EKONOMETRIA ECONOMETRICS (5) 06 Publishing House of Wrocław University of Economics Wrocław 06
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Contents Preface... 7 Marek Walesiak: Visualization of linear ordering results for metric data with the application of multidimensional scaling / Wizualizaca wyników porządkowania liniowego dla danych metrycznych z wykorzystaniem skalowania wielowymiarowego... 9 Radosław Mącik: Visualisation of nominal data practical and theoretical remarks / Wizualizaca danych mierzonych na skali nominalne uwagi praktyczne i teoretyczne... Marcin Pełka, Andrze Dudek: Regression analysis for interval-valued symbolic data versus noisy variables and outliers / Regresa liniowa danych symbolicznych a zmienne zakłócaące i obserwace odstaące... 35 Justyna Brzezińska: A polytomous item response theory models using R / Politomiczne modele teorii odpowiedzi na pozyce testowe w programie R. 43 Maria Straś-Romanowska, Jolanta Kowal, Magdalena Kapała: How to measure spiritual sensitivity at the IT user s workplace? The construction process and method of validation of Spiritual Sensitivity Inventory (SSI) / Jak zmierzyć duchową wrażliwość pracowników IT w miescu pracy? Proces konstrukci i metody walidaci inwentarza wrażliwości duchowe (SSI)... 53 Iwona Dittmann: Open-end debt investment funds and bank deposits in Poland 995-05. A comparison of the a posteriori probability (chance) of failure to achieve the level of aspiration / Dłużne otwarte fundusze inwestycyne oraz depozyty bankowe w Polsce w latach 995- -05. Porównanie a posteriori prawdopodobieństwa (szans) nieosiągnięcia poziomu aspiraci... 77 Jan Kaczmarzyk: Reflecting interdependencies between risk factors in corporate risk modeling using Monte Carlo simulation / Odzwierciedlanie współzależności pomiędzy czynnikami ryzyka w modelowaniu ryzyka działalności gospodarcze przedsiębiorstwa z wykorzystaniem symulaci Monte Carlo... 98 Eliza Khemissi: Problems of monotonicity of some popular risk measures / Problemy monotoniczności pewnych popularnych miar ryzyka... 08
Preface The fifty-second issue of the Econometrics contains eight articles. The first of them, by Marek Walesiak, is related to the visualization of linear ordering results with the application of multidimensional scaling. Radosław Mącik, in his article presents practical and theoretical remarks of the visualisation of nominal data. The article by Marcin Pełka and Andrze Dudek is dedicated to regression analysis for intervalvalued symbolic data. Justyna Brzezińska presents a polytomous item response theory models using R. The next article is related to the measure of spirituality and sensitivity at the workplace. This paper was written by Maria Straś-Romanowska, Jolanta Kowal and Magdalena Kapała. The article by Iwona Dittmann is related to open-end debt investment funds and bank deposits in Poland in the years from 995 to 05. Jan Kaczmarzyk presents in his article reflecting interdependencies between risk factors in corporate risk modelling using the Monte Carlo simulation. The last article, by Eliza Khemissi, concerns the problem of the monotonicity of some popular risk measures. Dear Authors and Reviewers, Econometrics is constantly at the forefront of scientific ournals in Poland. On December 3 rd 05, there appeared a Communication from the Polish Minister of Science and Higher Education regarding the list of scientific ournals. Econometrics obtained 4 points. Apart from that, as a result of the evaluation of scientific ournals conducted by Index Copernicus International, Econometrics achieves higher score from year to year. The grade earned for 04 was 75.77 points (a normalized value of 7.3 pts.). We are constantly making efforts to obtain an even higher evaluation in the years to come. Our purpose is to introduce Econometrics to the A list. For this reason, from next year most of the articles will be published in English. Jozef Dziechciarz Editor In-Chief
