Factor analysis for dummies. Factor analysis, its types and tasks

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 studied variables, 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, various kinds of errors and inaccuracies are possible, 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 that are linear combinations of the previous 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 received its own further development at 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 studied variables. The result of this step is a matrix of factor loading coefficients, which in the orthogonal case are correlation coefficients between variables and factors. When determining the number of 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 loads and residual variances, which makes it possible to meaningfully interpret the general factors and loads.

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, measures of subjects' opinions on a large number tests do not have a precisely established metric base. However, in general, it is assumed that many "ordinal variables" 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 the average responses (for example, for satisfaction) correspond to a value of 100, while below and above the average responses are lower and higher values, respectively. Two variables (responses to two different items) are correlated with each other. 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 one 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

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 a graphical 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 only "factorial scree" is located to the right of this point ("scree" is a geological term for rock fragments accumulating in the lower part of a 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 processes in business are interconnected. There are both direct and indirect links between them. Various economic parameters change under the action 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 multivariate technique that allows you to study the relationship 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.

Tasks

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 are determined. 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 interrelations between effective and factorial values.
• Determining the degree of influence of each of the factors.
• Analysis of the role of each of the values.
• Application of the factor model.

Each parameter that affects the final value must be investigated.

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 main factors of the company, to analyze their influence in quantitative terms. As a result of DA, you can understand what factors should be changed to improve the efficiency of the company. Advantages of the method: versatility, ease of use.

Stochastic analysis

Stochastic analysis allows you to analyze the existing indirect links. That is, there is a study of mediated factors. The method is used when it is impossible to find direct links. Stochastic analysis is considered optional. It is used only in some cases.

What is meant by indirect links? 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 of all. They allow you to get 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 the most complex calculations are performed in the process. For their implementation, special software is required.

Consider the stages of FA:

1. Establishing the purpose of the calculations.
2. Selection of values ​​that directly or indirectly affect the final result.
3. Classification of factors for a comprehensive study.
4. Detection of the relationship between the selected parameters and the final indicator.
5. Modeling the relationship between the result and the factors influencing it.
6. Determining the degree of influence of values ​​and assessing the role of each of the parameters.
7. The use of the formed factor table in the activities of the enterprise.

NOTE! Factor analysis involves the most complex calculations. Therefore, it is better to entrust its implementation to a professional.

IMPORTANT! It is extremely important when making calculations to correctly select the factors that affect the result of the enterprise. The choice of factors depends on the specific area.

Factor analysis of profitability

Profitability FA is carried out to analyze the rationality of resource allocation. As a result, you can determine which factors have the greatest impact on the final result. As a result, you can leave only those factors that the best way affect efficiency. Based on the data obtained, you can change the pricing policy of the company. The following factors can affect the cost of production:

• fixed costs;
• variable costs;
• profit.

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

Factor analysis is carried out using the following formula:

Rv \u003d ((Tue-Sat - KRB-URB) / W) - (VB-SB-KRB-URB) / WB, where:

WT - revenue for the current period;

SB - cost for the current period;

KRB - commercial expenses for the current period;

BDS - administrative expenses for the previous period;

WB - revenue for the previous period;

KRB - commercial expenses for the previous period.

Other formulas

Consider the formula for calculating the degree of impact of cost on profitability:

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

Cbot is the cost of production for the current period.

The formula for calculating the impact of management expenses:

Rur \u003d ((W-SB -KRB-URot) / W) - (W-SB-KRB-URB) / W,

URot is administrative expenses.

The formula for calculating the degree of impact of commercial costs:

Rk \u003d ((W-SB -KRO-URB) / W) - (W-SB-KRB-URB) / W,

KRo is the commercial expenses for the previous time.

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

Rob \u003d Rv + Rs + Rur + Rk.

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

Example

Consider the performance of the organization for two months (for two periods, in rubles). In July, the organization's income amounted to 10 thousand, the cost of production - 5 thousand, administrative expenses - 2 thousand, commercial expenses - 1 thousand. In August, the company's income amounted to 12 thousand, the cost of production - 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 profit of the organization increased by 14%.

