the overheating temperature of the winding in operating mode, and a number of others. During development of the technological process of manufacturing devices, it is necessary to determine how each of the listed factors affects the performance of the device.

Example 3

The trolleybus depot serves several trolleybus routes. They operate trolleybuses of various types, and 125 inspectors collect fares. The depot management is interested in the question: how to compare the economic indicators of the work of each controller (revenue) taking into account different routes and different types of trolleybuses? How to determine the economic feasibility of producing trolleybuses of a certain type on a particular route? How to establish reasonable requirements for the amount of revenue that a conductor brings in on each route in various types of trolleybuses?

The task of choosing a method is how to obtain maximum information regarding the influence of each factor on the final result, determine the numerical characteristics of such an influence, their reliability at minimal cost and in the shortest possible time. Methods of variance analysis allow solving such problems.

Univariate analysis

The purpose of the study is to assess the magnitude of the influence of a particular case on the analyzed review. Another purpose of univariate analysis may be to compare two or more circumstances with each other to determine the difference in their impact on recall. If the null hypothesis is rejected, then the next step is to quantify and construct confidence intervals for the obtained characteristics. In the case where the null hypothesis cannot be rejected, it is usually accepted and a conclusion is drawn about the nature of the influence.

One-way analysis of variance can become a nonparametric analogue of the Kruskal-Wallis rank method. It was developed by the American mathematician William Kruskal and economist Wilson Wallis in 1952. This criterion is designed to test the null hypothesis of the equality of effects on the studied samples with unknown but equal average values. In this case, the number of samples must be more than two.

The Jonckheere-Terpstra criterion was proposed independently by the Dutch mathematician T. J. Terpstra in 1952 and the British psychologist E. R. Jonckheere in 1954. It is used when it is known in advance that the existing groups of results are ordered by the growth of the influence of the factor under study, which is measured on an ordinal scale.

M - Bartlett's test, proposed by the British statistician Maurice Stevenson Bartlett in 1937, is used to test the null hypothesis about the equality of variances of several normal populations from which the samples under study are taken, generally having different sizes (the number of each sample must be at least four ).

G - Cochran's test, which was discovered by the American William Gemmell Cochran in 1941. It is used to test the null hypothesis about the equality of variances of normal populations in independent samples of equal size.

The nonparametric Levene test, proposed by the American mathematician Howard Levene in 1960, is an alternative to the Bartlett test in conditions where there is no confidence that the samples under study are subject to a normal distribution.

In 1974, American statisticians Morton B. Brown and Alan B. Forsythe proposed a test (Brown-Forsyth test) that is slightly different from Levene's test.

Two-factor analysis

Two-way analysis of variance is used for related normally distributed samples. In practice, complex tables of this method are often used, in particular those in which each cell contains a set of data (repeated measurements) corresponding to fixed level values. If the assumptions required to apply two-way analysis of variance are not met, then use the nonparametric Friedman rank test (Friedman, Kendall and Smith), developed by the American economist Milton Friedman in late 1930. This test does not depend on the type of distribution.

It is only assumed that the distribution of values is identical and continuous, and that they themselves are independent of each other. When testing the null hypothesis, the output data is presented in the form of a rectangular matrix, in which the rows correspond to the levels of factor B, and the columns correspond to levels of A. Each cell of the table (block) can be the result of measurements of parameters on one object or on a group of objects with constant values of the levels of both factors . In this case, the corresponding data are presented as the average values of a certain parameter for all dimensions or objects of the sample under study. To apply the output criterion, it is necessary to move from the direct results of measurements to their rank. Ranking is carried out for each row separately, that is, the values are ordered for each fixed value.

Page's test (L-test), proposed by American statistician E. B. Page in 1963, is designed to test the null hypothesis. For large samples, Page's approximation is used. They, subject to the reality of the corresponding null hypotheses, obey the standard normal distribution. In the case where the rows of the source table have the same values, it is necessary to use average ranks. In this case, the accuracy of the conclusions will be worse, the greater the number of such matches.

Q - Cochran's criterion, proposed by W. Cochran in 1937. It is used in cases where groups of homogeneous subjects are exposed to influences, the number of which exceeds two and for which two options for feedback are possible - conditionally negative (0) and conditionally positive (1) . The null hypothesis consists of equality of treatment effects. Two-way analysis of variance makes it possible to determine the existence of treatment effects, but does not make it possible to determine for which specific columns this effect exists. To solve this problem, the method of multiple Scheffe equations for related samples is used.

Multivariate analysis

The problem of multivariate analysis of variance arises when you need to determine the effect of two or more conditions on a certain random variable. The study involves the presence of one dependent random variable, measured on a difference or ratio scale, and several independent variables, each of which is expressed on a naming or rank scale. Variance analysis of data is a fairly developed section of mathematical statistics, which has a lot of options. The research concept is common for both single-factor and multifactor. Its essence lies in the fact that the total variance is divided into components, which corresponds to a certain grouping of data. Each data grouping has its own model. Here we will consider only the basic provisions necessary for understanding and practical use of its most used options.

