15.02.202427.05.2017

Analysis of variance

1. Concept of analysis of variance

Analysis of variance is an analysis of the variability of a trait under the influence of any controlled variable factors. In foreign literature, analysis of variance is often referred to as ANOVA, which is translated as analysis of variability (Analysis of Variance).

ANOVA problem consists in isolating variability of a different kind from the general variability of a trait:

a) variability due to the action of each of the independent variables under study;

b) variability due to the interaction of the independent variables being studied;

c) random variability due to all other unknown variables.

Variability due to the action of the variables under study and their interaction is correlated with random variability. An indicator of this relationship is Fisher's F test.

The formula for calculating the F criterion includes estimates of variances, that is, the distribution parameters of the attribute, therefore the F criterion is a parametric criterion.

The more the variability of a trait is due to the variables (factors) under study or their interaction, the higher empirical criterion values.

Zero the hypothesis in the analysis of variance will state that the average values of the studied effective characteristic are the same in all gradations.

Alternative the hypothesis will state that the average values of the resulting characteristic in different gradations of the factor under study are different.

Analysis of variance allows us to state a change in a characteristic, but does not indicate direction these changes.

Let's begin our consideration of variance analysis with the simplest case, when we study the action of only one variable (one factor).

2. One-way analysis of variance for unrelated samples

2.1. Purpose of the method

The method of one-factor analysis of variance is used in cases where changes in an effective characteristic are studied under the influence of changing conditions or gradations of a factor. In this version of the method, the influence of each of the gradations of the factor is different samples of subjects. There must be at least three gradations of the factor. (There may be two gradations, but in this case we will not be able to establish nonlinear dependencies and it seems more reasonable to use simpler ones).

A nonparametric version of this type of analysis is the Kruskal-Wallis H test.

Hypotheses

H 0: Differences between factor grades (different conditions) are no greater than random differences within each group.

H 1: Differences between factor grades (different conditions) are greater than random differences within each group.

2.2. Limitations of One-Way Analysis of Variance for Unrelated Samples

1. One-way analysis of variance requires at least three gradations of the factor and at least two subjects in each gradation.

2. The resulting characteristic must be normally distributed in the sample under study.

True, it is usually not indicated whether we are talking about the distribution of the characteristic in the entire surveyed sample or in that part of it that makes up the dispersion complex.

3. An example of solving a problem using the method of one-way analysis of variance for unrelated samples using the example:

Three different groups of six subjects were given lists of ten words. The words were presented to the first group at a low speed - 1 word per 5 seconds, to the second group at an average speed - 1 word per 2 seconds, and to the third group at a high speed - 1 word per second. Reproduction performance was predicted to depend on the speed of word presentation. The results are presented in Table. 1.

Number of words reproduced Table 1

Subject No.	low speed	average speed	high speed








total amount

H 0: Differences in word production span between groups are no more pronounced than random differences inside each group.

H1: Differences in word production volume between groups are more pronounced than random differences inside each group. Using the experimental values presented in Table. 1, we will establish some values that will be necessary to calculate the F criterion.

The calculation of the main quantities for one-way analysis of variance is presented in the table:

table 2

Table 3

Sequence of operations in one-way analysis of variance for unrelated samples

Often found in this and subsequent tables, the designation SS is an abbreviation for “sum of squares.” This abbreviation is most often used in translated sources.

SS fact means the variability of the characteristic due to the action of the factor under study;

SS generally- general variability of the trait;

S C.A.-variability due to unaccounted factors, “random” or “residual” variability.

MS- “mean square”, or the mathematical expectation of the sum of squares, the average value of the corresponding SS.

df - the number of degrees of freedom, which, when considering nonparametric criteria, we denoted by a Greek letter v.

Conclusion: H 0 is rejected. H 1 is accepted. Differences in word recall between groups were greater than random differences within each group (α=0.05). So, the speed of presentation of words affects the volume of their reproduction.

An example of solving the problem in Excel is presented below:

Initial data:

Using the command: Tools->Data Analysis->One-way ANOVA, we get the following results:

As already noted, the dispersion method is closely related to statistical groupings and assumes that the population under study is divided into groups according to factor characteristics, the influence of which should be studied.

Based on variance analysis, the following is produced:

1. assessment of the reliability of differences in group means for one or several factor characteristics;

2. assessing the reliability of factor interactions;

3. assessment of partial differences between pairs of means.

The application of variance analysis is based on the law of decomposition of variances (variations) of a characteristic into components.

