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Quantitative Methods in Economics METODY ILO ŚCIOWE W BADANIACH EKONOMICZNYCH QUANTITATIVE METHODS IN ECONOMICS Vol. XVI, No. 2 Warsaw University of Life Sciences – SGGW Faculty of Applied Informatics and Mathematics Department of Econometrics and Statistics METODY ILO ŚCIOWE W BADANIACH EKONOMICZNYCH QUANTITATIVE METHODS IN ECONOMICS Volume XVI, No. 2 Warsaw 2015 EDITORIAL BOARD Editor-in-Chief: Bolesław Borkowski Deputy Editor-in-Chief: Hanna Dudek Theme Editors: Econometrics: Bolesław Borkowski Multidimensional Data Analysis: Wiesław Szczesny Mathematical Economy: Zbigniew Binderman Analysis of Labour Market: Joanna Landmessser Financial Engineering: Grzegorz Koszela Statistical Editor: Wojciech Zieli ński Technical Editors: Jolanta Kotlarska, El żbieta Saganowska Language Editor: Agata Kropiwiec Native Speaker: Yochanan Shachmurove Editorial Assistant: Monika Krawiec SCIENTIFIC BOARD Peter Friedrich (University of Tartu, Estonia) Paolo Gajo (University of Florence, Italy) Vasile Glavan (Moldova State University, Moldova) Francesca Greselin (The University of Milano-Bicocca, Italy) Yuriy Kondratenko (Black Sea State University, Ukraine) Vassilis Kostoglou (Alexander Technological Educational Institute of Thessaloniki, Greece) Robert Kragler (University of Applied Sciences, Weingarten, Germany) Karol Kukuła (University of Agriculture in Krakow) Alexander N. Prokopenya (Brest State Technical University, Belarus) Yochanan Shachmurove (The City College of The City University of New York, USA) Mirbulat B. Sikhov (al-Farabi Kazakh National University, Kazakhstan) Ewa Syczewska (Warsaw School of Economics, Poland) Achille Vernizzi (University of Milan, Italy) Andrzej Wiatrak (University of Warsaw, Poland) Dorota Witkowska (University of Lodz, Poland) ISSN 2082 – 792X © Copyright by Department of Econometrics and Statistics WULS – SGGW (Katedra Ekonometrii i Statystyki SGGW) Warsaw 2015, Volume XVI, No. 2 The original version is the paper version Journal homepage: qme.sggw.pl Published by Warsaw University of Life Sciences Press QUANTITATIVE METHODS IN ECONOMICS Vol. XVI, No. 2, 2015 1 CONTENTS 2 Jerzy Korzeniewski – Determining the Number of Clusters 3 for Marketing Binary Data ............................................................................... 7 4 Monika Krawiec, Anna Górska – Granger Causality Tests for Precious Metals 5 Returns ............................................................................................................. 13 6 Justyna Kujawska – Measurement of Healthcare System Efficiency 7 in OECD Countries .......................................................................................... 23 8 Izabela Kurzawa, Jarosław Lira – The Application of Quantile Regression 9 to the Analysis of the Relationships between the Entrepreneurship Indicator 10 and the Water and Sewerage Infrastructure in Rural Areas 11 of Communes in Wielkopolskie Voivodeship ................................................... 33 12 Joanna M. Landmesser, Krzysztof Karpio, Piotr Łukasiewicz – Decomposition 13 of Differences Between Personal Incomes Distributions in Poland ................. 43 14 Jarosław Lira – A Comparison of the Methods of Relative Taxonomy 15 for the Assessment of Infrastructural Development 16 of Counties in Wielkopolskie Voivodeship ...................................................... 53 17 Rafik Nafkha – Assessment and Selection Model for Management System 18 Supporting Small and Medium-sized Enterprises ............................................ 63 19 Maria Parlińska, Iryna Petrovska – The Role of Information Systems 20 in Development of Voivodeships in Poland ..................................................... 73 21 Marcin Rudź – Precise Estimates of Ruin Probabilities ................................................. 80 22 Paweł Sakowski, Robert Ślepaczuk, Mateusz Wywiał – Cross-Sectional Returns 23 from Diverse Portfolio of Equity Indices with Risk Premia Embedded ........... 89 24 Victor Shevchuk – Determinants of the Demand for International Reserves 25 in Ukraine ......................................................................................................... 102 26 Iwona Skrodzka – Development of Knowledge-Based Economy in European Union 27 in 2000–2014..................................................................................................... 113 28 Emilia Tomczyk – Data Vintage in Testing Properties of Expectations ........................ 123 29 Alexandr Trunov – Modernization of Means for Analyses and Solution 30 of Nonlinear Programming Problems .............................................................. 133 31 Tomasz Wójtowicz – Macroeconomic Indicators Forecasts Accuracy 32 and Reaction of Investors on the WSE ............................................................. 