Visualisation of linear ordering results using multidimensional scaling – problems and an overview of studies

Marek Walesiak; Grażyna Dehnel; Andrzej Dudek

doi:10.15611/aoe.2025.1.12

Authors

Marek Walesiak Department of Econometrics and Computer Science, Wroclaw University of Economics and Business https://orcid.org/0000-0003-0922-2323
Grażyna Dehnel ✉️ Department of Statistics, Poznań University of Economics and Business
grazyna.dehnel@ue.poznan.pl https://orcid.org/0000-0002-0072-9681
Andrzej Dudek Department of Econometrics and Computer Science, Wroclaw University of Economics and Business https://orcid.org/0000-0002-4943-8703

DOI:

https://doi.org/10.15611/aoe.2025.1.12

Keywords:

aggregate measures, multidimensional scaling, R program, mdsOpt package, hybrid method

Abstract

Aim: The aim of the article was to review the methodological solutions proposed as part of the hybrid method, which combines linear ordering with multidimensional scaling for various types of data. In the first step, after applying multidimensional scaling, it was possible to visualise objects of interest in
a two-dimensional space. In the second step, the objects were linearly ordered according to an aggregate measure based on the Euclidean distance.

Methodology: The general procedure of the hybrid method, which can be used to visualise results of linear ordering for metric, ordinal and interval-valued data, was presented.

Results: The authors highlight the problems associated with the use of multidimensional scaling in linear ordering and how they can be solved. These problems with the application of multidimensional scaling in linear ordering are illustrated by an attempt to rank 27 EU countries in 2021 according to their progress towards reaching the sustainable development goal (SDG7). The article also contains an overview of studies involving the hybrid method.

Implications and recommendations: If the distribution of errors related to the arrangement of individual objects in the scaling space (stress-per-point values – spp) deviates significantly from the uniform distribution, the ranking of objects based on the results of multidimensional scaling is distorted. To solve this, the paper proposes to select the optimal multidimensional scaling procedure considering two criteria: Kruskal’s goodness-of-fit statistic and the Herfindahl-Hirschman index, calculated using spp values. The use of the hybrid method is facilitated by the mdsOpt package in R environment.

Originality/value: The hybrid method makes use of the concept of isoquants and the path of development (the shortest line connecting the pattern and anti-pattern of development) proposed by Hellwig (1981). By applying multidimensional scaling one can visualise the results of linear ordering for more than two variables, whereas other linear ordering methods cannot be used to visualise these results.

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