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Analysis of variance

Introductory overview

In this section, we will review the basic methods, assumptions, and terminology of ANOVA.

Note that in the English literature, analysis of variance is usually called the analysis of variation. Therefore, for brevity, below we will sometimes use the term ANOVA (An alysis o f va riation) for conventional ANOVA and the term MANOVA for multivariate analysis of variance. In this section, we will sequentially consider the main ideas of the analysis of variance ( ANOVA), analysis of covariance ( ANCOVA), multivariate analysis of variance ( MANOVA) and multivariate covariance analysis ( MANCOVA). After a brief discussion of the merits of contrast analysis and post hoc tests, let's look at the assumptions on which ANOVA methods are based. Towards the end of this section, the advantages of the multivariate approach for repeated measures analysis are explained over the traditional one-dimensional approach.

Key Ideas

The purpose of the analysis of variance. The main purpose of the analysis of variance is to study the significance of the difference between the means. Chapter (Chapter 8) provides a brief introduction to statistical significance testing. If you are just comparing the means of two samples, analysis of variance will give the same result as normal analysis. t- criterion for independent samples (if two independent groups of objects or observations are compared), or t- criterion for dependent samples (if two variables are compared on the same set of objects or observations). If you are not familiar with these criteria, we recommend that you refer to the introductory overview of the chapter (Chapter 9).

Where did the name come from Analysis of variance? It may seem strange that the procedure for comparing means is called analysis of variance. In fact, this is due to the fact that when we examine the statistical significance of the difference between the means, we are actually analyzing the variances.

Splitting the sum of squares

For a sample size of n, the sample variance is calculated as the sum of the squared deviations from the sample mean divided by n-1 (sample size minus one). Thus, for a fixed sample size n, the variance is a function of the sum of squares (deviations), denoted, for brevity, SS(from English Sum of Squares - Sum of Squares). The analysis of variance is based on the division (or splitting) of the variance into parts. Consider the following data set:

The means of the two groups are significantly different (2 and 6, respectively). Sum of squared deviations inside of each group is 2. Adding them together, we get 4. If we now repeat these calculations excluding group membership, that is, if we calculate SS based on the combined mean of the two samples, we get 28. In other words, the variance (sum of squares) based on within-group variability results in much smaller values ​​than when calculated based on total variability (relative to the overall mean). The reason for this is obviously the significant difference between the means, and this difference between the means explains the existing difference between the sums of squares. Indeed, if we use the module Analysis of variance, the following results will be obtained:

As can be seen from the table, the total sum of squares SS=28 divided into the sum of squares due to intragroup variability ( 2+2=4 ; see second row of the table) and the sum of squares due to the difference in the mean values. (28-(2+2)=24; see the first line of the table).

SS mistakes andSS effect. Intragroup variability ( SS) is usually called the variance errors. This means that it usually cannot be predicted or explained when an experiment is carried out. On the other side, SS effect(or intergroup variability) can be explained by the difference between the means in the studied groups. In other words, belonging to a certain group explains intergroup variability, because we know that these groups have different means.

Significance check. The main ideas of testing for statistical significance are discussed in the chapter Elementary concepts of statistics(Chapter 8). The same chapter explains the reasons why many tests use the ratio of explained and unexplained variance. An example of this use is the analysis of variance itself. Significance testing in ANOVA is based on comparing the variance due to between-group variation (called mean square effect or MSEffect) and dispersion due to within-group spread (called mean square error or MSerror). If the null hypothesis is true (equality of the means in the two populations), then we can expect a relatively small difference in the sample means due to random variability. Therefore, under the null hypothesis, the intra-group variance will practically coincide with the total variance calculated without taking into account the group membership. The resulting within-group variances can be compared using F- test that checks whether the ratio of variances is significantly greater than 1. In the above example, F- The test shows that the difference between the means is statistically significant.

Basic logic of ANOVA. Summing up, we can say that the purpose of analysis of variance is to test the statistical significance of the difference between the means (for groups or variables). This check is carried out using analysis of variance, i.e. by splitting the total variance (variation) into parts, one of which is due to random error (i.e., intragroup variability), and the second is associated with the difference in mean values. The last component of the variance is then used to analyze the statistical significance of the difference between the means. If this difference is significant, the null hypothesis is rejected and the alternative hypothesis that there is a difference between the means is accepted.

Dependent and independent variables. Variables whose values ​​are determined by measurements during an experiment (for example, a score scored on a test) are called dependent variables. Variables that can be manipulated in an experiment (for example, training methods or other criteria that allow you to divide observations into groups) are called factors or independent variables. These concepts are described in more detail in the chapter Elementary concepts of statistics(Chapter 8).

Multivariate analysis of variance

In the simple example above, you could immediately calculate the independent sample t-test using the appropriate module option Basic statistics and tables. The results obtained, of course, coincide with the results of the analysis of variance. However, the analysis of variance contains flexible and powerful technical tools that can be used for much more complex studies.

Lots of factors. The world is inherently complex and multidimensional. Situations where some phenomenon is completely described by one variable are extremely rare. For example, if we are trying to learn how to grow large tomatoes, we should consider factors related to the genetic structure of plants, soil type, light, temperature, etc. Thus, when conducting a typical experiment, you have to deal with a large number of factors. The main reason why using ANOVA is preferable to re-comparing two samples at different levels of factors using t- criterion is that the analysis of variance is more effective and, for small samples, more informative.

Factor management. Let's assume that in the example of two-sample analysis discussed above, we add one more factor, for example, Floor- Gender. Let each group consist of 3 men and 3 women. The design of this experiment can be presented in the form of a 2 by 2 table:

	Experiment. Group 1	Experiment. Group 2
Men	2	6
	3	7
	1	5
Average	2	6
Women	4	8
	5	9
	3	7
Average	4	8

Before doing the calculations, you can see that in this example the total variance has at least three sources:

(1) random error (within group variance),

(2) variability associated with membership in the experimental group, and

(3) variability due to the gender of the observed objects.

(Note that there is another possible source of variability - interaction of factors, which we will discuss later). What happens if we don't include floor –gender as a factor in the analysis and calculate the usual t-criterion? If we calculate sums of squares, ignoring floor -gender(i.e., combining objects of different sexes into one group when calculating the within-group variance, while obtaining the sum of squares for each group equal to SS=10, and the total sum of squares SS= 10+10 = 20), then we get a larger value of intragroup dispersion than with more accurate analysis with additional division into subgroups according to semi- gender(in this case, the intragroup means will be equal to 2, and the total intragroup sum of squares will be equal to SS = 2+2+2+2 = 8). This difference is due to the fact that the mean value for men - males less than the average for women -female, and this difference in means increases the total within-group variability if sex is not taken into account. Controlling the error variance increases the sensitivity (power) of the test.

This example shows another advantage of analysis of variance over conventional analysis. t-criterion for two samples. Analysis of variance allows you to study each factor by controlling the values ​​of other factors. This, in fact, is the main reason for its greater statistical power (smaller sample sizes are required to obtain meaningful results). For this reason, analysis of variance, even on small samples, gives statistically more significant results than a simple one. t- criterion.

Interaction effects

There is another advantage of using ANOVA over conventional analysis. t- criterion: analysis of variance allows you to detect interaction between the factors and therefore allows more complex models to be studied. To illustrate, consider another example.

Main effects, pairwise (two-factor) interactions. Let us assume that there are two groups of students, and psychologically the students of the first group are tuned in to the fulfillment of the assigned tasks and are more purposeful than the students of the second group, which consists of lazier students. Let's divide each group randomly in half and offer one half of each group a difficult task, and the other an easy one. After that, we measure how hard students work on these tasks. The averages for this (fictitious) study are shown in the table:

What conclusion can be drawn from these results? Is it possible to conclude that: (1) students work harder on a difficult task; (2) do motivated students work harder than lazy ones? None of these statements reflect the essence of the systematic nature of the averages given in the table. Analyzing the results, it would be more correct to say that only motivated students work harder on complex tasks, while only lazy students work harder on easy tasks. In other words, the nature of the students and the complexity of the task interacting each other affect the amount of effort required. That's an example pair interaction between the nature of students and the complexity of the task. Note that statements 1 and 2 describe main effects.

Interactions of higher orders. While pairwise interactions are relatively easy to explain, higher-order interactions are much more difficult to explain. Let us imagine that in the example considered above, one more factor is introduced floor -Gender and we got the following table of averages:

What conclusions can now be drawn from the results obtained? Mean plots make it easy to interpret complex effects. The analysis of variance module allows you to build these graphs with almost one click.

The image in the graphs below represents the three-way interaction under study.

Looking at the graphs, we can tell that there is an interaction between the nature and difficulty of the test for women: motivated women work harder on a difficult task than on an easy one. In men, the same interaction is reversed. It can be seen that the description of the interaction between factors becomes more confusing.

General way of describing interactions. In the general case, the interaction between factors is described as a change in one effect under the influence of another. In the example discussed above, two-factor interaction can be described as a change in the main effect of the factor characterizing the complexity of the task, under the influence of the factor describing the character of the student. For the interaction of the three factors from the previous paragraph, we can say that the interaction of two factors (the complexity of the task and the character of the student) changes under the influence of gender – Gender. If the interaction of four factors is studied, we can say that the interaction of three factors changes under the influence of the fourth factor, i.e. there are different types of interactions at different levels of the fourth factor. It turned out that in many areas the interaction of five or even more factors is not unusual.

Complex plans

Intergroup and intragroup plans (remeasurement plans)

When comparing two different groups, one usually uses t- criterion for independent samples (from module Basic statistics and tables). When two variables are compared on the same set of objects (observations), it is used t-criterion for dependent samples. For analysis of variance, it is also important whether or not the samples are dependent. If there are repeated measurements of the same variables (at different conditions or at different times) for the same objects, then they say about the presence repeated measures factor(also called an intragroup factor since the within-group sum of squares is calculated to evaluate its significance). If different groups of objects are compared (for example, men and women, three strains of bacteria, etc.), then the difference between the groups is described intergroup factor. The methods for calculating the significance criteria for the two types of factors described are different, but their general logic and interpretation are the same.