Variance analysis of factors requires a fairly careful attitude to the collection and presentation of input data, and especially to the interpretation of the results. Unlike a one-factor test, the results of which can be conditionally placed in a certain sequence, the results of a two-factor test require a more complex presentation. The situation becomes even more complicated when there are three, four or more circumstances. Because of this, it is quite rare to include more than three (four) conditions in a model. An example would be the occurrence of resonance at a certain value of capacitance and inductance of an electric circle; the manifestation of a chemical reaction with a certain set of elements from which the system is built; the occurrence of anomalous effects in complex systems under a certain coincidence of circumstances. The presence of interaction can radically change the model of the system and sometimes lead to a rethinking of the nature of the phenomena with which the experimenter is dealing.

Multivariate analysis of variance with repeated experiments

Measurement data can quite often be grouped not by two, but by a larger number of factors. Thus, if we consider the dispersion analysis of the service life of trolleybus wheel tires taking into account the circumstances (the manufacturing plant and the route on which the tires are operated), then we can single out as a separate condition the season during which the tires are operated (namely: winter and summer operation). As a result, we will have a problem of the three-factor method.

If there are more conditions, the approach is the same as in two-factor analysis. In all cases, they try to simplify the model. The phenomenon of interaction of two factors does not appear so often, and triple interaction occurs only in exceptional cases. Include those interactions for which there is previous information and good reasons to take it into account in the model. The process of identifying individual factors and taking them into account is relatively simple. Therefore, there is often a desire to highlight more circumstances. You shouldn't get carried away with this. The more conditions, the less reliable the model becomes and the greater the likelihood of error. The model itself, which includes a large number of independent variables, becomes quite complex to interpret and inconvenient for practical use.

General idea of analysis of variance

Analysis of variance in statistics is a method of obtaining observational results dependent on various simultaneously operating circumstances and assessing their influence. A controlled variable that corresponds to the method of influencing the object of study and acquires a certain value over a certain period of time is called a factor. They can be qualitative and quantitative. Levels of quantitative conditions acquire a certain meaning on a numerical scale. Examples are temperature, pressing pressure, amount of substance. Qualitative factors are different substances, different technological methods, devices, fillers. Their levels correspond to a scale of names.

Quality can also include the type of packaging material and storage conditions of the dosage form. It is also rational to include the degree of grinding of raw materials, the fractional composition of granules, which have quantitative significance, but are difficult to regulate if a quantitative scale is used. The number of qualitative factors depends on the type of dosage form, as well as the physical and technological properties of medicinal substances. For example, tablets can be obtained from crystalline substances by direct compression. In this case, it is enough to select sliding and lubricating substances.

Examples of quality factors for different types of dosage forms

Tinctures. Extractant composition, extractor type, raw material preparation method, production method, filtration method.
Extracts (liquid, thick, dry). Composition of the extractant, extraction method, type of installation, method of removing the extractant and ballast substances.
Pills. Composition of excipients, fillers, disintegrants, binders, lubricants and lubricants. Method of obtaining tablets, type of technological equipment. Type of shell and its components, film formers, pigments, dyes, plasticizers, solvents.
Injection solutions. Type of solvent, filtration method, nature of stabilizers and preservatives, sterilization conditions, method of filling ampoules.
Suppositories. Composition of the suppository base, method of producing suppositories, fillers, packaging.
Ointments. Composition of the base, structural components, method of preparing the ointment, type of equipment, packaging.
Capsules. Type of shell material, method of producing capsules, type of plasticizer, preservative, dye.
Liniments. Method of preparation, composition, type of equipment, type of emulsifier.
Suspensions. Type of solvent, type of stabilizer, dispersion method.

Examples of quality factors and their levels studied during the tablet manufacturing process

Baking powder. Potato starch, white clay, a mixture of sodium bicarbonate with citric acid, basic magnesium carbonate.
Binding solution. Water, starch paste, sugar syrup, methylcellulose solution, hydroxypropylmethylcellulose solution, polyvinylpyrrolidone solution, polyvinyl alcohol solution.
Sliding substance. Aerosil, starch, talc.
Filler. Sugar, glucose, lactose, sodium chloride, calcium phosphate.
Lubricant. Stearic acid, polyethylene glycol, paraffin.

Models of variance analysis in the study of the level of state competitiveness

One of the most important criteria for assessing the state of a state, by which the level of its well-being and socio-economic development is assessed, is competitiveness, that is, a set of properties inherent in the national economy that determine the state’s ability to compete with other countries. Having determined the place and role of the state in the world market, it is possible to establish a clear strategy for ensuring economic security on an international scale, because it is the key to positive relations between Russia and all players in the world market: investors, creditors, and governments.

To compare the level of competitiveness of states, countries are ranked using complex indices that include various weighted indicators. These indices are based on key factors influencing the economic, political, etc. situation. A set of models for studying state competitiveness involves the use of multivariate statistical analysis methods (in particular, analysis of variance (statistics), econometric modeling, decision making) and includes the following main stages:

Formation of a system of indicators.
Assessment and forecasting of state competitiveness indicators.
Comparison of indicators of the competitiveness of states.

Now let’s look at the content of the models of each of the stages of this complex.

At the first stage using expert study methods, a well-founded set of economic indicators for assessing the competitiveness of the state is formed, taking into account the specifics of its development based on international ratings and data from statistical departments, reflecting the state of the system as a whole and its processes. The choice of these indicators is justified by the need to select those that most fully, from a practical point of view, allow us to determine the level of the state, its investment attractiveness and the possibility of relative localization of existing potential and actual threats.