The total variation D o of the resulting characteristic during grouping can be decomposed into the following components:

1. to intergroup D m associated with a grouping characteristic;

2. for residual(intra-group) D B not related to the grouping characteristic.

The relationship between these indicators is expressed as follows:

D o = D m + D in. (1.30)

Let's look at the use of variance analysis with an example.

Let’s say you want to prove whether sowing dates affect wheat yields. The initial experimental data for analysis of variance are presented in table. 8.

Table 8

In this example, N = 32, K = 4, l = 8.

Let us determine the total total variation in yield, which is the sum of squared deviations of individual values of a trait from the overall average:

where N is the number of population units; Y i – individual yield values; Y o is the overall average yield for the entire population.

To determine the intergroup total variation, which determines the variation of the effective characteristic due to the factor being studied, it is necessary to know the average values of the effective characteristic for each group. This total variation is equal to the sum of the squared deviations of group averages from the overall average value of the trait, weighted by the number of population units in each group:

Within-group total variation is equal to the sum of squared deviations of individual values of a trait from group averages for each group, summed over all groups in the population.

The influence of a factor on the resulting characteristic is manifested in the relationship between Dm and Dv: the stronger the influence of the factor on the value of the characteristic being studied, the greater Dm and the less Dv.

To carry out analysis of variance, it is necessary to establish the sources of variation in a trait, the volume of variation by source, and determine the number of degrees of freedom for each component of variation.

The amount of variation has already been established; now it is necessary to determine the number of degrees of freedom of variation. Number of degrees of freedom is the number of independent deviations of individual values of a characteristic from its average value. The total number of degrees of freedom, corresponding to the total sum of squared deviations in ANOVA, is decomposed into components of variation. Thus, the total sum of squared deviations D o corresponds to the number of degrees of freedom of variation equal to N – 1 = 31. The group variation D m corresponds to the number of degrees of freedom of variation equal to K – 1 = 3. The intragroup residual variation corresponds to the number of degrees of freedom of variation equal to N – K = 28.

Now, knowing the sum of squared deviations and the number of degrees of freedom, we can determine the variances for each component. Let us denote these variances: d m - group and d in - intragroup.

After calculating these variances, we will proceed to establish the significance of the influence of the factor on the resulting attribute. To do this, we find the ratio: d M / d B = F f,

The quantity F f, called Fisher criterion , compared with the table, F table. As already noted, if F f > F table, then the influence of the factor on the effective attribute has been proven. If F f< F табл то можно утверждать, что различие между дисперсиями находится в пределах возможных случайных колебаний и, следовательно, не доказывает с достаточной вероятностью влияние изучаемого фактора.

The theoretical value is associated with probability, and in the table its value is given at a certain level of probability of the judgment. The appendix contains a table that allows you to set the possible F value for the probability of judgment, the most commonly used one: the probability level of the “null hypothesis” is 0.05. Instead of the “null hypothesis” probabilities, the table can be called the table for the probability of 0.95 of the significance of the factor's influence. Increasing the probability level requires a higher F value of the table for comparison.

The value of F table also depends on the number of degrees of freedom of the two dispersions being compared. If the number of degrees of freedom tends to infinity, then F table tends to unity.

The table of F table values is constructed as follows: the columns of the table indicate the degrees of freedom of variation for the larger dispersion, and the rows indicate the degrees of freedom for the smaller (within-group) dispersion. The value of F is found at the intersection of the column and row of the corresponding degrees of freedom of variation.

So, in our example, F f = 21.3/3.8 = 5.6. The tabulated value of F table for a probability of 0.95 and degrees of freedom, respectively equal to 3 and 28, F table = 2.95.

The value of F f obtained experimentally exceeds the theoretical value even for a probability of 0.99. Consequently, the experience with a probability of more than 0.99 proves the influence of the studied factor on the yield, i.e. the experience can be considered reliable, proven, and therefore the sowing time has a significant impact on the yield of wheat. The optimal sowing period should be considered the period from May 10 to 15, since it was during this sowing period that the best yield results were obtained.

We examined the method of analysis of variance when grouping by one characteristic and randomly distributing replicates within the group. However, it often happens that the experimental plot has some differences in soil fertility, etc. Therefore, a situation may arise that a larger number of plots of one of the options will fall on the best part, and its indicators will be overestimated, and of the other option - by the worst part, and the results in this case will naturally be worse, that is, underestimated.

To exclude variation that is caused by reasons not related to the experiment, it is necessary to isolate the variance calculated from replicates (blocks) from the within-group (residual) variance.

