*3.2. Statistical Methods*

A two-stage research procedure was used to study the relationship between indicators describing the degree of implementation of Goal 7 of the 2030 Agenda and Green growth strategy regarding the economy's environmental and resource productivity. At the first stage, synthetic measures were calculated based on the indicators in each area concerned, which allowed ordering the studied countries in terms of their performance. The TOPSIS method was used to determine synthetic measures. A detailed description of this method and examples of its application can also be found in the papers [78–86]. This method is often used in the literature to study customer preferences [78–82]. There are also a growing number of its application in research on the level and directions of regional development [84–86]. Its main advantage is the ability to determine the distance from the so-called pattern and the anti-pattern of development, enabling not only to study the similarity of development in relation to the pattern but also to identify objects similar to the so-called anti-pattern. It appears, which is also confirmed by the results of the research presented in this paper, that a large distance of an object from the pattern (in this case, a country) does not mean a high similarity to the so-called anti-pattern. This observation is important for determining the paths of development of the studied objects within the scope of the studied phenomenon or for comparing objects between one another.

The basis of linear ordering is a synthetic measure whose values are estimated based on observations of diagnostic variables describing the examined objects. TOPSIS is a computational technique that belongs to a group of reference methods for which there are two reference points for objects in multidimensional space, i.e., a pattern and an antipattern. The final result of the analysis is a synthetic indicator that creates a ranking of the surveyed objects (in this case, countries). The best object is considered the one with the shortest distance from the pattern and, at the same time, the longest from the anti-pattern.

The determination of synthetic measure in the TOPSIS method is as follows [78,79]:

1. Normalization of variables:

$$z\_{ij} = \frac{\mathbf{x}\_{ij}}{\sqrt{\sum\_{i=1}^{n} \mathbf{x}\_{ij}^{2}}} \tag{1}$$

2. Determination of the coordinates of the *z*+0*j* pattern and the distance of objects from it *<sup>d</sup>*+*i*0:

*z*+0*j* = ⎧⎪⎪⎨⎪⎪⎩ max *i zij* for stimulant variables min *i zij* for destimulant variables *d*+*i*0 = *m*∑*<sup>j</sup>*=<sup>1</sup>*zij* − *z*+*ij* 2 (2)

3. Determination of the coordinates of the *z*−0*j* anti-pattern and the distance of objects from it *<sup>d</sup>*<sup>−</sup>*i*0:

$$z\_{0j}^- = \left\{ \begin{array}{c} \underbrace{\min\limits\_{i}\{z\_{ij}\}}\_{i} \text{ for estimated variables} \\ \underbrace{\max\limits\_{i}\{z\_{ij}\}}\_{i} \text{ for destination variables} \end{array} \, d\_{i0}^- = \sqrt{\sum\_{j=1}^m \left(z\_{ij} - z\_{ij}^-\right)^2} \tag{3}$$

4. Determination of the synthetic measure:

$$\eta\_i = \frac{d\_{i0}^-}{d\_{i0}^+ + d\_{i0}^-} \,\prime \tag{4}$$

with: *qi* ∈ [0; 1], max *i*{*qi*}—best object, min *i*{*qi*}—worst object.

	- • Group I: *qi* ≥ *q* + *Sq*, containing objects (countries) with the highest values of the synthetic measure;
	- • Group II: *q* + *Sq* > *qi* ≥ *q*;
	- • Group III: *q* > *qi* ≥ *q* − *Sq*;
	- • Group IV: *qi* < *q* − *Sq*, containing objects (countries) with the lowest values of the synthetic measure.

The second stage was devoted to examining the relationship between the results obtained in the two areas analyzed. The first step is to set linear correlation coefficients: *r* Pearson (for the value of synthetic measures) and *τ* Kendall (for the positions occupied by the studied countries), describing the dependencies between the determined measures and the results obtained by individual countries [83–89]. The results allowed examining the relationships between the analyzed areas and, importantly, identifying countries where the assumed objectives of the green economy supporting sustainable development are being achieved.

