Approaches for Multiple Disease Mapping: MCAR and SANOVA

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1 Approaches for Multiple Disease Mapping: MCAR and SANOVA Dipankar Bandyopadhyay Division of Biostatistics, University of Minnesota SPH April 22, Adapted from Sudipto Banerjee s notes SANOVA vs MCAR

2 Outline Outline Introduction MCAR: Multivariate Conditionally Auto-Regressive model Jin, X., Banerjee, S. and Carlin, B.P. (2007). Order-free co-regionalized areal data models with application to multiple-disease mapping. Journal of the Royal Statistical Society, Series B, 65, SANOVA: Smoothed ANalysis Of VAriance Zhang, Y., Hodges, J.S. and Banerjee, S. (2009). Smoothed ANOVA with spatial effects as a competitor to MCAR in multivariate spatial smoothing. Annals of Applied Statistics, 3(4), MCAR vs SANOVA: modelling correlations vs means Simulation experiments and analysis of Minnesota 3-cancer data Discussion and future work 2 Adapted from Sudipto Banerjee s notes SANOVA vs MCAR

3 Introduction Disease mapping: Describe the geographic variation of disease Generate hypotheses about the possible causes for differences in risk of disease. 3 Adapted from Sudipto Banerjee s notes SANOVA vs MCAR

4 Introduction Disease mapping: Describe the geographic variation of disease Generate hypotheses about the possible causes for differences in risk of disease. Standard mortality ratio: SMR = Y i /E i Y i is the observed number of deaths in region i. E i is the expected number of deaths in region i. What factors are accountable for discrepancies between Y i s and E i s? 3 Adapted from Sudipto Banerjee s notes SANOVA vs MCAR

5 Single Disease Mapping Map of raw standard mortality ratios (SMR) of oesophagus cancer between 1991 and 1998 in counties of Minnesota, USA. (c) Esophagus Cancer Adapted from Sudipto Banerjee s notes SANOVA vs MCAR

6 Single Disease Mapping Background: Modelling of a single disease Let n be the number of counties For counts of (rare) disease, poisson regression model: Hierarchical spatial Poisson regression model Y i µ i ind P oisson {E i exp(µ i )} i = 1,..., n. µ i = x i β + φ i. The x i s are explanatory, region-level spatial covariates, having parameter coefficients β. How do we model φ i s? 5 Adapted from Sudipto Banerjee s notes SANOVA vs MCAR

7 Univariate CAR model Define: φ i : univariate spatial random variable, i = 1,, n; φ = (φ 1,, φ n ) ; i j = region i is a neighbor of region j; m i = number of neighbors of region i. Let D = diag{m i } W = adjacency matrix of the map. 6 Adapted from Sudipto Banerjee s notes SANOVA vs MCAR

8 Univariate CAR model Define: φ i : univariate spatial random variable, i = 1,, n; φ = (φ 1,, φ n ) ; i j = region i is a neighbor of region j; m i = number of neighbors of region i. Let D = diag{m i } W = adjacency matrix of the map. p(φ i φ j, j i, τ) = N α i j w ij w i+ φ j, 1 ; τm i Brook s Lemma: φ N ( 0, τ 1 (D αw ) 1). 6 Adapted from Sudipto Banerjee s notes SANOVA vs MCAR

9 Univariate CAR model Define: φ i : univariate spatial random variable, i = 1,, n; φ = (φ 1,, φ n ) ; i j = region i is a neighbor of region j; m i = number of neighbors of region i. Let D = diag{m i } W = adjacency matrix of the map. p(φ i φ j, j i, τ) = N α i j w ij w i+ φ j, 1 ; τm i Brook s Lemma: φ N ( 0, τ 1 (D αw ) 1). Aside: D αw is known as the Laplacian of a graph in combinatorics (Brualdi and Ryser, 1991). 6 Adapted from Sudipto Banerjee s notes SANOVA vs MCAR

