A new measure of consensus with reciprocal preference relations: The correlation consensus degree

A new measure of consensus with reciprocal preference relations: The correlation consensus degree Teresa Gonz´alez-Arteagaa, Roc´ıo de Andr´es Calleb, Francisco Chiclanac aPRESAD Research Group and Multidisciplinary Institute of Enterprise (IME), Faculty of Sciences University of Valladolid, Spain bBORDA Research Unit, PRESAD Research Group and Multidisciplinary Institute of Enterprise (IME), University of Salamanca, Spain cCentre for Computational Intelligence, School of Computer Science and Informatics, Faculty of Technology, De Montfort University, Leicester, UK Abstract The achievement of a ‘consensual’ solution in a group decision making problem depends on experts’ ideas, principles, knowledge, experience, etc. The measurement of consensus has been widely studied from the point of view of different research areas, and consequently different consensus measures have been formulated, although a common characteristic of most of them is that they are driven by the implementation of either distance or similarity functions. In the present work though, and within the framework of experts’ opinions modelled via reciprocal preference relations, a different approach to the measurement of consensus based on the Pearson correlation coefficient is studied. The new correlation consensus degree measures the concordance between the intensities of preference for pairs of alternatives as expressed by the experts. Although a detailed study of the formal properties of the new correlation consensus degree shows that it verifies important properties that are common either to distance or to similarity functions between intensities of preferences, it is also proved that it is different to traditional consensus measures. In order to emphasise novelty, two applications of the proposed methodology are also included. The first one is used to illustrate the computation process and discussion of the results, while the second one covers a real life application that makes use of data from Clinical Decision-Making. Keywords: Reciprocal preference relations, Consensus measure, Pearson correlation coefficient, Concordance opinions measure, Correlation consensus degree. 1. Introduction Consensus reaching is an important component in decision making processes, and indeed it plays a key role in the resolution process of group decision making problems. One of the most significant current discussion in consensus research concerns the measurement and achievement of consensus from both a theoretical and applied points of view. On the one hand, establishing and characterising different methodologies to measure consensus have been addressed from a Social Choice perspective [1, 3, 13]. On the other hand, within the Decision Making Theory framework, modelling group decision making problems in order to reach a higher level of cohesiveness has been managed successfully [15, 32, 34, 38, 39, 65]. Outside of these main areas, it is possible to find other methodologies that use the idea of consensus in different ways to the aforementioned ones, with [41, 46] being representative examples of these methodologies. Email addresses: [email protected] (Teresa Gonz´alez-Arteaga), [email protected] (Roc´ıo de Andr´es Calle), [email protected] (Francisco Chiclana) Preprint submitted to Knowledge-Based Systems. Revised version of KNOSYS-D-15-01671. May 31, 2016 Despite the productive research on this area, consensus measurement is still an open-ended research question because the methodology to use in each case is an essential component of the problem. Up to now most studies on consensus measurement have focused on the use of distance/similarity function based measures and association measures, respectively. Among the distance functions used, and worth highlighting, are the Kemeny, Mahalanobis, Mannhattan, Jacard, Dice and Cosine distance functions [1, 4, 6, 17, 19, 29, 31]. Association measures are less widely used than distance functions but it is also possible to find the use of some of them such as the Kendall’s coefficient, the Goodman-Kruskal’s index and the Spearman’s coefficient [18, 24, 35, 44, 58]. In this paper we focus on establishing a new consensus measure following the tradition of association measures. Our proposal is based on the original statistical correlation concept, the Pearson correlation coefficient. Therefore, this new measure is an alternative to the use of the aforementioned approaches. The Pearson correlation coefficient plays an important role in Statistics and Data Analysis and it is extensively used as a measure of the degree of linear dependence between two variables. It is easy to interpret as well as invariant to certain changes in the variables [52, 55, 57]. Specifically, in this paper the notion of dependence among elements from correlation coefficient as a measure of the cohesiveness between opinions is adopted. This seems natural because the measurement of consensus resembles the notion of a “measure of statistical correlation”, in the sense that the maximum value 1 captures the notion of unanimity as a perfect relationship among agents’ preferences (experts’ preferences follow the same direction), while the minimum value −1 captures the notion of total disagreement (experts’ preferences present a negative relationship). Furthermore, the higher the cohesiveness between experts’ preferences, the more positive correlated the preferences are. Similarly, the lower the cohesiveness between experts’ preferences, the more negative correlated the preferences are. This new consensus measure will be developed within assumptions of experts’ opinions or preferences being expressed by means of reciprocal preference relations, a framework that is currently of interest to the research community in decision theory under uncertainty [7, 27, 28, 45]. Under reciprocal preference relations, on the one hand and as it was mentioned above, the new proposed approach inherits advantages of