Factor analysis of economic activity - abstract. Factor analysis, its types and methods

are called factor analysis. Main varieties factor analysis are deterministic analysis and stochastic analysis.

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

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

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

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

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

Classification of factors affecting the effectiveness of the organization

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

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

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

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

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

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

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

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

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

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

Factor economic analysis

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

The combination of factor and resultant signs that are in the same causal relationship is called factor system. There is also the concept of a factor system model. It characterizes the relationship between the resultant feature, denoted as y, and factor features, denoted as . In other words, the factor system model expresses the relationship between general economic indicators and individual factors that affect this indicator. At the same time, other economic indicators act as factors, which are the reasons for the change in the generalizing indicator.

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

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

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

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

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

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

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

Methods used in factor analysis. Scheme No. 2

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

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

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

All business processes of enterprises are interconnected and interdependent. Some of them are directly related to each other, some are manifested indirectly. In this way, important issue in economic analysis is an assessment of the influence of a factor on a particular economic indicator, and for this, factor analysis is used.

Factor analysis of the enterprise. Definition. Goals. Kinds

Factor analysis refers in the scientific literature to the section of multivariate statistical analysis, where the assessment of observed variables is carried out using covariance or correlation matrices.

Factor analysis was first used in psychometrics and is currently used in almost all sciences, from psychology to neurophysiology and political science. The basic concepts of factor analysis were defined by the English psychologist Galton and then developed by Spearman, Thurstone, and Cattell.

Can be distinguished 2 goals of factor analysis:
- determination of the relationship between variables (classification).
— reduction of the number of variables (clustering).

Factor analysis of the enterprise – complex methodology systematic study and assessment of the impact of factors on the value of the effective indicator.

The following can be distinguished types of factor analysis:

Functional, where the effective indicator is defined as a product or an algebraic sum of factors.
Correlation (stochastic) - the relationship between the performance indicator and factors is probabilistic.
Direct / Reverse - from general to specific and vice versa.
Single stage / multi stage.
Retrospective / prospective.

Let's take a closer look at the first two.

In order to be able to factor analysis is necessary:
All factors must be quantitative.
- The number of factors is 2 times more than the performance indicators.
— Homogeneous sample.
— Normal distribution of factors.

Factor analysis carried out in several stages:
Stage 1. Selected factors.
Stage 2. Factors are classified and systematized.
Stage 3. The relationship between the performance indicator and factors is modeled.
Stage 4. Evaluation of the influence of each factor on the performance indicator.
Stage 5 Practical use of the model.

Methods of deterministic factor analysis and methods of stochastic factor analysis are singled out.

Deterministic factor analysis- a study in which factors affect the performance indicator functionally. Methods of deterministic factor analysis - the method of absolute differences, the method of logarithm, the method of relative differences. This type analysis is the most common due to its ease of use and allows you to understand the factors that need to be changed to increase / decrease the effective indicator.

Stochastic factor analysis- a study in which factors affect the performance indicator probabilistically, i.e. when a factor changes, there may be several values (or a range) of the resulting indicator. Methods of stochastic factor analysis - game theory, mathematical programming, multiple correlation analysis, matrix models.

Introduction to Factor Analysis

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

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

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

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

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

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

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

However, usually the factors obtained by the method of principal components do not lend themselves to a sufficiently visual interpretation. Therefore, the next step in factor analysis is the transformation (rotation) of the factors in such a way as to facilitate their interpretation. Rotation factors consists in finding the simplest factor structure, that is, such an option for estimating factor loadings and residual variances, which makes it possible to meaningfully interpret the general factors and loadings.

Most often, researchers use the varimax method as a rotation method. This is a method that allows, on the one hand, by minimizing the spread of squared loads for each factor, to obtain a simplified factor structure by increasing large and reducing small factor loads, on the other hand.

So, the main goals of factor analysis:

reduction number of variables (data reduction);

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

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

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

Issues of data preparation for application

factor analysis

Let's look at a series of questions and short answers as part of the use of factor analysis.