10 Univariate CAR model λ min and λ max are the minimum and maximum eigenvalues of D 1/2 W D 1/2. In fact, λ min < 0 and λ max = 1. α (λ 1 min, λ 1 max) ensures proper distribution for φ (Cressie, 1993; Sun et al., 2000). α = 1 implies improper (singular) distribution (Besag et al. 1991). { p(φ 1, φ 2,..., φ n τ) (τ) (n G)/2 exp τ } 2 φ (D W )φ, where G is the number of islands in the map. In fact, n G is the rank of D W. 7 Adapted from Sudipto Banerjee s notes SANOVA vs MCAR

11 Univariate CAR model Hierarchical Bayesian modeling: First stage: likelihood for observation data Second stage: prior distributions for fixed effects β and random effects ψ = (ψ 1,..., ψ N ). Estimate model using Markov chain Monte Carlo (MCMC) methods 8 Adapted from Sudipto Banerjee s notes SANOVA vs MCAR

12 Multiple Disease Mapping Maps of raw standard mortality ratios (SMR) of lung and esophagus cancer between 1991 and 1998 in Minnesota counties (a) (c) Lung Cancer Esophagus Cancer Adapted from Sudipto Banerjee s notes SANOVA vs MCAR

13 Multiple Disease Mapping Y ij is the disease count for disease j in county i. 10 Adapted from Sudipto Banerjee s notes SANOVA vs MCAR

14 Multiple Disease Mapping Y ij is the disease count for disease j in county i. x ij are explanatory, region-level spatial covariates for disease j having parameter coefficients β j. 10 Adapted from Sudipto Banerjee s notes SANOVA vs MCAR

15 Multiple Disease Mapping Y ij is the disease count for disease j in county i. x ij are explanatory, region-level spatial covariates for disease j having parameter coefficients β j. For counts of (rare) diseases, poisson regression model: Multiple disease model Y ij ind P oisson(e ij e x ijβ j + φ ij ), i = 1,..., n, j = 1,..., p. 10 Adapted from Sudipto Banerjee s notes SANOVA vs MCAR

16 Multiple Disease Mapping Y ij is the disease count for disease j in county i. x ij are explanatory, region-level spatial covariates for disease j having parameter coefficients β j. For counts of (rare) diseases, poisson regression model: Multiple disease model Y ij ind P oisson(e ij e x ijβ j + φ ij ), i = 1,..., n, j = 1,..., p. How do we model the φ ij s? 10 Adapted from Sudipto Banerjee s notes SANOVA vs MCAR

17 Multiple Disease Mapping Associations in multiple disease data: 11 Adapted from Sudipto Banerjee s notes SANOVA vs MCAR

18 Multiple Disease Mapping Associations in multiple disease data: Spatial association across regions now for each disease Dependence among multiple diseases (within the same region) Cross-spatial associations among multiple diseases in different regions 11 Adapted from Sudipto Banerjee s notes SANOVA vs MCAR

19 MCAR models M CAR(α, Λ) (Mardia, 1988) φ 2 Define φ j = (φ 1j,, φ nj ) ; φ = (φ 1., φ p). φ N ( 0, Λ 1 (D αw ) 1) Covariance matrix (p = 2): ( ) ( ( ) ( φ1 0 (D αw )Λ11 (D αw )Λ N, 12 0 (D αw )Λ 12 (D αw )Λ 22 Note: Common spatial smoothing parameter is used for all diseases. ) 1 ) 12 Adapted from Sudipto Banerjee s notes SANOVA vs MCAR

20 MCAR models MCAR(α 1,..., α p, Λ) (Gelfand and Vounatsou, 2003) Now every disease has its own smoothing parameter. Covariance matrix (p = 2). ( φ1 φ 2 ) N ( ( 0 0 ) ( (D α1 W )Λ, 11 (D α 2 W )Λ 22 ) 1 ), No additional parameter to control smoothness of cross-correlations. 13 Adapted from Sudipto Banerjee s notes SANOVA vs MCAR