previous approaches based on traditional distance/similarity and association measures. On the other hand, maximum consensus traditionally represents the case when experts provide the same preference intensities for each possible pair of alternatives. This, though, is not the only possible scenario of maximum consensus. Indeed, the proposal here put forward addresses this issue satisfactorily because maximum possible cohesiveness or consensus between experts’ opinions does not necessary imply that all reciprocal preference relations have to coincide, and therefore all experts do not necessary need to have the same preference intensities in all possible pairs of alternatives. It is sufficient, though, that experts rank alternatives in the same way. To support all these claims, a set of properties verified by the new proposed measure of consensus, the correlation consensus degree, are proved. These properties ensure the suitability of the correlation consensus degree. Furthermore, in order to emphasise novelty, two applications of the proposed methodology are also included. The first one is used to illustrate the computation process and discussion of the results, while the second one covers a real life application that makes use of data from Clinical Decision-Making. The rest of the paper is organised as follows. Section 2 contains a brief overview of the different approaches in literature to measure group cohesiveness. The basic notation and preliminaries are presented in Section 3. Section 4 provides the new approach to consensus measurement based on the Pearson correlation coefficient. In Section 5, properties of the new correlation consensus degree are studied. Section 6 presents two practical applications of the proposed methodology. Finally, some concluding remarks and future research are presented in Section 7. 2 2. Consensus measurement in the literature A considerable amount of literature has been published on measuring and reaching consensus in group decision making problems. Consensus measurement is a prominent and active research subject in several areas such as Social Choice Theory and Decision Making Theory. A brief overview of how this issue has been addressed in recent literature from the aforementioned research areas is provided. From the Social Choice Theory, the first serious discussions and analysis of consensus measurement from an Arrovian perspective emerged with Bosch’s PhD Thesis [13], where both absolute and intrinsic measures of consensus were proposed, analysed and axiomatically characterised. From the point of view of considering consensus among a family of voters, McMorris and Powers [48] characterised consensus rules defined on hierarchies, while Garc´ıa-Lapresta and P´erez-Rom´an [29] focused on how to measure consensus using complete preorders on alternatives and introduced a class of consensus measures based on seven well-known distances. Subsequently, Alcalde-Unzu and Vorstatz in [1] characterised a family of linear and additive consensus measures, whereas in [2] new ways to measure the similarity of preferences in a group of individuals were suggested. Alcantud, de Andr´es Calle and Casc´on [3] studied and characterised a class of consensus measure, called referenced consensus measure, that permits to produce a numerical social evaluation from purely ordinal individual information. This measure has to be specified by means of a voting mechanism and a measure of agreement between profiles of orderings and individual orderings. Moreover, Alcantud, de Andr´es Calle and Casc´on in [5] contributed to the formal and computational analysis of the aforementioned referenced consensus measure by focusing on two relevant and specific cases: the Borda and the Copeland rules under a Kemeny-type measure. There are, however, situations where each member of a population classifies a list of options as either acceptable or non-acceptable; either agree or disagree, etc., and therefore generating a dichotomous preference structure. Under this assumption, Alcantud, de Andr´es Calle and Casc´on [4] proposed the concept of approval consensus measure and gave axiomatic characterisations of two generic classes of such approval consensus measures. Alcantud, de Andr´es Calle and Gonz´alez-Arteaga [6] introduced the use of the Mahalanobis distance for the analysis of the cohesiveness of a group of complete preorders and proved that arbitrary codifications of the preferences are incompatible with their formulation although affine transformations permit to compare profiles on the basis of such proposal. Finally, it is worth mentioning a distance-based approach to measure the degree of consensus considering approval information about alternatives as well as the rankings of them suggested by Erdamar et al. in [25]. From the Decision Making Theory, a considerable amount of contributions have been made since the 1980’s. As such, it is worth mentioning the first preliminary work on reaching consensus and its measurements carried out by Kacprzyk and Fedrizzi [42], in which the concept of “degree of consensus” in the sense of expressing the degree to which “most of” the individuals in a group agree to “almost all of” the options. The point of departure of this paper being that the experts’ opinions are expressed by fuzzy preference relations. Within this framework of preference representation, different consensus measurement based on similarity measures have been put forward by Herrera-Viedma, et al. [37] and Wu and Chiclana [63] for both complete and incomplete information environments. The case when experts’ opinions are expressed by means of linguistic assessments has been extensively studied and it is worth mentioning the works of Ben-Arieh and Chen [12], Cabrerizo, Alonso and Herrera-Viedma [14], Garc´ıa- Lapresta, P´erez-Rom´an [30], Herrera, Herrera-Viedma and Verdegay [36], Herrera-Viedma, et al. [40], P´erez-Asurmendi and Chiclana [53] and Wu, Chiclana and Herrera-Viedma [65]. Finally, models to reach consensus where experts assess their preferences