Inter- and intra-group plans. In many cases, the experiment requires the inclusion of both an between-group factor and a repeated measures factor in the design. For example, the math skills of female and male students are measured (where floor -Gender-intergroup factor) at the beginning and at the end of the semester. The two dimensions of each student's skills form the within-group factor (repeated measures factor). The interpretation of the main effects and interactions for between-group and repeated measures factors is the same, and both types of factors can obviously interact with each other (for example, women gain skills during the semester, and men lose them).

Incomplete (nested) plans

In many cases, the interaction effect can be neglected. This occurs either when it is known that there is no interaction effect in the population, or when the implementation of the full factorial plan is impossible. For example, the effect of four fuel additives on fuel consumption is being studied. Four cars and four drivers are selected. Full factorial the experiment requires that each combination: supplement, driver, car, appear at least once. This requires at least 4 x 4 x 4 = 64 test groups, which is too time consuming. In addition, there is hardly any interaction between the driver and the fuel additive. With this in mind, you can use the plan latin squares, which contains only 16 groups of tests (four additives are designated by the letters A, B, C and D):

Latin squares are described in most experimental design books (eg Hays, 1988; Lindman, 1974; Milliken and Johnson, 1984; Winer, 1962) and will not be discussed in detail here. Note that Latin squares are Notnfull plans that do not include all combinations of factor levels. For example, driver 1 drives car 1 with additive A only, driver 3 drives car 1 with additive C only. Factor levels additives ( A, B, C and D) nested in table cells automobile x driver - like eggs in a nest. This mnemonic rule is useful for understanding the nature nested or nested plans. Module Analysis of variance provides simple ways analysis of plans of this type.

Covariance Analysis

main idea

In chapter Key Ideas there was a brief discussion of the idea of ​​controlling factors and how the inclusion of additive factors can reduce the sum of squared errors and increase the statistical power of the design. All this can be extended to variables with a continuous set of values. When such continuous variables are included as factors in the design, they are called covariates.

Fixed covariates

Suppose that we are comparing the mathematical skills of two groups of students who were taught from two different textbooks. Let's also assume that we have intelligence quotient (IQ) data for each student. We can assume that IQ is related to math skills and use this information. For each of the two groups of students, the correlation coefficient between IQ and math skills can be calculated. Using this correlation coefficient, it is possible to distinguish between the share of variance in groups explained by the influence of IQ and the unexplained share of variance (see also Elementary concepts of statistics(chapter 8) and Basic statistics and tables(Chapter 9)). The remaining fraction of the variance is used in the analysis as the error variance. If there is a correlation between IQ and math skills, then the error variances can be significantly reduced. SS/(n-1) .

Effect of covariates onF- criterion. F- the criterion evaluates the statistical significance of the difference between the mean values ​​in the groups, while the ratio of the intergroup variance is calculated ( MSeffect) to the error variance ( MSerror) . If MSerror decreases, for example, when taking into account the IQ factor, the value F increases.

Lots of covariates. The reasoning used above for a single covariate (IQ) easily extends to multiple covariates. For example, in addition to IQ, you can include the measurement of motivation, spatial thinking, etc. Instead of the usual correlation coefficient, a multiple correlation coefficient is used.

When the valueF -criteria decreases. Sometimes the introduction of covariates into the experimental design reduces the value F- criteria . This usually indicates that the covariates are not only correlated with the dependent variable (such as math skills) but also with factors (such as different textbooks). Assume that IQ is measured at the end of the semester, after two groups of students have spent almost a year studying two different textbooks. Although the students were divided into groups randomly, it may turn out that the difference in textbooks is so great that both IQ and math skills in different groups will vary greatly. In this case, the covariates not only reduce the error variance, but also the between-group variance. In other words, after controlling for the difference in IQ between groups, the difference in math skills will no longer be significant. It can be said otherwise. After “eliminating” the influence of IQ, the influence of the textbook on the development of mathematical skills is inadvertently excluded.

Adjusted averages. When the covariate affects the between-group factor, one should calculate adjusted averages, i.e. such means, which are obtained after removing all estimates of the covariates.

Interaction between covariates and factors. Just as interactions between factors are explored, interactions between covariates and between groups of factors can be explored. Suppose one of the textbooks is especially suitable for smart students. The second textbook is boring for smart students, and the same textbook is difficult for less smart students. As a result, there is a positive correlation between IQ and learning outcomes in the first group (smarter students, better results) and zero or little negative correlation in the second group (the smarter the student, the less likely it is to acquire mathematical skills from the second textbook). In some studies, this situation is discussed as an example of violation of the assumptions of the analysis of covariance. However, since the Analysis of Variance module uses the most common methods of analysis of covariance, it is possible, in particular, to evaluate the statistical significance of the interaction between factors and covariates.

Variable covariates

While fixed covariates are discussed quite often in textbooks, variable covariates are much less frequently mentioned. Usually, when conducting experiments with repeated measurements, we are interested in differences in measurements of the same quantities at different points in time. Namely, we are interested in the significance of these differences. If a covariate measurement is carried out at the same time as the dependent variable measurements, the correlation between the covariate and the dependent variable can be calculated.

For example, you can study interest in mathematics and math skills at the beginning and at the end of the semester. It would be interesting to check whether changes in interest in mathematics are correlated with changes in mathematical skills.

Module Analysis of variance V STATISTICS automatically assesses the statistical significance of changes in covariates in those plans, where possible.

Multivariate Designs: Multivariate ANOVA and Covariance Analysis

Intergroup plans

All examples considered earlier included only one dependent variable. When there are several dependent variables at the same time, only the complexity of the calculations increases, and the content and basic principles do not change.

For example, a study is being conducted on two different textbooks. At the same time, the success of students in the study of physics and mathematics is studied. In this case, there are two dependent variables and you need to find out how two different textbooks affect them simultaneously. To do this, you can use multivariate analysis of variance (MANOVA). Instead of a one-dimensional F criterion, multidimensional F test (Wilks l-test) based on comparison of error covariance matrix and intergroup covariance matrix.

If the dependent variables are correlated with each other, then this correlation should be taken into account when calculating the significance test. Obviously, if the same measurement is repeated twice, then nothing new can be obtained in this case. If a dimension that is correlated with it is added to an existing dimension, then some new information is obtained, but the new variable contains redundant information, which is reflected in the covariance between the variables.

Interpretation of results. If the overall multivariate criterion is significant, we can conclude that the corresponding effect (eg textbook type) is significant. However, the following questions arise. Does the type of textbook affect the improvement of only math skills, only physical skills, or both of them. In fact, after obtaining a meaningful multivariate criterion, for a single main effect or interaction, one-dimensional F criterion. In other words, dependent variables that contribute to the significance of the multivariate test are examined separately.

Plans with repeated measurements

If the mathematical and physical skills of students are measured at the beginning of the semester and at the end, then these are repeated measurements. The study of the criterion of significance in such plans is a logical development of the one-dimensional case. Note that multivariate ANOVA methods are also commonly used to investigate the significance of univariate repeated measures factors that have more than two levels. The corresponding applications will be discussed later in this part.

Summation of variable values ​​and multivariate analysis of variance

Even experienced users of univariate and multivariate ANOVA often get confused by getting different results when applying multivariate ANOVA to, say, three variables, and when applying univariate ANOVA to the sum of the three variables as a single variable.

Idea summation variables is that each variable contains some true variable, which is investigated, as well as a random measurement error. Therefore, when averaging the values ​​of the variables, the measurement error will be closer to 0 for all measurements and the averaged values ​​will be more reliable. In fact, in this case, applying ANOVA to the sum of variables is reasonable and a powerful technique. However, if the dependent variables are multivariate in nature, summing the values ​​of the variables is inappropriate.

For example, let the dependent variables consist of four measures success in society. Each indicator characterizes a completely independent side of human activity (for example, professional success, business success, family well-being, etc.). Adding these variables together is like adding an apple and an orange. The sum of these variables would not be a suitable univariate measure. Therefore, such data must be treated as multidimensional indicators in multivariate analysis of variance.

Contrast analysis and post hoc tests

Why are individual sets of means compared?

Usually hypotheses about experimental data are formulated not simply in terms of main effects or interactions. An example is the following hypothesis: a certain textbook improves mathematical skills only in male students, while another textbook is approximately equally effective for both sexes, but still less effective for men. It can be predicted that textbook performance interacts with student gender. However, this prediction also applies nature interactions. A significant difference between the sexes is expected for students in one book, and practically gender-independent results for students in the other book. This type of hypothesis is usually explored using contrast analysis.

Contrast Analysis

In short, contrast analysis allows us to evaluate the statistical significance of some linear combinations of complex effects. Analysis of contrasts main and required element any complex ANOVA plan. Module Analysis of variance has quite a variety of contrast analysis capabilities that allow you to select and analyze any type of comparison of averages.

a posteriori comparisons

Sometimes, as a result of processing an experiment, an unexpected effect is discovered. Although in most cases a creative researcher will be able to explain any result, this does not provide opportunities for further analysis and estimates for the forecast. This problem is one of those for which post hoc criteria, that is, criteria that do not use a priori hypotheses. To illustrate, consider the following experiment. Suppose that 100 cards contain numbers from 1 to 10. Having dropped all these cards into the header, we randomly select 20 times 5 cards, and calculate the average value for each sample (the average of the numbers written on the cards). Can we expect that there are two samples whose means are significantly different? This is very plausible! By choosing two samples with the maximum and minimum mean, one can obtain a difference in the means that is very different from the difference in the means, for example, of the first two samples. This difference can be investigated, for example, using contrast analysis. Without going into details, there are several so-called a posteriori criteria that are based exactly on the first scenario (taking extreme averages from 20 samples), i.e. these criteria are based on choosing the most different means to compare all means in the design. These criteria are applied in order not to get an artificial effect purely by chance, for example, to find a significant difference between the means when there is none. Module Analysis of variance offers a wide range of such criteria. When unexpected results are encountered in an experiment involving multiple groups, the a posteriori procedures for examining the statistical significance of the results obtained.

