Multivariate spatial modeling

Transcription

1 Multivariate spatial modeling Point-referenced spatial data often come as multivariate measurements at each location Chapter 7: Multivariate Spatial Modeling p. 1/21

2 Multivariate spatial modeling Point-referenced spatial data often come as multivariate measurements at each location Examples: Chapter 7: Multivariate Spatial Modeling p. 1/21

3 Multivariate spatial modeling Point-referenced spatial data often come as multivariate measurements at each location Examples: Environmental monitoring stations yield measurements on ozone, NO, CO, PM 2.5, etc. Chapter 7: Multivariate Spatial Modeling p. 1/21

4 Multivariate spatial modeling Point-referenced spatial data often come as multivariate measurements at each location Examples: Environmental monitoring stations yield measurements on ozone, NO, CO, PM 2.5, etc. In atmospheric modeling at a given site we observe surface temperature, precipitation and wind speed Chapter 7: Multivariate Spatial Modeling p. 1/21

5 Multivariate spatial modeling Point-referenced spatial data often come as multivariate measurements at each location Examples: Environmental monitoring stations yield measurements on ozone, NO, CO, PM 2.5, etc. In atmospheric modeling at a given site we observe surface temperature, precipitation and wind speed In real estate modeling for an individual property we observe selling price and total rental income Chapter 7: Multivariate Spatial Modeling p. 1/21

6 Multivariate spatial modeling Point-referenced spatial data often come as multivariate measurements at each location Examples: Environmental monitoring stations yield measurements on ozone, NO, CO, PM 2.5, etc. In atmospheric modeling at a given site we observe surface temperature, precipitation and wind speed In real estate modeling for an individual property we observe selling price and total rental income We anticipate dependence between measurements Chapter 7: Multivariate Spatial Modeling p. 1/21

7 Multivariate spatial modeling Point-referenced spatial data often come as multivariate measurements at each location Examples: Environmental monitoring stations yield measurements on ozone, NO, CO, PM 2.5, etc. In atmospheric modeling at a given site we observe surface temperature, precipitation and wind speed In real estate modeling for an individual property we observe selling price and total rental income We anticipate dependence between measurements at a particular location Chapter 7: Multivariate Spatial Modeling p. 1/21

8 Multivariate spatial modeling Point-referenced spatial data often come as multivariate measurements at each location Examples: Environmental monitoring stations yield measurements on ozone, NO, CO, PM 2.5, etc. In atmospheric modeling at a given site we observe surface temperature, precipitation and wind speed In real estate modeling for an individual property we observe selling price and total rental income We anticipate dependence between measurements at a particular location across locations Chapter 7: Multivariate Spatial Modeling p. 1/21

9 Basic issues Y(s) denotes a p 1 vector of random variables at s Chapter 7: Multivariate Spatial Modeling p. 2/21

10 Basic issues Y(s) denotes a p 1 vector of random variables at s We seek to model Y(s) : s D, again specifying a finite dimensional distribution for Y = (Y(s 1 ),...,Y(s n )) Chapter 7: Multivariate Spatial Modeling p. 2/21

11 Basic issues Y(s) denotes a p 1 vector of random variables at s We seek to model Y(s) : s D, again specifying a finite dimensional distribution for Y = (Y(s 1 ),...,Y(s n )) Crucial object: the cross-covariance, C(s, s ) = Cov(Y(s), Y(s )) a p p matrix that need not be symmetric, i.e., cov(y j (s),y j (s )) need not equal cov(y j (s),y j (s )) Chapter 7: Multivariate Spatial Modeling p. 2/21

12 Basic issues Y(s) denotes a p 1 vector of random variables at s We seek to model Y(s) : s D, again specifying a finite dimensional distribution for Y = (Y(s 1 ),...,Y(s n )) Crucial object: the cross-covariance, C(s, s ) = Cov(Y(s), Y(s )) a p p matrix that need not be symmetric, i.e., cov(y j (s),y j (s )) need not equal cov(y j (s),y j (s )) C(s, s ) is not positive definite except in a limiting sense: C(s, s) is the covariance matrix associated with Y(s). Chapter 7: Multivariate Spatial Modeling p. 2/21

