Weighted Rank Correlation measures in Hierarchical Cluster Analysis
Livia Dancelli, Marica Manisera, and Marika Vezzoli

Abstract When the aim is to group rankings, matching-type measures must be used in cluster analysis techniques. Among these, rank-based correlation coefficients, as the Spearman’s ρ , can be considered. To this regard, we think that Weighted Rank Correlation measures are remarkably useful, since they evaluate the agreement between two rankings emphasizing the concordance on top ranks. In this paper, we employ an appropriate Weighted Rank Correlation measure to evaluate the dissimilarity between rankings in a hierarchical cluster analysis, in order to segment subjects expressing their preferences by rankings. An illustrative example on selected rankings shows that the resulting groups contain subjects whose preferences are more similar on the most important ranks. The procedure is then applied to real data from an extensive 2011 survey carried out in the Italian McDonald’s restaurants. Key words: rank-based correlation coefficients, matching-type measures, hierarchical cluster analysis

1 Introduction
Cluster analysis aims at identifying groups of individuals or objects that are similar to each other but are different from individuals in other groups (among others, [4]). This is useful, for example, in market segmentation studies, also when consumers’ preferences are expressed by grades, leading to rankings of products or services provided by individuals. In this situations, standard clustering techniques are not appropriate, because they measure intersubject similarity by distance-type measures (including correlation coefficients) thus requiring quantitative data. Instead, when consumers express their preferences by means of rankings, ordinal data
Livia Dancelli, Marica Manisera, and Marika Vezzoli Department of Quantitative Methods, University of Brescia, C.da S. Chiara, 50 - 25122 Brescia, Italy, e-mail: {dancelli,manisera,vezzoli}@eco.unibs.it

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Livia Dancelli, Marica Manisera, and Marika Vezzoli

are obtained: ranking numbers convey the location/preference of the “product” in a given ordered list and distance-type measures must be replaced by matching-type measures. Among these, rank-based correlation coefficients, as the Spearman’s ρ , can be considered. In particular, we think that Weighted Rank Correlation (WRC) measures are remarkably useful, since they evaluate the agreement between two rankings emphasizing the concordance on top ranks ([1], [2], [6] and references therein). For example, consider 3 consumers A, B and C that ranked 6 aspects of a product attributing “1” to the most important aspect and “6” to the least important one. If the three rankings are: A: [1, 2, 3, 4, 5, 6], B: [1, 2, 3, 4, 6, 5], C: [2, 1, 3, 4, 5, 6] and we want to emphasize top ranks, A and B preferences are more similar than A and C. This is because, with respect to A, B exchanges the lowest ranks while C exchanges the top ones. The Spearman’s ρ gives the same result when computed between A and B and between A and C, while WRC measures capture the greater similarity between A and B. In this paper, we employ an appropriate WRC measure to evaluate the dissimilarity between rankings in a hierarchical cluster analysis, in order to segment consumers when they express their preferences by rankings (see also [5]). Section 2 briefly describes the procedure, while in Section 3 an illustrative example on selected rankings shows that the resulting groups contain subjects whose preferences are more similar on the top ranks. In the same Section, the procedure is then applied to real data involving 1, 909 customers of several Italian McDonald’s restaurants.

2 Hierarchical cluster analysis based on WRC measures
Among the various clustering techniques useful to group p subjects who expressed their preferences by ranking n objects, we focus on the agglomerative hierarchical method [4]. In order to evaluate the dissimilarity between rankings, we use a WRC measure able to emphasize the agreement on top ranks. In particular, we focus on WRC measures related to the Spearman’s ρ adopting weights that are function of both ranks and treat them symmetrically. As far as we know, the indices existing in the literature introduce weights in the simplified formula of ρ (based on the squared Euclidean distance between the rankings). Instead, we introduced weights in the unsimplified formula of ρ (i.e., the Pearson’s product-moment correlation index between ranks), obtaining a new class of indices leading to different results ([1], [2], [3]). In the previous studies, we compared the performance of indices of the two classes. From several points of view, the following WRC measure (attributed to Salama & Quade in [6]) performs well in measuring the agreement between two rankings R : r1 , r2 , · · · , ri , · · · , rn and Q : q1 , q2 , · · · , qi , · · · , qn : ( ) n 1 2 1 ∑ (ri − qi ) ri + qi ρw = 1 − 2 i=1 . (1) n [2i − (n + 1)]2 (n + 1) ∑ i=1 i(n − i + 1)