EKONOMETRIA ECONOMETRICS (5) 06 ISSN 507-3866 e-issn 449-9994 Marek Walesiak Wrocław University of Economics e-mail: marek.walesiak@ue.wroc.pl VISUALIZATION OF LINEAR ORDERING RESULTS FOR METRIC DATA WITH THE APPLICATION OF MULTIDIMENSIONAL SCALING WIZUALIZACJA WYNIKÓW PORZĄDKOWANIA LINIOWEGO DLA DANYCH METRYCZNYCH Z WYKORZYSTANIEM SKALOWANIA WIELOWYMIAROWEGO DOI: 0.56/ekt.06..0 JEL Classification: C38, C430 Summary: The article discusses the two-step research procedure allowing the visualization of linear ordering results for metric data. In the first step, as a result of the application of multidimensional scaling (see [Borg, Groenen 005; Mair et al. 06]) the visualization of obects in two-dimensional space is obtained. In the next step, the linear ordering of a set of obects is carried out based on the Euclidean distance from the pattern (ideal) obect. The suggested approach expanded the possibilities for the interpretation of the linear ordering results of a set of obects. The article applies the concept of isoquants and the path of development (the shortest way connecting a pattern and an anti-pattern obect) proposed by [Hellwig 98]. The graphical presentation of the linear ordering results based on this concept was possible for two variables only. The application of multidimensional scaling expanded the applicability of the results of linear ordering visualization for m variables. The suggested approach is illustrated by an empirical example with the application of R environment script. Keywords: linear ordering, multidimensional scaling, distance measures, composite measures, R environment. Streszczenie: W artykule zaproponowano dwukrokową procedurę badawczą pozwalaącą na wizualizacę wyników porządkowania liniowego. W pierwszym kroku w wyniku zastosowania skalowania wielowymiarowego (zob. [Borg, Groenen, 005; Mair i in. 06]) otrzymue się wizualizacę obiektów w przestrzeni dwuwymiarowe. W następnym kroku przeprowadza się porządkowanie liniowe zbioru obiektów na podstawie odległości Euklidesa od wzorca rozwou. Zaproponowane podeście rozszerzyło możliwości interpretacyne wyników porządkowania liniowego zbioru obiektów. W artykule wykorzystano koncepcę izokwant i ścieżki rozwou (osi zbioru nakrótsze drogi łączące wzorzec i antywzorzec rozwou) zaproponowaną w pracy [Hellwig 98]. Graficzna prezentaca wyników porządkowania liniowego w te koncepci możliwa była dla dwóch zmiennych. Zastosowanie ska-
0 Marek Walesiak lowania wielowymiarowego rozszerzyło możliwości zastosowania wizualizaci wyników porządkowania liniowego dla m zmiennych. Zaproponowane podeście zilustrowano przykładem empirycznym z zastosowaniem skryptu przygotowanego w środowisku R. Słowa kluczowe: porządkowanie liniowe, skalowanie wielowymiarowe, miary odległości, miary agregatowe, program R.. Introduction The article presents the proposal of the application of multidimensional scaling [Borg, Groenen 005] in linear ordering of a set of obects based on the pattern of development [Hellwig 968]. A two-step research procedure was suggested, which allows the visualization of linear ordering results for metric data. First, following the application of multidimensional scaling, the visualization of obects in twodimensional space is obtained. Next, the linear ordering of a set of obects is carried out based on the Euclidean distance from the pattern of development. The suggested approach is illustrated by an empirical example. The article applies the concept of isoquant and the path of development (the axis of the set the shortest way connecting a pattern and an anti-pattern obect ) proposed by [Hellwig 98]. The graphical presentation of the linear ordering results, based on this concept, was possible for two variables. The application of multidimensional scaling expanded the applicability of linear ordering visualization results for m variables.. The genesis of the concept of the pattern of development and measure of development The first research paper discussing the concept of the pattern of development and the measure of development in English was presented by Professor Zdzisław Hellwig at the UNESCO conference in Warsaw in 967 [Hellwig 967]. The study was published in English in a monograph edited by Z. Gostkowski [Hellwig 97]. The first article analyzing the pattern of development and the measure of development in Polish was published in the ournal Przegląd Statystyczny (The Statistical Review) in 968 [Hellwig 968]. These studies introduced the following terms: stimulants and destimulants, pattern of development, measure of development (distance from the pattern of development). There are two types of pattern obects: a pattern obect (upper pattern obect, ideal obect, upper pole) and an anti-pattern obect (lower pattern obect, anti-ideal obect, lower pole). The coordinates of a pattern obect cover the most preferred preference variable (stimulants, destimulants, nominants) values. The coordinates of an anti-pattern obect cover the least preferred preference variable values.