The total sum of squared deviations is divided in this case into 3 components:

D o = D m + D repeat + D rest. (1.33)

For our example, the sum of squared deviations caused by repetitions will be equal to:

Therefore, the actual random sum of squared deviations will be equal to:

D rest = D in – D repeat; D rest = 106 – 44 = 62.

For the residual dispersion, the number of degrees of freedom will be equal to 28 – 7 = 21. The results of the analysis of variance are presented in table. 9.

Table 9

Since the actual values of the F-criterion for a probability of 0.95 exceed the tabulated ones, the influence of sowing dates and repetitions on wheat yield should be considered significant. The considered method of constructing an experiment, when the site is preliminarily divided into blocks with relatively aligned conditions, and the tested options are distributed within the block in a random order, is called the method of randomized blocks.

Using analysis of variance, you can study the influence of not only one factor on the result, but two or more. Analysis of variance in this case will be called multivariate analysis of variance .

Two-way ANOVA differs from two single-factor ones in that it can answer the following questions:

1. 1what is the effect of both factors together?

2. What is the role of the combination of these factors?

Let's consider an analysis of variance of the experiment, in which it is necessary to identify the influence of not only sowing dates, but also varieties on wheat yield (Table 10).

Table 10. Experimental data on the influence of sowing dates and varieties on wheat yield

is the sum of squared deviations of individual values from the overall average.

Variation in the joint influence of sowing time and variety

is the sum of the squared deviations of the subgroup means from the overall mean, weighted by the number of replications, i.e. by 4.

Calculation of variation based on the influence of sowing time only:

Residual variation is defined as the difference between the total variation and the variation in the joint influence of the studied factors:

D rest = D o – D ps = 170 – 96 = 74.

All calculations can be presented in the form of a table (Table 11).

Table 11. Results of analysis of variance

The results of the analysis of variance show that the influence of the studied factors, i.e., sowing time and variety, on wheat yield is significant, since the actual F-criteria for each of the factors significantly exceed the tabulated ones found for the corresponding degrees of freedom, and at the same time with a fairly high probability (p = 0.99). The influence of a combination of factors in this case is absent, since the factors are independent of each other.

The analysis of the influence of three factors on the result is carried out according to the same principle as for two factors, only in this case there will be three variances for the factors and four variances for the combination of factors. With an increase in the number of factors, the volume of calculation work increases sharply and, in addition, it becomes difficult to arrange the initial information in a combination table. Therefore, it is hardly advisable to study the influence of many factors on the result using analysis of variance; it is better to take a smaller number, but choose the most significant factors from the point of view of economic analysis.

Often the researcher has to deal with so-called disproportionate dispersion complexes, i.e. those in which the proportionality of the numbers of variants is not observed.

In such complexes, the variation in the total effect of factors is not equal to the sum of the variation among factors and the variation in the combination of factors. It differs by an amount depending on the degree of connections between individual factors arising as a result of a violation of proportionality.

In this case, difficulties arise in determining the degree of influence of each factor, since the sum of individual influences is not equal to the total influence.

One of the ways to reduce a disproportionate complex to a single structure is to replace it with a proportional complex, in which the frequencies are averaged over groups. When such a replacement is made, the problem is solved according to the principles of proportional complexes.

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

Types and criteria of analysis of variance

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

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

Factors

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

Analysis of variance problems

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

Example 1

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

Example 2

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

Example 3

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

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

Univariate analysis

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

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

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

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

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

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

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

Two-factor analysis

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

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

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

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

Multivariate analysis

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

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

Multivariate analysis of variance with repeated experiments

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

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

General idea of analysis of variance

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

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

Examples of quality factors for different types of dosage forms

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

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

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

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

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

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

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

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

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

The main indicators of international rating systems are indices:

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

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

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

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

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

One-way analysis of variance.

Concept and models of variance analysis.

Topic 13. Analysis of variance

Lecture 1. Questions:

Analysis of variance, as a research method, appeared in the works of R. Fischer (1918-1935) in connection with research in agriculture to identify the conditions under which the tested variety of agricultural crop produces the maximum yield. Analysis of variance was further developed in the works of Yeats. Analysis of variance allows us to answer the question of whether certain factors have a significant influence on the variability of a factor, the values of which can be obtained as a result of experience. When testing statistical hypotheses, random variations in the factors being studied are assumed. In analysis of variance, one or more factors are changed in a given way, and these changes can affect the results of observations. The study of such influence is the purpose of analysis of variance.