Factor analysis of profit

P \u003d PP + RF + RVN, where:

P - profit or loss;

РР - profit from sales;

RF - results of financial activity;

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

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

РР = N - S1 -S2, where:

N - proceeds from the sale of goods at selling prices;

S1 - cost of goods sold;

S2 - commercial and administrative expenses.

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

NOTE! Factor analysis is extremely difficult to perform manually. For it, you can use special programs. The most simple program for calculations and automatic analysis - Microsoft Excel. It has analysis tools.

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 headcount workers;

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:

; Y = ; Y = ; Y = .

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 construct 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 detail 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:

y 0 = a 0 . b 0 . c 0 ;

y a = a 1 . b 0 . c 0 ;

y b = a 1 . b 1. c 0 ;

y 1 = a 1 . b1. c 1 ,

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 Dy=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:

Dy \u003d SDy (a, b, c) \u003d Dy a + Dy b + Dy c

Dy a \u003d y a - y 0; Dy b \u003d y c - y a; Dy s \u003d y 1 - y c.

Consider an example:

table 2

Initial data for factor analysis

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:

TP o \u003d H o. SW o \u003d 20. 146 = 2920 (thousand rubles).

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

TP conv 1 \u003d H 1. SW o \u003d 25. 146 = 3650 (thousand rubles),

DTPusl 1 \u003d TPusl 1 - TP o \u003d 3650 - 2920 \u003d 730 (thousand rubles).

TP 1 \u003d H 1. SW 1 \u003d 25. 136 = 3400 (thousand rubles),

DTP conv 2 = TP 1 – TP con 1 = 3400 – 3650 = - 250 (thousand rubles).

Thus, the change in the volume of marketable products positive influence had a change for 5 people. number of employees, which caused an increase in production by 730t. rub. and a negative impact was made 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 meanings. This is due to the fact that as a result of applying this method, a certain indecomposable residue is formed, which is added to the magnitude of the influence of the last factor. In practice, the accuracy of 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 define personal traits, signs and features of the studied phenomena (labor productivity, product quality, average working day, etc.).

Way absolute differences 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:

y 0 = a 0 . b 0 . c 0 ;

Dy a \u003d D a. b 0 . with 0 ;

Dy b \u003d Db. a 1 . with 0 ;

Dу s = Dс. a 1 . b1;

y 1 = a 1 . b1. with 1 ;

Dy \u003d Dy a + Dy b + Dy c.

Way relative differences 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 makes it possible to achieve complete decomposition effective indicator by factors and is of a universal 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.

Questions for self-control

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 simplest methods of 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.

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

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

The financial analysis is a method of evaluation financial condition and efficiency of the economic entity based on the study of the dependence and dynamics of indicators financial reporting.

Financial analysis has several goals:

• assessment of the financial situation;
• identification of changes in the financial condition in the spatio-temporal context;
• identification of the main factors that caused changes in the financial condition;
• forecast of the main trends in the financial condition.

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

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

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

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

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

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

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

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

normative models. Models of this type make it possible to compare the actual performance of enterprises with the expected ones calculated according to the budget. These models are mainly used in internal financial analysis. Their essence is reduced to the establishment of standards for each item of expenditure 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 rigidly determined factor models.

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

Basics of modeling.

The functioning of any socio-economic system (which includes an operating enterprise) occurs in a complex interaction of a complex of internal and external factors. Factor- this is the reason, the driving force of any process or phenomenon, which determines its nature or one of the 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 features. It allows you to better understand the reasons for the change in the phenomena under study, more accurately assess the place and role of each factor in the formation of the value of effective indicators.

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

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

Natural factors have big influence on performance in agriculture, forestry and other industries. Accounting for their influence makes it possible to more accurately assess the results of the work of business entities.

Socio-economic factors include the living conditions of workers, the organization of recreational work at 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 the use of the enterprise's production resources and the final results of its activities.

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

The factors are divided into internal and external, depending on whether they are affected by the activities of the enterprise or not. The analysis focuses on internal factors which the company can influence.

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

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

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

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

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

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

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

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

Factor systems

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

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

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

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

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

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

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

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

where is the effective sign;

Factor signs.

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

Factor analysis, its types and tasks.