21 MCAR models Gelfand and Vounatsou (2003) recognize the difficulties in specifying covariance matrices of the following form ( e.g. with (p = 2)): ( φ1 φ 2 ) N ( ( 0 0 ) ( (D α1 W )Λ, 11 (D α 3 W )Λ 12 (D α 3 W )Λ 12 (D α 2 W )Λ 22 ) 1 ) 14 Adapted from Sudipto Banerjee s notes SANOVA vs MCAR

22 MCAR models Gelfand and Vounatsou (2003) recognize the difficulties in specifying covariance matrices of the following form ( e.g. with (p = 2)): ( φ1 φ 2 ) N ( ( 0 0 ) ( (D α1 W )Λ, 11 (D α 3 W )Λ 12 (D α 3 W )Λ 12 (D α 2 W )Λ 22 ) 1 ) Kim, Sun and Tsutakawa (2001): Two-fold CAR model that allows smoothness parameters for cross-correlations for two diseases. Also work by Sain and Cressie (2002). Extremely difficult to generalize to p > Adapted from Sudipto Banerjee s notes SANOVA vs MCAR

23 MCAR models Gelfand and Vounatsou (2003) recognize the difficulties in specifying covariance matrices of the following form ( e.g. with (p = 2)): ( φ1 φ 2 ) N ( ( 0 0 ) ( (D α1 W )Λ, 11 (D α 3 W )Λ 12 (D α 3 W )Λ 12 (D α 2 W )Λ 22 ) 1 ) Kim, Sun and Tsutakawa (2001): Two-fold CAR model that allows smoothness parameters for cross-correlations for two diseases. Also work by Sain and Cressie (2002). Extremely difficult to generalize to p > 2. Jin et al. (2005): Conditional approach to building 2-CAR models: [φ 1 ][φ 2 φ 1 ]. 14 Adapted from Sudipto Banerjee s notes SANOVA vs MCAR

24 MCAR models county 1 county 2 county 3 α 1 α 1 φ 11 φ 21 φ 31 A α 3 α 3 α 0 α 3 α 0 α 3 α 0 α 2 α 2 φ 12 φ 22 φ 32 B 15 Adapted from Sudipto Banerjee s notes SANOVA vs MCAR

25 MCAR models through latent spatial effects Basic idea Borrow ideas from factor analysis and multivariate geostatistics? 16 Adapted from Sudipto Banerjee s notes SANOVA vs MCAR

26 MCAR models through latent spatial effects Basic idea Borrow ideas from factor analysis and multivariate geostatistics? For i = 1,..., n and j = 1,..., p: φ ij = a j1 u i1 + a j2 u i2 + + a jp u ip 16 Adapted from Sudipto Banerjee s notes SANOVA vs MCAR

27 MCAR models through latent spatial effects Basic idea Borrow ideas from factor analysis and multivariate geostatistics? For i = 1,..., n and j = 1,..., p: φ ij = a j1 u i1 + a j2 u i2 + + a jp u ip Letting φ j = (φ 1j,..., φ nj ), φ j = a j1 u 1 + a j2 u a jp u p 16 Adapted from Sudipto Banerjee s notes SANOVA vs MCAR

28 MCAR models through latent spatial effects Basic idea Borrow ideas from factor analysis and multivariate geostatistics? For i = 1,..., n and j = 1,..., p: φ ij = a j1 u i1 + a j2 u i2 + + a jp u ip Letting φ j = (φ 1j,..., φ nj ), φ j = a j1 u 1 + a j2 u a jp u p u j s are latent variables (no dimension reduction) a jk s are unknown coefficients (weights) not varying over regions (but could) 16 Adapted from Sudipto Banerjee s notes SANOVA vs MCAR