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

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

Should the researcher avoid using factor analysis when the metric basis of the variables is not well defined, i.e. Are the data presented in an ordinal scale?

Not necessary. Many variables representing, for example, measurements of subjects' opinions on a large number of tests do not have a well-established metric base. However, in general, it is assumed that many "ordinal variables" may contain numerical values that do not distort and even retain the basic properties of the feature under study. Tasks of the researcher: a) correctly determine the number of reflexively allocated orders (levels); b) take into account that the sum of the allowed distortions will be included in the correlation matrix, which is the basis of the input data of the factor analysis; c) correlation coefficients are fixed as "ordinal" distortions in measurements (Labovitz, 1967, 1970; Kim, 1975).

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

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

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

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

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

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

One more thing about this issue. The larger the sample size, the more reliable the criterion value. chi-square. Results are considered statistically significant if the sample includes at least 51 observations. In this way:

N-n-150,(3.33)

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

n is the number of variables (Lawley and Maxwell, 1971).

This, of course, is only a general rule.

What is the meaning of the factor load sign?

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

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

Principal Components and Factor Analysis

Factor analysis as a method of data reduction

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

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

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

An example in which two correlated variables are combined into one factor shows the main idea behind factor analysis, or more specifically principal component analysis. If the two-variable example is extended to include more variables, the calculations become more complex, but the basic principle of representing two or more dependent variables by a single factor remains valid.

Principal Component Method

Principal component analysis is a method of reducing or reducing data, i.e. method of reducing the number of variables. A natural question arises: how many factors should be singled out? Note that in the process of successive selection of factors, they include less and less variability. The decision as to when to stop the factor extraction procedure mainly depends on the point of view of what counts as small "random" variability. This decision is rather arbitrary, but there are some recommendations that allow you to rationally choose the number of factors (see section Eigenvalues and the number of distinguished factors).

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

Rice. 3.10. 3D feature scatterplot

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

After a line is found for which the dispersion is maximum, some data scatter remains around it, and it is natural to repeat the procedure. In principal component analysis, this is exactly what is done: after the first factor allocated, that is, after the first line is drawn, the next line is determined, maximizing the residual variation (scatter of data around the first line), and so on. Thus, the factors are sequentially allocated one after another. Since each subsequent factor is determined in such a way as to maximize the variability remaining from the previous ones, the factors turn out to be independent of each other (uncorrelated or orthogonal).

Eigenvalues and the number of distinguished factors

Let's look at some standard results of Principal Component Analysis. When recalculating, factors with less and less variance are distinguished. For simplicity, it is assumed that work usually begins with a matrix in which the variances of all variables are equal to 1.0. Therefore, the total variance is equal to the number of variables. For example, if there are 10 variables and the variance of each is 1, then the largest variance that can potentially be isolated is 10 times 1.

Assume that the Life Satisfaction Survey includes 10 items to measure various aspects of home and work satisfaction. The variance explained by successive factors is shown in Table 3.14:

Table 3.14

Table of eigenvalues

STATISTICA FACTOR ANALYSIS	Eigenvalues (factor.sta) Extraction: Principal Components
Meaning	Eigenvalues	% of total variance	Cumulate. own value	Cumulate. %

In the second column of table 3. 14. (Eigenvalues) the variance of a new, just isolated factor is presented. The third column for each factor gives the percentage of the total variance (10 in this example) for each factor. As you can see, factor 1 (value 1) explains 61 percent of the total variance, factor 2 (value 2) accounts for 18 percent, and so on. The fourth column contains the accumulated (cumulative) variance.

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

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

Criteria for selecting factors

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

Scree criterion is graphic method, first proposed by Cattell (Cattell, 1966). It allows you to display eigenvalues in a simple graph:

Rice. 3. 11. Scree criterion

Both criteria have been studied in detail by Brown (Browne, 1968), Cattell and Jaspers (Cattell, Jaspers, 1967), Hakstian, Rogers, and Cattell (Hakstian, Rogers, Cattell, 1982), Linn (Linn, 1968), Tucker, Koopman and Lynn (Tucker, Koopman, Linn, 1969). Cattell suggested finding a place on the graph where the decrease in eigenvalues from left to right slows down as much as possible. It is assumed that to the right of this point there is only a "factorial scree" ("scree" is a geological term for fragments of rocks accumulating in the lower part of the rocky slope). In accordance with this criterion, 2 or 3 factors can be left in the considered example.

