Ranking and Prioritization for Multi-indicator Systems

Chemicals can be as harmful to humans and the environment as they are useful. Therefore, it appears rather clear that only those chemicals should be used in the market that do not have an adverse impact on humans and the environment. The list of chemicals in the market of the European Union between 1971 and 1981 (EINECS list, http://​chemicalwatch.​com/​927) contains 1,00,000 chemicals and almost 1,000 chemicals newly enter the market yearly, see, e.g., Bruggemann and Drescher-Kaden (2003), van Leeuwen et al. (1996), and Ahlers (1999). How do we find out whether they are hazardous? There are many time-consuming and expensive investigations necessary to perform a risk assessment. Hence the question is: With which chemicals to begin at first? Thus ranking is needed to give the more involved investigations a reasonable operating sequence (Newman, 1995).

Suppose there are five objects. Think of them as sediment samples a, b, c, d, e which we would like to rank. The first and main question is: What is the aim of ranking? We can rank the five sediment samples according to their age, or according to their content of a mineral, etc. If we know the aim of ranking, we need to identify properties that are relevant. In the case of ranking according to their age, it may be simple. Just order the samples according to their age!

It is of interest to compare two partially ordered sets (more about this topic, see Chapter 10). We may, for example, ask whether all ≤ relations in one poset are reproduced in the other. In technical terms, we are asking whether or not

The fundamental basis of our ordinal analysis is the data matrix: The attributes define its columns and the objects its rows. We pose two questions: 1. What role does any single attribute play? Can we, for example, save time and money, because some attribute has little comparative power? 2. What can be said about the attribute set from the partial order point of view? Should we delete any attribute? Should we add more attributes to the data matrix? We attempt to answer these two questions in this chapter. In Section 4.2, we consider impacts to the partial order when reducing the information base (attribute-related sensitivity) and in Section 4.3, we introduce a measure for the ambiguity of partial order in response to the set of attributes.

1. Whether or not there is a messy system of lines.

2. Whether the Hasse diagram resembles a

a. triangular shape or

b. rectangular shape.

3. Whether there are different components or approximate components.

We have seen in Chapter 5 as to why and how the structure of partial order (X, IB) can be related to properties of the data matrix. However, partial order with many objects can lead to messy Hasse diagrams with too many lines hiding the structure. What may be the reason for complexity in such diagrams? The number of objects |X| is not necessarily causing messy Hasse diagrams because chains of height |X| or antichains of width |X| certainly allow clear visualizations. There is another reason for complexity: In partial orders, we obtain either x < y or x || y even if the numerical difference ɛ between attribute values is small:

In Chapter 6, we have shown as to how we can obtain simpler Hasse diagrams from messy ones. We transformed the data matrix so that incomparabilities or comparabilities disappear because objects become equivalent.

So far the core of all considerations was the partial order and its visualization by a Hasse diagram. On the one hand, the system of lines allowed us to identify comparabilities and on the other hand, it also revealed the status of objects relative to the others. In some cases, the Hasse diagram had a structure so that it was possible to explain as to why a certain relative position was obtained for an object. The concept of antagonistic indicators helped in clarifying the reasons for certain positions. The Hasse diagram is a graph focusing on the objects and their mutual relations. It will be extremely helpful, if we can construct a directed graph, where at the same time the constellation of the relevant attribute values responsible for the position of the object is exhibited. As we have seen in Chapters 6 and 7, we may perform ordinal modeling by focusing on object-related or attribute-related manipulations. In the theory of “formal concept analysis,” mutual relationship of the position of an object with the values of its attributes inducing its position is depicted into one single diagram (Davey and Priestley, 1990; Ganter and Wille, 1986; Wolff, 1993; Gugisch, 2001; Carpineto and Romano, 1994; Annoni and Bruggemann, 2008, 2009; Bartel and Nofz, 1997; Bartel, 1997; Kerber, 2006).

There are two reasons to compare different orders:

1. From (X, IB), a linear or weak order, called O _poset, can be derived, applying methods rendered in Chapter 9. By calculation of an index (Chapter 7), another linear or weak order can be obtained, which we call O _Γ. As there are the same set of indicators, IB, and the same set of objects, X, O _poset, and O _Γ should ideally be coincident. As can be suspected, this is not necessarily the case and we need measures to quantify the degree of coincidence between O _poset and O _Γ.

