Qual Quant (2015) 49:1145–1167 DOI 10.1007/s11135-014-0040-9
Assessing variation: a unifying approach for all scales of measurement Tamar Gadrich · Emil Bashkansky · Riˇcardas Zitikis
Published online: 1 June 2014 © Springer Science+Business Media Dordrecht 2014
Abstract Recent developments in the area of enterprise risk management, especially in the context of high impact events, their uncertainty and variability, have highlighted the need for developing a unified approach for variability measurement in qualitative and quantitative phenomena. In this paper we discuss such an approach, which is based on Gini’s seminal ideas and applicable for all types of data: nominal, ordinal, interval, and ratio. By establishing a general total-variation decomposition theorem, we provide a tool for decomposing the total variation into within (intra) and between (inter) components, and as a consequence introduce several indices of interest. We illustrate our general considerations using specially designed artificial data-sets as well as real-life examples pertaining to countries, their territorial units, and educational institutions. Keywords Stevens classification system · Gini mean difference · Variability · Segregation · Integration · Enterprise risk management
1 Introduction For various reasons such as assessing risk and uncertainty, quite often we wish or need to measure the variability of populations, or samples, which can be quantitative, qualitative, and quite often mixed. We may, for example, want to measure social inequality and mobility,
Research supported by the Natural Sciences and Engineering Research Council (NSERC) of Canada. T. Gadrich · E. Bashkansky Industrial Engineering and Management, ORT Braude College, 21982 Karmiel, Israel e-mail:
[email protected] E. Bashkansky e-mail:
[email protected] R. Zitikis (B) Statistical and Actuarial Sciences, University of Western Ontario, London, Ontario N6A 5B7, Canada e-mail:
[email protected]
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political consensus, homogeneity of some material, uncertainty of prediction, diversity or similarity of species, synchronization degree of biological rhythms, etc. These are complex tasks due to a number of reasons, not least because of the inherent heterogeneity of populations, which are usually made up of various groups and categories often requiring different scales of measurement. A number of variability measures have been developed to accommodate various scales of measurement, and according to Stevens’s (1946) classification system there are four of them: nominal, ordinal, interval, and ratio. (We shall discuss these scales later in the paper.) This variety of scales and, accordingly, various restrictions on possible arithmetical operations and order relationships carry serious challenges for researchers and decision makers. Most popular are measures designed to evaluate variability of numerical (interval and ratio) data, and among them are the range, interquartile distance, variance, and standard deviation, as well as measures based on mean absolute deviation/difference and also entropy-based measures (cf. Hart 1971). The common procedure when dealing with ordinal categorical scale is to convert the data into interval scale by assigning numerical values (ranks) to each category of the ordinal variable. This procedure, however, is undesirable since it can lead to illogical and distorted interpretations of results (cf., e.g., Bashkansky and Gadrich 2008; Blair and Lacy 2000; Franceschini et al. 2005). Hence, there is a necessity for developing variability measures based on legitimate arithmetical operations between categorical variables. For nominal data, the most popular measures are entropy-based, and the index of qualitative variation (IQV) introduced by Gibbs and Poston (1975), with a nontrivial modification for ordinal data proposed by Blair and Lacy (2000) which, essentially, replaces frequencies by cumulative frequencies. For the latter measure, its various features and in particular a possible decomposition into intra- and inter-components have been explored by Gadrich and Bashkansky (2012), Gadrich et al. (2013). The use of such a decomposition for testing homogeneity as well as for intra- and inter-laboratory comparisons (unbiased vs. biased estimators) of measurement results on the ordinal scale have been investigated by Bashkansky et al. (2012), Gadrich et al. (2013). In the present paper we go further than the aforementioned results and tackle all scales of measurement, as well as the corresponding decompositions, in a single unifying way. 1.1 Motivation: enterprise risk management The aforementioned variety of scales, arithmetical operations, and order relationships carry serious challenges in a variety of areas of application, and in particular in enterprise risk management (ERM) where risk metrics need to be aggregated into one enterprise-wide metric and then cascaded to individual business lines in the form of quotas of risk appetite or tolerance. We shall next elaborate on this ERM-based problem, which has given a serious impetus to our current research. The ERM is generally defined as “the process by which companies identify, measure, manage, and disclose all key risks to increase value to stakeholders” (Segal 2011, p. 3). There is a great variety of risks (cf., e.g., Huang et al. 2009), and all of them can be categorized into strategic, operational, financial, and insurance. The first two ones are predominantly qualitative, whereas the last two risks—financial and insurance—are usually quantitative. Segal (2011) develops a well thought out procedure for quantifying strategic and operational risks within the value-based ERM framework, but even in this case, the two types of risk can at best be turned into ordinal ones (e.g., pessimistic, baseline, and optimistic). Of course, severities and likelihoods can be and usually are attached to such risks using panels of experts.