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

In practice, an important additional question arises, namely, when the resulting solution can be meaningfully interpreted. Therefore, it is common to examine several solutions with more or less factors, and then choose the one that makes the most sense. This question will be further considered in terms of factor rotations.

communities

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

Main Factors and Main Components

Term factor analysis includes both principal component analysis and principal factor analysis. It is assumed that, in general, it is known how many factors should be distinguished. One can find out (1) the significance of factors, (2) whether they can be interpreted in a reasonable way, and (3) how to do this. To illustrate how this can be done, the steps are taken "in reverse", that is, starting with some meaningful structure and then seeing how it affects the results.

The main difference between the two factor analysis models is that Principal Component Analysis assumes that all variability of variables, while principal factor analysis uses only the variability of a variable that is common to other variables.

In most cases, these two methods lead to very close results. However, Principal Component Analysis is often preferred as a method of data reduction, while Principal Factor Analysis is best used to determine the structure of data.

Factor analysis as a data classification method

Correlation matrix

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

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

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

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

Deterministic factor analysis is a methodology for studying the influence of factors, the relationship of which with the performance indicator is of a functional nature.

The main properties of the deterministic approach to analysis:
building a deterministic model by logical analysis;
The presence of a complete (hard) connection between the indicators;
Impossibility of separating the results of the influence of simultaneously acting factors that cannot be combined in one model;
study of interrelations in the short term.

There are four types of deterministic models:

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

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

Multiplicative Models in a generalized form can be represented by the formula

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

where H- average number of employees;

CB is the average output per worker.

Multiple Models:

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

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

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

Examples of such models are cost indicators for 1 ruble. marketable products, profitability indicators, etc.

To study the relationship between indicators and to quantify the many factors that influenced the performance indicator, we present general model conversion rules to include new factor indicators.

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

If the original factorial model , and , then the model takes the form .

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

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

The detailing of factor analysis is largely determined by the number of factors whose influence can be quantitatively assessed, therefore, multifactorial multiplicative models are of great importance in the analysis. They are based on the following principles:
The place of each factor in the model should correspond to its role in the formation of the effective indicator;
The model should be built from a two-factor complete model by sequentially dividing the factors, usually qualitative ones, into components;
· when writing the formula of a multifactorial model, the factors should be arranged from left to right in the order of their replacement.

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

Method of chain substitutions consists in determining a number of intermediate values of the generalizing indicator by successively replacing the basic values of the factors with the reporting ones. This method is based on elimination. Eliminate- means to eliminate, exclude the influence of all factors on the value of the effective indicator, except for one. At the same time, based on the fact that all factors change independently of each other, i.e. first one factor changes, and all the others remain unchanged. then two change while the rest remain unchanged, and so on.

AT general view The application of the chain setting method can be described as follows:

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

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

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

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

Consider an example:

table 2

Initial data for factor analysis

Indicators	Conventions	Basic values	Actual values	Change
Indicators	Conventions	Basic values	Actual values	Absolute (+,-)	Relative (%)
The volume of marketable products, thousand rubles.
Number of employees, people
output per worker,

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

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

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

Advantages this method: universality of application, simplicity of calculations.

The disadvantage of the method is that, depending on the chosen order of factor replacement, the results of the factor expansion have different values. This is due to the fact that as a result of applying this method, a certain indecomposable residue is formed, which is added to the magnitude of the influence of the last factor. In practice, the accuracy of assessing factors is neglected, highlighting the relative importance of the influence of one or another factor. However, there are certain rules that determine the sequence of substitution:
If there are quantitative and qualitative indicators in the factor model, the change in quantitative factors is considered first of all;
· if the model is represented by several quantitative and qualitative indicators, the substitution sequence is determined by logical analysis.

