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How to do factor analysis in statistics 6. Factor and variance analysis in Excel with automated calculations

Statistica 6 q. Preparation of correlation matrix for factor analysis q. Creating a matrix for factor analysis q. Factor analysis q. Isolation of factor loadings q. Construction of a factor diagram

Preparing a correlation matrix for factor analysis in the Statistica program Since our ranks are ordinal scales, two coefficients will be adequate for this type of scale: Spearman and Kendall. We will consider Kendall, since he is more accurate. We enter our raw data into the Statistica program

We received a factor matrix calculated by the Kendall coefficient, since it is this that is adequate for our data, which are order scales.

Creating a matrix for calculating FA Now you need to create a matrix of such a structure that Statistica can use it to carry out factor analysis. It is necessary that the matrix, in addition to the values ​​of correlations between variables, include 4 more rows below them: 1) average values ​​of ranks, 2) standard deviations of ranks, 3) number of evaluated objects and 4) type of matrix. Click Analysis and select Basic statistics and tables

As a result, we received a correlation matrix for FA that Statistica can read. However, the correlation analysis here was carried out by the Pearson coefficient. Therefore, this correlation matrix (5 x 5) needs to be replaced with the one we calculated using the Kendall coefficient (copy and paste).

As can be seen, the Kendall correlation values ​​differ from the Pearson values. This is because our ranks are order scales for which the use of the Pearson coefficient is inadequate. Now we can proceed to factor analysis.

Variables → select all 5 variables Var 1 Var 5 → in the Data file field set Correlation matrix → OK

Max. We set the number of factors to 5 (since we have only 5 variables) → select the Centroid method (it was developed by Thurstone and implements a geometric approach to FA) → OK

The program identified 2 factors. To view factor loadings, click the Factor loadings button. To construct a factor diagram, click on the 2 M loading plot.

Statgraphics Centurion q. Factor analysis q. Isolation of factor loadings q. Constructing a factor diagram q. Building an object diagram

The program does not provide the ability to create your own correlation matrix, so we start immediately with an analysis of our ranks. We enter our ranks and select Analyze → Variable Data → Multivariate Methods → Factor Analysis

As a result, the program allocated us 2 factors with a level of explained variance of 82.468%. This means that these factors explain 82.468% (almost 4/5) of all our information across the five variables.

Scree plot (2 factors) The graph shows that all the information explained is in factors 1 and 2 (2 points above the red line)

Factor loadings Click Tables (second button from the left on the panel) Check the box next to Extraction Statistics → OK

As you can see, the factor loadings at the tenth level differ from those we obtained by manual calculation and in Statistica. This is explained by the fact that Statgraphics cannot include its own correlation matrix and the program always considers the Pearson coefficient, which is not adequate for data on order scales.

Factor chart Click Graphs (third button from the left on the panel) Check the box next to 2 D Factor Plot (if we had more than 2 factors, we would check the box next to 3 D Factor Plot to get a three-dimensional graph) → OK

We obtained a factor matrix after rotation. Segments (projections of points formed by factor loadings) 2 and 5 are located close to the y-axis (tend to 0) and are distant from the x-axis. This means that the x-axis coordinates of these points (which corresponds to the first factor) are represented by low values ​​(0, 6). Therefore, scales 2 and 5 represent 1 factor. By the same principle, segment 1 indicates that scales 1, 3 and 4 represent factor 2.

Object diagram Click Graphs (third button from the left on the panel) Check the box next to 2 D Scatterplot (if we had more than 2 factors, we would check the box next to 3 D Scatterplot to get a three-dimensional graph) → OK

Factor analysis is a statistical method that is used when processing large amounts of experimental data. The objectives of factor analysis are: reducing the number of variables (data reduction) and determining the structure of relationships between variables, i.e. classification of variables, so factor analysis is used as a data reduction method or as a structural classification method.

An important difference between factor analysis and all the methods described above is that it cannot be used to process primary, or, as they say, “raw” experimental data, i.e. obtained directly from the examination of subjects. The material for factor analysis is correlations, or more precisely, Pearson correlation coefficients, which are calculated between the variables (i.e. psychological characteristics) included in the survey. In other words, correlation matrices, or, as they are otherwise called, intercorrelation matrices, are subjected to factor analysis. The column and row names in these matrices are the same because they represent a list of variables included in the analysis. For this reason, intercorrelation matrices are always square, i.e. the number of rows in them is equal to the number of columns, and symmetrical, i.e. symmetrical places relative to the main diagonal have the same correlation coefficients.