using different preference representation structures (preference orderings, utility functions, multiplicative preference relations and fuzzy preference relations) have also been studied and proposed by Dong and Zhang 3 [23], Fedrizzi et al. [26] and Herrera-Viedma, Herrera and Chiclana [39]. The problem of measuring and reaching consensus with intuitionistic fuzzy preference relations and triangular fuzzy complementary preference relations have also been covered in detail by Wu and Chiclana in [62, 64]. To conclude, Table 1 summarises and classifies the approaches that have been reviewed in this Section. Consensus measures in Social Choice Theory Author(s)/Year Framework Measurement methodology Bosch [13], 2005 Ordinal Inf. Based on different distances McMorris and Powers [48], 2009 Ordinal Inf. Garc´ıa-Lapresta and P´erez-Rom´an [29], 2011 Ordinal Inf. Alcalde and Vorsatz [1], 2013 Ordinal Inf. Alcantud, de Andr´es Calle and Casc´on [3] [5], 2013 Ordinal Inf. Alcantud, de Andr´es Calle and Casc´on [4], 2013 Dichotomous Inf. Alcantud, de Andr´es Calle and Gonz´alez-Arteaga [6], 2013 Ordinal Inf. Erdamar, et al. [25], 2014 Ordinal Inf. Alcalde and Vorsatz [2], 2015 Ordinal Inf. Consensus measures in Decision Making Theory Author(s)/Year Framework Measurement methodology Kacprzyk and Fedrizzi [42], 1988 Fuzzy Inf. Based on collective solution Fedrizzi et al. [26], 2010 Fuzzy Inf. Herrera-Viedma et al. [37], 2007 Incomplete Fuz. Inf. Herrera, Herrera-Viedma and Verdegay [36], 1996 Linguistic Inf. Herrera-Viedma et al. [40], 2005 Linguistic Inf. Cabrerizo, Alonso and Herrera-Viedma [14], 2009 Linguistic Inf. Wu and Chiclana [62–64], 2014 Incomplete Fuz. and Ling. Inf. Garc´ıa-Lapresta, P´erez-Rom´an and Falc´o [30], 2015 Linguistic Inf. Wu, Chiclana and Herrera-Viedma [65], 2015 Incomplete Linguistic Inf. Herrera-Viedma, Herrera and Chiclana [39], 2002 Different Inf. Based on individual solution Ben-Arieh and Chen [12], 2006 Linguistic Inf. Dong and Zhang [23], 2014 Different Inf. Table 1: Summary table of studies related to consensus measures 3. Preliminaries This Section briefly presents the main concepts needed to make the paper self-contained, and as such a short review of the terminology and the concept of fuzzy binary relation are presented. The interested reader is advice to consult the following [7–9, 27, 28, 45, 50, 60]. Definition 1. Let Xbe a non empty set. A fuzzy binary relation Pon Xis a fuzzy subset of the Cartesian product X×Xcharacterised by its membership function µP:X×X−→ [0,1], where µP(x1, x2) = pij represents the strength of the relation between x1and x2. Henceforth, Xis a finite set X={x1...,xn}(n > 2), whose elements will be referred to as alternatives. Abusing notation, on occasions alternative xiwill be represented simply as i for convenience. Definition 2. Areciprocal preference relation on Xis a fuzzy binary relation Pwhere µP(xi, xj) = pij ∈[0,1] represents the partial preference intensity of element iover jand that verifies the following property: pij +pji = 1 ∀xi, xj∈X. In order to realise the meaning of a reciprocal preference relation, we suppose the following common situation: an expert compares two alternatives xiand xj. In this specific context, the 4 expert not only establishes that the alternative xiis preferred to the alternative xj, but also shows her/his intensity of preference between them by means of the value pij. So, the higher pij, the higher the preference intensity of alternative xiover alternative xj. Thus, 0 < pij <0.5 would indicate that xjis preferred to xi. If pij = 0.5 then alternatives xiand xjare equally preferred. When 0.5< pij <1, xiis preferred to xj. Moreover, pij = 0 (resp. pij = 1) indicates that xj(resp. xi) is absolutely preferred to xi(resp. xj). Let Pbe an n×nmatrix that contains all the partial intensity degrees of a reciprocal preference relation on the set X: P=      p11 p12 ··· p1n p21 p22 ··· p2n . . .. . ..... . . pn1pn2··· pnn      , verifying 0 ≤pij ≤1; pij +pji = 1 for i, j ∈ {1, . . . , n}. The set of all these matrices n×n is denoted by Pn×n. Here it is also noticed that a reciprocal preference relation can also be mathematically represented by means of a vector, namely the essential vector of preference intensities. Definition 3. The essential vector of preference intensities,VP, of a reciprocal preference relation P= (pij)n×n∈Pn×nis the vector made up with the n(n−1) 2elements above its main diagonal: VP=p12, p13, . . . , p1n, p23, . . . , p2n, . . . , p(n−1) n= =v1, . . . , vr, . . . , vn(n−1)/2. The reciprocity property of reciprocal preference relations allows the alternative definition of the essential vector of preference intensities of a reciprocal preference relation as the vector composed of the preference values below the main diagonal, VPt= (p21, p31, . . . , pn1, p32, . . . , pn2, . . . , pn(n−1)). 4. A novel measurement of consensus based on the Pearson correlation coefficient Based on the concept of correlation, specifically the Pearson correlation coefficient, this section introduces a new consensus measure for group decision making problems under reciprocal preference relations. First, we recall such a correlation coefficient and its properties as necessary to define the new correlation consensus degree and associated properties. 