1. 1 Poset Prioritization G. P. Patil October 2005

2. 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

3. 3

4. 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. 5 Ranking Possible Disease Clusters in the State of New York Data Matrix

6. 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. 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. 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. 9 Figure 3. The top two diagrams depict valid contours while the bottom two diagrams depict invalid contours.

10. 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. 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 69.0176.4627.381.7940.5730.1732.6328.3432.5634.6223.3521.5921.8419.439.8312.649.2535.2419.0563.9880.2529.8528.107.746.502.1014.296.893.200.001.075.041.131.99100981001001001001001001008210098100100100100100 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. 12 Hasse Diagram (all countries)

13. 13 Hasse Diagram (Western Europe)

14. 14

15. 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. 16 Figure 6. Bottom-up Hasse diagrams for the posets of Figure 5. Hasse diagrams for Posets A and B are unchanged.

17. 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. 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. 19 Figure 11. Hasse diagram for the 52 watersheds in the primary component. Labels are (arbitrary) row numbers in the data matrix.

20. 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. 21 Figure 13: Hasse diagrams (right) of the two possible rankings for the poset on the left.

22. 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. 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. 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. 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. 26 Figure 18. Histograms of the rank-frequency distributions for Poset B.

27. 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 Element123456 a91416161616 b71215161616 c0410161616 d026121616 e00141016 f0000616 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. 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. 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. 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. 31

32. 32 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.,

33. 33 Data Analysis of Environmental Air Pollutant Monitoring Systems in Europe Kristina Voigt, Gerhard Welzl, and Rainer Brüggemann Environmetrics 2004; 15: 577-596 Multiple Indicators, Partially Ordered Sets, and Linear Extensions: Multi-criterion Ranking and Prioritization G.P. Patil and Charles Taillie Environmental and Ecological Statistics 2004; 11: 199-228

34. 34 Evaluation Criteria and their Scores Number of chemicals monitored/NUScore <40 5-71 >72 Types and duration of measurements/ME Only daily values, yesterday’s values0 daily + past week, past 2 weeks1 daily + past weeks + past 6 months, past year(s)2 Measurement stations in capital/ST 1-20 3-51 >62 Method of presentation on the Internet/PR Table0 Graphic1 Table + Graphic2 Background material/BM Explanations, reports0 +limit values, laws and regulations1 direct communication, measures2

35. 35 15 European Monitoring Stations Evaluated by 5 Criteria Abb.NUMESTPRBM AUS12121 BEL12122 DEN22022 FIN11211 FRA12222 GER22221 GRE10200 IRE00011 ITA20211 LUX00010 NET11021 POR10201 SPA00110 SWE12010 UNK10212

36. 36 Hasse Diagram for the EEC 15*5 Dataset Abbr.Rank of EEC5 FRA1 GER2 BEL3 UNK4 DEN5 ITA6 AUS7 FIN8 NET9 POR10.5 SWE10.5 SPA12 GRE13 IRE14 LUX15

37. 37 Hasse Diagram : Levels

38. 38 Regions in the Coordinate Space of POSAC Y YpYp 0 P XpXp X (III) Region of points incomparable to P (II) Region of points greater than P (III) Region of points incomparable to P (I) Region of points smaller than P

39. 39 Two Profiles in a Cartesian Coordinate Space 2,2,2,2,1 GER 2,0,2,1,1 ITA 2 1 Y 1 X 2,2,2,2,1 GER 2,0,2,1,1 ITA 2 1 Y 1 X 2,2,2,2,1 GER 2,0,2,1,1 ITA 2 1Y 1 X 2 For these possible score assignments are: 2,0,2,1,1  112,0,2,1,1  112,0,2,1,1  11 2,2,2,2,1  122,2,2,2,1  212,2,2,2,1  22

40. 40 Three profiles in a two-dimensional coordinate space Y 3 2 1 0 3210 2,2,0,2,2 2,2,2,2,1 2,0,2,1,1 X

