1 Multi-criterion Ranking and Poset Prioritization G. P. Patil December 2004 – January 2005

2 We also present a prioritization innovation. It lies in the ability for prioritization and ranking of hotspots based on multiple indicator and stakeholder criteria without having to integrate indicators into an index, using Hasse diagrams and partial order sets. This leads us to early warning systems, and also to the selection of investigational areas. Prioritization Innovation Partial Order Set Ranking

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4 First stage screening First stage screening –Significant clusters by SaTScan and/or upper level sets upper level sets Second stage screening Second stage screening –Multicriteria noteworthy clusters by partially ordered sets and Hass diagrams Final stage screening Final stage screening –Follow up clusters for etiology, intervention based on multiple criteria using Hass diagrams based on multiple criteria using Hass diagrams Multiple Criteria Analysis, Multiple Indicators and Choices, Health Statistics, Disease Etiology, Health Policy, Resource Allocation

5 Ranking Possible Disease Clusters in the State of New York Data Matrix

6 Hotspot Prioritization and Poset Ranking Multiple hotspots with intensities significantly elevated relative to the rest of the region Multiple hotspots with intensities significantly elevated relative to the rest of the region Ranking based on likelihood values, and additional attributes: raw intensity values, socio-economic and demographic factors, feasibility scores, excess cases, seasonal residence, atypical demographics, etc. Ranking based on likelihood values, and additional attributes: raw intensity values, socio-economic and demographic factors, feasibility scores, excess cases, seasonal residence, atypical demographics, etc. Multiple attributes, multiple indicators Multiple attributes, multiple indicators Ranking without having to integrate the multiple indicators into a composite index Ranking without having to integrate the multiple indicators into a composite index

7 Regions of comparability and incomparability for the inherent importance ordering of hotspots. Hotspots form a scatterplot in indicator space and each hotspot partitions indicator space into four quadrants

8 Figure 2. Contour of index H passing through object a. A linear index is shown on the left and a non-linear index on the right.

9 Figure 3. The top two diagrams depict valid contours while the bottom two diagrams depict invalid contours.

10 Figure 4. The tradeoff or substitutability between height and weight in assessing the size of a person. The tradeoff is constant with a linear index (left) but varies across indicator space with a nonlinear index (right).

11 HUMAN ENVIRONMENT INTERFACE LAND, AIR, WATER INDICATORS RANK COUNTRY LANDAIRWATER 1 Sweden 2 Finland 3 Norway 5 Iceland 13 Austria 22 Switzerland 39 Spain 45 France 47 Germany 51 Portugal 52 Italy 59 Greece 61 Belgium 64 Netherlands 77 Denmark 78 United Kingdom 81 Ireland for land - % of undomesticated land, i.e., total land area-domesticated (permanent crops and pastures, built up areas, roads, etc.) for air - % of renewable energy resources, i.e., hydro, solar, wind, geothermal for water - % of population with access to safe drinking water

12 Hasse Diagram (all countries)

13 Hasse Diagram (Western Europe)

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15 Figure 5. Hasse diagrams for four different posets. Poset D has a disconnected Hasse diagram with two connected components {a, c, e} and {b, d}.

16 Figure 6. Bottom-up Hasse diagrams for the posets of Figure 5. Hasse diagrams for Posets A and B are unchanged.

17 Figure 7. Hasse diagram for the four countries of Table 1. Note that it has the same structure as Poset A in Figure 5.

18 Figure 10. Hasse diagram for Latin America. There are four connected components. Three of these components are isolates; the remaining component contains 13 countries.

19 Figure 11. Hasse diagram for the 52 watersheds in the primary component. Labels are (arbitrary) row numbers in the data matrix.

20 Figure 12. Map of the Mid-Atlantic region showing the primary Hasse component (shaded). Geographically, there are three connected components of which two are small and located near the periphery of the region.

21 Figure 13: Hasse diagrams (right) of the two possible rankings for the poset on the left.

22 Figure 14. Rank-intervals for all 106 countries. The intervals (countries) are labeled by their midpoints as shown along the horizontal axis. For each interval, the lower endpoint and the upper endpoint are shown vertically. The length of each interval corresponds to the ambiguity inherent in attempting to rank that country among all 106 countries.

23 Figure 15. Rank-intervals for all 106 countries, plotted against their HEI rank. The HEI rank appears as the 45-degree line. The HEI tends to be optimistic (closer to the lower endpoint) for better-ranked countries and pessimistic (closer to the upper endpoint) for poorer-ranked countries.

24 Figure 16. A ranking of a poset determines a linear Hasse diagram. The numerical rank assigned to each element is that element’s depth in the Hasse diagram.

25 Figure 17. Hasse diagram of Poset B (left) and a decision tree enumerating all possible linear extensions of the poset (right). Every downward path through the decision tree determines a linear extension. Dashed links in the decision tree are not implied by the partial order and are called jumps. If one tried to trace the linear extension in the original Hasse diagram, a “jump” would be required at each dashed link. Note that there is a pure-jump linear extension (path a, b, c, d, e, f) in which every link is a jump.

26 Figure 18. Histograms of the rank-frequency distributions for Poset B.

27 Cumulative Rank Frequency Operator – 5 An Example of the Procedure In the example from the preceding slide, there are a total of 16 linear extensions, giving the following cumulative frequency table. Rank Element a b c d e f Each entry gives the number of linear extensions in which the element (row label) receives a rank equal to or better that the column heading

28 Cumulative Rank Frequency Operator – 6 An Example of the Procedure 16 The curves are stacked one above the other and the result is a linear ordering of the elements: a > b > c > d > e > f

29 Cumulative Rank Frequency Operator – 7 An example where F must be iterated Original Poset (Hasse Diagram) a f eb c g d h a f e b ad c h g a f e b ad c h g F F 2

30 Cumulative Rank Frequency Operator – 8 An example where F results in ties Original Poset (Hasse Diagram) a cb d a b, c (tied) d F Ties reflect symmetries among incomparable elements in the original Hasse diagram Elements that are comparable in the original Hasse diagram will not become tied after applying F operator

31 Incorporating Judgment Poset Cumulative Rank Frequency Approach Certain of the indicators may be deemed more important than the others Certain of the indicators may be deemed more important than the others Such differential importance can be accommodated by the poset cumulative rank frequency approach Such differential importance can be accommodated by the poset cumulative rank frequency approach Instead of the uniform distribution on the set of linear extensions, we may use an appropriately weighted probability distribution , e.g., Instead of the uniform distribution on the set of linear extensions, we may use an appropriately weighted probability distribution , e.g.,

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