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

Qualitative Factors determine personal traits, signs and features of the studied phenomena (labor productivity, product quality, average working day, etc.).

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

Relative difference method is used to measure the influence of factors on the growth of the effective indicator in multiplicative and mixed models of the form y \u003d (a - c) . With. It is used in cases where the initial data contain previously defined relative deviations of factorial indicators in percent.

For multiplicative models like y = a . in . with the analysis technique is as follows:

find the relative deviation of each factor indicator:

determine the deviation of the effective indicator at for each factor

Example. Using the data in Table. 2, we will analyze by the method of relative differences. The relative deviations of the considered factors will be:

Let us calculate the impact on the volume of marketable output of each factor:

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

integral method allows you to avoid the disadvantages inherent in the method of chain substitution, and does not require the use of techniques for the distribution of the indecomposable remainder by factors, since it has a logarithmic law of redistribution of factor loadings. The integral method allows you to achieve a complete decomposition of the effective indicator by factors and is universal in nature, i.e. applicable to multiplicative, multiple, and mixed models. Calculation operation definite integral is solved with the help of a PC and is reduced to the construction of integrands that depend on the type of function or model of the factorial system.
1. What management tasks are solved through economic analysis?
2. Describe the subject of economic analysis.
3. What distinctive features characterize the method of economic analysis?
4. What principles underlie the classification of techniques and methods of analysis?
5. What role does the method of comparison play in economic analysis?
6. Explain how to build deterministic factor models.
7. Describe the algorithm for applying the most simple ways deterministic factor analysis: the method of chain substitutions, the method of differences.
8. Describe the advantages and describe the algorithm for applying the integral method.
9. Give examples of tasks and factor models to which each of the methods of deterministic factor analysis is applied.

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Identification of the relationship between performance indicators and indicators-factors, forms of dependence between them. Features of the application of the elimination method, integral and index methods. Mathematical Methods factor analysis.

Factors are the conditions of economic processes and the reasons that affect them.

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

All phenomena and processes of economic activity of enterprises are in interconnections, interdependence and interdependence. One of them directly interconnected, others indirectly . For example, the amount of profit from the main activities of the enterprise is directly affected by such factors as the volume and structure of sales, selling prices and production costs. All other factors affect this indicator indirectly. Each phenomenon can be considered both as a cause and as a result. For example, labor productivity can be considered, on the one hand, as the cause of a change in the volume of production, the level of its cost, and on the other hand, as a result of a change in the degree of mechanization and automation of production, improvement in the organization of labor, etc. If this or that indicator is considered as a consequence, as a result of the action of one or more causes and acts as an object of study, then when studying the relationships, it is called an effective indicator. Indicators that determine the behavior of the resulting feature are called factorial.

Each performance indicator depends on numerous and varied factors. The more detailed the influence of factors on the value of the effective indicator is studied, the more accurate the results of the analysis and assessment of the quality of work of enterprises. Hence, an important methodological issue in the analysis of economic activity is the study and measurement of the influence of factors on the magnitude of the studied economic indicators. Without a deep and comprehensive study of the factors, it is impossible to draw reasonable conclusions about the performance results, identify production reserves, justify plans and management decisions, predict performance results, assess their sensitivity to changes in internal and external factors.

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

There are the following types of factor analysis:

Deterministic (functional) and stochastic (probabilistic);

Direct (deductive) and reverse (inductive);

Single-stage and multi-stage;

Static and dynamic;

Retrospective and prospective (forecast).

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

Deterministic factor analysis is a technique for studying the influence of factors whose relationship with the performance indicator is functional in nature, i.e. the effective indicator can be represented as a product, private or algebraic sum of factors.

Stochastic factor analysis explores the influence of factors, the relationship of which with the performance indicator, in contrast to the functional one, is incomplete, probabilistic (correlation). If, with a functional (full) dependence, a corresponding change in the function always occurs with a change in the argument, then with a stochastic 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 not be the same at different enterprises. It depends on the optimal combination of all factors that form this indicator.

