**3. Results and Discussion**

Thus, within the framework of this study, a multiple econometric model of the relationship between biogas yield from different types of organic substance (m3/t)—*y* and dry organic substance content (kg/t)—*x*1, and the share of possible methane content in organic substance *x*<sup>2</sup> (m3/t) was built. The initial data were generally accepted indicators of biogas yield and methane content of substrates of plant and animal origin [37,38].

First of all, primary data are calculated, such as the average values of the factor and result characteristic and their dispersion (Tables 1 and 2).

The average values are used (Formula (1)) to generalize the signs of the set of significant signs, to compare these signs in different sets, and to study the patterns and development trends of phenomena. Since the individual values of the average sign for each unit of the population are known, simple arithmetic means are calculated.

$$y = \frac{\sum y}{n} = \frac{7664}{15} = 510.93\tag{1}$$

$$\chi = \frac{\sum x\_1}{n} = \frac{9984}{15} = 665.6$$

$$\infty = \frac{\sum x\_2}{n} = \frac{4538.34}{15} = 302.56$$

The study also used the common tool of dispersion, which is the average square of the deviations of all values of the variable sign from its arithmetic mean. This deviation characterizes the average fluctuations of the sign of the totality caused by individual characteristics of the totality from the average value of the sign (Formulas (2) and (3)):

$$
\delta\_y^2 = \frac{\sum \left(y - y\right)^2}{n} = \frac{2242936.93}{15} = 149529.13\tag{2}
$$

$$
\delta\_{\mathbf{x}\_1}^2 = \frac{\sum \left(\mathbf{x}\_1 - \overline{\mathbf{x}}\_1\right)^2}{n} \frac{1613657.6}{15} = 107577.17
$$

$$
\delta\_{\mathbf{x}\_2}^2 = \frac{\sum \left(\mathbf{x}\_2 - \overline{\mathbf{x}}\_2\right)^2}{n} = \frac{998865.18}{15} = 66591.01
$$

$$
\delta\_y = \sqrt[2]{\delta\_y} = \sqrt{149529.13} = 386.69
$$

$$
\delta\_{\mathbf{x}\_1} = \sqrt[2]{\delta\_{\mathbf{x}\_1}} = \sqrt{107577.17} = 327.99
$$

$$
\delta\_{\mathbf{x}\_2} = \sqrt[2]{\delta \mathbf{x}\_2} = \sqrt{66591.01}
$$

**Table 1.** Estimated data for modeling the relationship between biogas yield and different types of organic mass, source [37,38]; calculations of the authors.


**Table 2.** Estimated data for modeling the relationship between biogas yield and different types of organic mass, source [37,38]; calculations of the authors.



**Table 2.** *Cont.*

In turn, within the framework of this study, the preconditions of correlation analysis were determined:


In order to compare the set with different values of average arithmetic and standard deviation, we determined the coefficient of variation—the ratio of the square deviation to the average value of the variable sign (Formula (4)):

$$V\_y \frac{\delta\_y}{\overline{y}} \times 100 = \frac{386.69}{510.93} \times 100 = 75.68\tag{4}$$

$$V\_{x\_1} = \frac{\delta\_{x\_1}}{\overline{x\_1}} \times 100 = \frac{327.99}{665.6} \times 100 = 49.28$$

$$V\_{x\_2} = \frac{\delta\_{x\_2}}{\overline{x\_2}} \times 100 = \frac{258.05}{302.56} \times 100 = 85.29$$

Variations of the effective and the second factor sign are very large, with Vy, Vx2 > 50%; the variation of the first factor sign is large, as Vx1 is in the range of 21–50%.

