56
R. Bruggemann and L. Carlsen
The character of the resulting graph can easily be deduced by checking dh/dk:
The slope at k = 0 is 0, whereas the slope at k = 1 equals 2.
In all these three situations the behavior of U is calculated assuming an evolution
of uncertainty, corresponding to case 2a, i.e., while s is increasing and thus
extending the integral [gjmin, gjmax], the starting value of g(1) is always 0. It
should be noted that this is not a realistic assumption, following Nardo (2008)
recommending extreme weights (i.e. g1 near 0 or near 1) are to be avoided. However,
it is a simple consideration to see that the evolution of uncertainties around g1 = 0.5
is similar to that of Fig. 5, which is the outcome of a linearly decreasing distribution
of gc-values. Thus, env((0.5,0.5),s) sees many realizations for gc-values in the
neighborhood of g1 = 0.5, then increasing s and thus symmetrically increasing the
environments the number of realizations is diminishing. At g1 = 0.5 and s = 0 there
is exact knowledge of the weight, there is just one composite indicator and U = 0.
Extending s there are many gc-values with high realizations in the neighborhood of
g1 = 0.5 therefore U by Eq. 11 will strongly increase and then reach the final value
U = U0 asymptotically with a reduced slope.
3.2 Real World Example
3.2.1 Preliminaries
The following example is based on a report of UNICEF (UNICEF, Innocenti
Research Centre, Report card 7, 2007). A comprehensive assessment of well-being
of “children and young people in 21 nations of the industrial world” is given (quoted
from the report). The following part is an abridged version taken from Bruggemann
and Patil (2011).
The study provided 40 different indicators, which are aggregated through several
interim steps into 6 main indicators as follows together with their identifiers:
1. Material well-being, wb
2. Health and safety, hs
3. Educational well-being, ed.
4. Family and peer relationships, fa
5. Behaviors and risks, br
6. Subjective well-being, sub
The sixth indicator, sub, needs some explanation: It is the attempt to “reflect children’s own views and voices – for example, the surveys of reported family affluence,
experience of bullying, or the frequency of communications with parents.” (quoted
from the report).
From the data matrix UNICEF defines a composite indicator with equal weights
for all 6 indicators, i.e.:
CI (x) =
(1/6) ∗ R i (x) , weight vector = (1/6, 1/6, .., 1/6)
(13)
R. Bruggemann and L. Carlsen
The character of the resulting graph can easily be deduced by checking dh/dk:
The slope at k = 0 is 0, whereas the slope at k = 1 equals 2.
In all these three situations the behavior of U is calculated assuming an evolution
of uncertainty, corresponding to case 2a, i.e., while s is increasing and thus
extending the integral [gjmin, gjmax], the starting value of g(1) is always 0. It
should be noted that this is not a realistic assumption, following Nardo (2008)
recommending extreme weights (i.e. g1 near 0 or near 1) are to be avoided. However,
it is a simple consideration to see that the evolution of uncertainties around g1 = 0.5
is similar to that of Fig. 5, which is the outcome of a linearly decreasing distribution
of gc-values. Thus, env((0.5,0.5),s) sees many realizations for gc-values in the
neighborhood of g1 = 0.5, then increasing s and thus symmetrically increasing the
environments the number of realizations is diminishing. At g1 = 0.5 and s = 0 there
is exact knowledge of the weight, there is just one composite indicator and U = 0.
Extending s there are many gc-values with high realizations in the neighborhood of
g1 = 0.5 therefore U by Eq. 11 will strongly increase and then reach the final value
U = U0 asymptotically with a reduced slope.
3.2 Real World Example
3.2.1 Preliminaries
The following example is based on a report of UNICEF (UNICEF, Innocenti
Research Centre, Report card 7, 2007). A comprehensive assessment of well-being
of “children and young people in 21 nations of the industrial world” is given (quoted
from the report). The following part is an abridged version taken from Bruggemann
and Patil (2011).
The study provided 40 different indicators, which are aggregated through several
interim steps into 6 main indicators as follows together with their identifiers:
1. Material well-being, wb
2. Health and safety, hs
3. Educational well-being, ed.
4. Family and peer relationships, fa
5. Behaviors and risks, br
6. Subjective well-being, sub
The sixth indicator, sub, needs some explanation: It is the attempt to “reflect children’s own views and voices – for example, the surveys of reported family affluence,
experience of bullying, or the frequency of communications with parents.” (quoted
from the report).
From the data matrix UNICEF defines a composite indicator with equal weights
for all 6 indicators, i.e.:
CI (x) =
(1/6) ∗ R i (x) , weight vector = (1/6, 1/6, .., 1/6)
(13)