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ERM has recently developed into a very active area of research and practice, particularly since the financial crisis of 2007–2008. It has been explored from various viewpoints and applied to corporate and non-corporate entities (e.g., Fraser and Simkins 2010; Olson and Wu 2010; Segal 2011; Yau et al. 2011; Ferrari and Migliavacca 2014; Louisot and Ketcham 2014. For up-to-date and state-of-the-art contributions to ERM, we refer to Wu et al. (2011, 2014) and Wu and Olson (2013), and the extensive lists of references therein. For case studies, we refer to Wu and Olson (2010a,b) and Louisot and Ketcham (2014). Next, two illustrative examples follow. Example 1 In 2009, the University of British Columbia adopted an ERM framework to support strategic and operational decision-making (UBC 2014) and defined risk as “the possibility that an event, action or inaction could adversely affect the University’s ability to achieve its objectives and successfully execute its strategies.” As a set of risks that the University tackles, UBC (2014) notes: 1. 2. 3. 4. 5. 6.
Academic: faculty, students, research Financial Reputational: rankings, service Operational: processes, systems, staff Hazard: natural and other disasters Strategic: barriers to achieve strategic goals
Note that some of the risks can be directly measured but some may not, some are categorical but some are numerical. This diversity naturally and unavoidably poses challenges. Among the ERM objectives and benefits, UBC (2014) includes: 1. Management tool for faculties and administrative units to identify key areas of risk and prioritize resource allocation 2. Identify dependencies and key areas where coordination with other faculties/administrative units is required 3. Development of an on-going comprehensive risk database using a consistent methodology across all campuses of the University For further general details, we refer to UBC (2014). Specific details are confidential, as is usually the case with internal programs, and thus not available for our analysis. Example 2 The Government of British Columbia, Canada, has adopted the ERM as the framework for strategic and operational decision-making (GBC 2014a), and Core Policy Manual Chapter 14 (GBC 2014b) sets out the responsibility of ministries in establishing a risk management framework. It is instructive to note here the characteristic for the ERM bottom–up and top–down approaches to risk management, respectively: “In a culture of mature Enterprise Risk Management (ERM), every manager and employee is familiar with the principles of risk management, takes a role in the management of risk within their areas of responsibility, and escalates those risks beyond the scope of their authority or available resources.” (GBC 2014b, Sect. 14.2) and “Enterprise Risk Management (ERM)—the coordinated, ongoing application of risk management across all parts of an organization which flows from the strategic planning to the operational (service delivery) level.” (GBC 2014b, Sect. 14.2).
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The desire and necessity for a unified approach is explicitly stated in Section 14.3 of GBC (2014b): “This policy seeks to establish and confirm consistent and compatible risk management standards, processes and practice within ministries while reducing barriers to successful implementation.” For concrete illustrative data sets to which we have had access, in our following examples we shall rely on an Israeli college, whose name we need to withhold due to confidentiality reasons, and also on the territorial units of Israel, with data provided by the Israel Central Bureau of Statistics (CBS 2014). 1.2 ERM and dispersion analysis Coming now back to our general discussion, we note that dispersion analysis of all risks is particularly important (Segal 2011, pp. 149–150). Note, however, that different techniques are used for measuring the dispersion of qualitative and quantitative data. This is natural. Indeed, for example, the classical variance and its one-sided versions are not suited for categorical data for reasons such as the lack of well defined basic arithmetical operations (e.g., addition, multiplication, etc.). Naturally, therefore, different techniques have been proposed and used for measuring the dispersion of different types of data, but this diversity has in turn introduced serious challenges when, for example, developing and implementing value-based ERM, which is based on ten key criteria (Segal 2011, p. 25): 1. 2. 3. 4. 5. 6. 7. 8. 9. 10.