Currently, there is an increasingly widespread use of variance analysis in economics, sociology, biology, etc., especially after the advent of software that eliminated the problems of the cumbersomeness of statistical calculations.

In practical activities, in various fields of science, we are often faced with the need to evaluate the influence of various factors on certain indicators. Often these factors are of a qualitative nature (for example, a qualitative factor influencing the economic effect may be the introduction of a new production management system) and then variance analysis acquires particular value, since it becomes the only statistical method of research that gives such an assessment.

Analysis of variance makes it possible to determine whether one or another of the factors under consideration has a significant impact on the variability of a trait, as well as to quantify the “specific weight” of each source of variability in their totality. But analysis of variance allows us to give a positive answer only about the presence of a significant influence, otherwise the question remains open and requires additional research (most often, an increase in the number of experiments).

The following terms are used in analysis of variance.

Factor (X) is something that we believe should influence the result (resultative attribute) Y.

Factor level (or method of processing, sometimes literally, for example - method of tillage) - values (X, i = 1.2,...I) that the factor can take.

Response – the value of the measured characteristic (result value Y).

The ANOVA technique varies depending on the number of independent factors being studied. If the factors causing variability in the average value of a characteristic belong to one source, then we have a simple grouping, or one-factor analysis of variance and then, accordingly, a double grouping - two-factor analysis of variance, three-factor analysis of variance, ..., m-factor. Factors in multivariate analysis are usually denoted by Latin letters: A, B, C, etc.

The task of variance analysis is to study the influence of certain factors (or levels of factors) on the variability of the average values of observed random variables.

The essence of variance analysis. Analysis of variance consists of isolating and evaluating individual factors that cause variability. For this purpose, the total variance of the observed partial population (total variance of the trait), caused by all sources of variability, is decomposed into variance components generated by independent factors. Each of these components provides an estimate of the variance , ,..., caused by a particular source of variability, in the overall population. To test the significance of these component variance estimates, they are compared with the total variance in the population (Fisher's test).

For example, in two-factor analysis we get a decomposition of the form:

Total variance of the studied trait C;

The share of variance caused by the influence of factor A;

The share of variance caused by the influence of factor B;

The proportion of variance caused by the interaction of factors A and B;

The share of variance caused by unaccounted random causes (random variance);

In analysis of variance, the hypothesis is considered: H 0 - none of the factors under consideration has an effect on the variability of the trait. The significance of each variance estimate is checked by the value of its ratio to the random variance estimate and compared with the corresponding critical value, at significance level a, using tables of critical values of the Fisher-Snedecor F distribution (Appendix 4). Hypothesis H 0 regarding one or another source of variability is rejected if F calculated. >F cr. (for example, for factor B: S B 2 /S ε 2 >F cr.).

Variance analysis considers experiments of 3 types:

a) experiments in which all factors have systematic (fixed) levels;

b) experiments in which all factors have random levels;

c) experiments in which there are factors that have random levels, as well as factors that have fixed levels.

Cases a), b), c) correspond to three models that are considered in analysis of variance.

The input data for analysis of variance is usually presented in the form of the following table:

Observation number j	Factor levels
A 1	A 2	…	A r
	X 11	X 21	…	X p1
	X 12	X 22	…	Xp2
	X 13	X 23	…	X p3
.	.	.	…	…
.	.	.	…	…
.	.	.	…	…
n	X 1n	X2n	…	Xpn
RESULTS

Consider a single factor that takes p different levels, and assume that at each level n observations are made, giving N=np observations. (We will limit ourselves to considering the first model of variance analysis - all factors have fixed levels.)

Let the results be presented in the form X ij (i=1,2…,р; j=1,2,…,n).

It is assumed that for each level of n observations there is an average, which is equal to the sum of the overall average and its variation due to the selected level:

where m is the overall average;

A i - effect caused by the i – m level of the factor;

e ij – variation of results within an individual factor level. The term e ij takes into account all uncontrollable factors.

Let observations at a fixed factor level be normally distributed around the mean m + A i with a common variance s 2 .

Then (the dot instead of the index denotes the averaging of the corresponding observations over this index):

A.X ij – X.. = (X i . – X..) + (X ij – X i .). (12.3)

After squaring both sides of the equation and summing over i and j, we get:

since, but

Otherwise, the sum of squares can be written: S = S 1 + S 2. The value of S 1 is calculated from the deviations of p averages from the overall average X.., therefore S 1 has (p-1) degrees of freedom. The value of S 2 is calculated from the deviations of N observations from p sample means and, therefore, has N-р = np - p=p(n-1) degrees of freedom. S has (N-1) degrees of freedom. Based on the calculation results, a variance analysis table is constructed.