The main indicators of international rating systems are indices:

Global Competitiveness (GC).
Economic freedom (IES).
Human Development (HDI).
Perceptions of Corruption (CPC).
Internal and external threats (IETH).
International Influence Potential (IPIP).

Second phase provides for the assessment and forecasting of state competitiveness indicators according to international ratings for the 139 countries of the world under study.

Third stage provides for a comparison of the conditions of competitiveness of states using methods of correlation and regression analysis.

Using the results of the study, it is possible to determine the nature of the processes in general and for individual components of the state’s competitiveness; test the hypothesis about the influence of factors and their relationships at the appropriate level of significance.

The implementation of the proposed set of models will allow not only to assess the current situation of the level of competitiveness and investment attractiveness of states, but also to analyze management shortcomings, prevent errors of wrong decisions, and prevent the development of a crisis in the state.

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

Before we start talking about one of the types of financial analysis - factor analysis, let us 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 studying the dependence and dynamics of financial reporting indicators.

Financial analysis has several purposes:

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

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

horizontal analysis;
vertical analysis;
trend analysis;
method of financial ratios;
comparative analysis;
factor analysis.

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

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

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

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

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

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

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

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

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

Basics of modeling.

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

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 gain a deeper understanding of the reasons for changes in the phenomena under study, and to more accurately assess the place and role of each factor in the formation of the value of effective indicators.

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

Factor systems

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

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

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

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

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

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

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

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

where is the resultant sign;

Factor signs.

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

Factor analysis, its types and tasks.

Under factor analysis understands the methodology for a comprehensive and systematic study and measurement of the impact of factors on the value of performance indicators.

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

Setting the purpose of the analysis.
Selection of factors that determine the performance indicators under study.
Classification and systematization of factors in order to provide an integrated and systematic approach to the study of their influence on the results of economic activity.
Determination of the form of dependence between factors and the performance indicator.
Modeling the relationships 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 performance indicator.
Working with the factor model (its practical use for managing economic processes).

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

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

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

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

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

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

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

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

Stochastic analysis is a methodology for studying factors whose connection with a performance indicator, unlike a functional one, is incomplete and probabilistic (correlation). If with a functional (complete) dependence with a change in the argument there is always a corresponding change in the function, then with a correlation connection a change in the argument can give several values of the increase in the function depending on the combination of other factors that determine this indicator. For example, labor productivity at the same level of capital-labor ratio may be different at different enterprises. This depends on the optimal combination of other factors affecting this indicator.

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

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

it is necessary to study the influence of complex factors that cannot be combined in the same strictly determined model;
it is necessary to study the influence of complex factors that cannot be expressed by one quantitative indicator (for example, the level of scientific and technological progress).

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

the presence of a population;
sufficient volume of observations;
randomness and independence of observations;
uniformity;
the presence of a distribution of characteristics 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 purpose of the analysis, defining the population, determining the effective and factor characteristics, choosing the period for which the analysis is carried out, choosing the analysis method);
preliminary analysis of the simulated population (checking the homogeneity of the population, excluding anomalous observations, clarifying the required sample size, establishing distribution laws for the indicators being studied);
construction of a stochastic (regression) model (clarification of the list of factors, calculation of estimates of the parameters of the regression equation, enumeration of competing model options);
assessing the adequacy of the model (checking the statistical significance of the equation as a whole and its individual parameters, checking the compliance of the formal properties of the estimates with the objectives of the study);
economic interpretation and practical use of the model (determining the spatio-temporal stability of the constructed relationship, assessing the practical properties of the model).

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

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

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

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

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

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

Deterministic factor analysis.

Deterministic factor analysis has a fairly strict sequence of procedures:

construction of an economically sound deterministic factor model;
choosing a factor analysis technique and preparing conditions for its implementation;
implementation of counting procedures for model analysis;
formulating conclusions and recommendations based on the results of the analysis.

The first stage is especially important, since an incorrectly constructed model can lead to logically unjustified results. The meaning of this stage is as follows: any expansion of a strictly determined factor model should not contradict the logic of the “cause-effect” relationship. As an example, consider a model linking sales volume (P), headcount (H) and labor productivity (LP). 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. We will consider a brief comparative description of these methods below.

Types of deterministic factor models.

The following deterministic analysis models exist:

additive model, i.e., a model in which factors are included in the form of an algebraic sum; an example is the commodity balance model:

Where R- implementation;

Inventory at the beginning of the period;

P- receipt of goods;

Ending inventory;

IN- other disposal of goods;

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

Where R- implementation;

H- number;

PT- labor productivity;

multiple model, i.e., a model representing a relationship of factors, for example:

where is the capital-labor ratio;

H- number;

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

Where R- implementation;

Profitability;

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

A strictly deterministic model that has more than two factors is called multifactorial.

Typical problems of deterministic factor analysis.

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

Assessing the influence of relative changes in factors on the relative changes in the performance indicator.
Assessing the influence of an absolute change in the i-th factor on the absolute change in a performance indicator.
Determining the ratio of the change in the effective indicator caused by a change in the i-th factor to the base value of the effective indicator.
Determination of 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 to each of them using a specific simple example.

Example.

The volume of gross output (GP) depends on two main factors of the first level: the number of employees (NH) and average annual output (AG). We have a two-factor multiplicative model: . Let's consider a situation where both production and the number of workers in the reporting period deviated from the planned values.

Data for calculations are given in Table 1.

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

Task 1.