Enterprise-wide scope All risk categories included Key risk focus Integrated across risk types Aggregated metrics Includes decision making Balances risk and return management Appropriate risk disclosures Measures value impacts Primary stakeholder focus
In particular, paying attention to criteria 1, 2, 4, and 5, we see that all enterprise-wide (key) risks have to be integrated and their metrics aggregated. (This we have already encountered in Example 1 and 2 above.) However, if different risks use different dispersion measures due to reasons such as different scales of measurement, then aggregating individual metrics into one enterprise-wide metric becomes a challenge. Hence, in this paper we set out to develop a unifying measure of dispersion. We shall next briefly discuss two classical measures of dispersion – the variance and the Gini mean difference—that will naturally and clearly lead us to a solution of the above noted challenge. 1.3 The classical variance and the Gini mean difference As we have already noted above, a number of variability measures have been proposed in the literature. The classical variance VAR =
K (xk − μ)2 pk k=1
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(1)
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K with μ = k=1 xk pk is perhaps the most popular measure of variability, where K is the number of categories, xk ’s are either all possible values of the phenomenon under consideration or just some observed values, and pk ’s are the corresponding probabilities or proportions. In the latter case, that is, when we work with proportions and want to emphasize this fact, we shall use the notation pk instead of pk . The above formula of the variance, though widely used and classical, is somewhat misleading in that it interprets the variance as the average of squared deviations from the ‘midpoint’ (i.e., from the mean μ). But any good dispersion measure should not be tied to any ‘midpoint’—it should reflect just the variability of data. Fortunately, the classical variance allows for such an interpretation, frequently known as the Gini formula for the variance (e.g., Giorgi 2005 and references therein): VAR =
K K
L var (xi , x j ) pi p j ,
(2)
i=1 j=1
where L var (xi , x j ) = (1/2)(xi − x j )2 . One may of course argue about the wisdom of using the quadratic loss-of-similarity function L var (xi , x j ). Indeed, several modifications of the classical variance have arisen in various contexts. Among them we find, for example, the downside variance, which prominently features in the financial and insurance literature and, more generally, in ERM (e.g., Segal 2011). Another highly popular measure of variability is the Gini mean difference GMD =
K K
L gmd (xi , x j ) pi p j ,
(3)
i=1 j=1
where L gmd (xi , x j ) = |xi − x j |. Along with its numerous extensions and generalizations, the Gini mean difference has played a particularly prominent role in areas such as econometrics and social sciences. For details on the topic, we refer to insightful historical and other comments by Giorgi (2005), as well as to the recent Gini-centric monograph by Yitzhaki and Schechtman (2013). Note that the variability measures VAR and GMD are closely tied to the numerical scale, due to the form of their loss-of-similarity functions L var and L gmd . Hence, their applicability to measuring variability of categorical data (e.g., gender, race, or religion) becomes impossible, or at least controversial. In this paper, therefore, we seek for and offer a common approach to measuring variability on every scale. To facilitate a more self-contained presentation, and to also introduce some of our basic notations used throughout the paper, in the next section we shall recall Stevens’s (1946) classification system of measurement scales.
2 Classification of variables Measuring variability is based on two inter-twinned pillars: determining an appropriate measurement scale and a loss-of-similarity function. Note that the scale of measurement tends to narrow down the class of appropriate loss-of-similarity functions, and the latter ones are, to a large degree, determined by desirable properties that a variability measure should satisfy. For a set of such properties, we refer to Gadrich and Bashkansky (2012) and thus not delve into this topic here. Instead, we concentrate on Stevens’s (1946) four scales of measurement, which play a fundamental role in this paper.
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T. Gadrich et al. Categories
Groups
Total counts
1
···
m
···
M
c1
n 11
···
n 1m
···
n 1M
n 1+
.. .
.. .
..
.. .
..
.. .
.. .
.
.
ck
n k1
···
n km
···
nk M
n k+
. . .
. . .
..
. . .
..
. . .
. . .
cK
nK1
···
nKm
···
Total counts
n +1
···
n +m
···
.