13 Basic issues Y(s) denotes a p 1 vector of random variables at s We seek to model Y(s) : s D, again specifying a finite dimensional distribution for Y = (Y(s 1 ),...,Y(s n )) Crucial object: the cross-covariance, C(s, s ) = Cov(Y(s), Y(s )) a p p matrix that need not be symmetric, i.e., cov(y j (s),y j (s )) need not equal cov(y j (s),y j (s )) C(s, s ) is not positive definite except in a limiting sense: C(s, s) is the covariance matrix associated with Y(s). our primary focus: Gaussian processes and valid specification for C(s, s ) Chapter 7: Multivariate Spatial Modeling p. 2/21

14 Separable models A popular specification is the separable model C(s, s ) = ρ(s, s ) T where ρ is a valid (univariate) correlation function and T is a p p positive definite matrix Chapter 7: Multivariate Spatial Modeling p. 3/21

15 Separable models A popular specification is the separable model C(s, s ) = ρ(s, s ) T where ρ is a valid (univariate) correlation function and T is a p p positive definite matrix T is the non-spatial or local" covariance matrix Chapter 7: Multivariate Spatial Modeling p. 3/21

16 Separable models A popular specification is the separable model C(s, s ) = ρ(s, s ) T where ρ is a valid (univariate) correlation function and T is a p p positive definite matrix T is the non-spatial or local" covariance matrix ρ controls spatial association based upon proximity Chapter 7: Multivariate Spatial Modeling p. 3/21

17 Separable models A popular specification is the separable model C(s, s ) = ρ(s, s ) T where ρ is a valid (univariate) correlation function and T is a p p positive definite matrix T is the non-spatial or local" covariance matrix ρ controls spatial association based upon proximity Easy to verify that Σ Y = H T, where H ij = ρ(s i, s j ) and is the Kronecker product. Chapter 7: Multivariate Spatial Modeling p. 3/21

18 Separable models A popular specification is the separable model C(s, s ) = ρ(s, s ) T where ρ is a valid (univariate) correlation function and T is a p p positive definite matrix T is the non-spatial or local" covariance matrix ρ controls spatial association based upon proximity Easy to verify that Σ Y = H T, where H ij = ρ(s i, s j ) and is the Kronecker product. Σ Y is positive definite since H and T are Chapter 7: Multivariate Spatial Modeling p. 3/21

19 Separable models A popular specification is the separable model C(s, s ) = ρ(s, s ) T where ρ is a valid (univariate) correlation function and T is a p p positive definite matrix T is the non-spatial or local" covariance matrix ρ controls spatial association based upon proximity Easy to verify that Σ Y = H T, where H ij = ρ(s i, s j ) and is the Kronecker product. Σ Y is positive definite since H and T are Σ Y is convenient since Σ Y = H p T n and Σ 1 Y = H 1 T 1. Chapter 7: Multivariate Spatial Modeling p. 3/21

20 Application: Bivariate spatial regression A single covariate X(s) and a univariate response Y (s) Chapter 7: Multivariate Spatial Modeling p. 4/21

21 Application: Bivariate spatial regression A single covariate X(s) and a univariate response Y (s) To enable joint estimation and prediction, treat this as a bivariate process, ( ) X(s) Z(s) = N(µ(s),T) Y (s) Chapter 7: Multivariate Spatial Modeling p. 4/21

22 Application: Bivariate spatial regression A single covariate X(s) and a univariate response Y (s) To enable joint estimation and prediction, treat this as a bivariate process, ( ) X(s) Z(s) = N(µ(s),T) Y (s) Simplifying assumptions: Chapter 7: Multivariate Spatial Modeling p. 4/21

23 Application: Bivariate spatial regression A single covariate X(s) and a univariate response Y (s) To enable joint estimation and prediction, treat this as a bivariate process, ( ) X(s) Z(s) = N(µ(s),T) Y (s) Simplifying assumptions: separable cross-covariance for Z(s) Chapter 7: Multivariate Spatial Modeling p. 4/21