Sum of squares type I, II, III and IV

Multivariate regression and analysis of variance

There is a close relationship between the method of multivariate regression and analysis of variance (analysis of variations). In both methods, a linear model is studied. In short, almost all experimental designs can be explored using multivariate regression. Consider the following simple cross-group 2 x 2 plan.

DV	A	B	AxB
3	1	1	1
4	1	1	1
4	1	-1	-1
5	1	-1	-1
6	-1	1	-1
6	-1	1	-1
3	-1	-1	1
2	-1	-1	1

Columns A and B contain codes characterizing the levels of factors A and B, column AxB contains the product of two columns A and B. We can analyze these data using multivariate regression. Variable DV defined as a dependent variable, variables from A before AxB as independent variables. The study of significance for the regression coefficients will coincide with the calculations in the analysis of variance of the significance of the main effects of the factors A And B and interaction effect AxB.

Unbalanced and Balanced Plans

When calculating the correlation matrix for all variables, for example, for the data depicted above, it can be seen that the main effects of the factors A And B and interaction effect AxB uncorrelated. This property of effects is also called orthogonality. They say that the effects A And B - orthogonal or independent from each other. If all effects in the plan are orthogonal to each other, as in the example above, then the plan is said to be balanced.

Balanced plans have the “good property.” The calculations in the analysis of such plans are very simple. All calculations are reduced to calculating the correlation between effects and dependent variables. Since the effects are orthogonal, partial correlations (as in full multidimensional regressions) are not calculated. However, in real life plans are not always balanced.

Consider real data with an unequal number of observations in cells.

Factor A	Factor B
	B1	B2
A1	3	4, 5
A2	6, 6, 7	2

If we encode this data as above and calculate the correlation matrix for all variables, then it turns out that the design factors are correlated with each other. Factors in the plan are now not orthogonal and such plans are called unbalanced. Note that in this example, the correlation between the factors is entirely related to the difference in the frequencies of 1 and -1 in the columns of the data matrix. In other words, experimental designs with unequal cell volumes (more precisely, disproportionate volumes) will be unbalanced, which means that the main effects and interactions will mix. In this case, to calculate the statistical significance of the effects, you need to fully calculate the multivariate regression. There are several strategies here.

Sum of squares type I, II, III and IV

Sum of squares typeIAndIII. To study the significance of each factor in a multivariate model, one can calculate the partial correlation of each factor, provided that all other factors are already taken into account in the model. You can also enter factors into the model in a step-by-step manner, fixing all the factors already entered into the model and ignoring all other factors. In general, this is the difference between type III And typeI sums of squares (this terminology was introduced in SAS, see for example SAS, 1982; a detailed discussion can also be found in Searle, 1987, p. 461; Woodward, Bonett, and Brecht, 1990, p. 216; or Milliken and Johnson, 1984, p. 138).

Sum of squares typeII. The next “intermediate” model formation strategy is: to control all the main effects in the study of the significance of a single main effect; in the control of all main effects and all pairwise interactions, when the significance of a single pairwise interaction is examined; in controlling all main effects of all pairwise interactions and all interactions of three factors; in the study of a separate interaction of three factors, etc. The sums of squares for effects calculated in this way are called typeII sums of squares. So, typeII sums of squares controls all effects of the same order and below, ignoring all effects of a higher order.

Sum of squares typeIV. Finally, for some special plans with missing cells (incomplete plans), it is possible to calculate the so-called type IV sums of squares. This method will be discussed later in connection with incomplete plans (plans with missing cells).

Interpretation of the sum-of-squares conjecture of types I, II, and III

sum of squares typeIII easiest to interpret. Recall that the sums of squares typeIII examine the effects after controlling for all other effects. For example, after finding a statistically significant typeIII effect for the factor A in the module Analysis of variance, we can say that there is a single significant effect of the factor A, after introducing all other effects (factors) and interpret this effect accordingly. Probably in 99% of all applications of analysis of variance, this type of criterion is of interest to the researcher. This type of sum of squares is usually computed in the module Analysis of variance by default, regardless of whether the option is selected Regression Approach or not (standard approaches adopted in the module Analysis of variance discussed below).

Significant effects obtained using sums of squares type or typeII sums of squares are not so easy to interpret. They are best interpreted in the context of stepwise multivariate regression. If using the sum of squares typeI the main effect of factor B was found to be significant (after including factor A in the model, but before adding the interaction between A and B), it can be concluded that there is a significant main effect of factor B, provided that there is no interaction between factors A and B. (If at using the criterion typeIII, factor B also turned out to be significant, then we can conclude that there is a significant main effect of factor B, after introducing all other factors and their interactions into the model).

In terms of the marginal means of the hypothesis typeI And typeII usually do not have a simple interpretation. In these cases, it is said that one cannot interpret the significance of the effects by considering only the marginal means. rather presented p mean values ​​are related to complex hypothesis, which combines the means and the sample size. For example, typeII the hypotheses for factor A in the simple 2 x 2 design example discussed earlier would be (see Woodward, Bonett, and Brecht, 1990, p. 219):

nij- number of observations in a cell

uij- average value in a cell

n. j- marginal average

Without going into details (for more details see Milliken and Johnson, 1984, chapter 10), it is clear that these are not simple hypotheses and in most cases none of them is of particular interest to the researcher. However, there are cases where the hypotheses typeI may be of interest.

The default computational approach in the module Analysis of variance

Default if option is not checked Regression Approach, module Analysis of variance uses cell average model. It is characteristic of this model that the sums of squares for different effects are calculated for linear combinations of cell means. In a full factorial experiment, this results in sums of squares that are the same as the sums of squares discussed earlier as type III. However, in the option Scheduled Comparisons(in the window Analysis of variance results), the user can hypothesize about any linear combination of weighted or unweighted cell means. Thus, the user can test not only hypotheses typeIII, but hypotheses of any type (including typeIV). This general approach is particularly useful when examining designs with missing cells (so-called incomplete designs).

For full factorial designs, this approach is also useful when one wants to analyze weighted marginal means. For example, suppose that in the simple 2 x 2 design considered earlier, we want to compare the weighted (in terms of factor levels) B) marginal averages for factor A. This is useful when the distribution of observations over cells was not prepared by the experimenter, but was built randomly, and this randomness is reflected in the distribution of the number of observations by levels of factor B in the aggregate.

For example, there is a factor - the age of widows. A possible sample of respondents is divided into two groups: younger than 40 and older than 40 (factor B). The second factor (factor A) in the plan is whether or not widows received social support from some agency (while some widows were randomly selected, others served as controls). In this case, the age distribution of widows in the sample reflects the actual age distribution of widows in the population. Assessing the effectiveness of the social support group for widows all ages will correspond to the weighted average for the two age groups (with weights corresponding to the number of observations in the group).

Scheduled Comparisons

Note that the sum of the entered contrast ratios is not necessarily equal to 0 (zero). Instead, the program will automatically make adjustments so that the corresponding hypotheses do not mix with the overall average.

To illustrate this, let's go back to the simple 2 x 2 plan discussed earlier. Recall that the cell counts of this unbalanced design are -1, 2, 3, and 1. Let's say we want to compare the weighted marginal averages for factor A (weighted by the frequency of factor B levels). You can enter contrast ratios:

Note that these coefficients do not add up to 0. The program will set the coefficients so that they add up to 0, while maintaining their relative values, i.e.:

1/3 2/3 -3/4 -1/4

These contrasts will compare the weighted averages for factor A.

Hypotheses about the principal mean. The hypothesis that the unweighted principal mean is 0 can be explored using coefficients:

The hypothesis that the weighted principal mean is 0 is tested with:

In no case does the program correct the contrast ratios.

Analysis of plans with missing cells (incomplete plans)

Factorial designs containing empty cells (processing of combinations of cells in which there are no observations) are called incomplete. In such designs, some factors are usually not orthogonal and some interactions cannot be calculated. Doesn't exist at all best method analysis of such plans.

Regression Approach

In some older programs that are based on the analysis of ANOVA designs using multivariate regression, the factors in incomplete designs are set by default in the usual way (as if the plan were complete). A multivariate regression analysis is then performed for these dummy-coded factors. Unfortunately, this method leads to results that are very difficult, if not impossible, to interpret because it is not clear how each effect contributes to the linear combination of means. Consider the following simple example.

Factor A	Factor B
	B1	B2
A1	3	4, 5
A2	6, 6, 7	Missed

If multivariate regression of the form Dependent variable = Constant + Factor A + Factor B, then the hypothesis about the significance of factors A and B in terms of linear combinations of means looks like this:

Factor A: Cell A1,B1 = Cell A2,B1

Factor B: Cell A1,B1 = Cell A1,B2

This case is simple. In more complex plans, it is impossible to actually determine what exactly will be examined.

Mean cells, analysis of variance approach , type IV hypotheses

An approach that is recommended in the literature and seems to be preferable is the study of meaningful (in terms of research tasks) a priori hypotheses about the means observed in the cells of the plan. A detailed discussion of this approach can be found in Dodge (1985), Heiberger (1989), Milliken and Johnson (1984), Searle (1987), or Woodward, Bonett, and Brecht (1990). Sums of squares associated with hypotheses about a linear combination of means in incomplete designs, investigating estimates of part of the effects, are also called sums of squares. IV.