.
nK M n +M
nK+ N
It should be noted that while some phenomena can unambiguously be assigned to scales, e.g. gender or race to the nominal scale, the assignment of scales to other phenomena may carry elements of subjectivity. To illustrate the point with an ERM-type example, we may initially think of the downside risk as something to avoid but the upside risk to absorb, which would imply their ordinal nature. But Segal (2011) convincingly argues that the two— downside and upside—risks might be equally troublesome, thus effectively assigning them to the nominal scale. Hence, there might indeed be a fair amount of ambiguity and thus inevitable subjectivity in this research area, but we view this as a natural and unavoidable phenomenon. At the very basic level, all variables can be viewed as nominal by simply ignoring any order-type information that might be present. To reflect this fact, the most general notation K that we use for data is (ck )k=1 with ci = c j whenever i = j (cf. Sect. 2.1 below). When, however, data are viewed as numerical, and assuming that we want to emphasize this choice K with x = x of the scale, as we do in Sect. 2.2 below, then we use the ‘x’ notation (xk )k=1 i j whenever i = j. In both cases, ck ’s and xk ’s are always accompanied by either proportions K p ∈ [0, 1] or probabilities pk ∈ [0, 1], which of course add up to one: k=1 pk = 1 and k K p = 1. The next two subsections are devoted to categorical and numerical variables, k=1 k respectively. 2.1 Categorical variables The first two Stevens’s (1946) scales of measurement cover nominal and ordinal data, which make up two exhaustive subclasses of categorical data. We have depicted a general layout of such data in Table 1, which introduces some of our most frequently used notations in this paper. Specifically, we have N data points in total, and each of them falls into one of the K categories, which make up K rows in Table 1, and into one of the M groups, which make up M columns. k-th category contains n k+ data points, and the counts n k+ are The K random. Obviously, k=1 n k+ = N . The group sizes n +m are, however, non-random, which M n +m = N . is justified by our applications. Obviously, m=1 Nominal variables (=, =) These variables are coded using some labels or names that uniquely correspond to exhaustive and disjoint categories. If two variables carry the same code, then the variables belong to the same category, and this is the only significant feature of these two variables. Hence, the only two comparisons that we can make between these variables are their equality or difference. Examples of nominal variables are gender, race, religion. Furthermore,
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risks in ERM are categorized into financial, strategic, operational, and insurance, which are frequently viewed as nominal variables, although a more detailed view (e.g., Segal 2011) may suggest their ordinal nature, which we discuss next. Ordinal variables (=, =, >, , , x}d x
i=1 j=1
=2
F(x)(1 − F(x))d x,
(17)
K where F(x) = k=1 pk 1{xk ≤ x}. Comparing the right-hand sides of Eqs. (14) and (17), we see that the sum has turned into an integral. Note also that the right-hand side of Eq. (17) implies that VT is actually the Gini mean difference (e.g., Giorgi 2005 and references therein). We are now ready for decomposition arguments. T is given by the equation Decomposition The sample total-variation V
T = 2 F(x)(1 − F(x))d x, V
(18)
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K where the sample cdf is given by F(x) = k=1 pk 1{xk ≤ x}. Analogously to our considerT can be decomposed into the sum V W + C B ations in Sect. 5.2, the sample total-variation V of the within-variation W = V
M
m πm V
m=1
and the between-covariation B = 2 C
M
πm
2 m (x) − F(x)) (F d x,
m=1
m in the definition of V W is given by where the m-th group variation V
m (x))d x, m (x)(1 − F m = 2 F V m (x) = and the cdf of the m-th group is F
K k=1
pk|m 1{xk ≤ x}.
6.2 Ratio variables The loss-of-similarity function L(x, x ) = | log x − log x | is well suited for the ratio scale. By adopting this function, we effectively replace our considerations on the ratio scale by those on the interval scale, and thus work with the loss-of-similarity function L(y, y ) = |y − y |, where instead of the original x’s we now deal with their logarithms y = log x. Hence, all our earlier results pertaining to the interval scale can be utilized in a straightforward manner to establish analogous results on the ratio scale. Of course, there is an element of arbitrariness in our choice of the logarithmic transformation—there are indeed many alternatives. Nevertheless, our experience suggests that underlying problems and philosophies for tackling the problems usually restrict the class of loss-of-similarity functions as well as of transformations to just a few ‘reasonable’ ones, and certain axiomatic approaches may even produce unique choices.