24 Application: Bivariate spatial regression A single covariate X(s) and a univariate response Y (s) To enable joint estimation and prediction, treat this as a bivariate process, ( ) X(s) Z(s) = N(µ(s),T) Y (s) Simplifying assumptions: separable cross-covariance for Z(s) µ(s) = (µ 1,µ 2 ) is coordinate-free. Chapter 7: Multivariate Spatial Modeling p. 4/21

25 Application: Bivariate spatial regression A single covariate X(s) and a univariate response Y (s) To enable joint estimation and prediction, treat this as a bivariate process, ( ) X(s) Z(s) = N(µ(s),T) Y (s) Simplifying assumptions: separable cross-covariance for Z(s) µ(s) = (µ 1,µ 2 ) is coordinate-free. Then p(y (s) X(s)) N(β 0 + β 1 X(s),σ 2 ), where β 0 = µ 2 T 12 T 11 µ 1, β 1 = T 12 T 11, and σ 2 = T 22 T 2 12 T 11 Chapter 7: Multivariate Spatial Modeling p. 4/21

26 Bivariate spatial regression (cont d) Rearrangement of the components of Z to Z = (X(s 1 ),X(s 2 ),...,X(s n ),Y (s 1 ),Y (s 2 ),...,Y (s n )) yields ( X Y ) N (( µ 1 1 µ 2 1 ), T H (φ) ), Chapter 7: Multivariate Spatial Modeling p. 5/21

27 Bivariate spatial regression (cont d) Rearrangement of the components of Z to Z = (X(s 1 ),X(s 2 ),...,X(s n ),Y (s 1 ),Y (s 2 ),...,Y (s n )) yields ( X Y ) N (( µ 1 1 µ 2 1 ), T H (φ) Priors: Wishart for T 1, vague but proper normal for (µ 1,µ 2 ), uniform or other suitable choice for φ ), Chapter 7: Multivariate Spatial Modeling p. 5/21

28 Bivariate spatial regression (cont d) Rearrangement of the components of Z to Z = (X(s 1 ),X(s 2 ),...,X(s n ),Y (s 1 ),Y (s 2 ),...,Y (s n )) yields ( X Y ) N (( µ 1 1 µ 2 1 ), T H (φ) Priors: Wishart for T 1, vague but proper normal for (µ 1,µ 2 ), uniform or other suitable choice for φ Full conditionals for Gibbs sampler: again Wishart for T 1, bivariate normal for (µ 1,µ 2 ); nonconjugate for φ (so need Metropolis sampling here) ), Chapter 7: Multivariate Spatial Modeling p. 5/21

29 Example with dew-shrub data 1129 locations with UTM coordinates Chapter 7: Multivariate Spatial Modeling p. 6/21

30 Example with dew-shrub data 1129 locations with UTM coordinates Y (s) : shrub density (% of land cover) at location s Chapter 7: Multivariate Spatial Modeling p. 6/21

31 Example with dew-shrub data 1129 locations with UTM coordinates Y (s) : shrub density (% of land cover) at location s X(s) : dew duration (minutes from 8 am) at location s Chapter 7: Multivariate Spatial Modeling p. 6/21

32 Example with dew-shrub data 1129 locations with UTM coordinates Y (s) : shrub density (% of land cover) at location s X(s) : dew duration (minutes from 8 am) at location s Perform an illustrative Bayesian analysis assuming separability and assuming an exponential correlation function, ρ(h;φ) = e φh Chapter 7: Multivariate Spatial Modeling p. 6/21

33 Example with dew-shrub data 1129 locations with UTM coordinates Y (s) : shrub density (% of land cover) at location s X(s) : dew duration (minutes from 8 am) at location s Perform an illustrative Bayesian analysis assuming separability and assuming an exponential correlation function, ρ(h;φ) = e φh conjugate priors for µ, T as above; prior for φ has infinite variance and suggests a range (3/φ) of 125 km, roughly half the maximum pairwise distance in the region Chapter 7: Multivariate Spatial Modeling p. 6/21