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

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 workers and average annual output, i.e.

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

We can conclude: 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, relative changes in factor and performance indicators are related by the same relationship as the indicators in the original model. This problem is solved by answering questions like: “What will happen if the i-th indicator changes by n%, and the j-th indicator changes by k%?”

Task 2.

Is main task deterministic factor analysis; its general formulation has the form:

Let - a strictly determined model that characterizes the change in the performance indicator y from n factors; all indicators received an increase (for example, in dynamics, compared to the plan, compared to the standard):

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

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

The change in the performance indicator is influenced only by the factor.

Depending on which method of model analysis is chosen, factor decompositions may differ. Therefore, in the context of this task, let us consider the main methods of analyzing factor models.

Basic methods of deterministic factor analysis.

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

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

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

The chain substitution method is a very simple and visual method, the most universal 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 changes in the value of the performance indicator by gradually replacing the base value of each factor indicator in the scope of the performance indicator with the actual value in the reporting period. For this purpose, a number of conditional values of the performance indicator are determined, which take into account changes in one, then two, then three, etc. factors, assuming that the rest do not change. Comparing the value of an effective indicator before and after changing the level of a particular factor allows us to determine the impact of a specific factor on the increase in the effective indicator, excluding the influence of other factors. Using this method, complete decomposition is achieved.

Let us recall that when using this method, the order in which the values of the factors change is of great importance, since the quantitative assessment of the influence of each factor depends on this.

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

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

A sign that is directly related to the phenomenon under study and characterizes its quantitative aspect is called primary or quantitative. These signs are: a) absolute (volumetric); b) they can be summed up in space and time. Examples include sales volume, headcount, cost of working capital, etc.

Features that relate to the phenomenon under study not directly, but through one or more other features and characterize the qualitative side of the phenomenon being studied are called secondary or quality. These signs are: a) relative; b) they cannot be summed up in space and time. Examples include capital-labor ratio, profitability, etc. The analysis identifies secondary factors of the 1st, 2nd, etc. orders, obtained by sequential detailing.

A strictly 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, it is recommended to start replacing factors with a quantitative indicator. If there are several quantitative and several qualitative indicators, then you should first change the value of the factors of the first level of subordination, and then the lower one. Thus, the use of the chain substitution method requires knowledge of the relationship of factors, their subordination, and the ability to correctly classify and systematize them.

Now, using our example, let’s look at the procedure for applying the chain substitution method.

The calculation algorithm using the chain substitution method for this model is as follows:

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

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

Thus, exceeding the plan for gross output was the result of the influence of the following factors:

The 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, can achieve higher accuracy of calculations, but these methods have a more limited scope and require a large amount of calculations, which is inconvenient for conducting operational analysis.

Task 3.

In a certain sense, it is a consequence of the second standard problem, since it is based on the resulting factor decomposition. The need to solve this problem is due to the fact that the elements of factor decomposition are absolute values that are difficult to use for spatio-temporal comparisons. When solving problem 3, the factor decomposition is supplemented with relative indicators:

Economic interpretation: the coefficient shows by what percentage compared to the base level the performance indicator has changed under the influence of the i-th factor.

Let's calculate the coefficients α for our example, using the factor decomposition 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 was 50%.

Task 4.

It is also solved on the basis of basic problem 2 and comes down to calculating the indicators:

Economic interpretation: the coefficient shows the share of the increase in the performance indicator due to the change in the i-th factor. There is no question here if all factor characteristics change unidirectionally (either increase or decrease). If this condition is not met, solving the problem may be complicated. In particular, in the simplest two-factor model, in such a case, the calculation according to the given 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 characteristic, i.e., a characteristic that changes in the same direction as the effective indicator.

Let's calculate the coefficients γ for our example, using the factor decomposition obtained by the chain substitution method:

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

Variance multivariate analysis is a set of various statistical methods that are designed to test hypotheses and the relationship between the factors under study and certain characteristics that do not have a quantitative description. Also, such a technique allows us to determine the degree of interaction of factors and their influence on certain processes. All these definitions sound quite confusing, so let's understand them in more detail in our article.

Criteria and types of analysis of variance

The method of multivariate analysis of variance is most often used to find a relationship between a continuous quantitative variable and nominal qualitative characteristics. In essence, this technique is testing various hypotheses about the equality of various arithmetic samples. Thus, it can also be considered as a criterion for comparing several samples. However, the results will be identical if only two elements are used for comparison. A study of the t-test shows that such a technique allows us to study the problem of hypotheses in more detail than any other known method.

It is also impossible not to note the fact that some types of variance analysis are based on a certain law: the sum of squares of intergroup deviations and the sum of squares of intragroup deviations are absolutely equal. The study uses the Fisher test, which is used for a detailed analysis of within-group variances. Although this requires prerequisites for normal distribution, as well as homoscedasticity of samples - equality of variances. As for the type of variance analysis, the following are distinguished:

multivariate or multivariate analysis;
univariate or univariate analysis.

It is not difficult to guess that the second considers the dependence of one characteristic and the value under study, and the first is based on the analysis of several characteristics at once. In addition, multivariate dispersion does not allow identifying a stronger relationship between several elements, since the dependence of several quantities is studied at once (although the method is much simpler to carry out).