Here we demonstrate some of the tools outlined in Chapters 2, 3, 4, 5, 6, 7, 8, 9, and 10. The illustrative case studies are based on “real-life” data matrices. The sections are organized as follows (Table 11.1).

Starting point is the data matrix made of six indicators describing different aspects of child development and 21 nations, mainly of Europe. A ranking of the 21 nations is based on a composite indicator. Using several methods, we try to assess the construction of the composite indicator of UNICEF. Both the concordance analysis (Chapter 10) and the canonical order (Chapter 9) support the construction of the UNICEF. The comparison with the canonical order is naturally more detailed and we find some rank inversions. With the help of the local partial order model (Chapter 9), we explain these. Without partial order, there is no reason to define separated subsets. The partial order constructed from the six indicators and the 21 nations shows several separated subsets (see Chapter 5). Most striking is the separation between {It, Pt} and the residual set of nations. Which indicators explain this separation and with which values? The partial order identifies the indicators “family” and “education” as the responsible ones. When an aggregation to a composite indicator is performed, the single indicators lose their individuality as they are just summands contributing to the value of the composite indicator. Similarly, one can construct new orders (called the m ^r orders, see Chapter 7) which also do not take into account the individuality of the indicators. The resulting partial orders contain pretty long chains of nations which allow unambiguous ranking of many nations without crunching the indicators into one composite indicator.

Forty-nine bridge crossings were described by 13 indicators. As in Chapter 12, a composite indicator was suggested. The partial order analysis applied several tools in order to demonstrate its versatility. So, for example, formal concept analysis (Chapter 8) was applied resulting in a network of implications. These implications may be considered as hypotheses and should motivate further investigations concerning the problem of bridge stability in stream or channel crossings.

Twenty-one watersheds were characterized and ranked on the basis of two multiple indicator systems, level 1 (least expensive) and level 2 (expensive). Furthermore, level 3 indicators are defined which need investigations in the field and are pretty expensive and only six watersheds are characterized.

Composite indicators are defined on the basis of level 1 indicators, called LSI, and level 2 indicators, called SWR. LSI and SWR are thought of as two different means to rank the watersheds with respect to the environmental health. The indicators on the three levels are considered as proxies to describe the abstract and not measurable concept of “environmental health.”

In the EPI study, 16 indicators were introduced to analyze nations with respect to the human health and ecosystem vitality. Once again, indicators are used as proxies of an abstract principle and partial order shows how nations are ordered following these proxies.

It is always hard to try a positioning. Nevertheless, it may help interested readers to find their way through the jungle of concepts, relations, and equations of this text. Certainly, partial order has to do with graph theory in discrete mathematics, as its visualization is a digraph and questions like connectivity or identification of articulation points and of separated subsets are typical of graph theory; see, e.g., Wagner and Bodendiek 1989) and Patil and Taillie (2004). There is also a connection to the network domain, as partial order constitutes a directed graph, which is one of the characteristics of networks. In our applications here, there is always a matrix, which quantifies the multi-indicator system, the data matrix. With or without the interim step of deriving the rank matrix, we arrive at a partial order. Once, however, the poset is derived, it may be analyzed as a mathematical object on its own right.

Halfon (2006) reviews available software. We summarize and update the review for the convenience of the reader

RANA, Pavan (2003)

DART, Manganaro et al. (2008)

PRORANK, Pudenz (2005) and Voigt et al. (2006)

CORRELATION, Sørensen et al. (2005)

WHASSE, Bruggemann et al. (1999)

POSAC, Shye (1994) and Borg and Shye (1995); see also http://​ca.​huji.​ac.​il/​bf/​hudap-Info.​pdf

POSET, Patil et al. (2009, personal communication (POSET-ranking))

PyHasse, Bruggemann et al. (2008a, b), Bruggemann and Voigt (2009), and Voigt et al. (2008a, b)

VB-RAPID, Joshi et al. (2010)

We started with “Why prioritization, why ranking.” We showed that the comparability is a concept, which is suitable for performing prioritization or ranking. When we use computational support, concepts like “comparability” must be formalized. Partial order fits best into the powerful evaluation and utilization of multi-indicator systems.