4.1. Pearson correlation coefficient The measurement of the relationship strength among variables is an important issue in Statistical Analysis, and the Pearson correlation coefficient is a traditional tool used for that purpose [52, 55]. Definition 4. Given a sample of npairs of real values {(x1, y1),...,(xn, yn)}, the Pearson correlation coefficient of the two n-dimensional vectors x= (x1, . . . , xn)and y= (y1, . . . , yn), cor(x,y), is computed as cor(x,y) = n X i=1 (xi−x)(yi−y) v u u t n X i=1 (xi−x)2v u u t n X i=1 (yi−y)2 5 where x=1 n n X i=1 xiand y=1 n n X i=1 yiare the arithmetic means of xand y, respectively. The standard interpretation of the Pearson correlation coefficient states that positive coefficient values point out a positive tendency relationship between xand yi.e., xand yincrease (decrease) in the same direction. Negative correlation coefficient values point out towards a reverse direction between xand y. In addition, the nearer the absolute correlation coefficient value is to 1, the stronger and more linear the tendency is. The Pearson correlation coefficient verifies the following well-known properties [57]: 1. cor(x,y)∈[−1,1] ∀x,y∈Rn. 2. cor(x,y) = cor(y,x)∀x,y∈Rn. 3. cor(x,x)=1∀x∈Rn. 4. If cor(x,y) = 1 then there exists a perfect positive linear correlation between xand y, i.e. ∃a∈R, b ∈R+:y=a·1+b·xwhere 1= (1,...,1) is a vector of nones. Respectively, if cor(x,y) = −1 there exists a perfect negative linear correlation between xand y. 5. Let x0=a·1+b·xand y0=c·1+d·ybe two vectors with a, b, c, d ∈R,band dnon zero and of equal sign (both positive or both negative). Then, cor(x0,y0) = cor(x,y). 4.2. A new consensus measure: Correlation consensus degree From the Social Choice Theory perspective, the measurement of the degree of agreement in a group is associated the range [0,1], with 0 representing total lack of agreement and 1 unanimous agreement [1, 4, 13]. Also, as aforementioned in Section 1, the measurement of the degree of cohesiveness in a group has been based on the notion of distance or similarity between opinions or preferences of the members of such group. In this paper, a new way to measure the degree of consensus in a group based on the Pearson correlation coefficient, the correlation consensus degree (CCD), is proposed within the framework of opinions on a set of elements, alternatives or options being represented by reciprocal preference relations. A set of agents or experts will be represented by a finite subset E={1,2, ..., m}of natural numbers, m≥2. Assume that the mexperts provide their pairwise preferences on a finite set of nalternatives, n≥3, X={x1, ..., xn}using fuzzy preference relations {P(1), . . . , P(m)}. As per Definition 3, the essential vector of preference intensities associated to P(k)will be denoted by VP(k). Definition 5. The correlation consensus degree,CCD, for reciprocal preference relations is a mapping CCD :Pn×n×Pn×n→[0,1] that associates a pair of reciprocal preference relations (P(1), P(2)) the following [0,1]-value: CCD(P(1), P(2)) = 1 2(1 + cor(VP(1) , VP(2) )) .(1) Given P(1), P(2) ∈Pn×n, the elaborated expression of CCD(P(1), P(2)) is CCD(P(1), P(2)) = 1 2         1 + n(n−1)/2 X r=1 v(1) r−VP(1) v(2) r−VP(2)  v u u t n(n−1)/2 X r=1 v(1) r−VP(1) 2v u u t n(n−1)/2 X r=1 v(2) r−VP(2) 2         6 where VP(1) =1 n(n−1)/2 n(n−1)/2 X r=1 v(1) rand VP(2) =1 n(n−1)/2 n(n−1)/2 X r=1 v(2) r.1 Notice that the higher the value of CCD(P(1), P(2)), the more positive correlated the reciprocal preferences of P(1) and P(2) are. The maximum possible value CCD(P(1), P(2)) = 1 implies that cor(VP(1) , VP(2) ) = 1 which, contrary to previous consensus measures based on distance/similarity functions, does not necessarily implies that both reciprocal preference relations coincide. Consequently, CCD could be 1 even in cases when experts provide different preferences, although positive linearly correlated. On the other hand, the lower the value of CCD(P(1), P(2)), the more negative correlated the reciprocal preference intensities are, with CCD(P(1), P(2)) = 0 representing the case when preferences are negative linearly correlated. The following proposition reflects these limit cases: Proposition 1. Let P(1), P(2) ∈Pn×nbe two reciprocal preference relation matrices. Then CCD(P(1), P(2)) = 1 (resp. CCD(P(1), P(2)) = 0) if and only if ∃a∈R,b > 0(resp. b < 0) such that: p(2) ij =a+b·p(1) ij ∀i<j;p(2) ij = (1 −a−b) + b·p(1) ij ∀i>j. Proof. Using Equation (1), we have that CCD(P(1), P(2)) = 1 if and only if cor(VP(1) , VP(2) ) = 1. Property 4 of the Pearson correlation coefficient (Section 4.1) implies that ∃a∈R, b ∈R+such that VP(2) =a·1+b·VP(1) , being 1= (1,...,1) a vector of ones with suitable dimension, in this cases n(n−1)/2, i.e.: p(2) ij =a+b·p(1) ij ∀i < j. When j < i, reciprocity of preferences means that p(2) ij = 1 −p(2) ji = 1 −(a+b·p(1) ji ) = 1 −(a+b·(1 −p(1) ij )) = (1 −a−b) + b·p(1) ij . The proof for the case when CCD(P(1), P(2)) = 0 is obtained accordingly. Notice that if the set of alternatives is small, the experts can easily rank the alternatives and the possibility that the experts do it in a similar way (or opposite way) is high. Then, in this case the absolute value of the correlation coefficient tend to be close to 1. Meanwhile, when the set of alternatives is large, the experts may find it difficult to rank them (see [49]) and the possibility that the experts rank the alternatives in a similar way (or opposite way) is low. Then, in this case it is easy that the absolute value of the correlation coefficient becomes small. The following proposition provides the sufficient condition for the correlation consensus degree to coincide for different pairs of reciprocal preference relations. Proposition 2. Let P(1), P(2) ∈Pn×nbe reciprocal preference relation matrices such that CCD(P(1), P(2)) = 1, then CCD(P, P(1)) = CCD(P, P(2))∀P∈Pn×n. Proof. By Proposition 1, ∃a∈Rand b > 0 such that VP(2) =a·1+b·VP(1) . Applying Property 5 of the Pearson correlation coefficient (see Subsection 4.1) we have that cor(VP, VP(1) ) = cor(VP, VP(2) )∀P∈Pn×nand by Definition 5 it is equivalent to CCD(P, P(1)) = CCD(P, P(2))∀P∈Pn×n. The measurement of the degree of agreement among the preferences expressed by two or more experts can be captured by using a summary measure like the mean of all possible correlation consensus degrees between all different pairs of experts’ reciprocal preference relations. 1VP(i)summarizes the general level of uncertainty of the expert ion the set of alternatives. 