The homogeneity of the totality was checked using the Tao-criterion (τ) (Formula (5)):

$$\tau\_{y\_{\max}} = \frac{y\_{\max} - \overline{y}}{\delta\_{\overline{y}}} - \frac{1226 - 510.93}{386.69} = 1.85 \tag{5}$$

$$\tau\_{y\_{\min}} = \frac{|\, \overline{y\_{\min}} - \overline{y} \, |}{\delta\_{\overline{y}}} = \frac{|\, 75 - 510.93 \, |}{386.69} = 1.13$$

$$\tau\_{\mathbf{x}\_{1 \max}} = \frac{\mathbf{x}\_{1 \max} - \overline{\mathbf{x}\_{1}}}{\delta\_{\overline{\mathbf{x}}\_{1}}} = \frac{1000 - 665.6}{327.69} = 1.02$$

$$\tau\_{\mathbf{x}\_{1 \min}} = \frac{|\, \mathbf{x}\_{1 \min} - \mathbf{x}\_{1}|}{\delta\_{\overline{\mathbf{x}}\_{1}}} = \frac{|\, 150 - 665.6|}{327.99} = 1.57$$

$$\tau\_{\mathbf{x}\_{2 \max}} = \frac{\mathbf{x}\_{2 \max} - \overline{\mathbf{x}\_{2}}}{\delta\_{\overline{\mathbf{x}}\_{2}}} = \frac{833.68 - 302.56}{258.05} = 2.06$$

$$\tau\_{\mathbf{x}\_{2 \min}} = \frac{|\, \mathbf{x}\_{2 \min} - \overline{\mathbf{x}\_{2}}|}{\delta\_{\overline{\mathbf{x}}\_{2}}} = \frac{|\, 44 - 302.56|}{258.05} = 1.00$$

Since all *τ* < 3, the totality is defined as homogeneous.

The next step is to describe the relationship between the effective sign and the factors with the help of the matrix regression equation (Formula (6)):

$$y = a\_0 + a\_1 \mathbf{x}\_1 + a\_2 \mathbf{x}\_2 + u \tag{6}$$

Values of unknown parameters *a*0, *a*1, *a*<sup>2</sup> were determined based on the method of least squares (Formula (7))

$$S = \sum\_{i=1}^{n} \left( y - \overline{y} \right)^{2} \to \min \tag{7}$$

and based on the nest equation (Formula (8))

$$S = \sum\_{i=1}^{n} \left( y - (a\_0 + a\_1 x\_1 + \dots + a\_n x\_n) \right)^2 \to \min \tag{8}$$

Thus, based on the method of least squares, the following system of normal equations (Formula (9)) was obtained:

$$\begin{cases} a\_0 n + a\_1 \sum \mathbf{x}\_1 + a\_2 \sum \mathbf{x}\_2 = \sum y \\ a\_0 \sum \mathbf{x}\_1 + a\_1 \sum \mathbf{x}\_1^2 + a\_2 \sum \mathbf{x}\_1 \mathbf{x}\_2 = \sum \mathbf{x}\_1 y \\ a\_0 \sum \mathbf{x}\_2 + a\_1 \sum \mathbf{x}\_1 \mathbf{x}\_2 + a\_2 \sum \mathbf{x}\_2^2 = \sum \mathbf{x}\_2 y \end{cases} \tag{9}$$

The next step is to substitute the values of the unknowns into the system of equations and obtain its solution (Formula (10)):

$$\begin{cases} 15a\_0 + 9984a\_1 + 4538, 34a\_2 = 7664/15 \\ 9984a\_0 + 825000a\_1 + 4061655, 74a\_2 = 6810921/998, 4 \\ 4538, 34a\_0 + 4061655, 7a\_1 + 2371964, 15a\_2 = 379051, 984538, 4 \end{cases} \quad \text{(10)}$$

$$\begin{cases} a\_0 + 665, 6a\_1 + 302, 56a\_2 = 510, 93 \\ a\_0 + 827, 22a\_1 + 406, 82a\_2 = 682, 18 \\ a\_0 + 894, 97a\_1 + 522, 65a\_2 = 835, 12 \end{cases}$$