More advanced statistical analysis methods were used to study the relationships between the analyzed areas. The research used a multicriteria taxonomy described in the literature [90–92]. The mathematical algorithm of this method takes place in several stages. A detailed description of this method can be found in [93,94]. The first step requires a transformation of each indicator utilized in the analyses. The paper proposes that the clustering of countries should be carried out using the distances from the pattern in the TOPSIS method (*z*+0*j*) and the anti-pattern (*z*<sup>−</sup>0*j*), which replaces the normalized values of indicators used as a standard for this method. This approach allows countries to be grouped by their similarity in the distance to the pattern (the best object) and the similarity of distance to the anti-pattern (the worst object). In this paper, these distances are defined as baseline distances. The matrices containing information about the baseline distances determined for each indicator analyzed were used to determine two final distance matrices **D<sup>P</sup>** (based on distance from the pattern) and **DAP** (based on distance from the anti-pattern). Euclidean distance was used for this purpose. In the next step, based on the values in the

distance matrices, a threshold values *di* should be defined. The following formula can be utilized for this purpose:

$$d^{\vec{i}} = \min\_{\vec{i}} \max\_{\vec{j}} \{ d\_{\vec{i}\vec{j}} \} \tag{5}$$

The transformation of the **D<sup>A</sup>** and **DAP** distance matrices is carried out. For each indocator, the affinity matrix of (*n* × *n*) dimension is defined. The elements of this matrix: *<sup>c</sup>Kij*(*<sup>i</sup>*, *j* = 1, . . . , *n*) are equal to:

$$c\_{i\bar{j}}^K = 1 \text{ for } d\_{i\bar{j}} \le d \tag{6}$$

$$x\_{i\bar{j}}^K = 0 \text{ for } d\_{i\bar{j}} > d \tag{7}$$

If inequality (6) is satisfied, the objects designated as *i* and *j* are treated as similar. In opposite, if inequality (7) is satisfied, the analyzed objects are deemed as dissimilar. In the second case, the affinity measure of *cij* is equal to zero. Finally, **<sup>C</sup>A**(*n*×*n*) and **<sup>C</sup>AP**(*n*×*n*) affinity matrices are determined. In this case, the following formula is applied in which *cijl* elements of these matrices are equal to the product of relevant elements of **C<sup>A</sup>** and **CAP** matrices for all the analyzed indicators:

$$c\_{i\bar{j}l} = \prod\_{K=1}^{r} c\_{i\bar{j}}^{K} \tag{8}$$

If *cij* = <sup>1</sup>(*<sup>i</sup>*, *j* = 1, . . . , *<sup>n</sup>*), it means that each of *cKij* elements corresponding to it in **C<sup>K</sup>** matrices is equal to one, and if *cij* = 0, if one of the *cKij* elements corresponding to it are equal to zero.

Two analyzed objects (in the paper two countries) are considered to be similar to one another simultaneously on account of all the criteria if they are similar to one another separately on account of those criteria individually. Two objects are treated as dissimilar on account of all the examined criteria if they are not similar to one another, even in terms of one of such criteria. According to this assumption, sometimes it is challenging to find many similar objects in terms of every analyzed indicator.

In the following step, the analyzed objects are divided into typological groups. In the paper, for this purpose, the vector elimination method [87] is used. The procedure in this method, as in the multicriteria taxonomy, involves several stages. In the first step, the final **<sup>C</sup>**(*n*×*n*) affinity matrix is transformed into a **C\***(*<sup>n</sup>*×*n*) dissimilarity matrix. Next, based on **C\*** matrix, the *c*0 column vector is estimated with *n* components. In the second step, the row is eliminated from **C\*** matrix along with a corresponding column for which the*<sup>c</sup>*0 vector component has a maximum value. If the*<sup>c</sup>*0 vector contains several components whose value reaches the maximum, such a row and column are eliminated. The second step of the procedure is repeated until the*<sup>c</sup>*0 vector components are equal to zero. To the first sub-group, objects corresponding to the rows and columns that were not crossed off and remain in **C\*** matrix were assigned.