The main concept of factor analysis is factor. This is an artificial statistical indicator that arises as a result of special transformations of the table of correlation coefficients between the studied psychological characteristics, or the intercorrelation matrix. The procedure for extracting factors from an intercorrelation matrix is ​​called matrix factorization. As a result of factorization, a different number of factors can be extracted from the correlation matrix, up to a number equal to the number of original variables. However, the factors identified as a result of factorization, as a rule, are unequal in importance. (5)

With the help of identified factors, the interdependence of psychological phenomena is explained. (7)

Most often, as a result of factor analysis, not one, but several factors are determined that differently explain the matrix of intercorrelations of variables. In this case, factors are divided into general, general and individual. General factors are those all factor loadings of which differ significantly from zero (zero loading indicates that this variable is in no way connected with the others and does not have any influence on them in life). General are factors for which some of the factor loadings are different from zero. Single factors are factors in which only one of the loadings differs significantly from zero. (7)

Factor analysis may be appropriate if the following criteria are met.

  • 1. It is impossible to factorize qualitative data obtained on a scale of names, for example, such as hair color (black / chestnut / red), etc.
  • 2. All variables must be independent, and their distribution must approach normal.
  • 3. Relationships between variables should be approximately linear, or at least not clearly curvilinear.
  • 4. The initial correlation matrix should contain several correlations with a modulus higher than 0.3. Otherwise, it is quite difficult to extract any factors from the matrix.
  • 5. The sample of subjects must be large enough. Expert recommendations vary. The most stringent point of view recommends not using factor analysis if the number of subjects is less than 100, since the standard errors of correlation in this case will be too large.

However, if the factors are well defined (for example, with loadings of 0.7 rather than 0.3), the experimenter needs a smaller sample to isolate them. In addition, if the data obtained are known to be highly reliable (for example, valid tests are used), then data can be analyzed on a smaller number of subjects. (5).

Ministry of Agriculture of the Russian Federation

Federal State Educational Institution

Higher professional education

State University of Land Management

Department of Economic Theory and Management

Course work

In the discipline “Analysis and diagnostics of the financial activities of an enterprise”

On the topic: “Factor analysis of production elements.”

Performed:

student of group 34

Maksimova N.S.

Checked:

Chirkova L.L.

Moscow 2009

Introduction……………………………………………………………………………………….....3

Chapter 1. Factor analysis of production elements…………………………………………………………………………………..4

1.1. Factor analysis, its types and tasks…………………………………………………………………………………..4

1.2. Deterministic factor analysis. Modeling requirements…………………………………………………………………………………..8

1.3 Methods and types of deterministic factor analysis…………………..10

Chapter 2 . Practical part………………………………………………………..14

2.1. Methods for measuring the influence of factors in the analysis of economic activities………………………………………………………………………………….14

2.2. Factor analysis of the financial condition of the motor transport enterprise OJSC “Enterprise 1564”……………………………………………………………….….20

Conclusion……………………………………………………………………………….……..24

List of references………………………………………………………......25

Applications…………………………………………………………………………………..26

Introduction

Factor analysis- a set of methods of multivariate statistical analysis used to study the relationships between the values ​​of variables. Using factor analysis, it is possible to identify hidden (latent) variable factors responsible for the presence of linear statistical relationships (correlations) between observed variables.

Objectives of factor analysis:

  • reducing the number of variables;
  • determination of relationships between variables, their classification.

Factor analysis arose at the beginning of the 20th century, initially developed in problems of psychology. Charles Spearman and Raymond Cattell made great contributions to the development of factor analysis.

Factor analysis methods:

  • principal component method
  • correlation analysis
  • maximum likelihood method

Factor analysis - determining the influence of factors on the result - is one of the strongest methodological solutions in analyzing the economic activities of companies for decision making. For managers - an additional argument, an additional “angle of view”.

However, in practice it is rarely used for several reasons:

1) implementation of this method requires some effort and a specific tool (software product);

2) companies have other “eternal” priorities.

Chapter 1. Factor analysis of production elements

1.1 Factor analysis, its types and tasks.

Factor analysis is understood as a methodology for a comprehensive and systematic study and measurement of the impact of factors on the value of performance indicators.

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

1. Setting the purpose of the analysis.

2. Selection of factors that determine the performance indicators under study.

3. Classification and systematization of factors in order to provide an integrated and systematic approach to the study of their influence on the results of economic activity.

4. Determination of the form of dependence between factors and the performance indicator.

5. Modeling the relationships 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 the factor model (its practical use for managing economic processes).

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

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

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

The last stage of factor analysis is the practical use of the factor model to calculate reserves for the growth of an 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.

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

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

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

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

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

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

a) the presence of a population;

b) a sufficient volume of observations;

c) randomness and independence of observations;

d) homogeneity;

e) the presence of a distribution of characteristics close to normal;

f) the presence of a special mathematical apparatus.

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

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

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

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

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

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

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

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

1.2 Deterministic factor analysis. Modeling requirements.

Determinism(from Lat. determino - I determine) - the doctrine of the objective, natural and causal conditionality of all phenomena. Determination is based on the existence of causality, that is, on such a connection between phenomena in which one phenomenon (cause), under certain conditions, gives rise to another (effect). )