Automatic generation of type hypothesesIV. When multivariate designs have a complex missing cell pattern, it is desirable to define orthogonal (independent) hypotheses whose investigation is equivalent to the investigation of main effects or interactions. Algorithmic (computational) strategies (based on the pseudo-inverse design matrix) have been developed to generate appropriate weights for such comparisons. Unfortunately, the final hypotheses are not uniquely defined. Of course, they depend on the order in which the effects were defined and are rarely easy to interpret. Therefore, it is recommended to carefully study the nature of the missing cells, then formulate hypotheses typeIV, that are most relevant to the objectives of the study. Then explore these hypotheses using the option Scheduled Comparisons in the window results. The easiest way to specify comparisons in this case is to require the introduction of a vector of contrasts for all factors together in the window Scheduled comparisons. After calling the dialog box Scheduled Comparisons all groups of the current plan will be shown and those that are omitted will be marked.

Skipped Cells and Specific Effect Check

There are several types of plans in which the location of the missing cells is not random, but carefully planned, which allows a simple analysis of the main effects without affecting other effects. For example, when the required number of cells in a plan is not available, plans are often used. latin squares to estimate the main effects of several factors with a large number of levels. For example, a 4 x 4 x 4 x 4 factorial design requires 256 cells. At the same time, you can use Greco-Latin square to estimate the main effects, having only 16 cells in the plan (chap. Experiment planning, Volume IV, contains a detailed description of such plans). Incomplete designs in which the main effects (and some interactions) can be estimated using simple linear combinations of means are called balanced incomplete plans.

In balanced designs, the standard (default) method of generating contrasts (weights) for main effects and interactions will then produce a variance table analysis in which the sums of squares for the respective effects do not mix with each other. Option Specific Effects window results will generate missing contrasts by writing zero to the missing plan cells. Immediately after the option is requested Specific Effects for a user studying some hypothesis, a table of results appears with the actual weights. Note that in a balanced design, the sums of squares of the respective effects are computed only if those effects are orthogonal (independent) to all other principal effects and interactions. Otherwise, use the option Scheduled Comparisons to explore meaningful comparisons between means.

Missing Cells and Combined Error Effects/Members

If option Regression approach in the launch panel of the module Analysis of variance is not selected, the cell averages model will be used when calculating the sum of squares for the effects (default setting). If the design is not balanced, then when combining non-orthogonal effects (see above discussion of the option Missing cells and specific effect) one can obtain a sum of squares consisting of non-orthogonal (or overlapping) components. The results obtained in this way are usually not interpretable. Therefore, one must be very careful when choosing and implementing complex incomplete experimental designs.

There are many books that discuss plans in detail. different type. (Dodge, 1985; Heiberger, 1989; Lindman, 1974; Milliken and Johnson, 1984; Searle, 1987; Woodward and Bonett, 1990), but this kind of information is outside the scope of this textbook. However, later in this section we will show the analysis various types plans.

Assumptions and Assumption Violation Effects

Deviation from the assumption of normal distributions

Assume that the dependent variable is measured on a numerical scale. Let's also assume that the dependent variable has a normal distribution within each group. Analysis of variance contains a wide range of graphs and statistics to substantiate this assumption.

Violation effects. At all F the criterion is very resistant to deviation from normality (see Lindman, 1974 for detailed results). If the kurtosis is greater than 0, then the value of the statistic F may become very small. The null hypothesis is accepted, although it may not be true. The situation is reversed when the kurtosis is less than 0. The skewness of the distribution usually has little effect on F statistics. If the number of observations in a cell is large enough, then the deviation from normality does not matter much due to central limit theorem, according to which, the distribution of the mean value is close to normal, regardless of the initial distribution. Detailed discussion of sustainability F statistics can be found in Box and Anderson (1955), or Lindman (1974).

Homogeneity of dispersion

Assumptions. It is assumed that the variances of different groups of the plan are the same. This assumption is called the assumption dispersion homogeneity. Recall that at the beginning of this section, when describing the calculation of the sum of squared errors, we performed summation within each group. If the variances in two groups differ from each other, then adding them is not very natural and does not give an estimate of the total within-group variance (since in this case there is no general variance at all). Module Dispersion analysis -ANOVA/MANOVA contains a large set of statistical criteria for detecting deviation from the assumptions of homogeneity of variance.

Violation effects. Lindman (1974, p. 33) shows that F the criterion is quite stable with respect to the violation of the assumptions of homogeneity of the variance ( heterogeneity dispersion, see also Box, 1954a, 1954b; Hsu, 1938).

Special case: correlation of means and variances. There are times when F statistics can mislead. This happens when the mean values ​​in the design cells are correlated with the variance. Module Analysis of variance allows you to plot variance or standard deviation scatterplots against means to detect such a correlation. The reason why such a correlation is dangerous is as follows. Let's imagine that there are 8 cells in the plan, 7 of which have almost the same average, and in one cell the average is much larger than the rest. Then F the test can detect a statistically significant effect. But suppose that in a cell with a large mean value and the variance is much larger than the others, i.e. the mean and variance in the cells are dependent (the larger the mean, the greater the variance). In this case, the large mean is unreliable, as it may be caused by a large variance in the data. However F statistics based on united variance within cells will capture a large mean, although criteria based on variance in each cell will not consider all differences in the means to be significant.

This nature of the data (large mean and large variance) is often encountered when there are outlier observations. One or two outlier observations strongly shift the mean and greatly increase the variance.

Homogeneity of variance and covariance

Assumptions. In multivariate designs, with multivariate dependent measures, the homogeneity of variance assumptions described earlier also apply. However, since there are multivariate dependent variables, it is also required that their cross-correlations (covariances) be uniform across all plan cells. Module Analysis of variance offers different ways testing these assumptions.

Violation effects. Multidimensional analog F- criterion - λ-test of Wilks. Not much is known about the stability (robustness) of the Wilks λ-test with respect to the violation of the above assumptions. However, since the interpretation of module results Analysis of variance is usually based on the significance of one-dimensional effects (after establishing the significance general criterion), the discussion of robustness concerns mainly one-dimensional analysis of variance. Therefore, the significance of one-dimensional effects should be carefully examined.

Special case: analysis of covariance. Particularly severe violations of the homogeneity of variance/covariance can occur when covariates are included in the design. In particular, if the correlation between covariates and dependent measures is different in different cells of the design, misinterpretation of the results may follow. It should be remembered that in the analysis of covariance, in essence, a regression analysis is performed within each cell in order to isolate that part of the variance that corresponds to the covariate. The homogeneity of variance/covariance assumption assumes that this regression analysis is performed under the following constraint: all regression equations (slopes) for all cells are the same. If this is not intended, then large errors may occur. Module Analysis of variance has several special criteria to test this assumption. It may be advisable to use these criteria in order to make sure that the regression equations for different cells are approximately the same.

Sphericity and complex symmetry: reasons for using a multivariate repeated measures approach in analysis of variance

In designs containing repeated measures factors with more than two levels, the application of univariate analysis of variance requires additional assumptions: complex symmetry assumptions and sphericity assumptions. These assumptions are rarely met (see below). Therefore, in last years multivariate analysis of variance has gained popularity in such plans (both approaches are combined in the module Analysis of variance).

Complex symmetry assumption The complex symmetry assumption is that the variances (total within-group) and covariances (by group) for different repeated measures are uniform (the same). This is a sufficient condition for the univariate F test for repeated measures to be valid (ie, the F-values ​​reported are, on average, consistent with the F-distribution). However, in this case this condition is not necessary.

Assumption of sphericity. The assumption of sphericity is a necessary and sufficient condition for the F-criterion to be justified. It consists in the fact that within the groups all observations are independent and equally distributed. The nature of these assumptions, as well as the impact of their violations, are usually not well described in books on analysis of variance - this one will be described in the following paragraphs. It will also show that the results of the univariate approach may differ from the results of the multivariate approach and explain what this means.

The need for independence of hypotheses. The general way to analyze data in analysis of variance is model fit. If, with respect to the model corresponding to the data, there are some a priori hypotheses, then the variance is split to test these hypotheses (criteria for main effects, interactions). From a computational point of view, this approach generates some set of contrasts (set of comparisons of means in the design). However, if the contrasts are not independent of each other, the partitioning of the variances becomes meaningless. For example, if two contrasts A And B are identical and the corresponding part is selected from the variance, then the same part is selected twice. For example, it is silly and pointless to single out two hypotheses: “the mean in cell 1 is higher than the mean in cell 2” and “the mean in cell 1 is higher than the mean in cell 2”. So the hypotheses must be independent or orthogonal.

Independent hypotheses in repeated measurements. General algorithm implemented in the module Analysis of variance, will try to generate independent (orthogonal) contrasts for each effect. For the repeated measures factor, these contrasts give rise to many hypotheses about differences between the levels of the considered factor. However, if these differences are correlated within groups, then the resulting contrasts are no longer independent. For example, in training where learners are measured three times in one semester, it may happen that changes between 1st and 2nd dimensions are negatively correlated with the change between 2nd and 3rd dimensions of subjects. Those who have mastered most of the material between the 1st and 2nd dimensions master a smaller part during the time that has passed between the 2nd and 3rd dimensions. In fact, for most cases where analysis of variance is used in repeated measurements, it can be assumed that changes in levels are correlated across subjects. However, when this happens, the complex symmetry and sphericity assumptions are not met and independent contrasts cannot be computed.

The impact of violations and ways to correct them. When complex symmetry or sphericity assumptions are not met, analysis of variance can produce erroneous results. Before multivariate procedures were sufficiently developed, several assumptions were made to compensate for violations of these assumptions. (See, for example, Greenhouse & Geisser, 1959 and Huynh & Feldt, 1970). These methods are still widely used today (which is why they are presented in the module Analysis of variance).

Multivariate analysis of variance approach to repeated measures. In general, the problems of complex symmetry and sphericity refer to the fact that the sets of contrasts included in the study of the effects of repeated measures factors (with more than 2 levels) are not independent of each other. However, they do not have to be independent if one uses multidimensional a criterion for simultaneously testing the statistical significance of two or more repeated measures factor contrasts. This is the reason why multivariate analysis of variance methods have become increasingly used to test the significance of univariate repeated measures factors with more than 2 levels. This approach is widely used because it generally does not require the assumption of complex symmetry and the assumption of sphericity.

