26
F. Maggino et al.
typical aspect is its toxicity. Toxicity is usually measured as that concentration (of
the chemical) where p% of test organisms show an adverse effect (LC). In contrast
to accumulation a large value of LC implies a lower toxicity. Hence both indicators
together will be meaningless, a re-orientation is necessary to find a co-monotone
behaviour in both aspects. When a multi-indicator system is conceptualized in order
to support decisions in a complex system then the above requirements will select
out many candidates.
2.3 Indicator Systems
A final system of indicators leading to a Hasse diagram (or a poset if the
visualization is not in the focus of the study) can be seen as a system, where the
parts (the single indicators) are combined to a graph which indeed allows more
insight into the indicator system and the objects described by the indicators, then
an ensemble of single indicators. The Hasse diagram can be in two extremal states,
both visualized by extremely simple Hasse diagrams:
(i) No relation among the objects -AC (antichain)
(ii) All objects are related – CC (complete chain)
The first case (AC) may be a result of information noise or by an incorrect
orientation, the second case (CC) leads to a ranking, i.e. to an ordering of all objects
under all the indicators applied. In reality the Hasse diagram is in a state between the
two extremal cases and in the mathematically oriented literature there are attempts
to measure the complexity of Hasse diagrams (Luther et al. 2000; Restrepo 2014).
The degree, measuring the state of a Hasse diagram (or of the partial order it is
representing) with respect to AC or CC is of eminent importance and is certainly
not available if the indicator-system are seen only as an ensemble of many single
indicators.
The fact that a Hasse diagram is somewhere between the two states leads
immediately to the second mathematical component, namely combinatorics. The
question is, as to how far a ranking can be found without an aggregation of the
indicators, for example by weighted sums. The conceptual idea is very simple: Can
we find an order preserving map, by which a poset is mapped into an order, where
all objects are mutually comparable. For example: Three indicators leading to a
Hasse diagram of type (AC) imply that there are six such mappings! When a Hasse
diagram belongs to type (CC) then the requirement of “order preserving” implies
that only and only one mapping can be found. Generally, however one obtains a
number of order preserving mappings between 1 and 2 m , with m being the number
of indicators. Once such a set of mappings is obtained, statistical measures can be
applied to characterize this set of mappings. In the literature often just mean values
are used, to describe in the average the position each object has in the image of each
mapping (Rkav: average rank). Although numerical devices are known (Bubley and
Dyer 1999) and also a mathematically extremely elegant approach (De Loof et al.
F. Maggino et al.
typical aspect is its toxicity. Toxicity is usually measured as that concentration (of
the chemical) where p% of test organisms show an adverse effect (LC). In contrast
to accumulation a large value of LC implies a lower toxicity. Hence both indicators
together will be meaningless, a re-orientation is necessary to find a co-monotone
behaviour in both aspects. When a multi-indicator system is conceptualized in order
to support decisions in a complex system then the above requirements will select
out many candidates.
2.3 Indicator Systems
A final system of indicators leading to a Hasse diagram (or a poset if the
visualization is not in the focus of the study) can be seen as a system, where the
parts (the single indicators) are combined to a graph which indeed allows more
insight into the indicator system and the objects described by the indicators, then
an ensemble of single indicators. The Hasse diagram can be in two extremal states,
both visualized by extremely simple Hasse diagrams:
(i) No relation among the objects -AC (antichain)
(ii) All objects are related – CC (complete chain)
The first case (AC) may be a result of information noise or by an incorrect
orientation, the second case (CC) leads to a ranking, i.e. to an ordering of all objects
under all the indicators applied. In reality the Hasse diagram is in a state between the
two extremal cases and in the mathematically oriented literature there are attempts
to measure the complexity of Hasse diagrams (Luther et al. 2000; Restrepo 2014).
The degree, measuring the state of a Hasse diagram (or of the partial order it is
representing) with respect to AC or CC is of eminent importance and is certainly
not available if the indicator-system are seen only as an ensemble of many single
indicators.
The fact that a Hasse diagram is somewhere between the two states leads
immediately to the second mathematical component, namely combinatorics. The
question is, as to how far a ranking can be found without an aggregation of the
indicators, for example by weighted sums. The conceptual idea is very simple: Can
we find an order preserving map, by which a poset is mapped into an order, where
all objects are mutually comparable. For example: Three indicators leading to a
Hasse diagram of type (AC) imply that there are six such mappings! When a Hasse
diagram belongs to type (CC) then the requirement of “order preserving” implies
that only and only one mapping can be found. Generally, however one obtains a
number of order preserving mappings between 1 and 2 m , with m being the number
of indicators. Once such a set of mappings is obtained, statistical measures can be
applied to characterize this set of mappings. In the literature often just mean values
are used, to describe in the average the position each object has in the image of each
mapping (Rkav: average rank). Although numerical devices are known (Bubley and
Dyer 1999) and also a mathematically extremely elegant approach (De Loof et al.