With direct factor analysis research is conducted in a deductive way - from the general to the particular. Back factor analysis carries out a study of cause-and-effect relationships by the method of logical induction - from private, individual factors to general ones. It allows assessing the degree of sensitivity of performance results to changes in the studied factor.

Factor analysis can be single-stage and multi-stage. single stage is used to study the factors of only one level (one stage) of subordination without their detailing into component parts. For example, y = a b. With multi-stage factor analysis factors a and b are detailed on constituent elements in order to study their nature. Detailing factors can be continued. In this case, the influence of factors of different levels of subordination is studied.

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

Finally, factor analysis can be retrospective. , which studies the causes of changes in the results of economic activity for past periods, and prospective , which examines the behavior of factors and performance indicators in the future.

The main tasks of factor analysis

1. Selection of factors for the analysis of the studied indicators.

2. Classification and systematization of them in order to ensure a systematic approach.

3. Modeling the relationship between performance and factor indicators.

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

5. Working with a factor model (its practical use for managing economic processes).

To study the influence of factors on the results of management and calculation of reserves in the analysis, methods of deterministic and stochastic factor analysis, methods of optimization solution of economic problems(see picture).

Determining the magnitude of the influence of individual factors on the growth of performance indicators is one of the most important methodological tasks in AHD. In deterministic analysis, the following methods are used for this: chain substitution, absolute differences, relative differences, index, integral, proportional division, logarithm, balance, etc.

The main properties of the deterministic approach to analysis:

Building a deterministic model by logical analysis;

The presence of a complete (rigid) relationship between indicators;

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

The study of relationships in the short term.

Consider the possibility of using the main methods of deterministic analysis, summarizing the above in the form of a matrix

Matrix for applying methods of deterministic factor analysis

	Factor Models
	Multiplicative	Additive	mixed
Chain substitution
Absolute difference
Relative differences			y = a ∙ (b−c)
Integral

Designations: + is used;

- not used

There are four types of deterministic models:

Additive models are an algebraic sum of indicators and have the form:

Such models, for example, include cost indicators in conjunction with production cost elements and cost items; an indicator of the volume of production of goods in its relationship with the volume of output of individual products or the volume of output in individual divisions.

Multiplicative - this is a sequential division of the factors of the original system into factor factors. Models in a generalized form can be represented by the formula:

An example of a multiplicative model is a two-factor model of gross output: VP \u003d PR * CB

where CHR - the average number of employees;

CB - average annual output per worker.

Multiple models: y = x1 / x2.

An example of a multiple model is the indicator of the term of goods turnover (TOB.T) (in days): TOB.T \u003d WT / OR, (1.9)

where ST is the average stock of goods;

RR - one-day sales volume.

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

Examples of such models are cost indicators for 1 ruble. manufactured products, profitability indicators, etc.

1. The most universal method of deterministic analysis is the method of chain substitution.

It is used to calculate the influence of factors in all types of deterministic factor models: additive, multiplicative, multiple and mixed (combined). This method is based on elimination.

Elimination is the process of step-by-step exclusion of the influence of all factors on the value of the effective indicator, except for one. At the same time, based on the fact that all factors change independently of each other, i.e. first one factor changes, and all the others remain unchanged. Then two change while the rest remain unchanged, and so on.

This method allows you to determine the influence of individual factors on the change in the value of the effective indicator. The essence of this technique is to single out the main factors that have a decisive influence on the change in the indicator from all the existing factors. For this purpose, a number of conditional values of the performance indicator are determined, which take into account the change in one, then two, three and subsequent factors, assuming that the rest do not change. This means that in the calculations private planned indicators are consistently replaced by reporting ones, the results obtained are compared with the available previous data. Comparison of the values of the performance indicator before and after the change in the level of one or another factor makes it possible to eliminate the influence of all factors except one, and to determine the impact of the latter on the growth of the performance indicator.

When using the method of chain substitutions, the sequence of substitutions is of great importance: first of all, it is necessary to take into account the change in quantitative, and then qualitative indicators. The application of the reverse sequence of calculations does not give correct characteristics the influence of factors.