7 Segregation indices In the second part of this section, we shall introduce a segregation index whose idea and construction are based on the following general theorem that, under a null hypothesis of homogeneity (to be defined), relates the expected values of the sample total-variation, withinvariation and between-covariation. We note that the theorem is within the same general framework of Theorem 1 and therefore extends analogous results of Light and Margolin (1971) and Gadrich and Bashkansky (2012) obtained for the nominal and ordinal scales, respectively. For additional information, we refer to Bashkansky et al. (2012), Gadrich et al. (2013). Theorem 2 Assuming the null (homogeneity) hypothesis H0 that, for all m = 1, . . . , M, the conditional probabilities pk|m do not depend on m (and thus pk|m = pk for all k = 1, . . . , K ), we have the equations T ] W ] B ] E[V E[V E[C VT = = = , (19) df T df W df B N where df T = N − 1, df W = N − M, and df B = M − 1.
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The proof of the theorem is technical and thus relegated to Appendix 1. Next, we shall introduce an index S P. Namely, under the null hypothesis of homogeneity, H0 , that is, when the conditional proportions pk|m do not depend on m, we have from Theorem 2 that B ]/df B = E[V T ]/df T . This equation suggests using E[C = SP
B /df B C T /df T V
(20)
as an index of segregation power (SP) among the M groups. Of course, the index S P can be rewritten in other equivalent and perhaps more convenient ways, such as = PVE SP
M m df T V df T = 1− πm . T df B df B V
(21)
m=1
To develop a statistical procedure for testing segregation power based on the index S P, we need either a finite-sample or asymptotic distribution of the index so that critical values could be computed. For nominal variables and with the loss-of-similarity function given by Eq. (5), Light and Margolin (1971) have shown that, under the null hypothesis H0 , for large sample B is asymptotically independent of the total-variation V T sizes, the between-covariation C and proposed the statistic C = df × SP with df = (K − 1)(M − 1). In particular, Light is and Margolin (1971) have shown that, under the null hypothesis H0 , the distribution of C asymptotically close to that of χ 2 (df). Gadrich and Bashkansky (2012) utilized this fact in the binary case, that is, when K = 2. For ordinal data, determining critical values is a more complex task for which a special MATLAB program has been developed and is available upon request from the authors of this paper.
8 Back to the two illustrative examples: findings The above developed general and unifying methodology for measuring variation allows us to analyze different in nature phenomena. To illustrate, in this section we employ the variability and S and segregation indices PVE P to analyze our two real-life examples: college freshmen and the Israeli population. 8.1 College freshmen enrolments As noted earlier, 590 students joined six study programs at an Israeli college; the programs are groups in our terminology. The degrees of freedom are: df T = 589, df B = 5, and df W = 584. The relevant for our study summary statistics are reported in Table 7. The index S P has been calculated using formula (20) even in the third (i.e., interval) case by simply treating the interval from 500 to 800 as 300 categories. (In Sect. 9, we shall discuss an un-grouped treatment of this case.) The critical value of the test, denoted by Icv , has been calculated using the formula (df)−1 χ 2 (df) in the first (i.e., nominal) case, and using a special MATLAB program available upon request from the authors in the second and third cases. Following the usual practice, the confidence level in our research was set to 95 %, but the aforementioned program can certainly accommodate any level. By comparing S P with the corresponding critical values Icv , we conclude that:
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Table 7 Variability and segregation indices of the college freshmen S P
PVE
Parameter
Type
Domain
Icv
Place of residence
Nominal
7 categories
0.03
02.56
1.46
Level of physics
Ordinal
4 categories
0.22
26.42
1.98
Psychometric grade
Interval
[500–800]
0.02
01.96
1.07
Table 8 Variability and segregation indices of the Israeli population Parameter
Type
Domain
PVE
S P
Icv
Religion
Nominal
7 categories
0.12
97,531
1.42
Education
Ordinal
9 categories
0.02
12,288
1.36
Income
Interval
[0–100,000]
0.03
23,532
1.00
