236
M. Fattore and A. Arcagni
The formal development of the theory of synthetic indicators over posets is
not trivial (details can be found in Fattore 2017), but the basic idea is quite
simple and relies, again, on the equivalence between finite posets and their set
of linear extensions. Let π be an n-element poset generated by a MIS and let
p = (p 1 , . . . , p n ) be a relative frequency distribution over it (so that p i is the
fraction of statistical units associated to the i-th element of π ). Suppose, to set the
stage, that we want to measure how unequally p is distributed on π , by means of
a non-negative synthetic indicator G(p, π). Notice that G(·, ·) not only depends
upon the frequency vector p, but also upon the underlying order structure, i.e. upon
poset π . To see why this dependence is essential, just consider two posets on three
elements {a, b, c}, π 1 = a b c (a chain) and π 2 = a||b||c (an antichain),
and let p a = 0.5, p b = 0, p c = 0.5 be the frequency distribution defined on
both of them. Intuitively, in the chain case inequality is higher, since elements
a and c are at the “vertical extremes” of the poset, while in the antichain case
there is no such “vertical” dimension. Keeping fixed the frequency distribution p,
indicator G(·, ·) is thus required to depend upon the underlying ordinal relation
and, in particular, not to decrease as the poset gets extended. In addition, since all
finite posets can be reconstructed by their linear extensions, the value of G(·, ·) on
π is also required to be a function of the degree of inequality of p on the linear
extensions of π ; as derived in Fattore (2017), such a function must belong to the
class of power means. In summary, the inequality degree of a frequency distribution
on a poset π is computed as some power mean of its inequality degrees over the
linear extensions of π . But linear extensions are complete orders and, on them,
inequality can be measured by using any of the ordinal unidimensional indicators
available in literature Maggino and Fattore (2019). This way, the measurement
of inequality on a complex ordered structure gets reduced to an aggregation of
inequality indicators over simple linear orders. Needless to say, this very same
approach can be applied to many other kinds of synthetic indicators, for which
unidimensional ordinal counterparts are available.
3.4.1 Real Example
We apply the procedure outlined above to the measurement of inequality of
childhood poverty in the Democratic Republic of Congo (DRC). Data are taken
from Table 3 of Nanivazo (2015), here replicated in Table 5, and assess poverty in
terms of four binary attributes (1 – Deprivation; 0 – No deprivation):
1. Sanitation deprivation – Children with no access to any kind of improved latrines
or toilets.
2. Water deprivation – Children with only access to surface water for drinking or
for whom the nearest source of water is more than a 15 min walking distance
from their dwellings.
3. Shelter deprivation – Children living in dwellings with more than five people per
room or with no flooring material (e.g., a mud floor).
M. Fattore and A. Arcagni
The formal development of the theory of synthetic indicators over posets is
not trivial (details can be found in Fattore 2017), but the basic idea is quite
simple and relies, again, on the equivalence between finite posets and their set
of linear extensions. Let π be an n-element poset generated by a MIS and let
p = (p 1 , . . . , p n ) be a relative frequency distribution over it (so that p i is the
fraction of statistical units associated to the i-th element of π ). Suppose, to set the
stage, that we want to measure how unequally p is distributed on π , by means of
a non-negative synthetic indicator G(p, π). Notice that G(·, ·) not only depends
upon the frequency vector p, but also upon the underlying order structure, i.e. upon
poset π . To see why this dependence is essential, just consider two posets on three
elements {a, b, c}, π 1 = a b c (a chain) and π 2 = a||b||c (an antichain),
and let p a = 0.5, p b = 0, p c = 0.5 be the frequency distribution defined on
both of them. Intuitively, in the chain case inequality is higher, since elements
a and c are at the “vertical extremes” of the poset, while in the antichain case
there is no such “vertical” dimension. Keeping fixed the frequency distribution p,
indicator G(·, ·) is thus required to depend upon the underlying ordinal relation
and, in particular, not to decrease as the poset gets extended. In addition, since all
finite posets can be reconstructed by their linear extensions, the value of G(·, ·) on
π is also required to be a function of the degree of inequality of p on the linear
extensions of π ; as derived in Fattore (2017), such a function must belong to the
class of power means. In summary, the inequality degree of a frequency distribution
on a poset π is computed as some power mean of its inequality degrees over the
linear extensions of π . But linear extensions are complete orders and, on them,
inequality can be measured by using any of the ordinal unidimensional indicators
available in literature Maggino and Fattore (2019). This way, the measurement
of inequality on a complex ordered structure gets reduced to an aggregation of
inequality indicators over simple linear orders. Needless to say, this very same
approach can be applied to many other kinds of synthetic indicators, for which
unidimensional ordinal counterparts are available.
3.4.1 Real Example
We apply the procedure outlined above to the measurement of inequality of
childhood poverty in the Democratic Republic of Congo (DRC). Data are taken
from Table 3 of Nanivazo (2015), here replicated in Table 5, and assess poverty in
terms of four binary attributes (1 – Deprivation; 0 – No deprivation):
1. Sanitation deprivation – Children with no access to any kind of improved latrines
or toilets.
2. Water deprivation – Children with only access to surface water for drinking or
for whom the nearest source of water is more than a 15 min walking distance
from their dwellings.
3. Shelter deprivation – Children living in dwellings with more than five people per
room or with no flooring material (e.g., a mud floor).