41. 41

42. 42 POSAC Output Proportion of profile pairs correctly represented: 0.926 Label DIM 1 DIM 2 Joint Lateral Fit 22222 1.000 1.000 1.000 0.500 0.000 GER 0.935 0.866 0.901 0.535 0.726 FRA 0.901 0.935 0.918 0.483 0.000 BEL 0.707 0.901 0.804 0.403 0.000 DEN 0.612 0.968 0.790 0.322 0.234 AUS 0.661 0.829 0.745 0.416 0.000 FIN 0.829 0.707 0.768 0.561 0.000 UNK 0.866 0.661 0.764 0.602 0.651 ITA 0.968 0.559 0.764 0.705 0.492 NET 0.433 0.791 0.612 0.321 0.213 POR 0.750 0.354 0.552 0.698 0.076 SWE 0.354 0.750 0.552 0.302 0.108 GRE 0.791 0.250 0.520 0.770 0.076 IRE 0.500 0.612 0.556 0.444 0.415 SPA 0.559 0.500 0.530 0.530 0.234 LUX 0.250 0.433 0.342 0.408 0.000 00000 0.000 0.000 0.000 0.500 0.000

43. 43 PCA Output

44. 44

45. 45

46. 46 PCA chooses 3 variables ME, ST, and PR POSAC chooses 2 variables ST and PR Abbr.Rank of EEC5ME,ST, PRST, PR FRA11.5 GER21.5 BEL33.5 UNK47.56 DEN568.5 ITA67.56 AUS73.5 FIN856 NET99.58.5 POR10.51211 SWE10.59.514 SPA12 11 GRE131211 IRE1414.514 LUX1514.514 CORRELATION0.923450.90587

47. 47 Hasse Diagram of 15*4 Dataset (Leaving out the attribute ST) Abbr.Rank of EEC5 -ST FRA13 GER23 BEL33 UNK46.5 DEN51 ITA66.5 AUS75 FIN89.5 NET98 POR10.511.5 SWE10.59.5 SPA1214 GRE1314 IRE1411.5 LUX1514 CORRELATION0.912

48. 48 Leave Out One Variable at a Time Abbr.Rank of EEC5 -NU -ME -ST -PR -BM FRA111312 GER222321 BEL334336.5 UNK4436.548 DEN575.51 5 ITA685.56.573 AUS767.5586.5 FIN857.59.55.54 NET9998109.5 POR10.5 1011.5912 SWE10.5 13.59.5119.5 SPA12 111413.512 GRE13 121412 IRE14 13.511.513.514.5 LUX15 141514.5 CORRELATION0.96780.974020.911590.9720.891

49. 49 Sub-graph of the Object GER in the Super-graphic

50. 50 Evaluation of the W-matrix for All Elements

51. 51 Comparison of Results in Hasse Diagrams with Different Attribute Sets

52. 52 PCA Biplot of 15*5 Data Matrix

53. 53 POSAC Plot of Data Matrix 15*5

54. 54 Scatter Plots of the Variables ST and PR

55. 55 Hasse Diagram of Reduced Data Matrix 15*2 (LOVs from POSAC)

56. 56 Atlantic Slope Consortium

57. 57 Level 3 Watershed Data WATERSHEDBenthic IBIFish IBINO 3 LAB Back River1.5152.2151.318 Cattail Creek3.7563.2684.332 Gwynn Falls1.9382.4451.362 Saint Mary's A2.8583.8750.196 Southeast Creek2.6873.4443.008 Upper Patuxent3.7504.1132.836

58. 58 Description of Level 2 Indicators VariableDefinition BUFBuffer Score IRIncision Ratio BABasal Area of Trees INVInvasive Cover Class SHASHA Score STRM_STRESSNo. of Stream Stressors FP_WLSTRESNo. of Floodplain-wetland Stressors SWR_INDEXWeighted Average of Above Scores