34 Example with dew-shrub data 1129 locations with UTM coordinates Y (s) : shrub density (% of land cover) at location s X(s) : dew duration (minutes from 8 am) at location s Perform an illustrative Bayesian analysis assuming separability and assuming an exponential correlation function, ρ(h;φ) = e φh conjugate priors for µ, T as above; prior for φ has infinite variance and suggests a range (3/φ) of 125 km, roughly half the maximum pairwise distance in the region (µ 1,µ 2,T 11,T 12,T 22 ) updated directly; φ updated via Metropolis Chapter 7: Multivariate Spatial Modeling p. 6/21

35 Example with dew-shrub data 1129 locations with UTM coordinates Y (s) : shrub density (% of land cover) at location s X(s) : dew duration (minutes from 8 am) at location s Perform an illustrative Bayesian analysis assuming separability and assuming an exponential correlation function, ρ(h;φ) = e φh conjugate priors for µ, T as above; prior for φ has infinite variance and suggests a range (3/φ) of 125 km, roughly half the maximum pairwise distance in the region (µ 1,µ 2,T 11,T 12,T 22 ) updated directly; φ updated via Metropolis (β 0,β 1,σ 2 ) samples automatically determined as functions of the others Chapter 7: Multivariate Spatial Modeling p. 6/21

36 Dew and shrub data sites Eastings Northings Chapter 7: Multivariate Spatial Modeling p. 7/21

37 Parameter estimation, dew-shrub data Parameter 2.5% 50% 97.5% µ µ T T T φ β β σ T 12 / T 11 T surprising significant negative association between dew duration and shrub density! (also emerges with a nonspatial model) Chapter 7: Multivariate Spatial Modeling p. 8/21

38 Benefits and limitations of separability Benefits: Easy interpretation (decomposition of variance structure) Substantial computational benefits Chapter 7: Multivariate Spatial Modeling p. 9/21

39 Benefits and limitations of separability Benefits: Easy interpretation (decomposition of variance structure) Substantial computational benefits Limitations: Symmetry in cross-covariance matrix Imposes same spatial range for every measurement (only one correlation function) Chapter 7: Multivariate Spatial Modeling p. 9/21

40 Benefits and limitations of separability Benefits: Easy interpretation (decomposition of variance structure) Substantial computational benefits Limitations: Symmetry in cross-covariance matrix Imposes same spatial range for every measurement (only one correlation function) One possible solution: Coregionalization models!... Chapter 7: Multivariate Spatial Modeling p. 9/21

41 Linear Model of Coregionalization For point referenced data, assume Y(s) = Aw(s) where w(s) = (w 1 (s),w 2 (s),...,w p (s)). Chapter 7: Multivariate Spatial Modeling p. 10/21

42 Linear Model of Coregionalization For point referenced data, assume Y(s) = Aw(s) where w(s) = (w 1 (s),w 2 (s),...,w p (s)). p independent spatial processes with stationary correlation functions ρ j (s s ),j = 1, 2,...,p Chapter 7: Multivariate Spatial Modeling p. 10/21

43 Linear Model of Coregionalization For point referenced data, assume Y(s) = Aw(s) where w(s) = (w 1 (s),w 2 (s),...,w p (s)). p independent spatial processes with stationary correlation functions ρ j (s s ),j = 1, 2,...,p If ρ j = ρ for all j separable case with AA = T Chapter 7: Multivariate Spatial Modeling p. 10/21

44 Linear Model of Coregionalization For point referenced data, assume Y(s) = Aw(s) where w(s) = (w 1 (s),w 2 (s),...,w p (s)). p independent spatial processes with stationary correlation functions ρ j (s s ),j = 1, 2,...,p If ρ j = ρ for all j separable case with AA = T In general, the cross covariance matrix is C(s s ) = p j=1 ρ j (s s )a j a j where a j is the jth column of A. Chapter 7: Multivariate Spatial Modeling p. 10/21

