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Journal ArticleDOI

A cardinal dissensus measure based on the Mahalanobis distance

01 Jun 2016-European Journal of Operational Research (North-Holland)-Vol. 251, Iss: 2, pp 575-585

TL;DR: A new class of distance-based consensus model, the family of the Mahalanobis dissensus measures for profiles of cardinal values, is proposed.

AbstractIn this paper we address the problem of measuring the degree of consensus/dissensus in a context where experts or agents express their opinions on alternatives or issues by means of cardinal evaluations. To this end we propose a new class of distance-based consensus model, the family of the Mahalanobis dissensus measures for profiles of cardinal values. We set forth some meaningful properties of the Mahalanobis dissensus measures. Finally, an application over a real empirical example is presented and discussed.

Topics: Mahalanobis distance (58%)

Summary (3 min read)

1. Introduction

  • In Decision Making Theory and its applications, consensus measurement and its reaching in a society (i.e., a group of agents or experts) are relevant research issues.
  • Notwithstanding the use of different ordinal preference frameworks, the problem of how to measure consensus is an open-ended question in several research areas.
  • They evaluate two public goods with monetary amounts.
  • The Mahalanobis distance plays an important role in Statistics and Data Analysis.
  • In Section 3, the authors set forth the class of the Mahalanobis dissensus measures and their main properties.

2. Notation and definitions

  • This section is devoted to introduce some notation and a new concept in order to compare group cohesiveness: namely, dissensus measures.
  • The authors partially borrow notation and definitions from Alcantud et al. (2013b).
  • The authors consider that each expert evaluates each alternative by means of a quantitative value.
  • N×k A profile M ∈ MN×k is unanimous if the evaluations for all the alternatives are the same across experts.
  • The terms consensus and dissensus should not be taken as formal antonyms, especially because a universally accepted definition of consensus is not available and the authors do not intend to give an absolute concept of dissensus.

3. The class of Mahalanobis dissensus measures and its properties

  • The authors interest is to cover the specific characteristics in cardinal profiles, like possible differences in scales, and correlations among the issues.
  • Before providing their main definition, the authors recover the definition of the Mahalanobis distance on which their measure is based.
  • The off-diagonal elements of Σ permit to account for cross relations among the issues or alternatives.
  • The authors have only used the fact that the permutation matrix Pπ is orthogonal.

3.1. Some particular specifications

  • Some special instances of Mahalanobis dissensus measures have specific interpretations.
  • This ex- pression uses the square of the Euclidean distance between real-valued vectors, thus it recovers a version of the consensus measure for ordinal preferences based on this distance (Cook and Seiford (1982)).
  • Henceforth δI is called the Euclidean dissensus measure.
  • This particular specification of the dissensus measure allows to incorporate different weights to the alternatives.
  • This fact increases the richness of the analysis in comparison with the (square of the) Euclidean distance.

3.2. Some properties of the class of Mahalanobis dissensus measures

  • Measuring cohesiveness by means of the Mahalanobis dissensus measure ensures some interesting operational features.
  • The following properties hold true: 1. Neutrality .
  • If for a particular size N of a society the Mahalanobis dissensus measures associated with two matrices coincide for all possible profiles, then the corresponding dissensus measures are equal.
  • So an important question arises about if the scale choice disturbs the cohesiveness measures.
  • If the assessments of the new agent coincide with the average of the original agents’ evaluations for each alternative, then the minimal increment of the dissensus measure is obtained.

4. Comparison of Mahalanobis dissensus measures

  • In practical situations the authors could potentially use various Mahalanobis dissensus measures for profiles of cardinal information.
  • Theorems 1 and 2 below identify conditions on matrices that ensure consistent comparisons between Mahalanobis dissensus measures, whatever the number of agents.
  • Nevertheless, Theorem 2 below proves that a partial converse of Theorem 1 holds true under a technical restriction on the definite matrices.
  • Therefore it is not true that δΣ1(M) ≥ δΣ2(M) holds throughout.
  • Moreover, distance dΣ is always between the values of the corresponding distances dλ1I and dλkI .

5. Discussion on practical application using a real example

  • The authors are interested in assessing the cohesiveness of the forecasts of various magnitudes for the Spanish Economy in 2014: GDP (Gross Domestic Product), Unemployment Rate, Public Deficit, Public Debt and Inflation.
  • These forecasts have been published by different institutions and organizations, and each one was made at around the same time.
  • Next, the authors select a suitable reference matrix and finally they make the computations of the Mahalanobis dissensus measures.