Cases in which the multivariate analysis of variance approach cannot be used. There are examples (plans) when the multivariate analysis of variance approach cannot be applied. Usually these are cases where there is no a large number of subjects in the plan and many levels in the repeated measures factor. Then there may be too few observations to perform a multivariate analysis. For example, if there are 12 entities, p = 4 repeated measurements factor, and each factor has k = 3 levels. Then the interaction of 4 factors will “expend” (k-1)P = 2 4 = 16 degrees of freedom. However, there are only 12 subjects, hence a multivariate test cannot be performed in this example. Module Analysis of variance will independently detect these observations and calculate only one-dimensional criteria.

Differences in univariate and multivariate results. If the study includes a large number of repeated measures, there may be cases where the univariate repeated measures approach of ANOVA yields results that are very different from those obtained with the multivariate approach. This means that the differences between the levels of the respective repeated measurements are correlated across subjects. Sometimes this fact is of some independent interest.

Multivariate analysis of variance and structural modeling of equations

In recent years, structural equation modeling has become popular as an alternative to multivariate dispersion analysis (see, for example, Bagozzi and Yi, 1989; Bagozzi, Yi, and Singh, 1991; Cole, Maxwell, Arvey, and Salas, 1993). This approach allows you to test hypotheses not only about the means in different groups, but also about the correlation matrices of dependent variables. For example, you can relax the assumptions about the homogeneity of the variance and covariance and explicitly include errors in the model for each group of variance and covariance. Module STATISTICSStructural Equation Modeling (SEPATH) (see Volume III) allows for such an analysis.

The considered scheme of dispersion analysis is differentiated depending on: a) the nature of the feature, according to which the population is divided into groups (samples); b) the number of features, according to which the population is divided into groups (samples); c) on the method of sampling.

Feature values. which subdivides the population into groups may represent a general population or a population close to it in size. In this case, the scheme for conducting the analysis of variance corresponds to that considered above. If the values ​​of the attribute that forms different groups represent a sample from the general population, then the formulation of the null and alternative hypotheses changes. As a null hypothesis, it is assumed that there are differences between the groups, that is, the group means show some variation. An alternative hypothesis is that there is no volatility. Obviously, with such a formulation of hypotheses, there is no reason to concretize the results of comparison of variances.

With an increase in the number of grouping features, for example, up to 2, firstly, the number of null and, accordingly, alternative hypotheses increases. In this case, the first null hypothesis indicates the absence of differences between the means for the groups of the first grouping trait, the second null hypothesis indicates the absence of differences in the averages for the groups of the second grouping trait, and finally the third null hypothesis indicates the absence of the so-called effect of the interaction of factors (grouping traits).

By the effect of interaction is understood such a change in the value of the effective attribute, which cannot be explained by the total action of two factors. To test the three pairs of hypotheses put forward, it is necessary to calculate three actual values ​​of the F-Fisher criterion, which in turn implies the following version of the expansion of the total volume of variation

The dispersions required to obtain the F-criterion are obtained in a known manner by dividing the volumes of variation by the number of degrees of freedom.

As you know, samples can be dependent independent. If the samples are dependent, then in the total amount of variation, the so-called variation in repetitions should be distinguished
. If it is not singled out, then this variation can significantly increase the intragroup variation (
), which can distort the results of the analysis of variance.

Review questions

17-1. What is the specification of the results of the analysis of variance?

17-2. In what case is the Q-Tukey criterion used for concretization?

17-3. What are the differences of the first, second and so on orders?

17-4. How to find the actual value of Tukey's Q-criterion?

17-5. What are the hypotheses for each difference?

17-6. What does the tabular value of Tukey's Q-test depend on?

17-7. What will be the null hypothesis if the levels of the grouping feature represent a sample?

17-8. How is the total amount of variation decomposed when grouping data according to two criteria?

17-9. In what case is the variation in repetitions distinguished (
) ?

Summary

The considered mechanism for concretizing the results of dispersion analysis allows us to give it a finished form. Attention should be paid to limitations when using Tukey's Q-test. The material also outlined the basic principles for classifying models of variance analysis. It must be emphasized that these are just principles. A detailed study of the features of each model requires a separate deeper study.

Test tasks for the lecture

What statistical characteristics are hypothesized about in the analysis of variance?

Relative to two dispersions

Regarding one average

Regarding several averages

Relative to one variance

What is the content of the alternative hypothesis in the analysis of variance?

Comparable variances are not equal to each other

All compared averages are not equal to each other

At least two general means are not equal

Intergroup variance is greater than intragroup variance

What levels of significance are most often used in analysis of variance

If the within-group variation is greater than the between-group variation, should the analysis of variance be continued or should we immediately accept H0 or HA?

1. Should we continue by determining the necessary variances?

2. We must agree with H0

3. Should agree with NA

If the intra-group variance was equal to the inter-group variance, what should be the actions of the ANOVA?

Agree with the null hypothesis that the population means are equal

Agree with the alternative hypothesis about the presence of at least a pair of means unequal to each other

What variance should always be in the numerator when calculating the Fisher F test?

Only intragroup

In any case, intergroup

Intergroup, if it is greater than intragroup

What should be the actual value of the F-Fisher criterion?

Always less than 1

Always greater than 1

Equal or greater than 1

What does the tabular value of the F-Fisher criterion depend on?

1. From the accepted level of significance

2. On the number of degrees of freedom of the general variation

3. On the number of degrees of freedom of intergroup variation

4. On the number of degrees of freedom of intragroup variation

5. From the value of the actual value of the F-Fisher criterion?

Increasing the number of observations in each group with equal variances increases the likelihood of accepting ……

1. Null hypothesis

2.Alternative hypothesis

3. Does not affect the acceptance of both the null and alternative hypotheses

What is the point of concretizing the results of the analysis of variance?

Clarify whether the calculations of variances were carried out correctly

Determine which of the general averages turned out to be equal to each other

Clarify which of the general averages are not equal to each other

Is the statement true: “When concretizing the results of the analysis of variance, all the general averages turned out to be equal to each other”

Can be true and false

Not true, this may be due to errors in the calculations

Is it possible, when concretizing the analysis of variance, to come to the conclusion that all general averages are not equal to each other?

1. Quite possible

2. Possible in exceptional cases

3. Impossible in principle.

4. Possible only if there are errors in the calculations

If the null hypothesis was accepted according to the F-Fisher test, is it necessary to specify the analysis of variance?

1.Required

2.Not required

3.At the discretion of the ANOVA

In what case is Tukey's criterion used to concretize the results of the analysis of variance?

1. If the number of observations across groups (samples) is the same

2. If the number of observations by groups (samples) is different

3. If there are samples with both equal and unequal numbers

laziness

What is the NSR when concretizing the results of the analysis of variance based on the Tukey criterion?

1. The product of the mean error and the actual value of the criterion

2. The product of the mean error and the tabular value of the criterion

3. The ratio of each difference between the sample means to

average error

4. Difference between sample means

If the sample is divided into groups according to 2 features, how many sources should at least the total variation of the feature be divided into?

If observations by samples (groups) are dependent, how many sources should the total variation be divided into (grouping attribute one)?

What is the source (cause) of intergroup variation?

game of chance

Joint action of the game of chance and factor

Action of the factor(s)

It will become clear after the analysis of variance

What is the source (cause) of intragroup variation?

1. Game of chance

2.The joint action of the game of chance and factor

3. Action of the factor (factors)

4. It will become clear after the analysis of variance

What method of source data transformation is used if the characteristic values ​​are expressed in shares?

Logarithm

root extraction

Phi transformation

Lecture 8 Correlation

annotation

The most important method for studying the relationship between features is the correlation method. This lecture reveals the content of this method, approaches to the analytical expression of this connection. Particular attention is paid to such specific indicators as indicators of the closeness of communication

Keywords

Correlation. Least square method. Regression coefficient. Coefficients of determination and correlation.

Issues under consideration

Communication functional and correlation

Stages of construction of the correlation equation of communication. Interpretation of equation coefficients

Tightness indicators

Evaluation of sample indicators of communication

Modular unit 1 The essence of the correlation. Stages of construction of the correlation equation of communication, interpretation of coefficients of the equation.

The purpose and objectives of the study of the modular unit 1 consist in understanding the features of the correlation. mastering the algorithm for constructing a connection equation, understanding the content of the coefficients of the equation.

The essence of the correlation

In natural and social phenomena, there are two types of connections - a functional connection and a correlation connection. With a functional connection, each value of the argument corresponds to strictly defined (one or more) values ​​of the function. An example of a functional relationship is the relationship between circumference and radius, which is expressed by the equation
. Each radius value r corresponds to a single circumference value L . With a correlation, each value of a factor attribute corresponds to several not quite certain values ​​of the resultant attribute. Examples of a correlation can be the relationship between a person's weight (resultant trait) and his height (factorial trait), the relationship between the amount of fertilizer applied and yield, between the price and quantity of the goods offered. The source of the correlation is the fact that, as a rule, in real life the value of the effective feature depends on many factors, including those that have a random nature of their change. For example, the same weight of a person depends on age, gender, nutrition, occupation and many other factors. But at the same time, it is obvious that, in general, it is growth that is the decisive factor. In view of these circumstances, the correlation should be defined as an incomplete relationship, which can be established and evaluated only if there is a large number of observations, on average.

1.2 Stages of constructing the correlation equation of communication.

Like a functional connection, a correlation connection is expressed by a connection equation. To build it, you must sequentially go through the following steps (stages).

First, you should understand the cause-and-effect relationships, find out the subordination of signs, that is, which of them are causes (factorial signs), and which are consequences (effective signs). Cause-and-effect relationships between features are established by the theory of the subject where the correlation method is used. For example, the science of "human anatomy" allows you to say what is the source of the relationship between weight and height, which of these signs is a factor, which result, the science of "economics" reveals the logic of the relationship between price and supply, establishes what and at what stage is the cause and what is the effect . Without such a preliminary theoretical substantiation, the interpretation of the results obtained later is difficult, and sometimes can lead to absurd conclusions.