45 Linear Model of Coregionalization For point referenced data, assume Y(s) = Aw(s) where w(s) = (w 1 (s),w 2 (s),...,w p (s)). p independent spatial processes with stationary correlation functions ρ j (s s ),j = 1, 2,...,p If ρ j = ρ for all j separable case with AA = T In general, the cross covariance matrix is C(s s ) = p j=1 ρ j (s s )a j a j where a j is the jth column of A. Approach is constructive" and hence immediately valid, still stationary, and provides a distinct covariance function for each component Chapter 7: Multivariate Spatial Modeling p. 10/21

46 Linear Model of Coregionalization More frisky: Y(s) = A(s)w(s) = Spatially varying LMC! Chapter 7: Multivariate Spatial Modeling p. 11/21

47 Linear Model of Coregionalization More frisky: Y(s) = A(s)w(s) = Spatially varying LMC! model A(s) model T(s) = A(s)A (s) Chapter 7: Multivariate Spatial Modeling p. 11/21

48 Linear Model of Coregionalization More frisky: Y(s) = A(s)w(s) = Spatially varying LMC! model A(s) model T(s) = A(s)A (s) Possibilities for T(s): Chapter 7: Multivariate Spatial Modeling p. 11/21

49 Linear Model of Coregionalization More frisky: Y(s) = A(s)w(s) = Spatially varying LMC! model A(s) model T(s) = A(s)A (s) Possibilities for T(s): T(s) = g(x(s)) T Chapter 7: Multivariate Spatial Modeling p. 11/21

50 Linear Model of Coregionalization More frisky: Y(s) = A(s)w(s) = Spatially varying LMC! model A(s) model T(s) = A(s)A (s) Possibilities for T(s): T(s) = g(x(s)) T T(s) is a spatial process (e.g., T 1 (s) is a spatial Wishart process) Chapter 7: Multivariate Spatial Modeling p. 11/21

51 Other Approaches Moving average or kernel convolution of a process: Y j (s) = k j (u)z(s + u)du = k j (s s )Z(s )ds where Z(s) is a univariate spatial process and k j are kernel functions, j = 1, 2,...,p. Yields the cross covariance C ij (s s ) = k i (s s + u)k j (u )ρ(u u )dudu Chapter 7: Multivariate Spatial Modeling p. 12/21

52 Other Approaches Moving average or kernel convolution of a process: Y j (s) = k j (u)z(s + u)du = k j (s s )Z(s )ds where Z(s) is a univariate spatial process and k j are kernel functions, j = 1, 2,...,p. Yields the cross covariance C ij (s s ) = k i (s s + u)k j (u )ρ(u u )dudu Convolution of Covariance Functions: Suppose C 1,C 2,...C p are valid covariance functions. Define C ij (s) = C i (s t)c j (t)dt. Then the p p matrix C(s) = {C ij (s)} is a valid cross covariance function. Chapter 7: Multivariate Spatial Modeling p. 12/21

53 Multivariate Areal Data Models Now areal units (e.g., counties) instead of points Chapter 7: Multivariate Spatial Modeling p. 13/21

54 Multivariate Areal Data Models Now areal units (e.g., counties) instead of points Need to model dependence within and across units Chapter 7: Multivariate Spatial Modeling p. 13/21

55 Multivariate Areal Data Models Now areal units (e.g., counties) instead of points Need to model dependence within and across units As in univariate case, often handled using spatial random effects φ ij, where again i = 1,...,n indexes region but now j = 1,...,p indexes variable (e.g., cancer type) within region Chapter 7: Multivariate Spatial Modeling p. 13/21

56 Multivariate Areal Data Models Now areal units (e.g., counties) instead of points Need to model dependence within and across units As in univariate case, often handled using spatial random effects φ ij, where again i = 1,...,n indexes region but now j = 1,...,p indexes variable (e.g., cancer type) within region Suppose we observe Y i = (Y i1,y i2,...y ip ), Then g(e(y ij )) = x T ij β j + φ ij, where φ i = (φ i1,...,φ ip ) and φ = (φ 1,...,φ n ). Link function g useful for modeling rate (e.g., Poisson) or survival (e.g., Weibull or Cox model) data. Chapter 7: Multivariate Spatial Modeling p. 13/21