ANOVA table

If the hypothesis that the influence of all levels is equal is true, then both M 1 and M 2 (mean squares) will be unbiased estimates of s 2. This means that the hypothesis can be tested by calculating the ratio (M 1 / M 2) and comparing it with F cr. with ν 1 = (p-1) and ν 2 = (N-p) degrees of freedom.

If F calculated >F cr. , then the hypothesis about the insignificant influence of factor A on the result of observations is not accepted.

To assess the significance of differences at F calc. F table calculate:

a) experimental error

b) error of the difference of means

c) the smallest significant difference

Comparing the difference in average values for the options with the NSR, they conclude that the differences in the level of averages are significant.

Comment. The use of analysis of variance assumes that:

2) D(ε ij)=σ 2 = const,

3) ε ij → N (0, σ) or x ij → N (a, σ).

Analytical statistician

7.1 Analysis of variance. 2

In this version of the method, different samples of subjects are exposed to the influence of each of the gradations. There must be at least gradations of the factor three.

Example 1. Three different groups of six subjects were given lists of ten words. The words were presented to the first group at a low speed - 1 word per 5 seconds, to the second group at an average speed - 1 word per 2 seconds, and to the third group at a high speed - 1 word per second. Reproduction performance was predicted to depend on the speed of word presentation. The results are presented in table. 1.

Table 1. Number of words reproduced (by J. Greene, M D "Olivera, 1989, p. 99)

Subject No.	Group 1 low speed	Group 2 medium speed	Group 3 high speed






amounts
average	7,17	6,17	4,00
Total amount

Univariate analysis of variance allows you to test the hypotheses:

H 0 : differences in word production volume between groups are no more pronounced than random differences inside each group

H 1 : Differences in word production volume between groups are more pronounced than random differences inside each group.

Sequence of operations in one-way analysis of variance for unrelated samples:

1. let's count SS fact- variability of the trait due to the action of the factor under study. Common designation SS - abbreviation for "sum of squares" ( sum of squares ). This abbreviation is most often used in translated sources (see, for example: Glass J., Stanley J., 1976).

,(1)

where T c is the sum of individual values for each condition. For our example, 43, 37, 24 (see Table 1);

с – number of conditions (gradations) of the factor (=3);

n – number of subjects in each group (=6);

N – total number of individual values (=18);

Square of the total sum of individual values (=104 2 =10816)

Note the difference between , in which all individual values are first squared and then summed, and , where individual values are first summed to obtain a total sum, and then this sum is squared.

Using formula (1), having calculated the actual variability of the trait, we obtain:

2. let's count SS general– general variability of the trait:

(2)

3. calculate the random (residual) valueSS sl, caused by unaccounted factors:

(3)

4.number of degrees of freedom equals:

=3-1=2(4)

5."middle square" or the average value of the corresponding sums of squares SS is equal to:

(5)

6.meaning criterion statistics F em calculate using the formula:

(6)

For our example we have : F em =15.72/2.11=7.45

7.define F crit according to statistical tables Applications 3 for df 1 =k 1 =2 and df 2 =k 2 =15 the table value of the statistics is 3.68

8. if F em< F critical, then the null hypothesis is accepted, otherwise the alternative hypothesis is accepted. For our example F em> F crit (7.45>3.68), therefore p

Conclusion:differences in word recall between groups are more pronounced than random differences within each group (p<0,05). Т.о. скорость предъявления слов влияет на объем их воспроизведения.

7.1.2 Analysis of variance for related samples

The method of variance analysis for related samples is used in cases where the influence of different gradations of a factor or different conditions on the same sample of subjects. There must be at least gradations of the factor three.

In this case, differences between subjects are a possible independent source of differences. One-way ANOVA for related sampleswill allow us to determine what outweighs - the tendency expressed by the factor change curve, or individual differences between subjects. The factor of individual differences may be more significant than the factor of changes in experimental conditions.

Example 2.A group of 5 subjects was examined using three experimental tasks aimed at studying intellectual perseverance (Sidorenko E.V., 1984). Each subject was individually presented with three identical anagrams in succession: a four-letter, a five-letter, and a six-letter. Is it possible to assume that the length factor of an anagram influences the duration of attempts to solve it?

Table 2. Duration of solving anagrams (sec)

Subject code	Condition 1. four-letter anagram	Condition 2. Five-letter anagram	Condition 3. six-letter anagram	Amounts by subjects





amounts		1244		1342

Let's formulate hypotheses. In this case there are two sets of hypotheses.

