BackgroundBackgroundBackgroundBackground (cont'd)Background (cont'd)Background (cont'd)Crisp vs. Fuzzy Areal WomblingCrisp vs. Fuzzy Areal WomblingCrisp vs. Fuzzy Areal WomblingHierarchical modeling approachHierarchical modeling approachHierarchical modeling approachHierarchical modeling approachHierarchical modeling approachHierarchical modeling approachHierarchical modeling approachExample: Cancer late detection riskExample: Cancer late detection riskExample: Cancer late detection riskExample: Cancer late detection riskExample: Cancer late detection riskAlternative: Modeling adjacencyAlternative: Modeling adjacencyAlternative: Modeling adjacencyAlternative: Modeling adjacencyAlternative: Modeling adjacencyAlternative: Modeling adjacencyAlternative: Modeling adjacencyAlternative: Modeling adjacencyAlternative: Modeling adjacencyTwo-level CAR {ed (CAR2)} modelTwo-level CAR {ed (CAR2)} modelTwo-level CAR {ed (CAR2)} modelTwo-level CAR {ed (CAR2)} modelAnother Approach: SE-Ising modelAnother Approach: SE-Ising modelSE-Ising model (cont'd)SE-Ising model (cont'd)SE-Ising model (cont'd)SE-Ising model (cont'd)Spatial ZIP modelSpatial ZIP modelSpatial ZIP modelSpatial ZIP modelSpatial ZIP modelApplication: Hospices near Duluth, MNApplication: Hospices near Duluth, MNApplication: Hospices near Duluth, MNApplication: Hospices near Duluth, MNApplication: Hospices near Duluth, MNApplication: Hospices near Duluth, MNSt. Luke's Hospice Service AreaSt. Luke's Hospice Service AreaSt. Luke's Hospice Service AreaSt. Luke's Hospice Service AreaSt. Luke's Hospice Service AreaSMDC Hospice Service AreaSMDC Hospice Service AreaSMDC Hospice Service AreaSMDC Hospice Service AreaSMDC Hospice Service AreaEdge Correction and ThresholdingEdge Correction and ThresholdingEdge Correction and ThresholdingEdge Correction and ThresholdingSpatial ZIP results based on $omega _i$Spatial ZIP results based on $omega _i$Spatial ZIP results based on $omega _i$LC, ZIP, and SE-Ising for St. Luke'sLC, ZIP, and SE-Ising for St. Luke'sLC, ZIP, and SE-Ising for St. Luke'sLC, ZIP, and SE-Ising for St. Luke'sLC, ZIP, and SE-Ising for SMDCLC, ZIP, and SE-Ising for SMDCLC, ZIP, and SE-Ising for SMDCResidual boundaries for St. Luke'sResidual boundaries for St. Luke'sResidual boundaries for St. Luke'sResidual boundaries for St. Luke'sDiscussionDiscussionDiscussionOne more issue: {lue Multivariate} dataOne more issue: {lue Multivariate} dataApplication: St. Luke's/SMDC again!Application: St. Luke's/SMDC again!Application: St. Luke's/SMDC again!Application: St. Luke's/SMDC again!Bayesian Spatial Boundary Analysisfor Areal Health Outcome DataHaijun Ma, Bradley P. Carlin, and Sudipto [email protected], [email protected], and [email protected] of BiostatisticsSchool of Public HealthUniversity of MinnesotaBayesian Spatial Boundary Analysis for Areal Health Outcome Data – p. 1/25BackgroundSpatial data typically classified two ways:point-referenced (geostatistical): spatial locationsare points with known coordinatesareal (lattice): locations are geographic regions (e.g.,counties) with adjacency informationBayesian Spatial Boundary Analysis for Areal Health Outcome Data – p. 2/25BackgroundSpatial data typically classified two ways:point-referenced (geostatistical): spatial locationsare points with known coordinatesareal (lattice): locations are geographic regions (e.g.,counties) with adjacency informationImportant topic in spatial statistics: boundary analysis,wherein we seek to identify regions of abrupt change.Indicated by:steep gradients in a continuous surfaceregional boundaries separating regions withdrastically different measurements in a latticesurfaceBayesian Spatial Boundary Analysis for Areal Health Outcome Data – p. 2/25BackgroundSpatial data typically classified two ways:point-referenced (geostatistical): spatial locationsare points with known coordinatesareal (lattice): locations are geographic regions (e.g.,counties) with adjacency informationImportant topic in spatial statistics: boundary analysis,wherein we seek to identify regions of abrupt change.Indicated by:steep gradients in a continuous surfaceregional boundaries separating regions withdrastically different measurements in a latticesurfaceFocus in this talk: Boundary analysis for areal dataBayesian Spatial Boundary Analysis for Areal Health Outcome Data – p. 2/25Background (cont’d)Spatial boundary analysis techniques often calledwombling, after a foundational paper by Womble (1951).Bayesian Spatial Boundary Analysis for Areal Health Outcome Data – p. 3/25Background (cont’d)Spatial boundary analysis techniques often calledwombling, after a foundational paper by Womble (1951).In areal (or polygonal) wombling, a dissimilarity metricmeasures the difference between adjacent regions.Methods for choosing boundary elements:absolute (dissimilarity metrics greater than C)relative (dissimilarity metrics in the top k%)Bayesian Spatial Boundary Analysis for Areal Health Outcome Data – p. 3/25Background (cont’d)Spatial boundary analysis techniques often calledwombling, after a foundational paper by Womble (1951).In areal (or polygonal) wombling, a dissimilarity metricmeasures the difference between adjacent regions.Methods for choosing boundary elements:absolute (dissimilarity metrics greater than C)relative (dissimilarity metrics in the top k%)Problems:Relative (top k%) thresholding method always findsa fixed number of boundary elementsApproach is algorithmic, rather than stochastic: nomodel or likelihood, so statements about the“significance” of a detected boundary only relative topredetermined, often unrealistic null distributions.Bayesian Spatial Boundary Analysis for Areal Health Outcome Data – p. 3/25Crisp vs. Fuzzy Areal WomblingSuppose we have responses Zifor regions i = 1, . . . , nFor neighboring regions i and j and some distancemetric || · ||, assign the boundary likelihood value (BLV)Dij= ||Zi− Zj|| .Bayesian Spatial Boundary Analysis for Areal Health Outcome Data – p. 4/25Crisp vs. Fuzzy Areal WomblingSuppose we have responses Zifor regions i = 1, . . . , nFor neighboring regions i and j and some distancemetric || · ||, assign the boundary likelihood value (BLV)Dij= ||Zi− Zj|| .Crisp wombling: Boundary is those edges having BLV’sabove specified thresholds, i.e., for some c > 0,{(i, j) : Dij> c, i adjacent to j} .Bayesian Spatial Boundary Analysis for Areal Health Outcome Data – p. 4/25Crisp vs.