7 The use of aggregation functions to merge inputs into a single output has been extensively analysed in literature [11, 28, 33, 43]. In the Decision Making context, the use of aggregation functions to derive the degree of agreement among a group of experts has been justified (see for example [11, 29, 31, 43, 47]). Recall that the main aim of considering aggregation functions is to produce an overall output that can be considered representative of the aggregated values by incorporating desirable properties. The arithmetic mean has been widely investigated and it is considered the most common central tendency aggregation function. All these considerations are used to motivate the definition of the new correlation group consensus measure, CD, within a reciprocal preference relation framework. Definition 6. Let Ebe a group of mexperts with associated fuzzy preference relation relations P(1),...,,P(m)∈Pn×non a set of alternatives X. The group consensus degree among the set of experts is CD(E) = 2 m(m−1) m−1 X k=1 m X l=k+1 CCD(P(k), P(l)). 4.3. Consistency under maximum correlation consensus degree Given a reciprocal preference relation on a set of alternatives, the concept of non-dominance degree introduced by Orlovsky [51] has been extensively used to rank the alternatives [10, 23, 38, 61, 63, 65, 66]. In the following, and in order to improve the understanding of the proposed correlation consensus degree, the consistency of the correlation consensus degree with Orlovsky’s non-dominance degree is proved. Specifically, it is proved that when two reciprocal preference relations have a CCD equal to 1 then their Orlovsky’s non-dominance degree orderings of the set of alternatives coincide. First, the concept of non-dominance degree is provided. Given a reciprocal preference relation on a finite set of alternatives X,P= (pij)n×n∈Pn×n, when pji −pij >0 then alternative xiis dominated by alternative xj. Formally, it can be stated that alternative xiis dominated by alternative xjat degree d(xi, xj) = max{pji −pij,0}. Thus, the value 1 −d(xi, xj) = 1 −max{pji −pij,0}represents the degree of non-dominance of alternative xiby alternative xj. The degree up to which xiis not dominated by any of the elements of Xis known as the non-dominance degree of alternative xi. This is summarised in the following definition. Definition 7. Let P= (pij)n×n∈Pn×nbe a reciprocal preference relation on X. The nondominance degree is a mapping µND :X−→ [0,1] such that µND(xi) = min j:j6=i{1−d(xi, xj)}, where d(xi, xj) = max{pji −pij,0}. The aforementioned non-dominance degree can be used to provide a total ordering of alternatives by means of the following rule: xixj⇔µND(xi)≥µND(xj). Notice that pij −pji =−(pji −pij), and therefore to compute d(xj, xi) = max{pji −pij,0} when j > i, we use d(xj, xi) = max {−(pji −pij),0}. Now we are in disposition of introduce the following result. Proposition 3. Let P(1), P(2) ∈Pn×nbe two reciprocal preference relation matrices such that CCD(P(1), P(2)) = 1 and 2a+b= 1. The non-dominance based orderings of the set of alternatives derived from both reciprocal preference relation matrices are identical. 8 Proof. Let P(1), P(2) ∈Pn×nsuch that CCD(P(1), P(2)) = 1. By Proposition 1, ∃a∈Rand b > 0 such that p(2) ij =a+b·p(1) ij ∀i<jand p(2) ij = 1−p(2) ji = 1−(a+b·p(1) ji )=1−(a+b·(1−p(1) ij )) = (1 −a−b) + b·p(1) ij ∀i < j. 1. Notice that: (a) If i<jthen p(2) ji −p(2) ij = [(1 −a−b) + b·p(1) ji ]−[a+b·p(1) ij ] = (1 −2a−b) + b·(p(1) ji −p(1) ij ) = =b·(p(1) ji −p(1) ij ). (b) If i>jthen p(2) ji −p(2) ij = [a+b·p(1) ji ]−[(1 −a−b) + b·p(1) ij ] = −(1 −2a−b) + b·(p(1) ji −p(1) ij ) = =b·(p(1) ij −p(1) ji ). Thus: ∀i, j :p(2) ji −p(2) ij =b·(p(1) ij −p(1) ji ). 2. Let us denote by µND(1) (xi) and µND(2) (xi) the non-dominance choice degree associated to alternative xiobtained from P(1) and P(2), respectively. It is: µND(2) (xi) = min xj∈Xn1−max{p(2) ji −p(2) ij ,0}o. Because b > 0 we have that p(2) ji −p(2) ij and p(1) ij −p(1) ji are both negative, both positive or both equal to zero. Therefore, it is: max{p(2) ji −p(2) ij ,0}= max{b·(p(1) ij −p(1) ji ),0}=b·max{p(1) ji −p(1) ij ,0}.(2) Let lbe such that µND(1) (xi) = min j:j6=in1−max{p(1) ji −p(1) ij ,0}o= 1 −max{p(1) li −p(1) il ,0}. The following inequalities yield: 1−max{p(1) li −p(1) il ,0} ≤ 1−max{p(1) ji −p(1) ij ,0}for j= 1, . . . , n. They can be re-written equivalently as max{p(1) li −p(1) il ,0} ≥ max{p(1) ji −p(1) ij ,0}for j= 1, . . . , n. Consequently, 1−b·max{p(1) li −p(1) il ,0} ≤ 1−b·max{p(1) ji −p(1) ij ,0}for j= 1, . . . , n. Relation (2) implies that µND(2) (xi) = min j:j6=in1−max{p(2) ji −p(2) ij ,0}o= 1 −max{p(2) li −p(2) il ,0}.(3) 9 Intensities of preferences Diagnoses Patient p12 p13 p14 p15 p23 p24 p25 p34 p35 p45 Schizophrenia 1 0.3 0.2 0.1 0.1 0.4 0.3 0.1 0.5 0.1 0.1 2 0.4 0.3 0.2 0.1 0.5 0.4 0.1 0.5 0.1 0.1 3 0.2 0.2 0.2 0.1 0.3 0.2 0.1 0.3 0.1 0.1 4 0.6 1.0 0.8 0.7 0.9 0.6 0.7 0.3 0.4 0.6 Bipolar disorder 5 0.6 1.0 0.8 0.7 0.9 0.6 0.7 0.3 0.4 0.6 6 0.8 1.0 0.9 0.8 1.0 0.8 0.8 0.2 0.3 0.5 7 0.1 0.1 0.1 0.5 0.5 0.5 0.9 0.5 0.9 0.9 8 0.6 1.0 0.9 0.7 0.9 0.6 0.8 0.3 0.2 0.6 OCD 9 0.1 0.1 0.1 0.5 0.5 0.5 0.9 0.5 0.9 0.9 10 0.2 0.1 0.3 0.5 0.5 0.5 0.8 0.5 0.7 0.9 11 0.1 0.2 0.2 0.5 0.6 0.4 0.9 0.5 0.8 0.9 12 0.1 0.2 0.2 0.5 0.5 0.4 0.8 0.5 0.9 0.9 Table 3: Patients’ essential vector of preference intensities. OCD: Obsessive compulsive disorder. pij is the intensity of preference of alternative iversus alternative j. P(1) P(2) P(3) P(4) P(5) P(6) P(7) P(8) P(9) P(10) P(11) P(12) P(1) 1.00 0.98 0.95 0.97 0.39 0.30 0.43 0.41 0.36 0.35 0.36 0.33 P(2) 1.00 0.97 0.99 0.50 0.39 0.55 0.51 0.26 0.26 0.26 0.23 P(3) 1.00 0.93 0.54 0.42 0.56 0.56 0.24 0.25 0.26 0.23 P(4) 1.00 0.47 0.39 0.53 0.48 0.32 0.30 0.30 0.28 P(5) 1.00 0.95 0.96 0.98 0.28 0.28 0.34 0.29 P(6) 1.00 0.96 0.95 0.33 0.35 0.38 0.33 P(7) 1.00 0.96 0.23 0.25 0.28 0.22 P(8) 1.00 0.26 0.30 0.33 0.27 P(9) 1.00 0.98 0.99 0.99 P(10) 1.00 0.98 0.97 P(11) 1.00 0.99 P(12) 1.00 Table 4: Correlation consensus degree (CCD) between pairs of patients. 