Having established the presence of cause-and-effect relationships, these relationships should then be formalized, that is, expressed using a connection equation, while first choosing the type of equation. To select the type of equation, a number of methods can be recommended. You can turn to the theory of the subject where the correlation method is used, for example, the science of "agrochemistry" may have already received an answer to the question of which equation should express the relationship: yield - fertilizer. If there is no such answer, then to select an equation, one should use some empirical data, processing them accordingly. It should be said right away that having chosen the type of equation based on empirical data, one must clearly understand that this type of equation can be used to describe the relationship of the data used. The main technique for processing these data is the construction of graphs, when the values ​​of the factor attribute are plotted on the abscissa axis, and on the ordinate axis possible values resultant sign. Since, by definition, the same value of the factor attribute corresponds to a set of uncertain values ​​of the attribute of the effective one, as a result of the above actions, we will get a certain set of points, which is called the correlation field. The general form of the correlation field allows in a number of cases to make an assumption about the possible form of the equation. modern development in computer technology, one of the main methods for choosing an equation is to enumerate various kinds equations, while the equation that provides the highest coefficient of determination, which will be discussed below, is chosen as the best one. Before proceeding to the calculations, it is necessary to check whether the empirical data involved in constructing the equation satisfy certain requirements. The requirements apply to factor characteristics and to the totality of data. Factor signs, if there are several of them, must be independent of each other. As for the aggregate, it must first be homogeneous

(the concept of homogeneity was considered earlier), and Secondly big enough. For each factor sign, there should be at least 8-10 observations.

After choosing an equation, the next step is to calculate the coefficients of the equation. The calculation of the coefficients of the equation is most often made on the basis of the least squares method. From the point of view of correlation, the use of the method of least squares consists in obtaining such coefficients of the equation that
=min, that is, so that the sum of the squared deviations of the actual values ​​of the resultant feature ( ) from those calculated according to the equation ( ) was the minimum value. This requirement is realized by constructing and solving a well-known system of so-called normal equations. If, as an equation of the correlation between y And x the equation of a straight line is chosen
, where the system of normal equations is known to be:

Solving this system for a And b , we obtain the required values ​​of the coefficients. The correctness of calculating the coefficients is checked by the equality

The use of statistics in this note will be shown with a cross-cutting example. Let's say you're a production manager at Perfect Parachute. Parachutes are made from synthetic fibers supplied by four different suppliers. One of the main characteristics of a parachute is its strength. You need to make sure that all fibers supplied have the same strength. To answer this question, it is necessary to design an experiment in which the strength of parachutes woven from synthetic fibers from different suppliers is measured. The information obtained during this experiment will determine which supplier provides the most durable parachutes.

Many applications are related to experiments in which several groups or levels of one factor are considered. Some factors, such as ceramic firing temperature, may have multiple numerical levels (ie 300°, 350°, 400° and 450°). Other factors, such as the location of goods in a supermarket, may have categorical levels (eg, first supplier, second supplier, third supplier, fourth supplier). Single-factor experiments in which experimental units are randomly allocated to groups or factor levels are called fully randomized.

UsageF-criteria for evaluating the differences between several mathematical expectations

If the numerical measurements of a factor in groups are continuous and some additional terms, to compare the mathematical expectations of several groups, analysis of variance (ANOVA - An alysis o f Va riance). Analysis of variance using fully randomized designs is called one-way ANOVA. In a sense, the term analysis of variance is misleading because it compares the differences between the mean values ​​of the groups, not between the variances. However, the comparison of mathematical expectations is carried out precisely on the basis of the analysis of data variation. In the ANOVA procedure, the total variation of the measurement results is divided into intergroup and intragroup (Fig. 1). The intragroup variation is explained by experimental error, while the intergroup variation is explained by the effects of experimental conditions. Symbol With denotes the number of groups.

Rice. 1. Separation of Variation in a Fully Randomized Experiment

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Let's pretend that With groups are drawn from independent populations that have a normal distribution and the same variance. The null hypothesis is that the mathematical expectations of the populations are the same: H 0: μ 1 = μ 2 = ... = μ s. The alternative hypothesis states that not all mathematical expectations are the same: H 1: not all μ j are the same j= 1, 2, …, s).

On fig. Figure 2 presents the true null hypothesis about the mathematical expectations of the five compared groups, provided that the general populations have a normal distribution and the same variance. The five populations associated with different factor levels are identical. Therefore, they are superimposed one on another, having the same mathematical expectation, variation and form.

Rice. 2. Five populations have the same mathematical expectation: μ 1 = μ 2 = μ 3 = μ 4 = μ 5

On the other hand, suppose that in fact the null hypothesis is false, and the fourth level has the largest mathematical expectation, the first level has a slightly lower mathematical expectation, and the remaining levels have the same and even smaller mathematical expectations (Fig. 3). Note that, with the exception of the mean value, all five populations are identical (i.e., have the same variability and shape).

Rice. 3. The effect of the experimental conditions is observed: μ 4 > μ 1 > μ 2 = μ 3 = μ 5

When testing the hypothesis of equality of mathematical expectations of several general populations, the total variation is divided into two parts: intergroup variation, due to differences between groups, and intragroup variation, due to differences between elements belonging to the same group. The total variation is expressed as the total sum of squares (SST - sum of squares total). Since the null hypothesis is that the expectation of all With groups are equal to each other, the total variation is equal to the sum of the squared differences between individual observations and the total mean (mean of averages) calculated for all samples. Full variation:

Where - overall average, Xij - i-e watch in j-th group or level, nj- number of observations in j-th group, n- total number of observations in all groups (i.e. n = n 1 + n 2 + … + nc), With- number of studied groups or levels.

Intergroup variation, usually called the sum of squares among groups (SSA), is equal to the sum of squared differences between the sample mean of each group j and overall average multiplied by the volume of the corresponding group nj:

Where With- the number of groups or levels studied, nj- number of observations in j-th group, j- average value j-th group, - general average.

Intragroup variation, usually called the sum of squares withing groups (SSW), is equal to the sum of the squared differences between the elements of each group and the sample mean of this group j:

Where Xij - i-th element j-th group, j- average value j-th group.

Because they are compared With factor levels, the intergroup sum of squares has s - 1 degrees of freedom. Each of With levels has nj – 1 degrees of freedom, so the intragroup sum of squares has n- With degrees of freedom, and

In addition, the total sum of squares has n – 1 degrees of freedom, since each observation Xij compared with the overall average calculated over all n observations. If each of these sums is divided by the corresponding number of degrees of freedom, three kinds of dispersion will arise: intergroup(mean square among - MSA), intragroup(mean square within - MSW) and complete(mean square total - MST):

Despite the fact that the main purpose of the analysis of variance is to compare the mathematical expectations With groups to reveal the effect of experimental conditions, its name is due to the fact that the main tool is the analysis of variances of various types. If the null hypothesis is true, and between the expected values With groups there are no significant differences, all three variances - MSA, MSW and MST - are estimates of variance σ2 inherent in the analyzed data. So to test the null hypothesis H 0: μ 1 = μ 2 = ... = μ s and alternative hypothesis H 1: not all μ j are the same j = 1, 2, …, With), it is necessary to calculate the statistics F-criterion, which is the ratio of two variances, MSA and MSW. test F-statistics in univariate analysis of variance

Statistics F-criteria obeys F- distribution with s - 1 degrees of freedom in the numerator MSA And n - with degrees of freedom in the denominator MSW. For a given significance level α, the null hypothesis is rejected if the computed F FU inherent F- distribution with s - 1 n - with degrees of freedom in the denominator. Thus, as shown in fig. 4, the decision rule is formulated as follows: null hypothesis H 0 rejected if F > FU; otherwise, it is not rejected.

Rice. 4. Critical area of ​​analysis of variance when testing a hypothesis H 0

If the null hypothesis H 0 is true, computed F-statistics is close to 1, since its numerator and denominator are estimates of the same value - the variance σ 2 inherent in the analyzed data. If the null hypothesis H 0 is false (and there is a significant difference between the expectation values ​​of different groups), computed F-statistic will be much greater than one, because its numerator, MSA, in addition to the natural variability of the data, estimates the effect of experimental conditions or the difference between groups, while the denominator MSW estimates only the natural variability of the data. Thus, the ANOVA procedure is F is a test in which, at a given significance level α, the null hypothesis is rejected if the calculated F- statistics are greater than the upper critical value FU inherent F- distribution with s - 1 degrees of freedom in the numerator and n - with degrees of freedom in the denominator, as shown in Fig. 4.

To illustrate the one-way analysis of variance, let's return to the scenario outlined at the beginning of the note. The purpose of the experiment is to determine whether parachutes woven from synthetic fibers obtained from different suppliers have the same strength. Each group has five parachutes woven. Groups are divided by suppliers - Supplier 1, Supplier 2, Supplier 3 and Supplier 4. The strength of parachutes is measured using a special device that tests the fabric for tearing on both sides. The force required to break a parachute is measured on a special scale. The higher the breaking force, the stronger the parachute. Excel allows analysis F-Statistics with one click. Go through the menu Data → Data analysis, and select the line One-way analysis of variance, fill in the opened window (Fig. 5). The results of the experiment (gap strength), some descriptive statistics, and the results of one-way analysis of variance are shown in Figs. 6.

Rice. 5. Window One-Way ANOVA Analysis Package excel

Rice. Fig. 6. Strength indicators of parachutes woven from synthetic fibers obtained from different suppliers, descriptive statistics and results of one-way analysis of variance

An analysis of Figure 6 shows that there is some difference between the sample means. The average strength of fibers obtained from the first supplier is 19.52, from the second - 24.26, from the third - 22.84 and from the fourth - 21.16. Is this difference statistically significant? The rupture force distribution is shown in the scatter diagram (Fig. 7). It clearly shows the differences both between groups and within them. If the volume of each group were larger, they could be analyzed using a stem and leaf plot, a box plot, or a normal distribution plot.