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To transform the WRC measure (1) in a dissimilarity coefficient d to use in the cluster analysis, we adopt the transformation ([4]): √ ρw + 1 d = 1− . (2) 2 The resulting p× p dissimilarity matrix is then used as input in the agglomerative hierarchical method; in this study the intercluster dissimilarity is based on the furthest neighbor rule (known as complete linkage).

3 An illustrative example and the McDonald’s case study
We start from an illustrative example performed on the 10 rankings reported in Table 1 representing, for example, the judgements given by 10 consumers (or experts) about 6 products (or brands).
Consumer 1 2 3 4 5 Rankings 12345 21345 12346 21346 32145 Consumer 6 7 8 9 10 Rankings 12365 12364 12435 31245 12354

6 6 5 5 6

4 5 6 6 6

Table 1 Rankings of 6 products given by 10 consumers

Two hierarchical cluster analyses were performed on the selected data using: (a) the unweighted Spearman’s ρ , (b) the chosen WRC measure ρw (both transformed according to (2)). Figure 1 shows the dendrograms obtained in the two cases. The two classifications are quite different; note, for example, that consumers 1 and 3 (whose judgements differ only for the two lowest ranks) are clustered together at a very low height when ρw is used (Figure 1b) while they are grouped at a larger height when ρ is used (Figure 1a). Even more evidently, dendrograms highlight the different classification of consumers 1 and 2, differing only on the two top ranks. The hierarchical cluster analysis based on ρw was then applied to real data collected in a 2011 survey. The case study concerns 1909 customers (with children) of several Italian McDonald’s restaurants, who ranked (without ties) six aspects of products and service: neatness, atmosphere, price, ingredient quality, nutritional quality and accessories. For each respondent, personal data were also collected jointly with other information (e.g., customer satisfaction). We obtained four clusters (with 788, 334, 711, 76 members, respectively) grouping customers with rankings more similar on the top ranks. Subjects in cluster 1 pay much attention to nutritional quality, while the least important aspect is accessories; subjects in cluster 2 consider neatness and atmosphere most important, while the least important

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Fig. 1 Dendrograms obtained with hierarchical cluster analyses with dissimilarities based on (a) the unweighted Spearman’s ρ and (b) the WRC measure ρw

aspect is accessories; subjects in group 3 are more interested in ingredient quality and neatness and less interested in accessories and price; in the last group, the most important aspects is accessories and neatness while ingredient and nutritional quality are the least important ones. The 4th group is peculiar, with reference to personal data and other information: for example, they are the least satisfied consumers. More detailed results are available from Authors. Concluding, results from the illustrative example and the application to real data encourage the use of WRC measures instead of unweighted ones when the aim is to cluster subjects with alike preferences on the most important aspects.

References
1. Dancelli, L., Manisera, M., Vezzoli, M.: On some weighted rank correlation measures related to the Spearman’s ρ . WP 363, Dept. of Quantitative Methods, Univ. of Brescia, Italy (2011) 2. Dancelli L., Manisera M., Vezzoli M.: Weighting the Spearman’s Rank Correlation Index. In Book of short papers CLADAG 2011, pp. 1-4. Pavia University Press, Pavia (2011) 3. Dancelli L., Manisera M., Vezzoli M.: On two classes of Weighted Rank Correlation measures deriving from the Spearman’s ρ . Submitted (2012) 4. Kaufman, L., Rousseeuw, P.J.: Finding Groups in Data. New York: Wiley Interscience (1990) 5. Tarsitano, A. : Weighted rank correlation and hierarchical clustering. In Book of short papers CLADAG 2005, pp. 517-521. MUP Editore, Parma (2005) 6. Tarsitano, A.: Comparing the effectiveness of rank correlation statistics. Working Paper n.06/2009, Dept. of Economics and Statistics, Univ. of Calabria, Italy (2009)

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