1. Study program cannot serve as a good predictor of any of the three parameters (viewed as categories) because the unexplained variation dominates the explained one, that is, PVE is too small in all three cases. 2. Study program can serve as a distinguishing factor. Observe that the highest segregation (i.e., the value of S P) between various study programs exists in the physics study level. This can be rationalized by the fact that physics is among the prerequisites for some study programs. 3. We observe a large total polarization at the physics study level: 282 students are at level A, 15 at B, 44 at C, and 249 at D. The total ordinal variation is equal to 0.99, which is the sum of the within-variation and between-covariation, which are equal to 0.77 and 0.22, respectively. 4. There is some difference in the psychometric grade, but only a very small significant difference in the place of residence between study programs. It should be noted, however, that the last two parameters fluctuate from year to year, whereas the segregation by the physics level remains consistently significant. 8.2 Israeli population Recall that we deal with 4,945,592 Israeli citizens living in seven districts, which are seven groups in the terminology of this paper. The degrees of freedom are: df T = 4,945,591, df B = 6, and df W = 4,945,585. The relevant for our study summary statistics are given in Table 8, with ‘income’ meaning the average household income per capita in NIS. By with the corresponding critical values Icv calculated comparing the segregation index SP using the same methodology as described in previous Sect. 8.1, we conclude that: 1. District of residence cannot be a good predictor of any of the three parameters under consideration because the unexplained variation dominates the explained one for all of is too small for any meaningful prediction. them: every PVE 2. District of residence can be a distinguishing factor. Note that the highest segregation between districts exists with respect to the religion of residents. 3. The average household income per capita is in the second place in terms of S P values, due to the gap between the employment levels in the center part of Israel and its periphery.
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4. There is a significant segregation in the education levels of districts, although twice less than in the case of incomes and eight-fold less than in the case of religion.
9 Continuous variables In the two illustrative examples, we have so far treated their third parameters (i.e., the psychometric grade and household income) as categorical variables, but they can also, and very naturally, be viewed as continuous variables. That is, for example, in the case of Israeli incomes we can view the interval from 0 to 100,000 not as the collection of 100,000 groups but as a continuum. In this case, the notion of categories disappears. This section is designed to accommodate such arguments, and they are indeed useful when underlying populations are modeled by various parametric distributions, depending on the problem at hand. Hence, let X be random variable taking on real values, and let F denote its cdf. Given a symmetric, non-negative, two-argument loss-of-similarity function L(x, x ) such that L(x, x) = 0 for all x, we define the measure of variation of X by
VT = E L(X, X ) = L(x, x )d F(x)d F(x ), (22) where X denotes an independent copy of X . K pk 1{xk ≤ x}, then we have the ‘discrete’ total-variation VT = Note 1 When F(x) = k=1 K K L(x , x ) p p investigated earlier in the paper. Hence, our considerations in i j i j i=1 j=1 this section are indeed encompassing. πm = P[X ∈ G m ] Let G m , 1 ≤ m ≤ M, be non-intersecting groups. With the notations M and F(x|m) = P[X ≤ x|X ∈ G m ], we have the decomposition F(x) = m=1 πm F(x|m). Since F(x)F(x ) is equal to M
πm F(x|m)F(x |m) −
m=1
M
πm F(x|m) − F(x) F(x |m) − F(x ) ,
m=1
we use Eq. (22) and obtain the decomposition VT = VW + C B , where VW =
M
πm Vm
m=1
with
L(x, x )d F(x|m)d F(x |m)
Vm = and
CB =
L(x, x )dΘ(x, x ),
with the characteristic kernel Θ(x, x ) of the between-covariation defined by Θ(x, x ) = (−1)
M
πm F(x|m) − F(x) F(x |m) − F(x ) .
m=1
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10 Summary Not too distant and high impact events in the financial sector and beyond have convincingly demonstrated the need for departing from traditional risk management ways. Under holistic or integrated enterprise-wide approaches to risk management, qualitative risks are integrated with quantitative ones, and all risk metrics are aggregated and used for decision making at the enterprise level, as well as cascaded down to individual business lines. Responding to this progressive trend in risk management, in the present paper we have presented a unifying approach for assessing variation in populations and data sets that accommodates every scale of measurement: nominal, ordinal, interval, and ratio. In particular, we have put forward a general decomposition result for the total variation into within (intra) as the and between (inter) components. This has enabled us to introduce two indices: PVE proportion-of-variation-explained and S P as the segregation power. Our results extend and generalize the ORDANOVA method developed by Gadrich and Bashkansky (2012) in the case of categorical ordinal variables. Several real-life and artificial data sets have been analyzed and discussed to facilitate real-life uses of the herein developed results. Acknowledgments We are indebted to two anonymous referees for constructive criticism and insightful comments and suggestions that have guided our work on the revision of this paper.