Factors

Have you thought about methods for conducting multivariate correlation analysis? Then you should know that for a detailed study, you should study those factors that control the circumstances of the experiment and influence the final result. Factors can also mean methods and levels of processing values that characterize a specific manifestation of a certain condition. In this case, the numbers are given in an ordinal or nominal measurement system. If there is a problem with data grouping, you have to resort to using the same numeric values, which slightly changes the final result.

It should also be understood that the number of observations and groups cannot be excessively large, because this leads to excess data and the inability to complete the calculation. At the same time, the method of grouping depends not only on the volume, but also on the nature of the variation of certain values. The sizes and number of intervals in the analysis can be determined by the principle of equal frequencies, as well as equal intervals between them. As a result, all the studies obtained will be indicated in the statistics of multivariate analysis, which should be based on various examples. We will return to this in the following sections.

Purpose of ANOVA

So, sometimes situations may arise when it is necessary to compare two or more different samples. In this case, it would be most logical to apply multifactorial correlation and regression analysis, based on the study of the hypothesis and the relationship of various factors in the degree of regression. Also, the name of the technique indicates the fact that various components of variance are used in the research process.

What is the essence of the study? To begin with, two or more indicators are divided into separate parts, each of which corresponds to the action of a specific factor. After this, a series of research procedures are carried out to search for the relationship between various samples and connections between them. To understand such a complex but interesting technique in more detail, we recommend studying several examples of multifactor correlation analysis given in the following sections of our article.

Example one

In the production workshop there are several automatic machines, each of which is designed to produce a specific part. The size of the produced element is a random variable that depends not only on the settings of the machine itself, but also on the random deviations that will inevitably arise as a result of the production of parts. But how can a worker determine whether a machine is operating correctly if it initially produces defective parts? That's right, you need to purchase the same part on the market and compare its dimensions with what is obtained during production. The equipment can then be adjusted so that it produces parts of the required size. And it doesn’t matter at all that there is a manufacturing defect, because it is also taken into account in the calculations.

At the same time, if the machines have certain indicators that allow you to determine the intensity of adjustment (X and Y axes, depth, and so on), then the indicators on all machines will be completely different. If the measurements turn out to be exactly the same, then the manufacturing defect can be ignored altogether. However, this happens extremely rarely, especially if the errors are measured in millimeters. But if the produced part has the same dimensions as the standard purchased on the market, then there can be no talk of any defects, since in the production of the “ideal” a machine was also used, which gives certain errors, which were probably also taken into account by the workers.

Example two

To make a certain device that runs on electricity, it is necessary to use several types of different insulating paper: electrical, capacitor, and so on. In addition, the device can be impregnated with resin, varnish, epoxy compounds and other chemical elements that extend its service life. Well, various leaks under the vacuum cylinder at elevated pressure are easily eliminated using the method of heating or pumping out air. However, if the master has previously used only one element from each list, various difficulties may arise during the production process using the new technology. Moreover, it is almost certain that such a situation will be caused by one element. However, it will be almost impossible to calculate which factor influences the poor performance of the device. That is why it is recommended to use not a multifactor analysis method, but a single-factor one, in order to quickly understand the cause of the malfunction.

Of course, when using various tools and devices that monitor the influence of one or another factor on the final result, the research is simplified many times over, but a novice engineer will not be able to afford such units. That is why it is recommended to use one-way analysis of variance, which allows you to identify the cause of problems in a matter of minutes. To do this, it will be enough to set one of the most probable hypotheses, and then begin to prove it through experiments and analysis of the performance indicators of the device. Pretty soon the technician will be able to find the cause of the problem and fix it by replacing one of the samples with an alternative option.

Example three

Another example of multivariate analysis. Let's assume that a trolleybus depot can serve several routes during the day. Trolleybuses of completely different brands operate on these same routes, and fares are collected by 50 different controllers. However, depot management is interested in how several different indicators that affect overall revenue can be compared: trolleybus brand, route efficiency and employee skill. To see the economic feasibility, it is necessary to analyze in detail the influence of each of these factors on the final result. For example, some controllers may not perform their duties well, so you will have to hire more responsible employees. Most passengers do not like to ride old trolleybuses, so it is best to use a new brand. However, if both of these factors go along with the fact that most of the routes are in high demand, then is it worth changing anything at all?

The researcher’s task is to use one analytical method to obtain as much useful information as possible regarding the influence of each factor on the final result. To do this, it is necessary to put forward at least 3 different hypotheses, which will have to be proven in different ways. Dispersion analysis allows you to solve such problems in the shortest possible time and obtain maximum useful information, especially if a multiphase method is used. However, do not forget that univariate analysis gives much more confidence about the influence of a particular factor, since it examines the sample in more detail. For example, if the depot directs all its efforts to analyze the work of conductors, then it will be possible to identify many unscrupulous workers on all routes.

Univariate analysis

Univariate analysis is a set of research methods aimed at analyzing a specific factor for the final result in a particular case. Also quite often, a similar technique is used to compare the greatest influence between two factors. If we draw an analogy with the same depot, we should first analyze separately the impact of different routes and brands of trolleybuses on profitability, then compare the results obtained with each other and determine in which direction it would be best to develop the station.

In addition, we should not forget about such a concept as the null hypothesis - that is, a hypothesis that cannot be rejected and in any case is influenced by all the factors listed to one degree or another. Even if we compare only routes and brands of trolleybuses, there is still no escaping the influence of the conductors’ professionalism. Therefore, even if this factor cannot be analyzed, the influence of the null hypothesis should not be forgotten. For example, if you decide to study the dependence of profit on the route, use the same conductor on the flight so that the readings are as accurate as possible.

