BackgroundBackgroundBackgroundBackgroundBackgroundBut we can't retreat!But we can't retreat!But we can't retreat!But we can't retreat!Model-Based ApproachModel-Based ApproachModel-Based ApproachModel-Based ApproachModeling (cont'd)Modeling (cont'd)Modeling (cont'd)Really Brief Literature ReviewReally Brief Literature ReviewReally Brief Literature ReviewReally Brief Literature ReviewReally Brief Literature ReviewN Minnesota Breast Cancer DataN Minnesota Breast Cancer DataN Minnesota Breast Cancer DataN Minnesota Breast Cancer DataN Minnesota Breast Cancer DataModeling with Spatial CovariatesModeling with Spatial CovariatesModeling with Spatial CovariatesModeling with Spatial CovariatesModeling with Spatial CovariatesModeling with Spatial CovariatesModeling with Spatial CovariatesComputational ApproachComputational ApproachComputational ApproachComputational ApproachIntroducing Non-spatial CovariatesIntroducing Non-spatial CovariatesIntroducing Non-spatial CovariatesIntroducing Non-spatial CovariatesIntroducing Non-spatial CovariatesIntroducing Non-spatial CovariatesIntroducing Non-spatial CovariatesIntroducing Non-spatial CovariatesIntroducing Non-spatial CovariatesIntroducing Non-spatial CovariatesIntroducing Non-spatial CovariatesFull Likelihood SpecificationFull Likelihood SpecificationComputational IssuesComputational IssuesComputational IssuesComputational IssuesApplication to N Minn DataApplication to N Minn DataApplication to N Minn DataNon-spatial covariates & crude intensityPopulation and spatial covariatesModel fittingModel fittingModel fittingParameter EstimatesFitted intensity surfaces, full modelFitted log-relative intensity surfacesDiscussionDiscussionWombling for Spatial Point ProcessesWombling for Spatial Point ProcessesResidual wombling, areal caseResidual wombling, point-level caseAnalysis of Marked Point Patternswith Spatial and Non-spatialCovariate InformationShengde Liang, Bradley P. Carlin, and Alan E. [email protected], [email protected], and [email protected] of Biostatistics, School of Public Health, University of MinnesotaandInstitute of Statistics and Decision Sciences, Duke UniversityAnalysis of Marked Point Patterns with Spatial and Non-spatial Covariate Information – p. 1/27BackgroundIn disease mapping, data are typically presented asaggregated counts over areal regions (counties, zipcodes, etc.) ⇒ analyze with areal or lattice modelsAnalysis of Marked Point Patterns with Spatial and Non-spatial Covariate Information – p. 2/27BackgroundIn disease mapping, data are typically presented asaggregated counts over areal regions (counties, zipcodes, etc.) ⇒ analyze with areal or lattice modelsExample: Model regional counts as Poisson(Eieµi) andassume the µifollow an area-level conditionallyautoregressive (CAR) distributionAnalysis of Marked Point Patterns with Spatial and Non-spatial Covariate Information – p. 2/27BackgroundIn disease mapping, data are typically presented asaggregated counts over areal regions (counties, zipcodes, etc.) ⇒ analyze with areal or lattice modelsExample: Model regional counts as Poisson(Eieµi) andassume the µifollow an area-level conditionallyautoregressive (CAR) distributionBut if precise geographic coordinates are available, thedata are properly viewed as a spatial point patternAnalysis of Marked Point Patterns with Spatial and Non-spatial Covariate Information – p. 2/27BackgroundIn disease mapping, data are typically presented asaggregated counts over areal regions (counties, zipcodes, etc.) ⇒ analyze with areal or lattice modelsExample: Model regional counts as Poisson(Eieµi) andassume the µifollow an area-level conditionallyautoregressive (CAR) distributionBut if precise geographic coordinates are available, thedata are properly viewed as a spatial point patternUnder a non-homogeneous Poisson process, likelihoodfor the intensity surface generating the locations isknown, but complex...Analysis of Marked Point Patterns with Spatial and Non-spatial Covariate Information – p. 2/27BackgroundIn disease mapping, data are typically presented asaggregated counts over areal regions (counties, zipcodes, etc.) ⇒ analyze with areal or lattice modelsExample: Model regional counts as Poisson(Eieµi) andassume the µifollow an area-level conditionallyautoregressive (CAR) distributionBut if precise geographic coordinates are available, thedata are properly viewed as a spatial point patternUnder a non-homogeneous Poisson process, likelihoodfor the intensity surface generating the locations isknown, but complex...Spatial point process methods and computations bothmore challenging ⇒ retreat to the Poisson-CAR?...Analysis of Marked Point Patterns with Spatial and Non-spatial Covariate Information – p. 2/27But we can’t retreat!We often have individual-level covariates (either “ofinterest" or “nuisance") we want to incorporate:We seek to compare patterns across certaintreatment covariates that “mark" the point patternOther non-spatial covariates (e.g., patientcharacteristics or risk factors) are nuisancesStill other spatial covariates may be available ateither point or areal summary levelAnalysis of Marked Point Patterns with Spatial and Non-spatial Covariate Information – p. 3/27But we can’t retreat!We often have individual-level covariates (either “ofinterest" or “nuisance") we want to incorporate:We seek to compare patterns across certaintreatment covariates that “mark" the point patternOther non-spatial covariates (e.g., patientcharacteristics or risk factors) are nuisancesStill other spatial covariates may be available ateither point or areal summary levelWe propose modeling point patterns jointly overgeographic and non-spatial nuisance covariate space(precludes aggregation to counts)Analysis of Marked Point Patterns with Spatial and Non-spatial Covariate Information – p. 3/27But we can’t retreat!We often have individual-level covariates (either “ofinterest" or “nuisance") we want to incorporate:We seek to compare patterns across certaintreatment covariates that “mark" the point patternOther non-spatial covariates (e.g., patientcharacteristics or risk factors) are nuisancesStill other spatial covariates may be available ateither point or areal summary levelWe propose modeling point patterns jointly overgeographic and non-spatial nuisance covariate space(precludes aggregation to counts)Interest lies in certain covariate effects, and themarginal intensity