7. Concluding remarks and future research Research in the area of consensus measurement has advanced mainly in Social Choice Theory and Theory of Decision Making. In this work, a new consensus measure for reciprocal preference relations based on the classical definition of the Pearson correlation coefficient is studied. This new measure, the correlation consensus degree, pursues the measurement of the concordance between the intensities of pairwise preference values given by experts, decision makers or agents. This work open a new avenue to measure consensus. The correlation consensus degree between two reciprocal preference relations is neither a distance function nor a similarity function unlike the traditional consensus measures studied before. Nevertheless, the given correlation consensus degree verifies important properties that are common either to distances and/or similarities measures as well as additional properties that have been described in this work and that are different to traditional consensus measures properties. The novelty of the proposed correlation consensus measure as well as its application is shown with two examples. The first of the examples is used to illustrate the computation process and discussion of the results, while the second example covers a real life Clinical Decision-Making application. Both examples show the versatility and the applicability of the proposed measurement of consensus to a variety of real situations. 16 A future line of enquiry is the investigation of flexible consensus reaching processes based on the new correlation consensus degree. These processes would allow to produce a consensus solution by an iterative feedback mechanism accommodated to the this specific consensus measurement. We expect to conduct further investigations of these issues and report our findings in the future. Acknowledgment The authors thank the three anonymous reviewers for their valuable comments and recommendations. The authors acknowledge financial support by the Spanish Ministerio de Ciencia e Innovaci´on under Project Project ECO2012–32178 (R. de Andr´es Calle and T. Gonz´alez- Arteaga). References [1] J. Alcalde-Unzu and M. Vorsatz. Measuring the cohesiveness of preferences: An axiomatic analysis. Social Choice and Welfare, 41:965–988, 2013. [2] J. Alcalde-Unzu and M. Vorsatz. Do we agree? measuring the cohesiveness of preferences. Theory and Decision, 80:313–339, 2016. [3] J. C. R. Alcantud, R. de Andr´es Calle, and J. M. Casc´on. Consensus and the act of voting. Studies in Microeconomics, 1(1):1–22, 2013. [4] J. C. R. Alcantud, R. de Andr´es Calle, and J. M. Casc´on. On measures of cohesiveness under dichotomous opinions: Some characterizations of approval consensus measures. Information Sciences, 240:45–55, 2013. [5] J. C. R. Alcantud, R. de Andr´es Calle, and J. M. Casc´on. A unifying model to measure consensus solutions in a society. Mathematical and Computer Modelling, 57:1876–1883, 2013. [6] J. C. R. Alcantud, R. de Andr´es Calle, and T. Gonz´alez-Arteaga. Codifications of complete preorders that are compatible with Mahalanobis consensus measures. In B.De Baets, J. Fodor, and S. Montes, editors, EUROFUSE 2013 Workshop on Uncertainty and Imprecision Modelling in Decision Making, pages 19–26. Ediciones de la Universidad de Oviedo, 2013. [7] B. De Baets, B. V. de Walle, and E. Kerre. Fuzzy preference structures without incomparability. Fuzzy Sets and Systems, 76(3):333–348, 1995. [8] B. De Baets and J. Fodor. Twenty years of fuzzy preference structures. Rivista di Matematica per le Scienze Economiche e Sociali, 20, 1997. [9] B. De. Baets, H. De. Meyer, and B. De. Schuymer. Cyclic evaluation of transitivity of reciprocal relations. Social Choice and Welfare, 26(2):217–238, 2006. [10] E. Barrenechea, J. Fernandez, M. Pagola, F. Chiclana, and H. Bustince. Construction of interval-valued fuzzy preference relations from ignorance functions and fuzzy preference relations. application to decision making. Knowledge-Based Systems, 58:33–44, 2014. [11] G. Beliakov, T. Calvo, and S. James. Consensus measures constructed from aggregation functions and fuzzy implications. Knowledge-Based Systems, 55:1–8, 2014. 17 [12] D. Ben-Arieh and Z. Chen. Linguistic-labels aggregation and consensus measure for autocratic decision making using group recommendations. Systems, Man and Cybernetics, Part A: Systems and Humans, IEEE Transactions on, 36(3):558–568, 2006. [13] R. Bosch. Characterizations of Voting Rules and Consensus Measures. PhD thesis, Tilburg University, 2005. [14] F. J. Cabrerizo, S. Alonso, and E. Herrera-Viedma. A consensus model for group decision making problems with unbalanced fuzzy linguistic information. International Journal of Information Technology &Decision Making, 08(01):109–131, 2009. [15] F.J. Cabrerizo, F. Chiclana, R. Al-Hmouz, A. Morfeq, A.S. Balamash, and E. Herrera- Viedma. Fuzzy decision making and consensus: Challenges. Journal of Intelligent &Fuzzy Systems, 29(3):1109–1118, 2015. [16] S. Chen, B. Ma, and K. Zhang. On the similarity metric and the distance metric. Theoretical Computer Science, 410(24-25):2365 – 2376, 2009. [17] F. Chiclana, J. M. Tapia Garc´ıa, M. J. del Moral, and E. Herrera-Viedma. A statistical comparative study of different similarity measures of consensus in group decision making. Information Sciences, 221:110–123, 2013. [18] W. Cook and L. Seiford. On the Borda–Kendall consensus method for priority ranking problems. Management Science, 28:621–637, 1982. [19] W. D. Cook. Distance-based and ad hoc consensus models in ordinal preference ranking. European Journal of Operational Research, 172(2):369–385, 2006. [20] C. De las Cuevas, W. Pe˜nate, and L. de Rivera. Psychiatric patients’ preferences and experiences in clinical decision-making: Examining concordance and correlates of patients’ preferences. Patient Education and Counseling, 96(2):222–228, 2014. [21] L. F. Degner and J. A. Sloan. Decision making during serious illness: What role do patients really want to play? Journal of Clinical Epidemiology, 45(9):941 – 950, 1992. [22] M. M. Deza and E. Deza. Encyclopedia of Distances. Springer-Verlag, Berlin, 2009. [23] Y. Dong and H. Zhang. Multiperson decision making with different preference representation structures: A direct consensus framework and its properties. Knowledge-Based Systems, 58(0):45 – 57, 2014. [24] E. J. Emond and D. W. Mason. A new rank correlation coefficient with application to the consensus ranking problem. Journal of Multi-Criteria Decision Analysis, 11(1):17–28, 2002. [25] B. Erdamar, J. L. Garc´ıa-Lapresta, D. P´erez-Rom´an, and M. R. Sanver. Measuring consensus in a preference-approval context. Information Fusion, 17:14–21, 2014. [26] M. Fedrizzi, M. Fedrizzi, R. A. Marques Pereira, and M. Brunelli. Preferences and Decisions: Models and Applications, chapter The Dynamics of Consensus in Group Decision Making: Investigating the Pairwise Interactions between Fuzzy Preferences, pages 159–182. Springer Berlin Heidelberg, Berlin, Heidelberg, 2010. [27] J. Fodor and M. Roubens. Fuzzy Preference Modeling and Multicriteria Decision Support. Kluwer, The Netherlands, 1994. 