$$\text{II} - \text{I}, \text{III} -- \text{II} \rightarrow \begin{cases} 162a\_1 + 104a\_2 = 171/162 \\ 67, 74a\_1 + 115, 83a\_2 = 152, 94/67, 74 \end{cases}$$

$$\begin{cases} a\_1 + 0, 65a\_2 = 1, 06 \\ a\_1 + 1, 71a\_2 = 2, 26 \end{cases}$$

$$\text{II} - \text{I} \rightarrow \text{ } \begin{cases} 1, 2 \\ 1, 06a\_1 = \frac{1, 2}{1, 06} \end{cases}$$

$$a\_2 = 1, 13$$

$$\text{Therefore, the values of unknown parameters were obtained (Formally, (11)).}$$

Therefore, the values of unknown parameters were obtained (Formula (11)):

$$a\_2 = 1.13\tag{11}$$

$$a\_1 = 1.06 = 0.65 \times 1.13 = 0.33$$

$$a\_0 = 835.12 - 894.97 \times 0.33 - 522.65 = -51.63$$

According to Formula (11), *a*0—free member of the regression, which has no economic interpretation but contains everything that is not taken into account in the created dependence; *a*1,2—regression coefficients that characterize the proportion of the factor's influence on the result.

Thus, we obtained the regression equation of the following formula (Formula (12)):

$$y = -51.63\_{a\_0} + 0.33\_{a\_1} + 1.13\_{a\_2} \tag{12}$$

The correctness of the regression equation was checked by the following regularity (Formula (13)):

$$u = \sum \overline{y} - \sum y = 0 \tag{13}$$

$$u = 7664 - 7664 = 0$$

Based on the regression equation (Figure 1), we can calculate the coefficient of elasticity *Ei* (Formula (14)), which shows by what percentage the value of the resultant trait will change when the factor trait changes by 1%.

$$E\_i = a\_i \times \frac{\overline{x\_i}}{\overline{y}} \tag{14}$$

$$E\_1 = a\_1 \times \frac{\overline{x\_1}}{\overline{y}} = 0.33 \times \frac{665.6}{510.93} = 0.43\%$$

$$E\_2 = a\_2 \times \frac{\overline{x\_2}}{\overline{y}} = 1.13 \times \frac{302.56}{510.93} = 0.67\%$$

**Figure 1.** Direct regression equitation on the scatter diagram. **Figure 1.** Direct regression equitation on the scatter diagram.

Linear correlation analysis is about determining the closeness or density of the relationship between factors and performance. The closeness of the relationship in the correlation analysis is characterized by the correlation coefficient. Accordingly, the density of the relationship between the factors was estimated using simple correlation coefficients (Formula (15)), partial correlation coefficients (Formula (16)), and multiple correlation coefficients (Formula (17)). Linear correlation analysis is about determining the closeness or density of the relationship between factors and performance. The closeness of the relationship in the correlation analysis is characterized by the correlation coefficient. Accordingly, the density of the relationship between the factors was estimated using simple correlation coefficients (Formula (15)), partial correlation coefficients (Formula (16)), and multiple correlation coefficients (Formula (17)).

× ∑ 1<sup>2</sup> − ∑ <sup>1</sup> × <sup>2</sup>

√1 − <sup>2</sup>

<sup>2</sup> × √ × ∑ <sup>2</sup>

<sup>1</sup> − <sup>2</sup> × 1<sup>2</sup>

<sup>2</sup> × √1 − 1<sup>2</sup>

2

Simple correlation coefficients (15): Simple correlation coefficients (15):

Partial correlation coefficients (16):