Visualization of linear ordering results for metric data with the application... It can be stated, without any exaggeration, that Hellwig s idea initiated an avalanche of proposals for the development of linear ordering methods. These modifications aimed at (see [Borys, Strahl, Walesiak 990; Pociecha, Zaąc 990]): a) differentiating the method for the normalization of variable values, b) introducing nominant variables in a set, c) determining the pattern of development (comparative base) in a different way, d) applying various constructions of the composite measure, e) applying fuzzy sets in the construction of the composite measure. Recently the concepts using fuzzy numbers were developed in linear ordering based on the pattern of development (see e.g. [Chen 000; Wysocki 00; Jefmański, Dudek 06]) and taking into account spatial dependencies (see [Antczak 03; Pietrzak 04]) and interval symbolic data (see [Młodak 04]). 3. Linear ordering for metric data based on a pattern obect general procedure The general procedure in linear ordering of the set of obects based on a pattern obect (or an anti-pattern obect) and metric data takes the following form: P A X SDN T N d R, () where: P choice of a complex phenomenon the overriding phenomenon for ordering A set obects, which is not subect to direct measurement; A choice of obects; X selection of variables. Collecting data and the construction of data matrix [ x ] ( x the value of the -th variable on i-th obect); SDN identifying preferential variables (stimulants, destimulants, nominants). M variable is a stimulant (see [Hellwig 98, p. 48]), when for every two of S S S S its observations x, x k referring to obects Ai, A k take x > xk Ai Ak ( means A i obect domination over A k obect). M variable is a destimulant (see [Hellwig 98, p. 48]), when for every two of its observations D D D D x, x k referring to obects Ai, A k take x > xk Ai Ak ( means A k obect domination over A i obect). Therefore, M variable represents a unimodal nominant (see [Borys 984, p. 8]), when for every two of its observations x, x referring to obects A, A ( nom means the nominal level N N of k -th variable): if x, x nom, then N N k w i k N N k N N k i k i N N k i k x > x A A ; if x, x > nom, then x > x A A, T w transformation of nominants into stimulants (required for an anti-pattern obect only). Transformation formulas can be found for example in the study by [Walesiak 0, p. 8]; N normalization of variable values. The review of methods for the normali-
Marek Walesiak zation of variable values is presented in the study [Walesiak 04a]; d i aggregated measure (composite measure) calculation for i-th obect the application of distance measures from a pattern obect using weights; R ordering of obects in accordance with d i value. Table. Selected distance measures from a pattern obect for metric data Name Distance d i Interval Measure of development I [Hellwig 968] Measure of development + d i + d + sd + d i m α z+ z = II [Hellwig 98] ( ) TOPSIS measure [Hwang, Yoon 98] GDM distance [Walesiak, 00] GDM_TOPSIS TOPSIS measure with GDM distance [Walesiak, 04b] + di + di + di GDM + i = m m n α ( z zw )( zw z ) + α ( z zl )( zw zl ) = = l= l iw, m n m n α ( z zl ) α ( zw zl ) = l= = l= GDMi + GDMi + GDMi ( ;] [0; ] [0; ] [0; ] [0; ] i, l =,..., n obect number, =,..., m variable number, z ( zl, z w ) the normalized value of -th variable for the i-th (l-th, w-th) obect, z+ ( z ) the normalized -th coordinate of pattern obect + m (anti-pattern obect), d ( ) i = α z z+ weighted Euclidean distance between i-th obect and = m pattern obect, d ( ) i = α z z = weighted Euclidean distance between i-th obect and anti- + n + n + + pattern obect, d = d i= i, sd = ( di d ) n n, α weight of -th variable ( α [0; ] i= m m and α = or α [0; m] and = α = m ), GDM i ( GDM + i ) GDM distance between = i-th obect and anti-pattern obect (pattern obect), z ( z w = z ) for GDM + i ( GDM w = z+ i ). Source: author s compilation.