Under factor analysis refers to the methodology of complex and systematic study and measurement of the impact of factors on the magnitude of performance indicators.

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

1. Setting the goal of the analysis.
2. Selection of factors that determine the studied performance indicators.
3. Classification and systematization of factors in order to provide an integrated and systematic approach to the study of their impact on the results of economic activity.
4. Determination of the form of dependence between factors and the performance indicator.
5. Modeling the relationship between performance and factor indicators.
6. Calculation of the influence of factors and assessment of the role of each of them in changing the value of the effective indicator.
7. Working with a factor model (its practical use for managing economic processes).

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

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

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

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

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

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

is a methodology for studying the influence of factors whose relationship with the performance indicator is functional, i.e. when the performance indicator of the factor model is presented as a product, private or algebraic sum of factors.

This type factor analysis is the most common, because, being quite simple to use (compared to stochastic analysis), it allows you to understand the logic of the main factors in the development of an enterprise, quantify their influence, understand which factors and in what proportion it is possible and expedient to change to improve production efficiency . Deterministic factor analysis will be discussed in detail in a separate chapter.

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

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

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

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

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

1. the presence of a population;
2. sufficient volume of observations;
3. randomness and independence of observations;
4. homogeneity;
5. the presence of a distribution of signs close to normal;
6. the presence of a special mathematical apparatus.

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

• qualitative analysis (setting the goal of the analysis, determining the population, determining the effective and factor signs, choosing the period for which the analysis is carried out, choosing the analysis method);
• preliminary analysis of the simulated population (checking the homogeneity of the population, excluding anomalous observations, clarifying the required sample size, establishing the laws of distribution of the studied indicators);
• building a stochastic (regression) model (refinement of the list of factors, calculation of estimates of the parameters of the regression equation, enumeration of competing models);
• assessing the adequacy of the model (checking the statistical significance of the equation as a whole and its individual parameters, checking the correspondence of the formal properties of the estimates to the objectives of the study);
• economic interpretation and practical use of the model (determination of the spatio-temporal stability of the constructed dependence, assessment of the practical properties of the model).

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

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

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

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

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

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

Deterministic factor analysis.

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

• building an economically sound deterministic factor model;
• choice of method of factor analysis and preparation of conditions for its implementation;
• implementation of computational procedures for model analysis;
• formulation of conclusions and recommendations based on the results of the analysis.

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

All three formulas are correct from the standpoint of arithmetic, however, from the standpoint 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. Brief comparative characteristic we will discuss these methods below.

Types of deterministic factor models.

There are the following models of deterministic analysis:

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

where R- implementation;

Stocks at the beginning of the period;

P- receipt of goods;

Stocks at the end of the period;

AT- other disposal of goods;

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

where R- implementation;

H- number;

Fri- labor productivity;

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

where - capital-labor ratio;

OS

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- the cost of working capital.

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

Typical problems of deterministic factor analysis.

There are four typical tasks in deterministic factor analysis:

1. Evaluation of the influence of the relative change in factors on the relative change in the performance indicator.
2. Assessment of the influence of the absolute change of the i-th factor on the absolute change of the effective indicator.
3. Determination of the ratio of the magnitude of the change in the effective indicator caused by the change in the i-th factor to the base value of the effective indicator.
4. Determining the share of the absolute change in the performance indicator caused by the change in the i-th factor in the total change in the performance indicator.

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

Example.

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

The data for calculations are given in Table 1.

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

Task 1.

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

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

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

;

.

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

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

.

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

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

Task 2.

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

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

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

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

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

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

Basic methods of deterministic factor analysis.

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

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

Let's give brief description the most common ways.

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

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

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

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

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

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

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

Now let's look at our example, the procedure for applying the method of chain substitutions.

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

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

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

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

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

The absence of such equality indicates errors in the calculations.

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

Task 3.

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

.

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

Calculate the coefficients α for our example, using the factorial expansion obtained earlier by the method of chain substitutions:

;

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

Task 4.

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

.