Set A.

H 0 (A): Differences in the duration of attempts to solve anagrams of different lengths are no more pronounced than differences due to random reasons.

H 1 (A): Differences in the duration of attempts to solve anagrams of different lengths are more pronounced than differences due to random reasons.

Set B.

N about (B): Individual differences between subjects are no more pronounced than differences due to random causes.

H 1 (B): Individual differences between subjects are more pronounced than differences due to random reasons.

Sequence of operations in one-way analysis of variance for related samples:

1. let's count SS fact- variability of the trait due to the action of the factor under study according to formula (1).

where T c is the sum of individual values for each of the conditions (columns). For our example, 51, 1244, 47 (see Table 2); с – number of conditions (gradations) of the factor (=3); n – number of subjects in each group (=5); N – total number of individual values (=15); - square of the total sum of individual values (=1342 2)

2. let's count SS isp- variability of the sign due to the individual values of the subjects.

Where T and is the sum of individual values for each subject. For our example, 247, 631, 100, 181, 183 (see Table 2); с – number of conditions (gradations) of the factor (=3); N – total number of individual values (=15);

3. let's count SS general– general variability of the trait according to formula (2):

4. calculate the random (residual) valueSS sl, caused by unaccounted factors according to formula (3):

5. number of degrees of freedom equals (4):

; ; ;

6. "middle square" or mathematical expectation of the sum of squares, the average value of the corresponding sums of squares SS is equal to (5):

;

7. criterion statistic value F em calculate using formula (6):

;

8. Let's determine F crit from the statistical tables of Appendix 3 for df 1 =k 1 =2 and df 2 =k 2 =8 table value of statistics F crit_fact =4.46, and for df 3 =k 3 =4 and df 2 =k 2 = 8 F crit_exp =3.84

9. F em_fact> F critical_fact (6.872>4.46), therefore p an alternative hypothesis is accepted.

10. F em_use < F крит_исп (1,054<3,84), следовательно пThe null hypothesis is accepted.

Conclusion:differences in the volume of word reproduction in different conditions are more pronounced than differences due to random reasons (p<0,05).Индивидуальные различия между испытуемыми являются не более выраженными, чем различия, обусловленные случайными причинами.

7.2 Correlation analysis

7.2.1 Concept of correlation

A researcher is often interested in how two or more variables are related to each other in one or more samples being studied. For example, can students with high levels of anxiety demonstrate stable academic achievements, or is the length of time a teacher works in a school related to the size of his salary, or what is more related to the level of mental development of students - their performance in mathematics or literature, etc. .?

This kind of dependence between variables is called correlation, or correlation. Correlation connection- this is a coordinated change in two characteristics, reflecting the fact that the variability of one characteristic is in accordance with the variability of the other.

It is known, for example, that on average there is a positive relationship between the height of people and their weight, and such that the greater the height, the greater the person’s weight. However, there are exceptions to this rule, when relatively short people are overweight, and, conversely, asthenic people with high stature have low weight. The reason for such exceptions is that each biological, physiological or psychological sign is determined by the influence of many factors: environmental, genetic, social, environmental, etc.

Correlation connections are probabilistic changes that can only be studied on representative samples using the methods of mathematical statistics. “Both terms,” writes E.V. Sidorenko, - correlation connection and correlation dependence- are often used as synonyms. Dependency implies influence, connection - any coordinated changes that can be explained by hundreds of reasons. Correlation connections cannot be considered as evidence of a cause-and-effect relationship; they only indicate that changes in one characteristic are usually accompanied by certain changes in another.

Correlation dependence - these are changes that introduce the values of one characteristic into the probability of the appearance of different values of another characteristic (E.V. Sidorenko, 2000).

The task of correlation analysis comes down to establishing the direction (positive or negative) and form (linear, nonlinear) of the relationship between varying characteristics, measuring its closeness, and, finally, checking the level of significance of the obtained correlation coefficients.

Correlations vary in form, direction and degree (strength).

By shapethe correlation relationship can be linear or curvilinear. For example, the relationship between the number of training sessions on the simulator and the number of correctly solved problems in the control session may be straightforward. For example, the relationship between the level of motivation and the effectiveness of a task may be curvilinear (see Fig. 1). As motivation increases, the effectiveness of completing a task first increases, then the optimal level of motivation is achieved, which corresponds to the maximum effectiveness of completing the task; A further increase in motivation is accompanied by a decrease in efficiency.