18 [28] J. L. Garc´ıa-Lapresta and B. Llamazares. Aggregation of fuzzy preferences: Some rules of the mean. Social Choice and Welfare, 17(4):673–690, 2000. [29] J. L. Garc´ıa-Lapresta and D. P´erez-Rom´an. Measuring consensus in weak orders. In E. Herrera-Viedma, J. L. Garc´ıa-Lapresta, J. Kacprzyk, M. Fedrizzi, H. Nurmi, and S. Zadrozny, editors, Consensual Processes, volume 267 of Studies in Fuzziness and Soft Computing, pages 213–234. Springer Berlin Heidelberg, 2011. [30] J. L. Garc´ıa-Lapresta, D. P´erez-Rom´an, and E. Falc´o. Consensus reaching processes under hesitant linguistic assessments. Advances in Intelligent Systems and Computing, 322:257– 268, 2015. [31] T. Gonz´alez-Arteaga, J. C. R. Alcantud, and R. de Andr´es Calle. A cardinal dissensus measure based on the Mahalanobis distance. European Journal of Operational Research, 251(2):575 – 585, 2016. [32] J. Gonz´alez-Pach´on and C. Romero. Distance-based consensus methods: a goal programming approach. Omega, 27(3):341–347, 1999. [33] J. Gonz´alez-Pach´on and C. Romero. Aggregation of partial ordinal rankings: an interval goal programming approach. Computers &Operations Research, 28(8):827–834, 2001. [34] J. Gonz´alez-Pach´on and C. Romero. Properties underlying a preference aggregator based on satisficing logic. International Transactions in Operational Research, 22(2):205–215, 2015. [35] L. Goodman and W. Kruskal. Measures of association for cross classifications. In Measures of Association for Cross Classifications, Springer Series in Statistics, pages 2–34. Springer New York, 1979. [36] F. Herrera, E. Herrera-Viedma, and J. L. Verdegay. A model of consensus in group decision making under linguistic assessments. Fuzzy Sets and Systems, 78:73–87, 1996. [37] E. Herrera-Viedma, S. Alonso, F. Chiclana, and F. Herrera. A consensus model for group decision making with incomplete fuzzy preference relations. IEEE Transactions on Fuzzy Systems, 15(5):863–877, 2007. [38] E. Herrera-Viedma, F. J. Cabrerizo, J. Kacprzyk, and W. Pedrycz. A review of soft consensus models in a fuzzy environment. Information Fusion, 17:4–13, 2014. [39] E. Herrera-Viedma, F. Herrera, and F. Chiclana. A consensus model for multiperson decision making with different preference structures. IEEE Transactions on Systems, Man, and Cybernetics - Part A: Systems and Humans, 32(3):394–402, 2002. [40] E. Herrera-Viedma, L. Martinez, F. Mata, and F. Chiclana. A consensus support system model for group decision-making problems with multigranular linguistic preference relations. IEEE Transactions on Fuzzy Systems, 13(5):644–658, 2005. [41] L. Gonz´alez Jaime, E. E. Kerre, M. Nachtegael, and H. Bustince. Consensus image method for unknown noise removal. Knowledge-Based Systems, 70:64 – 77, 2014. [42] J. Kacprzyk and M. Fedrizzi. A “soft” measure of consensus in the setting of partial (fuzzy) preferences. European Journal of Operational Research, 34(3):316 – 325, 1988. [43] J. Kacprzyk, M. Fedrizzi, and H. Nurmi. Group decision making and consensus under fuzzy preferences and fuzzy majority. Fuzzy Sets and Systems, 49(1):21 – 31, 1992. 19 [44] M. Kendall and J. D. Gibbons. Rank Correlation Methods. Edward Arnold, London, Melbourne, Auckland, 1990. [45] B. Llamazares and B. De Baets. Fuzzy strict preference relations compatible with fuzzy orderings. International Journal of Uncertainty, Fuzziness and Knowledge-Based Systems, 18(1):13–24, 2010. [46] C. Lopez-Molina, B. De Baets, and H. Bustince. Twofold consensus for boundary detection ground truth. Knowledge-Based Systems, 98:162 – 171, 2016. [47] F. Mata, L. G Prez, S.-M. Zhou, and F. Chiclana. Type-1 OWA methodology to consensus reaching processes in multi-granular linguistic contexts. Knowledge-Based Systems, 58:11– 22, 2014. [48] F. R. McMorris and R. C. Powers. Consensus rules based on decisive families: The case of hierarchies. Mathematical Social Sciences, 57:333–338, 2009. [49] G. A. Miller. The magical number seven plus or minus two: some limits on our capacity of processing information. Psychological Review, 63:81–97, 1956. [50] H. Nurmi. Approaches to collective decision making with fuzzy preference relations. Fuzzy Sets and Systems, 6(3):249 – 259, 1981. [51] S. A. Orlovsky. Decision-making with a fuzzy preference relation. Fuzzy Sets and Systems, 1(3):155 – 167, 1978. [52] K. Pearson. Notes on the history of correlation. Biometrika, 13(1):25–45, 1920. [53] P. P´erez-Asurmendi and F. Chiclana. Linguistic majorities with difference in support. Applied Soft Computing, 18:196 – 208, 2014. [54] A. Robinson and R. Thomson. Variability in patient preferences for participating in medical decision making: implication for the use of decision support tools. Quality in Health Care, 10(Suppl I):i34–i38, 2001. [55] J. L. Rodgers and W. A. Nicewander. Thirteen ways to look at the correlation coefficient. The American Statistician, 42(1):59–66, 1998. [56] K. L. Rodr´ıguez, C. J. Appelt, G. E. Switzer, A. F. Sonel, and R. M. Arnold. Veterans’ decision-making preferences and perceived involvement in care for chronic heart failure. Heart &Lung: The Journal of Acute and Critical Care, 37(6):440–448, 2008. [57] G. W. Snedecor and W. G. Cochran. Statistical Methods, 6th Edition. The Iowa State University Press, Ames, Iowa, 1967. [58] C. Spearman. The proof and measurement of association between two things. American Journal of Psycology, (15):72–101, 1904. [59] S. T. Tang, F-H. Wen, C-H. Hsieh, W-C. Chou, W.