Two-factor analysis

Most often, this technique is also called the comparison method and is used to identify the dependence of two factors on each other. In practice, you will have to use various tables with exact indicators so as not to get confused in your own calculations and the influence of factors on them. For example, you can run two completely different trolleybuses on two identical routes at the same time, neglecting the null hypothesis factor (choose two responsible conductors). In this case, the comparison of the two situations will be of the highest quality, since the experiment takes place at the same time.

Multivariate analysis with repeated experiments

This method is used in practice much more often than others, especially when it comes to a group of novice researchers. Repeated experience allows you not only to verify the influence of one or another factor on the final result, but also to find errors that were made during the study. For example, most inexperienced analysts forget that there is one or more null hypotheses, which leads to inaccurate results during the study. Continuing the example with the depot, we can analyze the influence of certain factors in different seasons of the year, since the number of passengers in winter is very different from summer. In addition, repeated experience can lead the researcher to new ideas and new hypotheses.

Video and conclusion

We hope our article helped you understand what the method of multivariate correlation analysis is based on. If you still have any questions about this topic, we recommend watching a short video. It describes in detail the methods of dispersion research using a specific example.

As you can see, multifactor analysis is a rather complex, but very interesting process that allows you to identify the dependence of certain factors on the final result. This technique can be applied in absolutely all areas of life and can be used effectively for doing business. Also, the multivariate analysis model can be used to achieve breakthrough goals using simple methods.

One of the main tools of economic research is factor analysis, which is a section of multivariate statistical analysis that combines methods for estimating the dimension of many observed variables by examining the structure of covariance or correlation matrices. Unlike other analysis methods, it allows analysts to decide two main tasks: compactly and comprehensively describe the subject of measurement and identify the factors responsible for the presence of linear statistical correlations between the observed variables.

Justifiably applying the method of principal components, intended to replace correlated factors with uncorrelated ones, and also limiting itself to the study of the most significant informative factors and excluding the rest from the analysis, thereby simplifying the interpretation of the results, factor analysis appears as a technique for a comprehensive and systematic study of the dependence of other factors on the value of the criterion performance indicator .

Main types of factor analysis are: deterministic, functional(resultative criterion indicator, which is a product of partial or algebraic sum of factors); stochastic, correlation(if there is an incomplete or probabilistic connection between the resultant and factor indicators); direct, deductive(From general to specific); reverse, inductive(from particular to general); static and dynamic; retrospective and prospective; single-stage and multi-stage.

Factor analysis begins with checking its mandatory conditions, according to which: all signs are quantitative; the number of features is twice the number of variables; the sample is homogeneous; the distribution of the original variables is symmetrical; the study of factors is carried out using correlating variables. Factor analysis is carried out in several stages: selection of factors; classification and systematization of factors; modeling relationships 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; practical use of the factor model (calculation of reserves for growth of the effective indicator). Based on the nature of the relationship between indicators, methods of deterministic and stochastic factor analysis are distinguished (Table 1.5).

Factor analysis methods

Table 1.5

Methods	a brief description of
Deterministic factor analysis	Deterministic factor analysis- this is a technique for the influence of factors that are functionally related to the criterion performance indicator, which allows us to present the criterion indicator of the factor model as a quotient, product or algebraic sum of variables. Deterministic factor analysis is characterized by the following methods:chain substitutions; absolute differences; relative differences; integral; logarithms
Stochastic	Stochastic analysis- a methodology for studying factors whose connection with the criterion performance indicator is, in contrast to the functional connection, incomplete, probabilistic (correlation) in nature. With a correlation connection, by changing the argument depending on the combination of other variables that influence the value of the performance indicator, you can obtain a number of values for the increase in the function, while with a functional (complete) dependence, a change in the argument always leads to corresponding changes in the function. Stochastic analysis is carried out using the following methods factor analysis: pair correlation; multiple correlation analysis; matrix model; mathematical programming; game theory
Static and dynamic	Static factor analysis is practiced in order to assess the influence of factors on criterion performance indicators on a specific date, and dynamic - to identify the dynamics of cause-and-effect relationships
Retrospective and prospective	Factor analysis can be used as retrospective character (identify the reasons for changes in the value of the performance indicator over the past period), and perspective(to study the influence of factors on the value of the criterion indicator in the future)

For economic analysis, it is important to use deterministic modeling and different types of deterministic factor models designed to model correlations between the criterion effective factor and other variable factor indicators. The essence of this modeling is to present the relationship of the indicator under study with factors as a specific mathematical equation expressing a functional or correlation relationship.

Deterministic factor models make it possible to study the functional relationship between the studied indicators if the following requirements are met when constructing a factor model: the factors included in the model must be real and not abstract; factors must be in a cause-and-effect relationship with the performance indicator being studied; indicators of the factor model must be quantitatively measurable; it must be possible to measure the influence of individual factors; First, quantitative factors are written into the factor model, then qualitative ones; If there are several quantitative or qualitative factors in a factor model, then factors of a higher order are recorded first, and then lower ones.

The most widely used in factor analysis are the following: types of deterministic factor models(Table 1.6).