59. 59 Level 2 Scored Watershed Data WATERSHEDBUFIRBAINVSHASTRM_STRESSFP_WLSTRESSWR_INDEX Back River0.6050.5750.4280.2030.5220.2190.7040.454 Cattail Creek0.3580.7970.5690.6530.7020.4100.2790.590 Gwynn Falls0.3230.7680.5640.6830.6770.3110.1650.548 Saint Mary's A0.7770.6890.5990.8380.6970.6350.6690.658 Southeast Creek0.7700.8190.5810.5080.6110.5710.7060.661 Upper Patuxent0.6830.6370.7250.6130.8120.6900.5420.704 Ahoskie0.5630.3620.5190.5980.4770.4000.5890.425 Buffalo Creek0.3520.8750.5820.8600.6310.3890.2450.634 Chickahominy0.5830.6600.6430.7600.5430.4530.4980.566 Christian Creek0.0950.5950.2480.9200.7040.3890.4670.522 Clearfield Creek0.6470.8370.1560.9800.7030.3950.6790.557 Conodoguinet A0.3230.7240.3640.6470.6930.3670.2150.555 Grindle Creek0.4560.3350.4680.5900.4810.4480.6160.415 Little Contentnea0.5200.7270.6280.8230.6880.6280.7440.705 Mantua0.5230.8930.6360.7590.6880.6410.3450.704 Middle Creek0.3610.7900.4520.7950.6710.3790.2620.583 Middle River0.1910.5240.2660.8350.6190.2500.3070.459 Pamunkey0.7010.5730.7040.9100.6710.5450.6770.626 Repaupo0.5310.8480.5680.6710.7110.7590.4370.717 White Deer Creek0.8600.8610.8020.9750.9360.7950.7200.864 Wisconisco0.5800.8070.5250.8400.7170.4360.4300.698

60. 60 Description of Level 1 Indicators VariableDefinition FOREST% Forest in Watershed LDILandscape Density Index in Watershed IMP% Impervious Surface in Watershed MPATMean Forest Patch Size in Watershed CORFOR% Total Forest That is Core Forest In Watershed LAND_INDEXWeighted Average of Above Scores

61. 61 Level 1 Scored Watershed Data WATERSHEDForestLDIIMPMPATCORFORLand Index Back River0.1410.0660.0220.0790.0700.087 Cattail Creek0.3160.6610.6890.2970.2410.420 Gwynn Falls0.2450.1540.0440.1820.1240.166 Saint Mary's A0.6950.7610.3070.6340.5740.611 Southeast Creek0.3240.5290.6660.4200.2630.421 Upper Patuxent0.3850.6740.6440.3660.4000.476 Ahoskie0.7020.7500.4720.6640.7250.669 Buffalo Creek0.3040.6040.5600.3930.6540.470 Chickahominy0.4460.4420.1230.3840.3670.368 Christian Creek0.2990.6360.4460.3860.3150.397 Clearfield Creek0.7410.8390.5540.7170.6140.701 Conodoguinet A0.3170.5190.2040.2450.2830.314 Grindle Creek0.5750.6800.6310.5420.6660.611 Little Contentnea0.5850.6560.5110.5360.5430.569 Mantua0.3680.4170.1390.2480.1200.277 Middle Creek0.4280.6770.5380.4320.5790.514 Middle River0.3090.6040.3380.2990.2780.356 Pamunkey0.6190.7650.6000.6700.5890.644 Repaupo0.3550.6250.5680.3790.3070.432 White Deer Creek0.9450.9610.6161.0000.8660.887 Wisconisco0.8460.8880.4800.7620.7190.757

62. 62 Hasse Diagram and POSET Ranking for Level 3 WATERSHEDL3-Rank Upper Patuxent1 Cattail Creek2.5 Southeast Creek2.5 Gwynn Falls4.5 Saint Mary's4.5 Back River6 Cattail CreekSoutheast CreekUpper Patuxent Gwynn FallsSaint Mary’s Back River

63. 63 POSAC Profile Plot

64. 64 PCA Output for Level 2

65. 65

66. 66

67. 67 Rank Correlation Between Level 3 and Level 2 Variables L2-Rank L3-Rank WATERSHEDBUF(1)IR(2)BA(3)INV(4)SHA(5)STRM(6)FP(7) Back River42636636 Cattail Creek56422462.5 Gwynn Falls63564544.5 Saint Mary's A14253214.5 Southeast Creek25345322.5 Upper Patuxent31111151 Correlation0.14712-0.088270.735610.588490.73561 -0.470791.00000

68. 68 Comparison of Indices for Level 2 WATERSHEDL2- rankL2-3456L2-356SWR-rankL3-rank Back River66666 Cattail Creek3.53342.5 Gwynn Falls3.54554.5 Saint Mary's A3.51234.5 Southeast Creek15422.5 Upper Patuxent3.52111 Correlation0.6092720.4119430.7356120.853311 P- VALUE.199.417.096.031*