Visualization of linear ordering results for metric data with the application... 3 Table presents the chosen distance measures from the pattern of development characterized by the normalized variability interval. The subect literature also offers other distance measures from the pattern (ideal) obect (see e.g. [Grabiński 984; Pawełek 008]). 4. Research procedure allowing the visualization of linear ordering results for the set of obects for metric data The research procedure allowing the visualization of linear ordering results for a set of obects covers the following steps:. The choice of a complex phenomenon in linear ordering which is not subect to an immediate measurement (e.g. the economic development level of countries worldwide, tourism attractiveness level of counties).. Determining the set of obects and the set of variables substantively related to the analyzed complex phenomenon. The variables used to describe the obects are measured on metric scale. Following data collection a data matrix is constructed [ x ] n xm ( x -th variable value for i-th obect; i=,, n obect number, =,, m variable number). 3. Among the variables the following preferential variables are distinguished: stimulants, destimulants, nominants. Nominants are transformed into stimulants. 4. A pattern obect (upper pole of development) and an anti-pattern obect (lower pole of development) are added to the set of obects and result in a data matrix [ x ] nxm ( n= n + ). 5. If the variables describing obects are measured on an interval or ratio scale, they should be comparable using normalization (see [Walesiak, 04a]) and receive a normalized data matrix [ z ] nxm. 6. The distances between obects are calculated and arranged in a distance matrix [ δ ik ]. The following distance measures can be applied in this case (measures taking into account weights of variables) e.g. city-block, Euclidean, GDM (see [Walesiak 0, pp. 3 4]). Multidimensional scaling: f : dik dik is carried out. Multidimensional scaling is the method representing the distance matrix between the obects in m-dimensional space [ ] ik δ into the distance matrix between the obects in q-dimensional space [ ] (q < m) for the purposes of the graphical visualization of relations occurring between the analyzed obects and to specify (interpret) the content of q dimensions. The dimensions cannot be observed directly. They represent latent type of variables, which allow explaining the similarities and differences between the analyzed obects. Due to the possibility for the graphical presentation of linear ordering results, q equals. The iterative procedure in the smacof algorithm was presented in the study [Borg, d ik
4 Marek Walesiak Groenen 005, pp. 04-05]. Finally, the data matrix in two-dimensional space [ v ] nx is obtained. 7. The graphical presentation and the interpretation of the results in a twodimensional (multidimensional scaling results) and one-dimensional space (linear ordering results): in the figure a straight line connects the points determining a pattern and an antipattern obect in the so-called axis of the set in a two-dimensional space (multidimensional scaling results). Isoquants of development are determined based on a pattern obect, e.g. dividing the set axis into four parts allows determining four isoquants. The obects between isoquants present a similar development level. The same development level can be achieved by the obects placed in different points on the same isoquant of development (due to a different configuration of variable values). Such a presentation of the results expands the interpretation of the linear ordering results; + normalized di distances of i-th obect from the pattern of development are calculated in accordance with the formula (cf. [Hellwig 98, p. 6]): d + i = ( v v+ ) = ( v+ v ) =, d + i [0; ], () where: ( v v + ) Euclidean distance between i-th obect and pattern ob- = ect (ideal point co-ordinates), ( v+ v ) Euclidean distance between pattern obect and anti- = pattern obect (anti-ideal point co-ordinates). The obects of the study are ordered by the growing values of distance measure (). The linear ordering results are graphically presented in the figure. 5. Empirical results The empirical study uses the statistical data presented in the article [Gryszel, Walesiak 04], referring to the attractiveness level of 9 Lower Silesian counties. The evaluation of the touristic attractiveness of Lower Silesian counties was performed using 6 metric variables (measured on a ratio scale):
Visualization of linear ordering results for metric data with the application... 5 x beds in hotels per km of a county area, x number of nights spent daily by resident tourists (Poles) per 000 inhabitants of a county, x 3 number of nights spent daily by foreign tourists per 000 inhabitants of a county, x 4 gas pollution emission in tons per km of a county area, x 5 number of criminal offences and crimes against life and health per 000 inhabitants of a county, x 6 number of property crimes per 000 inhabitants of a county, x 7 number of historical buildings per 00 km of a county area, x 8 % of a county forest cover, x 9 % share of legally protected areas within a county area, x 0 number of events as well as cultural and tourist ventures in a county, x number of natural monuments calculated per km of a county area, x number of tourist economic entities per 000 inhabitants of a county (natural and legal persons), x 3 expenditure of municipalities and counties on tourism, culture and national heritage protection as well as physical culture per inhabitant of a county in PLN, x 4 cinema goers per 000 inhabitants of a county, x 5 museum visitors per 000 inhabitants of a county, x 6 number of construction permits (hotels and accommodation