In this way, the application of the method of chain substitution requires knowledge of the relationship of factors, their subordination, the ability to correctly classify and systematize them.

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

y0 = a0 ∙ b0 ∙ c0 ;

ya = a1 ∙ b0 ∙ c0 ;

yb = a1 ∙ b1 ∙ c0 ;

y1 = a1 ∙ b1 ∙ c1 ;

where a0, b0, c0 - basic values of factors influencing the generalizing indicator y;

a1, b1, c1 - actual values of factors;

ya, yb, - intermediate values of the resulting indicator associated with the change in factors a and b, respectively.

The total change Δy = y1 - y0 is the sum of the changes in the resulting indicator due to changes in each factor with fixed values of the other factors. Those. the sum of the influence of individual factors should be equal to the overall increase in the performance indicator.

∆y = ∆ya + ∆yb + ∆yc = y1– y0

∆ya = ya – y0 ;

∆yb = yb – ya;

∆yc = y1 – yb.

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

The disadvantage of the method is that, depending on the chosen order of factor replacement, the results of the factor expansion have different values.

2. The absolute difference method is a modification of the chain substitution method.

The method of absolute differences is used to calculate the influence of factors on the growth of the effective indicator in deterministic analysis, but only in multiplicative models (Y = x1 ∙ x2 ∙ x3 ∙∙∙∙∙ xn) and models of multiplicative-additive type: Y = (a - b) ∙c and Y = a∙(b - c). And although its use is limited, but due to its simplicity, it has been widely used in AHD.

The essence of the method of the method is that the value of the influence of factors is calculated by multiplying the absolute increase in the value of the factor under study by the base (planned) value of the factors that are to the right of it, and by the actual value of the factors located in the model to the left of it.

y0 = a0 ∙ b0 ∙ c0

∆ya = ∆a ∙ b0 ∙ c0

∆yb = a1 ∙ ∆b ∙ c0

∆yс = a1 ∙ b1 ∙ ∆с

y1 = a1 ∙ b1 ∙ c1

The algebraic sum of the increase in the effective indicator due to individual factors should be equal to its total change Δy = y1 - y0.

∆y = ∆ya + ∆yb + ∆yc = y1 – y0

Consider the algorithm for calculating factors in this way in multiplicative-additive models. For example, let's take a factorial model of profit from the sale of products:

P \u003d VRP ∙ (C - C),

where P - profit from the sale of products;

VRP - sales volume of products;

P is the price of a unit of production;

C - unit cost of production.

The increase in the amount of profit due to changes in:

sales volume ∆PVRP = ∆VRP ∙ (P0 − С0);

sales price ∆PC = VRP1 ∙ ∆C;

production costs ∆PS = VRP1 ∙ (−∆С);

3. The method of relative differences It is used in cases where the source data contain previously defined relative deviations of factor indicators in percent. It is used to measure the influence of factors on the growth of the effective indicator only in multiplicative models. Here, relative increases in factor indicators are used, expressed as coefficients or percentages. Consider the methodology for calculating the influence of factors in this way for multiplicative models of the type Y = abc.

The change in the performance indicator is determined as follows:

According to this algorithm to calculate the influence of the first factor, it is necessary to multiply the base value of the effective indicator by the relative growth of the first factor, expressed as a decimal fraction.

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

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

The calculation results are the same as for the previous methods.

The method of relative differences is convenient to use in cases where it is required to calculate the influence of a large complex of factors (8-10 or more). Unlike the previous methods, the number of computational procedures is significantly reduced here, which determines its advantage.

4. The integral method for estimating factor influences makes it possible to avoid the disadvantages inherent in the chain substitution method and does not require the use of techniques for distributing the indecomposable residue over factors, since it has a logarithmic law of redistribution of factor loadings. The integral method allows you to achieve a complete decomposition of the effective indicator by factors and is universal in nature, i.e. applicable to multiplicative, multiple, and mixed models. The operation of calculating a definite integral is carried out using the computing capabilities of personal computers and is reduced to the construction of integrands that depend on the type of function or model of the factorial system.