Types of deterministic factor models

Table 1.6

Factorial

models

a brief description of

Additive

They are used if the criterion performance indicator is presented in the form of an algebraic sum of a number of factor parameters of the indicators:

The developed factor model can be subjected to additional transformations when the ongoing research deepens, using a number of methods and techniques for these purposes. The final results of the economic analysis of the organization’s business depend on how realistically and accurately the developed models reflect the relationship between the indicators being studied. Modeling additive factor systems involves the implementation of a sequential decomposition of the factors of the original factor system into component variables:

at= a + b.

Thus, the first level factors a and b depend, in turn, on a number of other factors:

a= c + d, b= e+ m,

y = c+ d+ e+m.

Factorial models	a brief description of
Multiplicative models	They are used in cases where the criterion performance indicator is expressed as a product of a number of factor indicators: The essence of modeling multiplicative factor systems lies in the detailed sequential decomposition of the complex factors of the original factor system into factor factors: at= I X b. The magnitude of the first level factors a and b, in turn, depend on a number of other factors: a = c X, b = e X T, y=cxdexm*
Multiple models	If a criterion performance indicator can be defined as the ratio of one factor indicator to another, then The following are distinguished: methods for transforming factorial multiple models: 1)elongation(transforms the numerator by replacing one factor or a number of factors with the sum of homogeneous indicators): 2) formal decomposition(extends the denominator by replacing one or a number of factors with the sum or product of homogeneous indicators):
	3) extension(transforms the original factor model by multiplying the numerator and denominator of the ratio by one indicator or several new indicators):

Criteria-based performance indicators can be decomposed into factors in various ways and presented as different types of deterministic factor models. The modeling method is chosen depending on the object of study and the goals set, as well as on the professional knowledge and skills of the analyst.

Most methods for assessing factors in determination models are based on elimination, the most universal method of which is chain substitutions, used to measure the influence of factors in all types of factor determination models: multiplicative, additive, multiple and mixed (combined). Thanks to this method, it is possible to assess how individual factors influence the value of the criterion performance indicator, gradually replacing the basic value of each factor of the indicator as part of the criterion indicator with the actual value in the reporting period. To do this, a number of conditional values of the criterion performance indicator are calculated, taking into account the sequential change of one, two or more factors, with the remaining values remaining unchanged. A comparative assessment of the change in the value of a criterion parameter before and after a change in the level of a particular factor makes it possible to exclude (eliminate) the influence of all factors, except for the one whose impact on the increase in the performance indicator is determined.

The influence of one or another indicator is assessed by sequential subtraction: from the second calculation of the first, from the third - the second, etc. In the first calculation, all values are planned, in the last - actual. For example, the calculation algorithm for a three-factor multiplicative model is as follows:

In algebraic form, the sum of the influence of factors is equivalent to the total increase in the criterion performance indicator:

If this equality is not observed, the analyst should look for errors in his calculations. Based on this, a rule has been developed according to which it follows that the number of calculations per unit is greater than the number of indicators of the given equation.

When using the chain substitution method, it is assumed ensuring adherence to a strict substitution sequence, because its arbitrary change is fraught with distortion of the results of the analysis. IN process of analytical procedures It is advisable to identify the influence of quantitative indicators first, then qualitative ones. For example, it is required to assess the impact of the number of employees and labor productivity on the volume of industrial production. To do this, the impact of a quantitative indicator (number of employees) is first assessed, and then a qualitative indicator (labor productivity).

The chain substitution method has a significant drawback since when using it, it should be assumed that the values of the factors change independently of each other. Although in reality they change simultaneously and in interrelation, which entails an additional increase in the effective indicator, as a rule, attached to the last of the factors under study. Thus, the magnitude of the influence of factors on the change in the performance indicator depends on the location of a particular factor in the scheme of the analytical model. This explains the difference in calculations when changing the substitution sequence. Thus, the degree of influence of factors on changes in the criterion indicator varies depending on the place of the factor in the determination model. This disadvantage of deterministic factor analysis is eliminated by using a more complex integral method, allowing to evaluate the influence of factors in multiplicative, multiple and mixed models of multiple additive type.

Absolute difference method- this is a modification of the chain substitution method, in which the change in the criterion indicator due to each factor by the method of absolute differences is defined as the product of the deviation of the studied factor by the basic or reporting value of another factor, depending on the selected substitution sequence:

Relative difference method is intended to assess the influence of factors on the growth of a criterion indicator in multiplicative and mixed models of the form:

It involves finding the relative deviation of each factor indicator and determining the direction and size of the influence of factors as a percentage by sequential subtraction (from the first - always 100%).

When using abbreviated substitution method indicators for calculation are intermediate products with sequential accumulation of influencing factors. Substitutions are made, and then, by sequential subtraction, the influence of the factors is found.

Integral method allows you 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 change in the criterion indicator is measured over infinitely small periods of time by summing the increment of the result, defined as partial products multiplied by the increments of factors over infinitely small intervals.

The use of the integral method provides higher accuracy in calculating the influence of factors compared to the methods of chain substitution, absolute and relative differences, making it possible to eliminate ambiguous assessment of the influence, because in this case the results do not depend on the location of the factors in the model, and the additional increase in the effective indicator arising from for the interaction of factors, is distributed evenly between them.

To distribute additional growth, it is not enough to take its part corresponding to the number of factors, since factors can act in different directions. Therefore, the change in the effective indicator is measured over infinitely small periods of time by summing the increment of the result, defined as partial products multiplied by the increments of factors over infinitely small intervals. The operation of calculating a definite integral is reduced to constructing integrands that depend on the type of function or model of the factor system.