Rice. 7. Strength spread diagram of parachutes woven from synthetic fibers obtained from four suppliers

The null hypothesis states that there are no significant differences between the mean strength values: H 0: μ 1 = μ 2 = μ 3 = μ 4. An alternative hypothesis is that there is at least one supplier whose average fiber strength differs from others: H 1: not all μ j are the same ( j = 1, 2, …, With).

Overall Average (See Figure 6) = AVERAGE(D12:D15) = 21.945; to determine, you can also average all 20 original numbers: \u003d AVERAGE (A3: D7). Variance values ​​are calculated Analysis package and are reflected in the table Analysis of variance(see Fig. 6): SSA = 63.286, SSW = 97.504, SST = 160.790 (see column SS tables Analysis of variance figure 6). Averages are calculated by dividing these sums of squares by the appropriate number of degrees of freedom. Because the With= 4, and n= 20, we obtain the following values ​​of the degrees of freedom; for SSA: s - 1= 3; for SSW: n–c= 16; for SST: n - 1= 19 (see column df). Thus: MSA = SSA / ( c - 1)= 21.095; MSW=SSW/( n–c) = 6.094; MST = SST / ( n - 1) = 8.463 (see column MS). F-statistics = MSA / MSW = 3.462 (see column F).

Upper critical value FU, characteristic for F-distribution, is determined by the formula = F. OBR (0.95; 3; 16) = 3.239. Function parameters =F.OBR(): α = 0.05, the numerator has three degrees of freedom, and the denominator is 16. Thus, the calculated F-statistic equal to 3.462 exceeds the upper critical value FU= 3.239, the null hypothesis is rejected (Fig. 8).

Rice. 8. Critical region of analysis of variance at a significance level of 0.05 if the numerator has three degrees of freedom and the denominator is -16

R-value, i.e. the probability that under a true null hypothesis F- statistics not less than 3.46, equal to 0.041 or 4.1% (see column p-value tables Analysis of variance figure 6). Since this value does not exceed the significance level α = 5%, the null hypothesis is rejected. Moreover, R-value indicates that the probability of finding such or a large difference between the mathematical expectations of the general populations, provided that they are actually the same, is 4.1%.

So. There is a difference between the four sample means. The null hypothesis was that all the mathematical expectations of the four populations are equal. Under these conditions, a measure of the total variability (i.e. total SST variation) of the strength of all parachutes is calculated by summing the squared differences between each observation Xij and overall average . Then the total variation was divided into two components (see Fig. 1). The first component was the intergroup variation in SSA, and the second was the intragroup variation in SSW.

What explains the variability in the data? In other words, why aren't all observations the same? One reason is that different firms supply fibers with different strengths. This partly explains why the groups have different expected values: the stronger the effect of the experimental conditions, the greater the difference between the mean values ​​of the groups. Another reason for data variability is the natural variability of any process, in this case the production of parachutes. Even if all the fibers were purchased from the same supplier, their strength would not be the same, all other things being equal. Since this effect appears in each of the groups, it is called within-group variation.

The differences between the sample means are called the intergroup variation of the SSA. Part of the intragroup variation, as already mentioned, is explained by the fact that the data belong to different groups. However, even if the groups were exactly the same (i.e., the null hypothesis would be true), there would still be intergroup variation. The reason for this lies in the natural variability of the parachute manufacturing process. Since the samples are different, their sample means differ from each other. Therefore, if the null hypothesis is true, both between-group and within-group variability are estimates of population variability. If the null hypothesis is false, the between-group hypothesis will be larger. It is this fact that underlies F-criteria for comparing the differences between the mathematical expectations of several groups.

After performing one-way ANOVA and finding a significant difference between firms, it remains unknown which of the suppliers is significantly different from the others. We only know that the mathematical expectations of populations are not equal. In other words, at least one of the mathematical expectations differs significantly from the others. To determine which provider is different from the others, you can use Tukey procedure, which uses pairwise comparison between providers. This procedure was developed by John Tukey. Subsequently, he and C. Cramer independently modified this procedure for situations in which sample sizes differ from each other.

Multiple comparison: Tukey-Kramer procedure

In our scenario, one-way analysis of variance was used to compare the strength of parachutes. Having found significant differences between the mathematical expectations of the four groups, it is necessary to determine which groups differ from each other. Although there are several ways to solve this problem, we will only describe the Tukey-Kramer multiple comparison procedure. This method is an example of post hoc comparison procedures, since the hypothesis to be tested is formulated after data analysis. The Tukey-Kramer procedure allows you to simultaneously compare all pairs of groups. At the first stage, the differences are calculated Xj – Xj ’ , Where j ≠j’ , between mathematical expectations s(s – 1)/2 groups. Critical Span Tukey-Kramer procedure is calculated by the formula:

Where Q U- the upper critical value of the distribution of the studentized range, which has With degrees of freedom in the numerator and n - With degrees of freedom in the denominator.

If the sample sizes are not the same, the critical range is calculated for each pair of mathematical expectations separately. At the last stage, each s(s – 1)/2 pairs of mathematical expectations is compared with the corresponding critical range. The elements of a pair are considered to be significantly different if the modulus of the difference | Xj – Xj ’ | between them exceeds the critical range.

Let us apply the Tukey-Cramer procedure to the problem of the strength of parachutes. Since the parachute company has four suppliers, 4(4 – 1)/2 = 6 pairs of suppliers should be tested (Figure 9).

Rice. 9. Pairwise comparisons of sample means

Since all groups have the same volume (i.e. all nj = nj ’ ), it is sufficient to calculate only one critical range. To do this, according to the table ANOVA(Fig. 6) we determine the value of MSW = 6.094. Then we find the value Q U at α = 0.05, With= 4 (number of degrees of freedom in the numerator) and n- With= 20 – 4 = 16 (the number of degrees of freedom in the denominator). Unfortunately, I did not find the corresponding function in Excel, so I used the table (Fig. 10).

Rice. 10. Critical value of studentized range Q U

We get:

Since only 4.74 > 4.47 (see bottom table in Figure 9), a statistically significant difference exists between the first and second supplier. All other pairs have sample means, which do not allow us to talk about their difference. Consequently, the average strength of parachutes woven from fibers purchased from the first supplier is significantly less than that of the second.

Necessary conditions for one-way analysis of variance

When solving the problem of the strength of parachutes, we did not check whether the conditions are met under which one can use the one-factor F-criterion. How do you know if you can apply single-factor F-criterion in the analysis of specific experimental data? Single factor F The -test can only be applied if three basic assumptions are met: the experimental data must be random and independent, have a normal distribution, and their variances must be the same.

The first guess is randomness and data independence- should always be done, since the correctness of any experiment depends on the randomness of the choice and / or the randomization process. To avoid distorting the results, it is necessary that the data be extracted from With populations randomly and independently of each other. Similarly, the data should be randomly distributed over With levels of the factor of interest to us (experimental groups). Violation of these conditions can seriously distort the results of the analysis of variance.

The second guess is normality- means that the data is drawn from normally distributed populations. As for t-criterion, one-way analysis of variance based on F-criterion is relatively insensitive to the violation of this condition. If the distribution is not too far from normal, the significance level F-criterion changes little, especially if the sample size is large enough. If the condition of the normal distribution is seriously violated, it should be applied.

The third guess is uniformity of dispersion- means that the variances of each general population are equal to each other (i.e. σ 1 2 = σ 2 2 = … = σ j 2). This assumption allows one to decide whether to separate or pool the within-group variances. If the volumes of the groups are the same, the condition of homogeneity of the variance has little effect on the conclusions obtained using F-criteria. However, if the sample sizes are not the same, violation of the condition of equality of variances can seriously distort the results of the analysis of variance. Thus, one should strive to ensure that the sample sizes are the same. One of the methods for checking the assumption about the homogeneity of the variance is the criterion Levenay described below.

If, of all three conditions, only the uniformity of dispersion condition is violated, a procedure analogous to t-criterion using separate variance (see details). However, if the assumptions of normal distribution and homogeneity of variance are violated at the same time, it is necessary to normalize the data and reduce the differences between the variances or apply a nonparametric procedure.

Leveney's criterion for checking the homogeneity of the variance

Although F- the criterion is relatively resistant to violations of the condition of equality of variances in groups, a gross violation of this assumption significantly affects the level of significance and power of the criterion. Perhaps one of the most powerful is the criterion Levenay. To check the equality of variances With general populations, we will test the following hypotheses:

H 0: σ 1 2 = σ 2 2 = ... = σj 2

H 1: Not all σ j 2 are the same ( j = 1, 2, …, With)

The modified Leveney test is based on the assertion that if the variability in groups is the same, analysis of the variance of the absolute values ​​of the differences between observations and group medians can be applied to test the null hypothesis of equality of variances. So, first you should calculate the absolute values ​​of the differences between the observations and the medians in each group, and then perform a one-way analysis of variance on the obtained absolute values ​​of the differences. To illustrate the Levenay criterion, let us return to the scenario outlined at the beginning of the note. Using the data presented in Fig. 6, we will carry out a similar analysis, but with respect to the modules of the differences in the initial data and medians for each sample separately (Fig. 11).

Analysis of variance

Course work discipline: "System Analysis"

Performer student gr. 99 ISE-2 Zhbanov V.V.

Orenburg State University

Faculty information technologies

Department of Applied Informatics

Orenburg-2003

Introduction

The purpose of the work: to get acquainted with such a statistical method as analysis of variance.

Dispersion analysis (from the Latin Dispersio - dispersion) - statistical method, which allows you to analyze the influence of various factors on the variable under study. The method was developed by the biologist R. Fisher in 1925 and was originally used to evaluate experiments in crop production. Later, the general scientific significance of dispersion analysis for experiments in psychology, pedagogy, medicine, etc., became clear.

