In Walesiak [1993], pp. 44-45 the distance measure was proposed, which can be used for the ordinal data. In the paper the proposal of the general distance measure is given. This measure can be used for data measured in ratio, interval and ordinal scale. The proposal is based on the idea of the generalised correlation coefficient.
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 2005; Mair et al. 2016]) the visualization of objects in two-dimensional space is obtained. In the next step, the linear ordering of a set of objects is carried out based on the Euclidean distance from the pattern (ideal) object. The suggested approach expanded the possibilities for the interpretation of the linear ordering results of a set of objects. The article applies the concept of isoquants and the path of development (the shortest way connecting a pattern and an anti-pattern object) proposed by [Hellwig 1981]. 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.
The choice of the normalization method is one of the steps for constructing a composite indicator for metric data (see, e.g. Nardo et al., 2008, pp. 19-21). Normalization methods lead to different rankings of the set of objects based on composite indicator values. In the article 18 normalization methods and 5 aggregation measures (composite indicators) were taken into account. In the first step the groups of normalization methods, leading to identical rankings of the set of objects, were identified. The considerations included in Table 3 reduce this number to 10 normalization methods. Next, the article discusses the procedure which allows separating groups of normalization methods leading to similar rankings of the set of objects separately for each composite indicator formula. The proposal, based on Kendall's tau correlation coefficient (Kendall, 1955) and cluster analysis, can reduce the problem of choosing the normalization method. Based on the suggested research procedure the simulation results for five composite indicators and ten normalization methods were presented. Moreover, the proposed approach was illustrated by an empirical example. Based on the analysis of the dendrograms three groups of normalization methods were separated. The biggest differences in the results of linear ordering refer to methods n2, n9a against the other normalization methods.
In multidimensional scaling (MDS) carried out on the basis of a metric data matrix (interval, ratio), the main decision problems relate to the selection of the method of normalization of the values of the variables, the selection of distance measure and the selection of MDS model. The article proposes a solution that allows choosing the optimal multidimensional scaling procedure according to the normalization methods, distance measures and MDS model applied. The study includes 18 normalization methods, 5 distance measures and 3 types of MDS models (ratio, interval and spline). It uses two criteria for selecting the optimal multidimensional scaling procedure: Kruskal's Stress-1 fit measure and Hirschman-Herfindahl HHI index calculated based on Stress per point values. The results are illustrated by an empirical example.
scite is a Brooklyn-based organization that helps researchers better discover and understand research articles through Smart Citations–citations that display the context of the citation and describe whether the article provides supporting or contrasting evidence. scite is used by students and researchers from around the world and is funded in part by the National Science Foundation and the National Institute on Drug Abuse of the National Institutes of Health.