57 Univariate vs. Multivariate Modeling We can combine CAR and traditional models to handle a broad range of problems. Chapter 7: Multivariate Spatial Modeling p. 14/21

58 Univariate vs. Multivariate Modeling We can combine CAR and traditional models to handle a broad range of problems. Example: For breast cancer control model in county i, let Y 1i = observed age-adjusted mortality rate / 100,000 Y 2i = observed age-adjusted incidence rate / 100,000 Y 3i = observed percent of late diagnoses Y 4i = % of surveyed women over the age of 40 who have not had a mammogram in two years Chapter 7: Multivariate Spatial Modeling p. 14/21

59 Univariate vs. Multivariate Modeling We can combine CAR and traditional models to handle a broad range of problems. Example: For breast cancer control model in county i, let Y 1i = observed age-adjusted mortality rate / 100,000 Y 2i = observed age-adjusted incidence rate / 100,000 Y 3i = observed percent of late diagnoses Y 4i = % of surveyed women over the age of 40 who have not had a mammogram in two years ind Assume Y ki N(θ ki,σki 2 ), k = 1,...,4, i = 1,...,I = 87, and set σki 2 = σ2 k /n ki, where n ki is the number of persons at risk for event k in county i. Chapter 7: Multivariate Spatial Modeling p. 14/21

60 Univariate vs. Multivariate Modeling Writing θ k = (θ k1,...,θ ki ), let θ k CAR(λ k ). Chapter 7: Multivariate Spatial Modeling p. 15/21

61 Univariate vs. Multivariate Modeling Writing θ k = (θ k1,...,θ ki ), let θ k CAR(λ k ). We then borrow strength across cancer indicators: λ k iid G(a,b) and σ 2 k iid IG(a,b), where G and IG denote the gamma and inverse gamma distributions, respectively. Chapter 7: Multivariate Spatial Modeling p. 15/21

62 Univariate vs. Multivariate Modeling Writing θ k = (θ k1,...,θ ki ), let θ k CAR(λ k ). We then borrow strength across cancer indicators: λ k iid G(a,b) and σ 2 k iid IG(a,b), where G and IG denote the gamma and inverse gamma distributions, respectively. Parameter of interest: For a set of weights α k such that 4 k=1 α k = 1, form the breast cancer control variable, η i = 4 k=1 α k θ ki. Chapter 7: Multivariate Spatial Modeling p. 15/21

63 Univariate vs. Multivariate Modeling A sample from the posterior distribution of η = (η 1,...,η I ), p(η y), is easily obtained in WinBUGS. Chapter 7: Multivariate Spatial Modeling p. 16/21

64 Univariate vs. Multivariate Modeling A sample from the posterior distribution of η = (η 1,...,η I ), p(η y), is easily obtained in WinBUGS. Using α k = 1/4 for all k, compare the posterior medians of the η i with naive averages: Chapter 7: Multivariate Spatial Modeling p. 16/21

65 Univariate vs. Multivariate Modeling A sample from the posterior distribution of η = (η 1,...,η I ), p(η y), is easily obtained in WinBUGS. Using α k = 1/4 for all k, compare the posterior medians of the η i with naive averages: Note smoothed control variates cover a narrower range, and clarify the overall spatial pattern in the state! Chapter 7: Multivariate Spatial Modeling p. 16/21

66 Multivariate CAR (MCAR) models Again, local or neighbor idea, conditioning, CAR Two strategies: Chapter 7: Multivariate Spatial Modeling p. 17/21

67 Multivariate CAR (MCAR) models Again, local or neighbor idea, conditioning, CAR Two strategies: multivariate CAR (MCAR): p(φ i φ j,j i) Chapter 7: Multivariate Spatial Modeling p. 17/21

68 Multivariate CAR (MCAR) models Again, local or neighbor idea, conditioning, CAR Two strategies: multivariate CAR (MCAR): p(φ i φ j,j i) two-fold CAR: p(φ ij φ (ij) ) Chapter 7: Multivariate Spatial Modeling p. 17/21