-C. Chang, J-S. Chen, and M-C. Chiang. Preferences for life-sustaining treatments and associations with accurate prognostic awareness and depressive symptoms in terminally ill cancer patients’ last year of life. Journal of Pain and Symptom Management, 51(1):41–51, 2016. [60] T. Tanino. Fuzzy preference orderings in group decision making. Fuzzy Sets and Systems, 12(2):117 – 131, 1984. 20 [61] R. Ure˜na, F. Chiclana, H. Fujita, and E. Herrera-Viedma. Confidence-consistency driven group decision making approach with incomplete reciprocal intuitionistic preference relations. Knowledge-Based Systems, 89:86–96, 2015. [62] J. Wu and F. Chiclana. Multiplicative consistency of intuitionistic reciprocal preference relations and its application to missing values estimation and consensus building. Knowledge- Based Systems, 71:187–200, 2014. [63] J. Wu and F. Chiclana. A social network analysis trust-consensus based approach to group decision-making problems with interval-valued fuzzy reciprocal preference relations. Knowledge-Based Systems, 59:97 – 107, 2014. [64] J. Wu and F. Chiclana. Visual information feedback mechanism and attitudinal prioritisation method for group decision making with triangular fuzzy complementary preference relations. Information Sciences, 279(0):716 – 734, 2014. [65] J. Wu, F. Chiclana, and E. Herrera-Viedma. Trust based consensus model for social network in an incomplete linguistic information context. Applied Soft Computing, 35:827– 839, 2015. [66] Z. Xu and H. Liao. A survey of approaches to decision making with intuitionistic fuzzy preference relations. Knowledge-Based Systems, 80:131–142, 2015. 21 ● ● ● ● ● ● ● ● ●● 0.0 0.2 0.4 0.6 0.8 1.0 0.0 0.2 0.4 0.6 0.8 1.0 VP(1) p12 p13 p14 p15 p23 p24 p25 p34 p35 p45 VP(2) CCD= 0.98 ●●● ● ● ● ● ● ●● 0.0 0.2 0.4 0.6 0.8 1.0 0.0 0.2 0.4 0.6 0.8 1.0 VP(1) p12 p13 p14 p15 p23 p24 p25 p34 p35 p45 VP(3) CCD= 0.95 ● ● ●● ●● ● ● ●● 0.0 0.2 0.4 0.6 0.8 1.0 0.0 0.2 0.4 0.6 0.8 1.0 VP(1) p12 p13 p14 p15 p23 p24 p25 p34 p35 p45 VP(4) CCD= 0.97 ●●● ● ● ● ● ● ●● 0.0 0.2 0.4 0.6 0.8 1.0 0.0 0.2 0.4 0.6 0.8 1.0 VP(2) p12 p13 p14 p15 p23 p24 p25 p34 p35 p45 VP(3) CCD= 0.97 ● ● ●● ●● ● ● ●● 0.0 0.2 0.4 0.6 0.8 1.0 0.0 0.2 0.4 0.6 0.8 1.0 VP(2) p12 p13 p14 p15 p23 p24 p25 p34 p35 p45 VP(4) CCD= 0.99 ● ● ●● ●● ● ● ●● 0.0 0.2 0.4 0.6 0.8 1.0 0.0 0.2 0.4 0.6 0.8 1.0 VP(3) p12 p13 p14 p15 p23 p24 p25 p34 p35 p45 VP(4) CCD= 0.93 Figure 3: Plots of essential vectors of intensities of preferences corresponding to patients diagnosed Schizophrenia (Subsection 6.2) and the best adjusted line. ● ● ● ● ● ● ● ● ● ● 0.0 0.2 0.4 0.6 0.8 1.0 0.0 0.2 0.4 0.6 0.8 1.0 VP(5) p12 p13 p14 p15 p23 p24 p25 p34 p35 p45 VP(6) CCD= 0.95 ● ● ● ● ● ● ● ● ● ● 0.0 0.2 0.4 0.6 0.8 1.0 0.0 0.2 0.4 0.6 0.8 1.0 VP(5) p12 p13 p14 p15 p23 p24 p25 p34 p35 p45 VP(7) CCD= 0.96 ● ● ● ● ● ● ● ● ● ● 0.0 0.2 0.4 0.6 0.8 1.0 0.0 0.2 0.4 0.6 0.8 1.0 VP(5) p12 p13 p14 p15 p23 p24 p25 p34 p35 p45 VP(8) CCD= 0.98 ● ● ● ● ● ● ● ● ● ● 0.0 0.2 0.4 0.6 0.8 1.0 0.0 0.2 0.4 0.6 0.8 1.0 VP(6) p12 p13 p14 p15 p23 p24 p25 p34 p35 p45 VP(7) CCD= 0.96 ● ● ● ● ● ● ● ● ● ● 0.0 0.2 0.4 0.6 0.8 1.0 0.0 0.2 0.4 0.6 0.8 1.0 VP(6) p12 p13 p14 p15 p23 p24 p25 p34 p35 p45 VP(8) CCD= 0.95 ● ● ● ● ● ● ● ● ● ● 0.0 0.2 0.4 0.6 0.8 1.0 0.0 0.2 0.4 0.6 0.8 1.0 VP(7) p12 p13 p14 p15 p23 p24 p25 p34 p35 p45 VP(8) CCD= 0.96 Figure 4: Scatterplots of essential vectors of intensities of preferences corresponding to patients diagnosed Bipolar disorder (subsection 6.2) and the best adjusted line. 22 ● ● ● ●●● ● ● ● ● 0.0 0.2 0.4 0.6 0.8 1.0 0.0 0.2 0.4 0.6 0.8 1.0 VP(9) p12 p13 p14 p15 p23 p24 p25 p34 p35 p45 VP(10) CCD= 0.98 ● ●● ● ● ● ● ● ● ● 0.0 0.2 0.4 0.6 0.8 1.0 0.0 0.2 0.4 0.6 0.8 1.0 VP(9) p12 p13 p14 p15 p23 p24 p25 p34 p35 p45 VP(11) CCD= 0.99 ● ●● ●● ● ● ● ●● 0.0 0.2 0.4 0.6 0.8 1.0 0.0 0.2 0.4 0.6 0.8 1.0 VP(9) p12 p13 p14 p15 p23 p24 p25 p34 p35 p45 VP(12) CCD= 0.99 ● ● ● ● ● ● ● ● ● ● 0.0 0.2 0.4 0.6 0.8 1.0 0.0 0.2 0.4 0.6 0.8 1.0 VP(10) p12 p13 p14 p15 p23 p24 p25 p34 p35 p45 VP(11) CCD= 0.98 ● ● ● ●● ● ● ● ● ● 0.0 0.2 0.4 0.6 0.8 1.0 0.0 0.2 0.4 0.6 0.8 1.0 VP(10) p12 p13 p14 p15 p23 p24 p25 p34 p35 p45 VP(12) CCD= 0.97 ● ●● ● ● ● ● ● ● ● 0.0 0.2 0.4 0.6 0.8 1.0 0.0 0.2 0.4 0.6 0.8 1.0 VP(11) p12 p13 p14 p15 p23 p24 p25 p34 p35 p45 VP(12) CCD= 0.99 Figure 5: Plots of essential vectors of intensities of preferences corresponding to patients diagnosed OCD (Subsection 6.2) and the best adjusted line. 23 ● ● ● ● ● ● ● ● ● ● 0.0 0.2 0.4 0.6 0.8 1.0 0.0 0.2 0.4 0.6 0.8 1.0 VP(1) p12 p13 p14 p15 p23 p24 p25 p34 p35 p45 VP(5) CCD= 0.39 ● ● ● ● ● ● ● ● ● ● 0.0 0.2 0.4 0.6 0.8 1.0 0.0 0.2 0.4 0.6 0.8 1.0 VP(2) p12 p13 p14 p15 p23 p24 p25 p34 p35 p45 VP(6) CCD= 0.39 Figure 6: Plots of essential vectors of intensities of preferences corresponding to patients diagnosed Schizophrenia versus Bipolar disorder (Subsection 6.2) and the best adjusted line. ●●● ● ●● ● ● ●● 0.0 0.2 0.4 0.6 0.8 1.0 0.0 0.2 0.4 0.6 0.8 1.0 VP(1) p12 p13 p14 p15 p23 p24 p25 p34 p35 p45 VP(9) CCD= 0.36 ● ● ● ● ●● ● ● ● ● 0.0 0.2 0.4 0.6 0.8 1.0 0.0 0.2 0.4 0.6 0.8 1.0 VP(2) p12 p13 p14 p15 p23 p24 p25 p34 p35 p45 VP(10) CCD= 0.26 Figure 7: Plots of essential vectors of intensities of preferences corresponding to patients diagnosed Schizophrenia versus OCD (Subsection 6.2) and the best adjusted line. ● ●● ● ●● ● ● ● ● 0.0 0.2 0.4 0.6 0.8 1.0 0.0 0.2 0.4 0.6 0.8 1.0 VP(5) p12 p13 p14 p15 p23 p24 p25 p34 p35 p45 VP(9) CCD= 0.28 ● ● ● ●●● ● ● ● ● 0.0 0.2 0.4 0.6 0.8 1.0 0.0 0.2 0.4 0.6 0.8 1.0 VP(6) p12 p13 p14 p15 p23 p24 p25 p34 p35 p45 VP(10) CCD= 0.35 Figure 8: Plots of essential vectors of intensities of preferences corresponding to patients diagnosed Bipolar disorder versus OCD (Subsection 6.2) and the best adjusted line. 24