Appendix 1: proofs and technical notes Proof of Theorem 1 We start with the equations
pk =
M M M 1 n +m n km n km = = πm pk|m . N N n +m m=1
m=1
m=1
M πm pi|m p j|m + pi p j . Multiplying both sides of the latter equation Hence, θi j = − m=1 by L(ci , c j ) and then summing up with respect to i and j, we establish the decomposition T = V W + C B and complete the proof of Theorem 1 . V Thinking about straightforward and thus faster computational implementation of Theorem it is useful to introduce additional matrix-based 1, in addition to the characteristic kernel Θ, B is notations. Namely, with the notation Λ = [L(ci , c j )]i,K j=1 , the between-covariation C = tr(ΛΘ) of the matrices Λ and Θ. We can also view C B as the Frobenius product Λ : Θ of the two matrices Λ and Θ. These the sum of the entries of the Hadamard product Λ Θ interpretations also bring additional insights into the meaning of the between-covariation B and connects it with other research areas such as “lossy compression” in information C technology. Normalization Finding the maximal value VTmax of VT ≡ VT (p) with respect to all probability distributions K is a classical quadratic programming problem. Namely, with the vector 1 = p = ( pi )i=1
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K whose all elements are equal to 1, we need to solve the problem: (1)k=1
Maximize p Λ p
subject to p 1 = 1 p ≥ 0. With p0 denoting the solution to this maximization problem, we have VTmax = p0 Λ p
0. Since this maximization problem has a direct link to Markowitz’s (1959) mean-variance portfolio theory, many software packages are available for producing numerical solutions, and perhaps even exact theoretical ones, depending on the loss-of-similarity function. For some special loss-of-similarity functions, the upper bound VTmax can be derived using specialized techniques. Two illustrative examples follow: – In the nominal case and with loss-of-similarity function (5), the normalizing constant is VTmax =
K −1 , K
(23)
K whose which is the upper bound of p → VT (p) achieved at the vector p0 = (1/K )k=1 all elements are equal to 1/K . – When the scale is ordinal and the loss-of-similarity function is given by Eq. (6), then
VTmax =
K −1 , 2
(24)
K whose first and last elements are p 0 = 1/2 which is achieved at the vector p0 = ( pk0 )k=1 1 0 and p K = 1/2, respectively, and the other ones are equal to 0.
Proof of Theorem 2 To calculate the three expectations featuring in Eq. (19), we need preliminary results. First, the probability that a data point ends up in the category k is pk . Due to the multinomial nature of the problem, we have the following formulas pk (1 − pk ) pk pk pk ] = (−1) and Cov[ pk , N N whenever k = k . We are now ready to calculate the three expectations. E[ pk ] = pk , Var[ pk ] =
T ], we need to know E[ Part 1 To calculate the ‘total’ expectation E[V pi p j ], but only for the pairs i = j because L(ck , ck ) = 0 for every k. Using the above displayed equations, we have E[ pi p j ] = (1 − 1/N ) pi p j . Consequently,
1 df T (25) E[VT ] = 1 − VT . VT = N N W ], we need to calculate E[ Part 2 For the ‘within’ expectation E[V pi|m p j|m ] for every pair i = j. Note that we are currently working within the m-th group only. Hence, E[ pk|m ] = pk and Cov[ pi|m , p j|m ] = (−1) pi p j /n +m whenever i = j. Consequently, E[ pi|m p j|m ] = (1 − 1/n +m ) pi p j and thus in view of Eq. (9) we have
M n +m df W 1 W ] = (26) E[V VT = VT . 1− N n +m N m=1
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Part 3 Finally, we calculate the ‘between’ expectation. For this, we first use Theorem 1 and B ] = E[V T ] − E[V W ]. Employing Eqs. (25) and (26), we obtain have the equation E[C B ] = E[C
df T df W df B VT − VT = VT . N N N
This concludes the proof of Theorem 2.
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