69 Multivariate CAR (MCAR) models Again, local or neighbor idea, conditioning, CAR Two strategies: multivariate CAR (MCAR): p(φ i φ j,j i) two-fold CAR: p(φ ij φ (ij) ) For MCAR, p(φ i φ j i, Σ i ) = N j B ij φ j, Σ i, i = 1,...,n Chapter 7: Multivariate Spatial Modeling p. 17/21

70 Multivariate CAR (MCAR) models Again, local or neighbor idea, conditioning, CAR Two strategies: multivariate CAR (MCAR): p(φ i φ j,j i) two-fold CAR: p(φ ij φ (ij) ) For MCAR, p(φ i φ j i, Σ i ) = N j B ij φ j, Σ i, i = 1,...,n As earlier, Brook s Lemma yields p(φ), improper, etc. Chapter 7: Multivariate Spatial Modeling p. 17/21

71 Multivariate CAR (MCAR) models Again, local or neighbor idea, conditioning, CAR Two strategies: multivariate CAR (MCAR): p(φ i φ j,j i) two-fold CAR: p(φ ij φ (ij) ) For MCAR, p(φ i φ j i, Σ i ) = N j B ij φ j, Σ i, i = 1,...,n As earlier, Brook s Lemma yields p(φ), improper, etc. Simplification: B ij = b ij I,b ij = w ij /w i+, Σ i = ( 1 w i+ )Σ Chapter 7: Multivariate Spatial Modeling p. 17/21

72 Multivariate CAR (MCAR) models Again, local or neighbor idea, conditioning, CAR Two strategies: multivariate CAR (MCAR): p(φ i φ j,j i) two-fold CAR: p(φ ij φ (ij) ) For MCAR, p(φ i φ j i, Σ i ) = N j B ij φ j, Σ i, i = 1,...,n As earlier, Brook s Lemma yields p(φ), improper, etc. Simplification: B ij = b ij I,b ij = w ij /w i+, Σ i = ( 1 w i+ )Σ To make proper, add ρ or perhaps ρ j, j = 1,...,p Chapter 7: Multivariate Spatial Modeling p. 17/21

73 Application to spatial frailty modeling From the NCI s SEER (Surveillance, Epidemiology, and End Results) database (seer.cancer.gov), we can obtain information on patients suffering from possibly multiple cancers by county for several U.S. states. Chapter 7: Multivariate Spatial Modeling p. 18/21

74 Application to spatial frailty modeling From the NCI s SEER (Surveillance, Epidemiology, and End Results) database (seer.cancer.gov), we can obtain information on patients suffering from possibly multiple cancers by county for several U.S. states. For instance, cancers of 5 GI-related organs: colorectal, gall bladder, pancreas, small intestine, and stomach. Chapter 7: Multivariate Spatial Modeling p. 18/21

75 Application to spatial frailty modeling From the NCI s SEER (Surveillance, Epidemiology, and End Results) database (seer.cancer.gov), we can obtain information on patients suffering from possibly multiple cancers by county for several U.S. states. For instance, cancers of 5 GI-related organs: colorectal, gall bladder, pancreas, small intestine, and stomach. t ijk, the time to death/censoring for patient k with primary cancer j in Iowa county i for diagnosis years Chapter 7: Multivariate Spatial Modeling p. 18/21

76 Application to spatial frailty modeling From the NCI s SEER (Surveillance, Epidemiology, and End Results) database (seer.cancer.gov), we can obtain information on patients suffering from possibly multiple cancers by county for several U.S. states. For instance, cancers of 5 GI-related organs: colorectal, gall bladder, pancreas, small intestine, and stomach. t ijk, the time to death/censoring for patient k with primary cancer j in Iowa county i for diagnosis years δ ijk, the corresponding death indicator Chapter 7: Multivariate Spatial Modeling p. 18/21