142 QUANTITATIVE METHODS IN ECONOMICS Vol. XVI, No. 2, 2015 1 Wojciech Zieliński – A Confidence Interval for Proportion in Finite Population 2 Divided into Two Strata: a Numerical Study ................................................... 152 3 Joanna Żyra – The Influence of Public and Private Higher Education 4 in Poland on the Economic Growth of the Country ......................................... 159 5 6 7 8 QUANTITATIVE METHODS IN ECONOMICS Vol. XVI, No. 2, 2015, pp. 7 – 12 1 DETERMINING THE NUMBER OF CLUSTERS 2 FOR MARKETING BINARY DATA 3 Jerzy Korzeniewski 4 Department of Statistical Methods, University of Lodz 5 e-mail: [email protected] 6 Abstract: In the article a new way of determining the number of clusters was 7 proposed focused on data made up of binary variables. An important 8 application aspect is that the data sets on which the new formula was 9 investigated were generated in the way characteristic for the marketing data 10 following the work of Dimitriadou et al. [2002]. The new formula is a 11 modification of the Ratkowsky-Lance index and proved to be better in some 12 respects than this index, which was the best in the mentioned research. The 13 modification proposed is based on measuring the quality of grouping into the 14 predicted number of clusters and running the same index on the twice smaller 15 set of objects comprising dense regions of the original data set. 16 Keywords: cluster analysis, binary data, number of clusters index, market 17 segmentation 18 INTRODUCTION 19 Predicting of the number of clusters 20 One of very important parts of cluster analysis (unsupervised learning) is to 21 find out how many clusters there should be in a data set. Obviously, this task is 22 closely related to other cluster analysis tasks e.g. selection of variables and grouping 23 of objects, however, the subject of selecting the proper number of clusters has 24 attracted much interest which resulted in dozens of different proposals of indices or 25 stopping rules. Milligan [1985] was probably the first to carry out a thorough 26 investigation of more than two dozens of different indices but the research was 27 concentrated on continuous variables data sets and it took place 30 years ago. Since 28 that time many new proposals were published and the task has been directed to 29 different targets related to e.g. different variable measuring scales. As far the binary 30 variables are concerned a good examination was carried out by Dimitridou et al. 8 Jerzy Korzeniewski 1 [2002]. The conclusion from this research is in favour of the Ratkowsky-Lance index 2 which turned out to be better than other indices. Therefore, in order to carry out 3 a new research on similar data sets this index was applied as the reference point. 4 From a couple of newer proposals, the Fang and Wang index [2012] was also used 5 in this article. 6 Binary marketing data 7 Binary marketing data specificity consists in a number of variables being 8 correlated (or not) to create separate groups of variables. The whole data set consists 9 of a couple of groups of such variables. In this research we followed the scheme 10 suggested by Dimitraidou et al. [2002] in which every data set is described by twelve 11 binary variables composed into four groups of different or equal numbers 12 of variables. An example of such data pattern is presented in Table 1. 13 Table 1. An example of binary marketing data pattern, twelve variables in four groups Group1 Group2 Group3 Group4 V1 V2 V3 V4 V5 V6 V7 V8 V9 V10 V11 V12 Cluster1 H H H H H H L L L L L L Cluster2 L L L L L L H H H H H H Cluster3 L L L H H H H H H L L L Cluster4 H H H L L L L L L H H H Cluster5 L L L H H H L L L H H H Cluster6 H H H L L L H H H L L L 14 Source: Dimitriadou et al. [2002] 15 The idea of this example is to present connections between groups 16 of respondents and groups of questions in a questionnaire. The symbol H stands for 17 the high probability of value 1 on a given variable and the symbol L stands for the 18 low probability of 1. Obviously, the number of variables in each group, their 19 correlation within the group, the level of H and L will be varied (see experiment 20 description for details). 21 INDICES OF THE NUMBER OF CLUSTERS 22 Out of the multitude of the number of clusters indices which one can find in 23 the literature we picked up as the reference point one that came the best in the 24 Dimitriadou research i.e. the Ratkowsky-Lance index given by the formula meanB 25 RL T , (1) k 26 where B stands for the sum of squares between the clusters for each variable, T stands 27 for the total sum of squares for each variable and k stands for the number of groups 28 into which the data has to be previously grouped by means of some grouping method. Determining the number of clusters for
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