Economic interpretation: the coefficient shows the share of the increase in the effective indicator due to the change in the i-th factor. There is no question here if all factor signs change in the same direction (either increase or decrease). If this condition is not met, the solution of the problem can be complicated. In particular, in the simplest two-factor model, in such a case, the calculation according to the above formula is not performed and it is considered that 100% of the increase in the effective indicator is due to a change in the dominant factor attribute, i.e., a sign that changes unidirectionally with the effective indicator.

Calculate the coefficients γ for our example, using the factorial expansion obtained by the method of chain substitutions:

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

Perform factorial analysis of the phenomenon according to the multiplier model using the method of relative differences, absolute differences, the method of chain substitutions and formalization of the irreducible remainder and the logarithmic method.

a) absolute change: b) relative change:

Calculations

3,62*5,02*2,92*5,82=308,829

76,7807

=0,00

Examination

У4.52*5.02*4.02*5.72=521.7521

3,62*5,02*2,92*5,82=308,829

521,721-308,829=212,92

CONCLUSION: factor analysis calculations show that under the influence of all independent factors A, B, C, D, the effective factor Y increased by 212.92 units. At the same time, factors such as B and D also had a negative impact on the effective factor Y. Of these, factor D had the greatest influence, and its change caused a decrease in the effective factor Y by 9.12 units. At the same time, factors A and C had a positive impact on factor Y, of which factor C had the greatest influence, its change caused an increase in the effective factor Y by 145.264 units.

2) the method of "indecomposable remainder"

Isolated influence of factors

For factor A \u003d 0.9 * 5.02 * 2.92 * 5.82 \u003d 76.7807

B \u003d 0.00 * 3.62 * 2.92 * 5.82 \u003d 0.00

C \u003d 1.1 * 3.62 * 5.02 * 5.82 \u003d 116.3397

D \u003d -0.10 * 3.62 * 5.02 * 5.82 \u003d -10.5763

"Indecomposable residue" is determined by the formula

NO \u003d No \u003d 212.92-182.5441 \u003d 30.38

CONCLUSION: factor analysis calculations show that under the influence of all independent factors A, B, C, D, the effective factor Y increased by 182.5441 units. At the same time, factors such as B and D also had a negative impact on the effective factor Y. Of these, factor D had the greatest influence, and its change caused a decrease in the effective factor Y by 10.5763 units. At the same time, factors A and C had a positive impact on factor Y, of which factor C had the greatest influence, its change caused an increase in the effective factor Y by 116.3397 units. The error was 30.38.

3) Logarithmic method.

I Lg (i) i /Lg (i) y

For factor A = 0.09643*212.92/0.22775=90.151

For factor B = 0.00*212.92/0.22775=0.00

For factor С = 0.13884*212.92/0.22775=129.8

For factor D = -0.00753*212.92/0.22775=-7.0397

90,151+0,00+129,8+(-7,0397)= 212,9113

CONCLUSION: factor analysis calculations show that under the influence of all independent factors A, B, C, D, the effective factor U increased by 212.9113 units (the error in the calculations is associated with rounding off the change in the factor). At the same time, factor D had a negative impact on the effective factor Y , and its change caused a decrease in the effective factor Y by 7.03997 units. At the same time, factors A and C had a positive impact on factor Y, of which factor C had the greatest influence, its change caused an increase in the effective factor Y by 129.8 units.

4) The method of absolute differences. Y= A*B*S*D

b) general change in the results of factors

Solution

0,9*5,02*2,92*5,82=76,781

4,52*0,00*2,92*5,82=0,00

4,52*5,02*1,1*5,82=145,2639

4,52*5,02*4,02*(-0,1)= -9,1215

76,781+0,00+145,2639+(-9,1215)= 212,923

Checking the results:

У4.52*5.02*4.02*5.72=521.7521

3,62*5,02*2,92*5,82=308,829

521,721-308,829=212,92

CONCLUSION: factor analysis calculations show that under the influence of all independent factors A, B, C, D, the effective factor Y increased by 212.923 units. At the same time, factor D had a negative impact on the effective factor Y, and its change caused a decrease in the effective factor Y by 9.12 units. At the same time, factors A and C had a positive impact on factor Y, of which factor C had the greatest influence, its change caused an increase in the effective factor Y by 145.2639 units.

5) the method of chain substitutions.