buildings, commercial and service buildings, transport and communication buildings, civil and water engineering constructions) issued in a county in 0-0 per km of a county area. The statistical data were collected in 0 and come from the Local Data Bank of the Central Statistical Office of Poland, the data for x 7 variable only were obtained from the Regional Conservation Officer. R environment script was used in this article and prepared in accordance with the research procedure discussed in section 4, which applied the following methodology: x 4, x 5 and x 6 variables take the form of destimulants, x 9 is a nominant (50% level was adopted as the optimal one). The other variables represent stimulants, whereas x 9 nominant was transformed into a stimulant. a pattern and an anti-pattern obect were added to the set of 9 counties. Therefore, the data matrix covers 3 obects described by 6 variables. due to the fact that all variables are metrical, the normalization of variable values was performed using the following method (see [Walesiak 04a; Walesiak, Dudek 06]):
6 Marek Walesiak z = n x i= med ( x ) med, (3) where: x ( z ) value (normalized value) of -th variable for i-th obect, med = med( x ) median for -th variable. i distance matrix [ δ ik ] between obects was calculated using GDM, for which the where: same weights were used (see [Walesiak 0, p. 47]): m m n a a b a a b ik k il kl = = l= l ik, m n m n a ail a bkl = l= = l= δ, ik = δ ik GDM distance measure, ikl,, =,, n obect number, =,, m variable number, aip = z z p for p = k, l, b = z z for r = i, l kr k r + z ( zk, z l ) normalized i-th (k-th, l-th) observation for -th variable, m α weight of -th variable ( α [0; m] and α = m or α [0; ] = m and α = ). = The multidimensional scaling of 3 obects was carried out (9 Lower Silesian counties plus the pattern and the anti-pattern obect) in terms of tourist attractiveness level using the smacofsym function of the smacof package [Mair et al. 06], as a result of which the configuration of 3 obects (points) in a twodimensional space was obtained. Figure illustrates the graphical presentation of the multidimensional scaling results for 3 obects. The pattern obect (30) and the anti-pattern obect (3) were connected by a straight line and the so-called axis of the set was obtained. Four isoquants of development were determined by dividing the axis into four equal parts. The distances of each obect (county) from a pattern obect were calculated in accordance with formula (). Counties were ordered by the growing measure values () and the next four classes of similar counties, regarding their tourist attractiveness, were distinguished. The ordering of 3 obects referring to 9 counties, (4)
Visualization of linear ordering results for metric data with the application... 7 the pattern obect (30) and the anti-pattern obect (3) regarding tourist attractiveness, presented by the growing measure values () are as follows (see Table ). Figure. Graphical presentation of multidimensional scaling results in a two-dimensional space of 3 obects containing 9 counties, pattern obect (30) and anti-pattern obect (3) referring to the Lower Silesian counties tourist attractiveness Source: author s compilation using R program. Table. The ordering of 3 obects regarding tourist attractiveness Obect Name Distance 3 30 Pattern 0.0000000 3 Jeleniogórski 0.94865 5 Kłodzki 0.37578 7 Jelenia Góra 0.343470 9 Wrocław 0.466754
8 Marek Walesiak Table, cont. 3 7 Wałbrzyski 0.5598 6 Świdnicki 0.600063 3 Polkowicki 0.64773 8 Legnica 0.648797 Bolesławiecki 0.6805073 5 Lubański 0.68667 0 Oleśnicki 0.737595 Lubiński 0.7493 4 Trzebnicki 0.7436534 8 Ząbkowicki 0.7476896 6 Lwówecki 0.756893 Jaworski 0.7606505 4 Dzierżoniowski 0.79836 9 Milicki 0.798040 0 Górowski 0.7997443 Legnicki 0.8066449 9 Głogowski 0.808935 5 Wołowski 0.80554 4 Kamiennogórski 0.8837 7 Zgorzelecki 0.836544 Strzeliński 0.83859 Oławski 0.8438694 3 Średzki 0.8767444 6 Wrocławski 0.9558 8 Złotoryski 0.934035 3 Anti-pattern.0000000 Source: author s compilation using R program. The graphical results of the linear ordering for 3 obects covering 9 counties, the pattern obect (30) and the anti-pattern obect (3), in terms of tourist attractiveness, presented by the growing measure values () are presented in Figure. Such a form of presenting the results allows for: the presentation of counties ordering in terms of their tourist attractiveness in accordance with measure () values and in the form of graphical presentation in Figure,
Visualization of linear ordering results for metric data with the application... 9 Figure. Graphical presentation of linear ordering of 3 obects containing 9 counties, pattern obect (30) and anti-pattern obect (3) referring to the Lower Silesian counties tourist attractiveness by the growing measure values () Source: author s compilation using R program. distinguishing the classes of counties (counties between isoquants) presenting the similar level of tourist attractiveness (see Figure ), identifying counties characterized by a similar level of tourist attractiveness, but different regarding their location on the isoquant of development (see Figure ). For example, Zgorzelecki County (7) and Strzeliński County () have a similar level of touristic attractiveness, but a different location on the isoquant of development. A similar situation occurs for Polkowicki County (3) and Legnica County (8). Therefore these counties achieved a similar level of development, however they are characterized by quite different configurations of variable values. 6. Final remarks The article presents the proposal of a research procedure allowing the visualization of linear ordering results for the set of obects by applying multidimensional scaling for this purpose.