5.1. Reference matrix

  • Once the profiles have been fixed, the following step to compute their Mahalanobis dissensus measures is to avail oneself of a suitable reference matrix Σ.
  • This matrix contains the variances and covariances among the statistical variables, therefore, those characteristics are brought into play in this distance.
  • One exception is the unlikely case when the data are generated by a known multivariate probability distribution.
  • It seems natural to produce such a matrix from historical macroeconomic data corresponding to the issues under inspection.
  • The ellipses slant upward (resp., downward) show a positive (resp., negative) correlation.

5.2. Computation of the dissensus

  • Now the authors calculate the Mahalanobis dissensus measures associated with Σ for the profiles of the forecasts for the Spanish Economy, namely, M (S), M (A) and M (lS).
  • Table 6 provides these items for comparison.

5.3. Other simpler approaches: Drawbacks or limitations

  • The choice of the reference matrix is a key point in the application of the Mahalanobis dissensus measure.
  • However the choice of the identity matrix as the reference matrix discards much relevant information.
  • The authors remove the effects of the interdependence among the economic magnitudes on the dissensus measure.
  • Table 7 shows the dissensus measures derived from the three matrices mentioned above, Σ, I and Σσ.

6. Concluding remarks

  • The authors use the general concept of dissensus measure and introduce one particular formulation based on the Mahalanobis distance for numerical vectors, namely the Mahalanobis dissensus measure.
  • The authors provide some properties which make their proposal appealing.
  • T. González-Arteaga acknowledges financial support by the Spanish Ministerio de Economı́a y Competitividad (Project ECO2012-32178).
  • The authors define the permutation matrix Pπ whose rows are eπ(i).

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A cardinal dissensus measure based on the Mahalanobis
distance
T. Gonz´alez-Arteaga
a,
, J. C. R. Alcantud
b
, R. de Andr´es Calle
c
a
Presad Research Group, Facultad de Ciencias
Universidad de Valladolid, E47011 Valladolid, Spain
b
Borda Research Unit and Multidisciplinary Institute of Enterprise (IME)
Universidad de Salamanca, E37007 Salamanca, Spain
c
Borda Research Unit, Presad Research Group and IME
Universidad de Salamanca, E37007 Salamanca, Spain
Abstract
In this paper we address the problem of measuring the degree of con-
sensus/dissensus in a context where experts or agents express their opinions
on alternatives or issues by means of cardinal evaluations. To this end we
propose a new class of distance-based consensus model, the family of the
Mahalanobis dissensus measures for profiles of cardinal values. We set forth
some meaningful properties of the Mahalanobis dissensus measures. Finally,
an application over a real empirical example is presented and discussed.
Keywords: Decision analysis, consensus/dissensus, cardinal profile,
Mahalanobis distance, correlation
1. Introduction
In Decision Making Theory and its applications, consensus measurement
and its reaching in a society (i.e., a group of agents or experts) are relevant
research issues. Many studies investigating the aforementioned subjects have
been carried out under several frameworks (see Herrera-Viedma et al. (2002),
Fedrizzi et al. (2007), Dong et al. (2008), Cabrerizo et al. (2010), Dong et al.
(2010), Fu and Yang (2012), Dong and Zhang (2014), Palomares et al. (2014),
Corresponding author
Email addresses: teresag@eio.uva.es (T. Gonz´alez-Arteaga), jcr@usal.es (J. C.
R. Alcantud ), rocioac@usal.es ( R. de Andr´es Calle )
Preprint submitted to European Journal of Operational Research October 16, 2015

