Computational procedures, Computer calculations...

Computational procedures

Modern applications of the variance analysis method involve the use of modern computing tools equipped with modern statistical processing packages or spreadsheets. However, this does not mean that a small amount of data relating to simple experimental plans can not be handled manually or with the help of a simple pocket or desktop calculator. In this case, however, the formulas that were presented above for clarifying the meaning of one-way analysis of variance are not entirely suitable, since they somewhat complicate the already not very simple calculations. In manual it makes sense to use the formulas obtained from the simple algebraic transformations already resulted as a result.

So, it is better to convert formula (3.1) to the following form:


It makes sense to rewrite formula (3.2) in this way:


Finally, the formula (3.4) will be better expressed as:


It is not difficult to see that three new formulas (3.6) - (3.8) have common elements. Therefore, it makes no sense to perform all the calculations for each of them separately. It is better to start counting with these common elements, there are only three of them. As a result, the calculations will be much simpler.

In Table. 3.4 summarizes the logic of a single-factor experimental plan and its computational procedures.

Table 3.4

Computational procedures for one-way analysis of variances

Independent variable levels















X 1n


X jn


X kn



Elements of formulas


Dispersion Source

Accepted designation



Degrees of freedom

Experimental error


(2) - (3)

( k n - 1)

Experimental Impact


(3) - (1)

k - 1



(2) - (1)

kn - 1

As you can see, Table. 3.4 consists of three blocks. The upper block summarizes the logic of the single-factor experimental plan, which involves the allocation of one independent variable with an arbitrary number of levels equal to k. Based on this experimental plan, a sample of subjects is formed, including k independent experimental groups for n subjects in each. Based on the results of the experiment with this sample of subjects, the values ​​of the dependent variable X ij are estimated and their total sum and sum of values ​​for each group are calculated. These data are needed at the next stage of the work, presented in the middle block of Table. 3.4, which contains computational symbols or elements of formulas for calculations. They are used in the next stage of work. Finally, the lower block generalizes the main results of the calculations required by the dispersion analysis. The leftmost column contains a list of additive (independent) parts of the total variance into which it decomposes. The second column contains the English-language index names of these sources adopted in mathematical statistics. The third and fourth columns contain formulas for calculating the total squares and the number of degrees of freedom for each dispersion source, and the formulas for the total squares are given in terms of the computational symbols given in the middle part of Table. 3.4. Based on these data, the average squares for each dispersion source are calculated and F -statistics are constructed.

Computer calculations

It is much more efficient to compare two or more samples using standard ANOVA procedures using statistical packages. As an example, consider how to compare several independent samples using the IBM statistical package of SPSS Statistics. A more detailed description of this procedure using specific experimental data is given in paragraph 3.7. Here we confine ourselves to a description of the general scheme of analysis.

First of all, you need to prepare the data correctly. To do this, after running the program, go to the Variables and create only two variables - independent and dependent. For example, if we wanted to process the data presented in Table. 3.3, we would need to create the variables Method (or Group ) and Estimate & quot ;. Variable Method This is an independent variable, and it takes three different meanings. These values ​​must be entered in the Values ​​ as we have done, considering a practical example of using the t -test in paragraph 2.7 (see Figure 2.8). Variable Estimate is a dependent variable. Its values ​​must be entered for each subject of each group by returning to the Data tab.

After all the data has been correctly entered, to perform an analysis of variance in the Analysis you can select the Medium Comparison section, and there is One-way ANOVA analysis ... & quot ;. Since the variance analysis is assigned to the class of general linear models, for carrying out one-way analysis of variance for disjoint samples in the "Analysis" menu, you can select the section General Linear Models & quot ;, and in it - OLM-one-dimensional ... & quot ;. This option provides us with several more possibilities for data analysis, but for the sake of simplicity of the further presentation, we assume that we have chosen the first, simpler version.

So, choosing the Analysis & quot ;, Medium Comparison & quot ;, Univariate Analysis & quot ;, we open the window for selecting variables. It contains three fields. In the large left field, a complete list of variables will be presented. In our case, these are the variables Method and Rate & quot ;. On the right, we see two fields: Dependent Variables List and under it - the "Factor". In the first field, you must enter all the replicas of the dependent variable that are used in the experiment. In our case, this is one variable - the variable Estimate & quot ;. In the second field, you need to specify an independent variable. In our case, this is the Methodic variable.

Click the OK button. After a short pause, a window opens with the results of the variance analysis. These results should be presented in the form of a table in which (in the corresponding columns):

• Dispersion sources - between groups and within a group (first column);

• results of calculating the total squares for these dispersion sources (second column);

• number of degrees of freedom corresponding to these total squares (third column);

• the results of dividing the total squares by the corresponding number of degrees of freedom - the average squares (the fourth column),

• values ​​ F -statistics (fifth column);

• the magnitude of the quantile cut by this statistic, which indicates the level of significance of the result (sixth column).

Similarly, a statistical package STATISTICA can be used for variance analysis.

It should be noted that standard procedures for one-way variance analysis can be applied using Excel's MS spreadsheet. To do this, you need to install a special add-on called the Analysis Pack. & Quot ;. This feature, however, is not available on computers and tablets running Windows RT. It should also be noted that, unlike full statistical packages, the "Analysis package" MS Excel implements only basic analysis of variance analysis, and in particular, it is not possible to evaluate contrasts and perform preliminary statistical tests, which are discussed below.

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