0 Marek Walesiak The concept of isoquants and the path of development suggested in the study [Hellwig 98], allow for the graphical presentation of linear ordering results for two variables only. The application of multidimensional scaling extends the possibilities of visualizing linear ordering results for m variables. Following such a solution, the interpretation of linear ordering results was expanded. The proposed approach was illustrated by an empirical example using R environment script [R Development Core Team 06]. It should be borne in mind that the application of multidimensional scaling results in a partial loss of information about the obects. A set of obects is initially presented in the space of m variables. As a result of multidimensional scaling application, the graphical presentation of obects in a two-dimensional space is obtained. In the smacofsym function of the smacof package STRESS- Kruskal s fit measure is used [Borg, Groenen 005, pp. 50 54]. Bibliography Antczak E., 03, Przestrzenny taksonomiczny miernik rozwou [Spatial taxonomic measure of development], Wiadomości Statystyczne, 7, pp. 37 53. Borg I., Groenen P.J.F., 005, Modern Multidimensional Scaling. Theory and Applications, nd Edition, Springer Science+Business Media, New York. Borys T., 984, Kategoria akości w statystyczne analizie porównawcze [Category of Quality in Statistical Comparative Analysis], Prace Naukowe Akademii Ekonomiczne we Wrocławiu, nr 84, Seria: Monografie i Opracowania nr 3, Wydawnictwo Akademii Ekonomiczne we Wrocławiu, Wrocław. Borys T., Strahl D., Walesiak M., 990, Wkład ośrodka wrocławskiego w rozwó teorii i zastosowań metod taksonomicznych [The contribution of the center of Wroclaw in the development of the theory and application of taxonomic methods], [in:] Pociecha J. (red.), Taksonomia teoria i zastosowania, Wydawnictwo Akademii Ekonomiczne w Krakowie, Kraków, pp. 3. Chen C.T., 000, Extensions of the TOPSIS for group decision-making under fuzzy environment, Fuzzy Sets and Systems, 4(), pp. 9. Grabiński T., 984, Wielowymiarowa analiza porównawcza w badaniach dynamiki zawisk ekonomicznych [Multivariate Comparative Analysis in Research Over the Dynamics of Economic Phenomena], Zeszyty Naukowe Akademii Ekonomiczne w Krakowie, Seria specalna: Monografie nr 6, Wydawnictwo Akademii Ekonomiczne w Krakowie, Kraków. Gryszel P., Walesiak M., 04, Zastosowanie uogólnione miary odległości GDM w ocenie atrakcyności turystyczne powiatów Dolnego Śląska [The application of the general distance measure (GDM) in the evaluation of Lower Silesian districts attractiveness], Folia Turistica, nr 3, pp. 7 47. Hellwig Z., 967, Procedure of Evaluating High-Level Manpower Data and Typology of Countries by Means of the Taxonomic Method, COM/WS/9, Warsaw, 9 December, 967 (unpublished UNESCO working paper). Hellwig Z., 968, Zastosowanie metody taksonomiczne do typologicznego podziału kraów ze względu na poziom ich rozwou i strukturę wykwalifikowanych kadr [Procedure of evaluating high level manpower data and typology of countries by means of the taxonomic method], Przegląd Statystyczny, tom 5, z. 4, pp. 307 37. Hellwig Z., 97, Procedure of Evaluating High-Level Manpower Data and Typology of Countries by Means of the Taxonomic Method, [in:] Gostkowski Z. (ed.), Towards a system of Human Re-
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