Wu and Chiclana (2014b,a), Liu et al. (2015) and Wu et al. (2015) among
others) and based on different methodologies (Gonz´alez-Pacon and Romero
(1999), Cook (2006), Eklund et al. (2007), Fedrizzi et al. (2007), Eklund
et al. (2008), Chiclana et al. (2013), Fu and Yang (2010, 2011), Palomares
and Mart´ınez (2014), Gong et al. (2015) and Liu et al. (2015) among others).
Since the seminal contribution by Bosch (2005) several authors have ad-
dressed the consensus measurement topic from an axiomatic perspective.
Earlier analyses can be mentioned, e.g., Hays (1960) or Day and McMorris
(1985). This issue is also seen as the problem of combining a set of ordinal
rankings to obtain an indicator of their ‘consensus’, a term with multiple
possible meanings (Mart´ınez-Panero (2011)).
Generally speaking, the usual axiomatic approaches assume that each
individual expresses his or her opinions through ordinal preferences over the
alternatives. A group of agents is characterized by the set of their preferences
–their preference profile. Then a consensus measure is a mapping which
assigns to each preference profile a number between 0 and 1. The assumption
is made that the higher the values, the more consensus in the profile.
Technical restrictions on the preferences provide various approaches in
the literature. In most cases the agents are presumed to linearly order the
alternatives (see Bosch (2005) or Alcalde-Unzu and Vorsatz (2013)). Since
this assumption seems rather demanding (especially as the number of alter-
natives grows), an obvious extension is to allow for ties. This is the case
where the agents have complete preorders on the alternatives (e.g., Garc´ıa-
Lapresta and erez-Rom´an (2011)). Alcantud et al. (2013a, 2015) take a
different position. They study the case where agents have dichotomous opin-
ions on the alternatives, a model that does not necessarily require pairwise
comparisons.
Notwithstanding the use of different ordinal preference frameworks, the
problem of how to measure consensus is an open-ended question in several
research areas. This fact is due to that methodology used in each case is a rel-
evant element in the problem addressed. To date various methods have been
developed to measure consensus under ordinal preference structures based
on distances and association measures like Kemeny’s distance, Kendall’s co-
efficient, Goodman-Kruskal’s index and Spearman’s coefficient among others
(see e.g., Spearman (1904), Kemeny (1959), Goodman and Kruskal (1979),
Cook and Seiford (1982) and Kendall and Gibbons (1990)).
2

In this paper we first tackle the analysis of coherence that derives from
profiles of cardinal rather than ordinal evaluations. Modern convention
applies the term cardinal to measurements that assign significance to diffe-
rences (cf., Basu (1982), High and Bloch (1989), Chiclana et al. (2009)). By
contrast ordinal preferences only permit to order the alternatives from best
to worst without any additional information. To see how this affects the
analysis of our problem, let us consider a naive example of a society with two
agents. They evaluate two public goods with monetary amounts. One agent
gives a value of 1e for the first good and 2e for the second good. The other
agent values these goods at 10e and 90e respectively. If we only use the
ordinal information in this case, we should conclude that there is unanimity
in the society: all members agree that ‘good 2 is more valuable than good
1’. However the agents disagree largely. Therefore, the subtleties of cardi-
nality clearly have an impact when we aim at measuring the cohesiveness of
cardinal evaluations.
Unlike previous references, we adopt the notion of dissensus measure as
the fundamental concept. This seems only natural because it resembles more
the notion of a “measure of statistical dispersion”, in the sense that 0 captures
the natural notion of unanimity as total lack of variability among agents, and
then increasingly higher numbers mean more disparity among evaluations in
the profile.
1
In order to build a particular dissensus measure we adopt a distance-based
approach. Firstly, one computes the distances between each pair of indivi-
duals. Then all these distances are aggregated. In our present proposal the
distances (or similarities) are computed through the Mahalanobis distance
(Mahalanobis (1936)). We thus define the class of Mahalanobis dissensus
measures.
The Mahalanobis distance plays an important role in Statistics and Data
Analysis. It arises as a natural generalization of the Euclidean distance.
A Mahalanobis distance accounts for the effects of differences in scales and
associations among magnitudes. Consequently, building on the well-known
performance of the Mahalanobis distance, our novel proposal seems especially
fit for the cases when the measurement units of the issues are different, e.g.,
1
As a remote antecedent of this position, we note that statistically variance-based
methods are commonly employed to measure consensus of verbal opinions (cf., Hoffman
(1994) and Mejias et al. (1996).
3

