Page 320 - Becoming Metric Wise

312 Becoming Metric-Wise

The theory proposed above is just the basic theory for fixed N. When
N is variable things become much more complicated and divergent opi-
nions exist. As these aspects are mostly studied in a diversity context we
continue using the terminology of diversity measures.
Stirling (2007) and Leinster and Cobbold (2012) have pointed out that
the notion of diversity has three components variety, balance, and dispar-
ity. Each of them, considered separately, is necessary, but not sufficient, to
measure diversity in an adequate manner.
Variety is the number of nonempty categories to which system ele-
ments are assigned. In particular, it is the answer to the question: how
many types of things do we have? In information science it may be the
answer to the question: In how many different journals has this author
published? Assuming that all things are equal, the greater the variety, the
greater the diversity. In the previous section variety was kept fixed and
equal to N.
Balance is a function of the pattern of the assignment of elements
across categories. It is the answer to the question: What is the relative
number of items of each type? Balance is also called evenness (in ecology)
and concentration (in economics and the information sciences). Evenness
can be represented by the Lorenz curve (Nijssen et al., 1998): it is the
theory explained in Chapter 4, Statistics. In information science one may
consider, for instance, how many articles an author has published in each
journal. All else being equal, the more balanced the distribution, the
larger the diversity.
Disparity refers to the manner and the degree in which things may be
distinguished. It is the answer to the question: How different from each
other are the types of things that we observe? For instance, publishing
only in LIS journals shows less disparity than publishing in LIS and man-
agement and economics journals. All else being equal, the higher the dis-
parity, the greater the diversity.
Mathematically speaking, variety is a positive natural number as cate-
gories are numbered in sequence, balance is a function of fractions sum-
ming up to one, and disparity is a function of a matrix of distances (or
dissimilarities). Sometimes one starts from a matrix of similarities or
proximities, which are then converted to a matrix of disparities.
The problem now is how to find a single index that can aggregate
properties of variety, balance and disparity in a meaningful way and with-
out much loss of information. The term meaningful includes, among
others, notions such as sensitivity to changes, perspective and context.