Due to the complexity of calculating some definite integrals and additional difficulties associated with the possible action of factors in opposite directions, in practice specially formed working formulas are used:

1. View model

2. View model

3. View model

4. View model

The main methods of elimination, which are based on relative indicators of dynamics, spatial comparisons, plan implementation (assessed by the ratio of the actual level of the indicator under study with the one being compared), include index method.

Index models make it possible to construct a quantitative assessment of the role of individual factors in the trends in the dynamics of changes in general indicators in statistics, planning and economic analysis. The calculation of any index involves comparing the measured value with the base value. If the index is reflected in the form of a ratio of directly comparable quantities, then it is called individual, and if the index represents the ratio of complex phenomena, then it is called group or total. There are several forms of indices (aggregate, arithmetic, harmonic).

The basis of any form of general index is aggregate index, allowing to assess the degree of influence of various factors on changes in the level of criterion indicators in multiplicative and multiple models. The correctness of determining the size of each factor is influenced by: the number of decimal places (at least four); the number of factors themselves (the relationship is inversely proportional).

Principles for constructing aggregate indexes are: a change in one factor while keeping all others constant. Moreover, if a generalizing economic indicator is the product of quantitative (volume) and qualitative indicators of factors, then when determining the influence of a quantitative factor, the qualitative indicator is fixed at the basic level, and when determining the influence of a qualitative factor, the quantitative indicator is fixed at the level of the reporting period.

Let's assume that Y - a * b * c x d,

Factor index showing how the indicator changes b etc.;

The so-called “general index of changes in the resulting indicator” depending on all factors.

Wherein

Using the index method, it is possible to decompose into factors not only relative, but also absolute deviations of the generalizing indicator, while determining the influence of individual factors using the difference between the numerator and denominator of the corresponding indices, i.e. when calculating the influence of one factor, eliminating the influence of another:

Using the index method of factor analysis, it is possible to decompose into factors not only relative, but also absolute deviations in the general indicator. In other words, the influence of an individual factor can be determined using the difference between the numerator and denominator of the corresponding indices, i.e. when calculating the influence of one factor, eliminating the influence of another.

Let's say:

Where A - quantitative factor, and b- qualitative,

indicator due to factor A;

Absolute increase in the resulting

indicator due to factor b

- absolute increase in the resulting

indicator due to the influence of all factors.

It is advisable to apply the considered principle of decomposing the absolute growth of a generalizing indicator into factors if the number of factors is equal to two (one of them is quantitative, the other is qualitative), and the analyzed indicator is presented as their product, since the theory of indices does not provide a general method for decomposing the absolute deviations of a generalizing indicator into factors when the number of factors is more than two. To solve this problem, the method of chain substitutions is used.

Factor analysis methods are successfully applied in order to objectively assess the influence of factors on the criterion indicator of the organization’s performance. As one example of this approach, consider how changes in the volume of product sales affect the financial results of an organization. As a rule, a change in sales revenue occurs due to: 1) a change in sales volume (in physical terms); 2) changes in selling prices. The total change in sales revenue can be presented as the sum of factor deviations:

Where N x - revenue for the reporting year;

N 0 - base year revenue;

A N- change in revenue as a result of changes in sales volume;

A Np- change in revenue as a result of changes in selling prices for products;

A N c- change in revenue as a result of changes in the structure of product sales.

Let's imagine the revenue (N) as the product of the selling price (R) on sales volume ( Q):

N 0 = P 0 X Q 0 - base year revenue;

jV, = P, x (2, - revenue of the reporting year.

The impact of changes in product sales volume (at constant prices) on changes in revenue is assessed as follows:

The impact of a change in sales price (with a constant volume) on a change in revenue is assessed as follows:

In the process of analysis, the influence of factors such as changes in the sales structure is determined, as well as the share of individual assortment items in the total sales volume in the base and analyzed periods, and then the impact of structural changes on the total sales volume is calculated. Lost revenue as a result of changes in the range of products sold is assessed negatively, while excess revenue is assessed positively.

Factor analysis statistics for dummies. Factor analysis of sales profit: example of calculation of indicators. Criteria and types of analysis of variance

The concept of factor analysis

Objectives of the

Factor analysis techniques

Deterministic Analysis

Stochastic analysis

Stages and features of factor analysis

Factor analysis of profitability

Other formulas

Example

Factor analysis of profit

Types and criteria of analysis of variance

Factors

Analysis of variance problems

Univariate analysis

Two-factor analysis

Multivariate analysis

Multivariate analysis of variance with repeated experiments

General idea of ​​analysis of variance

Examples of quality factors for different types of dosage forms

Examples of quality factors and their levels studied during the tablet manufacturing process

Models of variance analysis in the study of the level of state competitiveness

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

Basics of modeling.

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

Factor systems

Factor analysis, its types and tasks.

Deterministic factor analysis.

Types of deterministic factor models.

Typical problems of deterministic factor analysis.

Basic methods of deterministic factor analysis.

Criteria and types of analysis of variance

Factors

Purpose of ANOVA

Example one

Example two

Example three

Univariate analysis

Two-factor analysis

Multivariate analysis with repeated experiments

Video and conclusion

General idea of analysis of variance