performance appraisal processes when employees are evaluated attending to
their productivity and their leadership capacity; or where the issues are cor-
related. For example, evaluation of related public projects. An antecedent
for the weaker case of profiles of preferences has been provided elsewhere, cf.
Alcantud et al. (2013b), and an application to comparisons of real rankings
on universities worldwide is developed. Here we apply our new indicator to
a real situation, namely, economic forecasts made by several agencies. Since
the forecasts concern economic quantities, they have an intrinsic value which
is naturally cardinal and also there are relations among them.
The paper is structured as follows. In Section 2, we introduce basic nota-
tion and definitions. In Section 3, we set forth the class of the Mahalanobis
dissensus measures and their main properties. Section 4 provides a compari-
son of several Mahalanobis dissensus measures. Next, a practical application
with discussion is given in Section 5. Finally, we present some concluding
remarks. Appendices contain proofs of some properties and a short review
in matrix algebra.
2. Notation and definitions
This section is devoted to introduce some notation and a new concept in
order to compare group cohesiveness: namely, dissensus measures. Then, a
comparison with the standard approach is made. We partially borrow nota-
tion and definitions from Alcantud et al. (2013b). In addition, we use some
elements of matrix analysis that we recall in the AppendixB to make the
paper self-contained.
Let X = {x
1
, ..., x
k
} be the finite set of k issues, options, alternatives,
or candidates. It is assumed that X contains al least two options, i.e., the
cardinality of X is at least 2. Abusing notation, on occasions we refer to
issue x
s
as issue s for convenience. A population of agents or experts is a
finite subset N = {1, 2, ..., N } of natural numbers. To avoid trivialities we
assume N > 1.
We consider that each expert evaluates each alternative by means of a
quantitative value. The quantitative information gathered from the set of N
experts on the set of k alternatives is summarized by an N × k numerical
matrix M:
M =
M
ij
N×k
4

We write M
i
to denote the evaluation vector of agent i over the issues (i.e.,
row i of M) and M
j
to denote the vector with all the evaluations for issue
j (i.e., column j of M ). For convenience, (1)
N×k
denotes the N × k matrix
whose cells are all equal to 1 and 1
N
denotes the column vector whose N
elements are equal to 1. We write M
N×k
for the set of all N × k real-valued
matrices. Any M M
N×k
is called a profile.
Any permutation σ of the experts {1, 2, ..., N} determines a profile M
σ
by permutation of the rows of M: row i of the profile M
σ
is row σ(i) of
the profile M . Similarly, any permutation π of the alternatives {1, 2, ..., k}
determines a profile
π
M by permutation of the columns of M : column i of
the profile
π
M is column π(i) of the profile M.
For each profile M M
N×k
, its restriction to subprofile on the issues in
I X, denoted M
I
, arises from exactly selecting the columns of M that
are associated with the respective issues in I (in the same order). And for
simplicity, if I = {j} then M
I
= M
{j}
= M
j
is column j of M . Any partition
{I
1
, . . . , I
s
} of {1, 2, . . . , k}, that we identify with a partition of X, generates
a decomposition of M into subprofiles M
I
1
, . . . , M
I
s
.
2
A profile M M
N×k
is unanimous if the evaluations for all the alterna-
tives are the same across experts. In matrix terms, the columns of M M
N×k
are constant, or equivalently, all rows of the profile are coincident.
An expansion of a profile M M
N×k
of N on X = {x
1
, ..., x
k
} is a profile
¯
M M
¯
N×k
of
¯
N = {1, ..., N, N + 1, . . . ,
¯
N} on X = {x
1
, ..., x
k
}, such that
the restriction of
¯
M to the first N experts of N coincides with M .
Finally, a replication of a profile M M
N×k
of the society N on
X = {x
1
, ..., x
k
} is the profile M ] M M
2N×k
obtained by duplicating
each row of M, in the sense that rows t and N + t of M ] M are coincident
and equal to row t of M, for each t = 1, . . . , N .
We now define a dissensus measure as follows:
Definition 1. A dissensus measure on M
N×k
is a mapping defined by
δ : M
N×k
[0, ) with the property:
i) Unanimity: for each M M
N×k
, δ(M ) = 0 if and only if the profile
M M
N×k
is unanimous.
2
A partition of a set S is a collection of pairwise disjoints non-empty subsets of S whose
union is S.
5

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Abstract: Innovative methodology for measuring consensus based on the Pearson correlation coefficient is proposed.Experts express their opinions on alternatives or issues by means of reciprocal preference relations.We provide interesting properties for the new consensus measure proposed.An illustrative example with discussion is presented. 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.

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Q1. What contributions have the authors mentioned in the paper "A cardinal dissensus measure based on the mahalanobis distance" ?

In this paper the authors address the problem of measuring the degree of consensus/dissensus in a context where experts or agents express their opinions on alternatives or issues by means of cardinal evaluations. To this end the authors propose a new class of distance-based consensus model, the family of the Mahalanobis dissensus measures for profiles of cardinal values. Finally, an application over a real empirical example is presented and discussed.