Fig.1. The relationship between the effectiveness of solving a problem

and the strength of the motivational tendency (according to J. W. A t k in son, 1974, p 200)

Towardsthe correlation can be positive (“direct”) and negative (“inverse”). With a positive linear correlation, higher values of one characteristic correspond to higher values of another, and lower values of one characteristic correspond to low values of another. With a negative correlation, the relationships are reversed. With a positive correlation, the correlation coefficient has a positive sign, for exampler =+0.207, with a negative correlation - a negative sign, for exampler = -0.207.

Degree, strength or tightness correlation connection is determined by the value of the correlation coefficient.

The strength of the connection does not depend on its direction and is determined by the absolute value of the correlation coefficient.

Maximum possible absolute value of the correlation coefficientr =1.00; minimum r =0.00.

General classification of correlations (according to Ivanter E.V., Korosov A.V., 1992):

strong, or tight with correlation coefficientr >0.70;

average at 0,50< r<0,69 ;

moderate at 0,30< r<0,49 ;

weak at 0,20< r<0,29 ;

very weak at r<0,19 .

Variables X and Y can be measured on different scales, this is what determines the choice of the appropriate correlation coefficient (see Table 3):

Table 3. Use of correlation coefficient depending on the type of variables

Scale type		Measure of connection
Variable X	Variable Y
Interval or relationship	Interval or relationship	Pearson coefficient
	Rank, interval or ratio	Spearman coefficient
Ranked	Ranked	Kendall coefficient
Dichotomous	Dichotomous	Coefficient "j"
Dichotomous	Ranked	Rank-biserial
Dichotomous	Interval or relationship	Biserial

7.2.2 Pearson correlation coefficient

The term “correlation” was introduced into science by the outstanding English naturalist Francis Galton in 1886. However, the exact formula for calculating the correlation coefficient was developed by his student Karl Pearson.

The coefficient characterizes the presence of only a linear relationship between characteristics, usually denoted by the symbols X and Y. The formula for calculating the correlation coefficient is constructed in such a way that if the relationship between characteristics is linear, the Pearson coefficient accurately establishes the closeness of this relationship. Therefore, it is also called the Pearson linear correlation coefficient. If the connection between the variables X and Y is not linear, then Pearson proposed the so-called correlation relation to assess the closeness of this connection.

The value of the Pearson linear correlation coefficient cannot exceed +1 and be less than -1. These two numbers +1 and -1 are the boundaries for the correlation coefficient. When the calculation results in a value greater than +1 or less than -1, an error has therefore occurred in the calculations.

The sign of the correlation coefficient is very important for interpreting the resulting relationship. Let us emphasize once again that if the sign of the linear correlation coefficient is plus, then the relationship between the correlated features is such that a larger value of one feature (variable) corresponds to a larger value of another feature (another variable). In other words, if one indicator (variable) increases, then the other indicator (variable) increases accordingly. This dependence is called a directly proportional dependence.

If a minus sign is received, then a larger value of one characteristic corresponds to a smaller value of another. In other words, if there is a minus sign, an increase in one variable (sign, value) corresponds to a decrease in another variable. This dependence is called inversely proportional dependence.

In general, the formula for calculating the correlation coefficient is:

(7)

Where X i- values taken in sample X,

y i- values accepted in sample Y;

Average for X, - average for Y.

The calculation of the Pearson correlation coefficient assumes that variables X and Y are distributed Fine.

Formula (7) contains the quantity when divided by n (the number of values of the variable X or Y) it is called covariance. Formula (7) also assumes that when calculating correlation coefficients, the number of values of the variable X is equal to the number of values of the variable Y.

Number of degrees of freedom k = n -2.

Example 3. 1 0 schoolchildren were given tests for visual-figurative and verbal thinking. The average time for solving test tasks was measured in seconds. The researcher is interested in the question: is there a relationship between the time it takes to solve these problems? Variable X denotes the average time for solving visual-figurative tasks, and variable Y denotes the average time for solving verbal test tasks.

Solution. Let us present the initial data in the form of Table 4, which contains additional columns necessary for calculation using formula (7).