The purpose of the analysis of variance is to test the significance of the difference between the means by comparing the variances. The variance of the measured attribute is decomposed into independent terms, each of which characterizes the influence of a particular factor or their interaction. The subsequent comparison of such terms allows us to evaluate the significance of each factor under study, as well as their combination /1/.

If the null hypothesis is true (about the equality of means in several groups of observations selected from the general population), the estimate of the variance associated with intragroup variability should be close to the estimate of intergroup variance.

When conducting market research, the question of comparability of results often arises. For example, when conducting surveys on the consumption of a certain product in different regions of the country, it is necessary to draw conclusions on how the survey data differ or do not differ from each other. It does not make sense to compare individual indicators, and therefore the procedure for comparison and subsequent assessment is carried out according to some average values ​​and deviations from this average assessment. The variation of the trait is being studied. Variance can be taken as a measure of variation. Dispersion σ 2 is a measure of variation, defined as the average of the deviations of a feature squared.

In practice, tasks of a more general nature often arise - the tasks of checking the significance of differences in the averages of several sample samples. For example, it is required to evaluate the effect of various raw materials on the quality of products, to solve the problem of the effect of the amount of fertilizers on the yield of agricultural products.

Sometimes analysis of variance is used to establish the homogeneity of several populations (the variances of these populations are the same by assumption; if the analysis of variance shows that the mathematical expectations are the same, then the populations are homogeneous in this sense). Homogeneous populations can be combined into one and thus obtain more complete information about it, and therefore more reliable conclusions /2/.

1 Analysis of variance

1.1 Basic concepts of analysis of variance

In the process of observing the object under study, the qualitative factors change arbitrarily or in a predetermined way. A particular implementation of a factor (for example, a specific temperature regime, selected equipment or material) is called the factor level or processing method. An ANOVA model with fixed levels of factors is called model I, a model with random factors is called model II. By varying the factor, one can investigate its effect on the magnitude of the response. Currently general theory analysis of variance developed for models I.

Depending on the number of factors that determine the variation of the resulting feature, analysis of variance is divided into single-factor and multi-factor.

The main schemes for organizing initial data with two or more factors are:

Cross-classification, characteristic of models I, in which each level of one factor is combined with each gradation of another factor when planning an experiment;

Hierarchical (nested) classification, characteristic of model II, in which each randomly chosen value of one factor corresponds to its own subset of values ​​of the second factor.

If the dependence of the response on qualitative and quantitative factors is simultaneously investigated, i.e. factors of mixed nature, then covariance analysis is used /3/.

Thus, these models differ from each other in the way of choosing the levels of the factor, which, obviously, primarily affects the possibility of generalizing the experimental results obtained. For analysis of variance in single-factor experiments, the difference between these two models is not so significant, but in multivariate analysis of variance it can be very important.

When conducting an analysis of variance, the following statistical assumptions must be met: regardless of the level of the factor, the response values ​​have a normal (Gaussian) distribution law and the same variance. This equality of dispersions is called homogeneity. Thus, changing the processing method affects only the position random variable response, which is characterized by the mean or median. Therefore, all response observations belong to the shift family of normal distributions.

The ANOVA technique is said to be "robust". This term, used by statisticians, means that these assumptions can be violated to some extent, but despite this, the technique can be used.

When the law of distribution of response values ​​is unknown, nonparametric (most often rank) methods of analysis are used.

The analysis of variance is based on the division of the variance into parts or components. The variation due to the influence of the factor underlying the grouping is characterized by the intergroup dispersion σ 2 . It is a measure of the variation of partial means over groups

around the overall average and is determined by the formula: ,

where k is the number of groups;

n j is the number of units in the j-th group;

- private average for the j-th group; - the total average for the population of units.

The variation due to the influence of other factors is characterized in each group by the intragroup dispersion σ j 2 .

.

Between total variance σ 0 2 , intragroup variance σ 2 and intergroup variance

1.2 One-way analysis of variance

The one-factor dispersion model has the form:

x ij = μ + F j + ε ij , (1)

where x ij is the value of the variable under study, obtained on i-th level factor (i=1,2,...,m) c j-th ordinal number (j=1,2,...,n);

F i is the effect due to the influence of the i-th level of the factor;

ε ij is a random component, or a disturbance caused by the influence of uncontrollable factors, i.e. variation within a single level.

Basic prerequisites for analysis of variance:

The mathematical expectation of the perturbation ε ij is equal to zero for any i, i.e.

M(ε ij) = 0; (2)

Perturbations ε ij are mutually independent;

The variance of the variable x ij (or perturbation ε ij) is constant for

any i, j, i.e.

D(ε ij) = σ 2 ; (3)

The variable x ij (or perturbation ε ij) has a normal law

distributions N(0;σ 2).

The influence of factor levels can be either fixed or systematic (Model I) or random (Model II).

Let, for example, it is necessary to find out whether there are significant differences between batches of products in terms of some quality indicator, i.e. check the impact on the quality of one factor - a batch of products. If all batches of raw materials are included in the study, then the influence of the level of such a factor is systematic (model I), and the findings are applicable only to those individual batches that were involved in the study. If we include only a randomly selected part of the parties, then the influence of the factor is random (model II). In multifactorial complexes, a mixed model III is possible, in which some factors have random levels, while others are fixed.

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

Based on the analysis of variance, the following is performed:

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

2. assessment of the reliability of factor interactions;

3. estimation of partial differences between pairs of means.

The application of dispersion analysis is based on the law of decomposition of dispersions (variations) of a feature into components.

The general variation D o of the effective feature during grouping can be decomposed into the following components:

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

2. for residual(intragroup) D B , not associated with a grouping feature.

The ratio between these indicators is expressed as follows:

D o \u003d D m + D in. (1.30)

Let's consider application of dispersion analysis on an example.

Suppose you want to prove whether the timing of sowing affects the yield of wheat. 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 yield variation, which is the sum of the squared deviations of the individual trait values ​​from the overall average:

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

To determine the intergroup total variation, which determines the variation of the resulting trait due to the studied factor, it is necessary to know the average values ​​of the resulting trait for each group. This total variation is equal to the sum of the squared deviations of the group means from the total mean of the trait, weighted by the number of population units in each of the groups:

The intragroup total variation is equal to the sum of the squared deviations of the individual trait values ​​from the group averages for each group, summed over all groups of the population.

The influence of the factor on the resulting trait is manifested in the ratio between Dm and Dv: the stronger the influence of the factor on the value of the studied attribute, the more Dm and less Dv.

To conduct an analysis of variance, it is necessary to establish the sources of variation of a feature, the amount of variation by source, and determine the number of degrees of freedom for each component of the variation.

The volume 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 the individual values ​​of a feature from its mean value. Total number degrees of freedom, corresponding to the total sum of squared deviations in the analysis of variance, is decomposed into the components of the variation. Thus, the total sum of squared deviations D о 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 sums of squared deviations and the number of degrees of freedom, we can determine the variances for each component. Let's designate these variances: dm - group and dv - intragroup.

After calculating these variances, we 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 value of 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 feature is 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 judgment probability. The appendix contains a table that allows you to set the possible value of F with the most commonly used probability of judgment: the probability level of the “null hypothesis” is 0.05. Instead of the probabilities of the "null hypothesis", the table can be called a table for the probability of 0.95 of the significance of the influence of the factor. Increasing the level of probability requires a comparison of a higher value of F table.

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

The table of values ​​F table is constructed as follows: the columns of the table indicate the degrees of freedom of variation for a larger variance, and the rows indicate the degrees of freedom for a smaller (intragroup) variance. The value of F is at the intersection of the column and row of the corresponding degrees of freedom of variation.

So, in our example, F f \u003d 21.3 / 3.8 \u003d 5.6. The tabular 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 in the experiment 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, which means that the sowing time has a significant impact on the yield of wheat. The optimal sowing time should be considered the period from May 10 to May 15, since it was during this sowing time that the best yield results were obtained.

We have considered the method of analysis of variance when grouping according to one attribute and random distribution repetitions within the group. However, it often happens that the experimental plot has some differences in soil fertility, etc. Therefore, a situation may arise that more plots of one of the options fall into the best part, and its indicators will be overestimated, and the other option - by the worst part, and the results in this case, of course, will be worse, i.e., underestimated.

In order to exclude variation that is caused by reasons not related to experience, it is necessary to isolate the variance calculated from repetitions (blocks) from the intragroup (residual) variance.

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

D o \u003d D m + D rep + 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 ost \u003d D in - D rep; D rest \u003d 106 - 44 \u003d 62.

For the residual dispersion, the number of degrees of freedom will be 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 values, the effect 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 equal conditions, and the tested options are distributed inside the block in a random order, is called the method of randomized blocks.

With the help of dispersion analysis, it is possible to 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 analysis of variance differs from the two single-factor ones in that it can answer the following questions:

1. 1What is the influence of both factors together?

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

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

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

is the sum of the squared deviations of the individual values ​​from the overall mean.

Variation in the combined influence of sowing time and variety

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

Calculation of variation by the influence of sowing dates only:

Residual variation is defined as the difference between the total variation and the variation in the combined influence of the factors under study:

D rest \u003d D about - D ps \u003d 170 - 96 \u003d 74.

All calculations can be made 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 dates and varieties, on the yield of wheat is significant, since the actual F-criteria for each of the factors significantly exceed the tabular ones found for the corresponding degrees of freedom, and at the same time with a fairly high probability (p = 0.99). The influence of the 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 amount of computational 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 of them, but choose the most significant factors from the point of view of economic analysis.

Often a researcher has to deal with the so-called non-proportional dispersion complexes, i.e., those in which the proportionality of the number of options is not respected.

In such complexes, the variation of the total action of factors is not equal to the sum of the variation by factors and the variation of the combination of factors. It differs by an amount depending on the degree of links between the individual factors that arise 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 particular influences is not equal to the total influence.

One way to bring 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.