1<sup>2</sup> =

$$r\_{yx\_1} = \frac{n \times \sum y \mathbf{x\_1} - \sum \mathbf{x\_1} \times \sum y}{\sqrt{n \times \sum \mathbf{x\_1}^2 - \left(\sum \mathbf{x\_1}\right)^2} \times \sqrt{n \times \sum y^2 - \left(\sum y\right)^2}} = 0.899\tag{15}$$

$$r\_{yx\_2} = \frac{n \times \sum y \mathbf{x\_2} - \sum \mathbf{x\_2} \times \sum y}{\sqrt{n \times \sum \mathbf{x\_2}^2 - \left(\sum \mathbf{x\_2}\right)^2} \times \sqrt{n \times \sum y^2 - \left(\sum y\right)^2}} = 0.983$$

<sup>2</sup> − (∑ 2)

2

= 0.881

= 0.82

(16)

1×<sup>2</sup> =

$$\sigma\_{\mathbf{x}\_1 \mathbf{x}\_2} = \frac{n \times \sum y \mathbf{x}\_1 \mathbf{x}\_2 - \sum \mathbf{x}\_1 \times \mathbf{x}\_2}{\sqrt{n \times \sum \mathbf{x}\_1^2 - \left(\sum \mathbf{x}\_1\right)^2} \times \sqrt{n \times \sum \mathbf{x}\_2^2 - \left(\sum \mathbf{x}\_2\right)^2}} = 0.82$$

Partial correlation coefficients (16):

$$r\_{yx\_1 \times x\_2} = \frac{\delta\_{yx\_1} - \delta\_{x\_2} \times \delta\_{x\_1 x\_2}}{\sqrt{1 - r\_{yx\_2}^2 \times \sqrt{1 - r\_{x\_1 x\_2}^2}}} = 0.881 \tag{16}$$

$$r\_{yx\_2 \times x\_1} = \frac{\delta\_{yx\_2} - \delta\_{x\_1} \times \delta\_{x\_1 x\_2}}{\sqrt{1 - r\_{yx\_1}^2 \times \sqrt{1 - r\_{x\_1 x\_2}^2}}} = 0.98$$

Multiple correlation coefficients (17):

$$R\_{yx\_1x\_2} = \sqrt{\frac{r\_{yx\_1}^2 + r\_{yx\_2}^2 - 2r\_{yx\_1} \times r\_{yx\_2} \times r\mathbf{x}\_{1x\_2}}{1 - r\_{x\_1x\_2}^2}}\tag{17}$$

To determine the extent to which the constructed econometric model is consistent with the empirical information on the basis of which it was constructed, the coefficient of determination, namely, multiple, was used (Formula (18)):

$$D\_{yx\_1x\_2} = R\_{yx\_1x\_2}^2 \times 100 = 0.966^2 \times 100 = 99.24\% \tag{18}$$

Partial was also used (Formula (19)):

$$d\_{yx\_1} = a\_1 \times r\_{yx\_1} \times \frac{\delta\_{x\_1}}{\delta\_y} \times 100 = 0.33 \times 0.899 \times \frac{327.99}{386.69} \times 100 = 25.44\% \tag{19}$$

$$d\_{yx\_2} = a\_2 \times r\_{yx\_2} \times \frac{\delta\_{x\_2}}{\delta\_y} \times 100 = 1.13 \times 0.983 \times \frac{258.05}{386.69} \times 100 = 73.8\%$$

The correctness of the calculation of partial coefficients of determination is checked by regularity (Formula (20)):

$$D\_{yx\_1x\_2} = \sum d\_{yx\_n}\tag{20}$$

$$99.24\% = 25.44\% + 73.8\%$$

In turn, the multiple factor was checked for materiality. To do this, we formed a null hypothesis H0: *R* <sup>2</sup> = 0 (insignificant) and hypothesis Ha: *R* <sup>2</sup> <sup>6</sup><sup>=</sup> 0 (significant). For checking H0, F-criteria (Fisher's) was used (Formula (21)):

$$F\_{R^2} = \frac{\frac{R^2}{p-1}}{\frac{1-R^2}{n-p}} = \frac{\frac{0.992}{3-1}}{\frac{1-0.992}{15-3}} = 788.6\tag{21}$$