77 Application to spatial frailty modeling From the NCI s SEER (Surveillance, Epidemiology, and End Results) database (seer.cancer.gov), we can obtain information on patients suffering from possibly multiple cancers by county for several U.S. states. For instance, cancers of 5 GI-related organs: colorectal, gall bladder, pancreas, small intestine, and stomach. t ijk, the time to death/censoring for patient k with primary cancer j in Iowa county i for diagnosis years δ ijk, the corresponding death indicator x ijk, a vector of patient-specific covariates Chapter 7: Multivariate Spatial Modeling p. 18/21

78 Application to spatial frailty modeling From the NCI s SEER (Surveillance, Epidemiology, and End Results) database (seer.cancer.gov), we can obtain information on patients suffering from possibly multiple cancers by county for several U.S. states. For instance, cancers of 5 GI-related organs: colorectal, gall bladder, pancreas, small intestine, and stomach. t ijk, the time to death/censoring for patient k with primary cancer j in Iowa county i for diagnosis years δ ijk, the corresponding death indicator x ijk, a vector of patient-specific covariates z ijk = ( z ijk1,z ijk2,...,z ijkn ) T, where zijkl = 1 if patient ijk suffers from cancer type l, and 0 if not. Chapter 7: Multivariate Spatial Modeling p. 18/21

79 Application to spatial frailty modeling Hierarchical model: The likelihood is p i=1 n sij { ( )} δijk j=1 k=1 h tijk ; x ijk ( ) ( )} exp { H 0i tijk exp x T ijk β + zt ijk θ + φ ij, where h ( t ijk ; x ijk ) = h0i ( tijk ) exp (x T ijk β + zt ijk θ + φ ij H 0i ( tijk ) = tijk 0 h 0i (u) du. φ i = (φ i1,φ i2,...,φ in ) T ( ) T and Φ = φ T 1,...,φ T p MCAR (α, Λ). ) Chapter 7: Multivariate Spatial Modeling p. 19/21

80 Application to spatial frailty modeling We take flat priors on β and θ, and model the county-specific baseline hazard semiparametrically Chapter 7: Multivariate Spatial Modeling p. 20/21

81 Application to spatial frailty modeling We take flat priors on β and θ, and model the county-specific baseline hazard semiparametrically Alternate approach: Random coefficients model: x T ijk β + zt ijk θ i, β flat, but now Θ ( θ T 1,...,θ T p ) T MCAR (α, Λ). Chapter 7: Multivariate Spatial Modeling p. 20/21

82 Application to spatial frailty modeling We take flat priors on β and θ, and model the county-specific baseline hazard semiparametrically Alternate approach: Random coefficients model: x T ijk β + zt ijk θ i, β flat, but now Θ ( θ T 1,...,θ T p ) T MCAR (α, Λ). Next slide maps the fitted spatially varying coefficients Chapter 7: Multivariate Spatial Modeling p. 20/21

83 Application to spatial frailty modeling We take flat priors on β and θ, and model the county-specific baseline hazard semiparametrically Alternate approach: Random coefficients model: x T ijk β + zt ijk θ i, β flat, but now Θ ( θ T 1,...,θ T p ) T MCAR (α, Λ). Next slide maps the fitted spatially varying coefficients Not residuals, but the effects of the presence of the primary cancer on the death rate Chapter 7: Multivariate Spatial Modeling p. 20/21

84 Application to spatial frailty modeling We take flat priors on β and θ, and model the county-specific baseline hazard semiparametrically Alternate approach: Random coefficients model: x T ijk β + zt ijk θ i, β flat, but now Θ ( θ T 1,...,θ T p ) T MCAR (α, Λ). Next slide maps the fitted spatially varying coefficients Not residuals, but the effects of the presence of the primary cancer on the death rate Strong spatial pattern: SW Iowa counties have high fitted values for pancreatic and stomach cancer, while SE counties have high rates of colorectal and small intestinal cancer. Chapter 7: Multivariate Spatial Modeling p. 20/21

85 Fitted spatial effects, Iowa SEER data Colorectal Cancer Stomach Cancer Pancreas Cancer Small Intestine Cancer Chapter 7: Multivariate Spatial Modeling p. 21/21