Its use allows you to get more accurate results of calculating the influence of factors compared to the methods of chain substitution, absolute and relative differences, since the additional increase in the effective indicator from the interaction of factors is not added to the last factor, but is divided equally between them.

Consider the algorithms for calculating the influence of factors for different models:

1) View model: y = a ∙ b

2) View model: y = a ∙ b ∙ c

3) View model:

If there are more than two factors in the denominator, then the procedure continues.

Thus, the use of the integral method does not require knowledge of the entire integration process. It is enough to substitute the necessary numerical data into these ready-made working formulas and do not really complex calculations using a calculator or other computer technology.

The results of calculations by the integral method differ significantly from those obtained by the method of chain substitutions or modifications of the latter. The larger the change in factors, the greater the difference.

5. The index method allows you to identify the impact on the studied aggregate indicator various factors. By calculating the indices and constructing a time series that characterizes, for example, output in value terms, one can judge the dynamics of production volume in a qualified manner.

It is based on relative indicators of dynamics, expressing the ratio of the level of the analyzed indicator in the reporting period to its level in the base period. The index method can

Any index is calculated by comparing the measured (reporting) value with the base value. For example, the index of production volume: Ivvp = VVP1 / VVP0

Indexes expressing the ratio of directly commensurate quantities are called individual , and the characterizing ratios of complex phenomena - group , or total . The statistics name a few forms indices that are used in analytical work - aggregate, arithmetic, harmonic, etc.

Applying the aggregate form of the index and observing the established computational procedure, it is possible to solve the classical analytical problem: determining the influence of the quantity factor and the price factor on the volume of produced or sold products. The calculation scheme will be as follows:

It should be recalled here that the aggregate index is the basic form of any general index; it can be converted to both the arithmetic mean and harmonic mean indices.

The dynamics of turnover in the sale of industrial products should be characterized, as is known, by time series built over a number of past years, taking into account changes in prices (this naturally applies to procurement, wholesale and retail turnover).

The index of the volume of sales (turnover), taken in prices relevant years, looks like:

General price index:

General indexes- relative indicators obtained as a result of comparing phenomena covering heterogeneous product groups.

General turnover index (value of marketable products);

where p1q1 is the turnover of the reporting period

p0q0 − turnover of the base period

p - prices, q - quantity

General price index: Ip =

Average indices are relative indicators used to analyze structural changes. They are used only for homogeneous goods.

Price index of variable composition (average prices):

Fixed composition price index:

6. The method of proportional division can be used in some cases to determine the magnitude of the influence of factors on the growth of the effective indicator . This applies to those cases when we are dealing with additive models Y=∑хi and models of a multiply additive type:

In the first case, when we have a single-level model of the type Y = a + b + c, the calculation is carried out as follows:

In models of a multiply additive type, it is first necessary to determine by the method of chain substitution how much the effective indicator has changed due to the numerator and denominator, and then to calculate the influence of second-order factors by the method of proportional division according to the above algorithms.

For example, the level of profitability increased by 8% due to an increase in the amount of profit by 1000 thousand rubles. At the same time, profit increased due to an increase in sales by 500 thousand rubles, due to an increase in prices - by 1,700 thousand rubles, and due to an increase in the cost of production decreased by 1,200 thousand rubles. Let's determine how the level of profitability has changed due to each factor:

7. To solve this type of problem, you can also use the method of equity participation. . To do this, first determine the share of each factor in the total amount of their growth (coefficient of equity participation), which is then multiplied by the total growth of the effective indicator (Table 4.2):

Calculation of the influence of factors on the performance indicator by the equity method

	Change in profit, thousand rubles	Factor share in changing the overall profit amounts	Change in the level of profitability, %
Volume of sales			8 ∙ 0,5 = +4,0
			8 ∙1,7 = +13,6
Cost price			8 ∙ (-1,2)= -9,6
Total

8. Based on the method of sequential isolation of factors lies the method of scientific abstraction, which makes it possible to investigate a large number of combinations with a simultaneous change in all or part of the factors.