Table 4

No. of subjects	x		x i -	(x i - ) 2	y i -	(y i -) 2
			16,7	278,89		51,84	120,24
				13,69	17,2	295,84	63,64
				7,29		51,84	19,44
				68,89		14,44	31,54
				59,29		7,84	21,56
				0,49		46,24	4,76
				10,89		17,64	13,86
				10,89		51,84	23,76
				68,89	10,8	116,64	89,64
				68,89	18,8	353,44	156,04
Sum	357	242		588,1		1007,6	416,6
Average	35,7	24,2

We calculate the empirical value of the correlation coefficient using formula (7):

We determine the critical values for the obtained correlation coefficient according to the table in Appendix 3. When finding the critical values for the calculated Pearson linear correlation coefficient, the number of degrees of freedom is calculated as k = n – 2 = 8.

k crit = 0.72 > 0.54, therefore, hypothesis H 1 is rejected and hypothesis is accepted H 0 , in other words, the connection between the time of solving visual-figurative and verbal test tasks has not been proven.

7.3 Regression analysis

This is a group of methods aimed at identifying and mathematically expressing those changes and dependencies that take place in a system of random variables. If such a system models a pedagogical one, then, consequently, through regression analysis, psychological and pedagogical phenomena and the dependencies between them are identified and mathematically expressed. The characteristics of these phenomena are measured on different scales, which imposes restrictions on the ways of mathematical expression of changes and dependencies that are studied by the teacher-researcher.

Regression analysis methods are designed primarily for the case of a stable normal distribution, in which changes from trial to trial appear only in the form of independent trials.

Various formal problems of regression analysis are identified. They can be simple or complex in terms of formulation, mathematical means and labor intensity. Let us list and consider with examples those that seem to be the main ones.

The first task is identify the fact of variability the phenomenon being studied under certain, but not always clearly fixed, conditions. In the previous lecture, we already solved this problem using parametric and nonparametric criteria.

Second task - identify a trend as a periodic change in a characteristic. This feature itself may or may not be dependent on the condition variable (it may depend on conditions unknown or uncontrollable by the researcher). But this is not important for the task under consideration, which is limited only to identifying the trend and its features.

Testing hypotheses about the absence or presence of a trend can be performed using the Abbe criterion . Abbe criterion designed to test hypotheses about the equality of average values established for 4

The empirical value of the Abbe criterion is calculated by the formula:

(8)

where is the arithmetic mean of the sample;

P– number of values in the sample.

According to the criterion, the hypothesis of equality of means is rejected (the alternative hypothesis is accepted) if the value of the statistic is . The tabular (critical) value of statistics is determined from the table for Abbe’s q-criterion, which, with abbreviations, is borrowed from the book by L.N. Bolysheva and N.V. Smirnova (see Appendix 3).

Such quantities for which the Abbe criterion is applicable can be sample shares or percentages, arithmetic averages and other statistics of sample distributions if they are close to normal (or previously normalized). Therefore, the Abbe criterion can find wide application in psychological and pedagogical research. Let's consider an example of identifying a trend using the Abbe criterion.

Example 4.In table 5 shows the dynamics of the percentage of students IV course, who passed exams in winter sessions with “excellence” during 10 years of work at one of the university faculties. It is required to establish whether there is a tendency to increase academic performance.

Table 5. Dynamics of the percentage of fourth-year excellent students over 10 years of work of the faculty

Academic year
1995-96	10,8
1996-97	16,4
1997-98	17,4
1998-99	22,0
1999-00	23,0
2000-01	21,5
2001-02	26,1
2002-03	17,2
2003-04	27,5
2004-05	33,0

As null We test the hypothesis about the absence of a trend, i.e. about the equality of percentages.

We average the percentages given in table. 5, we find that =21.5. We calculate the differences between subsequent and previous values in the sample, square them and sum them up:

Similarly calculates the denominator in formula (8), summing the squares of the differences between each measurement and the arithmetic mean:

Now using formula (8) we get:

In the Abbe criterion table from Appendix 3, we find that with n = 10 and a significance level of 0.05, the critical value is greater than the 0.41 we obtained, therefore the hypothesis about the equality of the percentage of “excellent students” must be rejected, and we can accept the alternative hypothesis about the presence of a trend .

The third task is identifying a pattern expressed in the form of a correlation equation (regression).

Example 5.Estonian researcher J. Mikk, studying the difficulties of understanding text, established a “readability formula”, which is a multiple linear regression:

Assessing the difficulty of understanding the text,

where x 1 is the length of independent sentences in the number of printed characters,

x 2 - percentage of different unfamiliar words,

x 3 - abstractness of repeating concepts expressed by nouns .

Comparing the regression coefficients expressing the degree of influence of factors, one can see that the difficulty of understanding a text is determined primarily by its abstractness. The difficulty of understanding the text depends half as much (0.27) on the number of unfamiliar words and practically does not depend at all on the length of the sentence.