29 MCAR models through latent spatial effects Basic idea Case 1: p independent and identical latent CAR effects Assume p latent independent CAR effects u j (one for each disease): u j N n (0, (D αw ) 1), j = 1,..., p. 17 Adapted from Sudipto Banerjee s notes SANOVA vs MCAR

30 MCAR models through latent spatial effects Basic idea Case 1: p independent and identical latent CAR effects Assume p latent independent CAR effects u j (one for each disease): u j N n (0, (D αw ) 1), j = 1,..., p. Let φ = (φ 1,..., φ p), A = {a jk } and Σ = AA. 17 Adapted from Sudipto Banerjee s notes SANOVA vs MCAR

31 MCAR models through latent spatial effects Basic idea Case 1: p independent and identical latent CAR effects Assume p latent independent CAR effects u j (one for each disease): u j N n (0, (D αw ) 1), j = 1,..., p. Let φ = (φ 1,..., φ p), A = {a jk } and Σ = AA. The joint distribution of φ is ( φ N 0, Σ (D αw ) 1). 17 Adapted from Sudipto Banerjee s notes SANOVA vs MCAR

32 MCAR models through latent spatial effects Basic idea Case 1: p independent and identical latent CAR effects Assume p latent independent CAR effects u j (one for each disease): u j N n (0, (D αw ) 1), j = 1,..., p. Let φ = (φ 1,..., φ p), A = {a jk } and Σ = AA. The joint distribution of φ is ( φ N 0, Σ (D αw ) 1). This is the separable MCAR distribution M CAR(α, Σ) distribution. 17 Adapted from Sudipto Banerjee s notes SANOVA vs MCAR

33 MCAR models through latent spatial effects Basic idea Case 2: p independent but not identical latent CAR effects Assume that the latent CAR effects u j are independent and not identical. ( u j N 0, (D α j W ) 1), j = 1,..., p. 18 Adapted from Sudipto Banerjee s notes SANOVA vs MCAR

34 MCAR models through latent spatial effects Basic idea Case 2: p independent but not identical latent CAR effects Assume that the latent CAR effects u j are independent and not identical. ( u j N 0, (D α j W ) 1), j = 1,..., p. The joint distribution of φ is φ N ( 0, (A I n n )Γ 1 (A I n n ) ), 18 Adapted from Sudipto Banerjee s notes SANOVA vs MCAR

35 MCAR models through latent spatial effects Basic idea Case 2: p independent but not identical latent CAR effects Assume that the latent CAR effects u j are independent and not identical. ( u j N 0, (D α j W ) 1), j = 1,..., p. The joint distribution of φ is φ N ( 0, (A I n n )Γ 1 (A I n n ) ), Γ is block diagonal with diagonal entries Γ j = D α j W, j = 1,..., p. 18 Adapted from Sudipto Banerjee s notes SANOVA vs MCAR

36 MCAR models through latent spatial effects Basic idea Case 2: p independent but not identical latent CAR effects Assume that the latent CAR effects u j are independent and not identical. ( u j N 0, (D α j W ) 1), j = 1,..., p. The joint distribution of φ is φ N ( 0, (A I n n )Γ 1 (A I n n ) ), Γ is block diagonal with diagonal entries Γ j = D α j W, j = 1,..., p. This is the non-separable MCAR, MCAR(α 1,..., α p, Σ) distribution. 18 Adapted from Sudipto Banerjee s notes SANOVA vs MCAR

37 MCAR models through latent spatial effects Basic idea Case 3: p dependent and not identical latent CAR effects Assume that the latent CAR effects u = (u 1,..., u p) are such that: u N ( 0, (I p p D B W ) 1), where B = {b jk } is a p p symmetric matrix. 19 Adapted from Sudipto Banerjee s notes SANOVA vs MCAR

38 MCAR models through latent spatial effects Basic idea Case 3: p dependent and not identical latent CAR effects Assume that the latent CAR effects u = (u 1,..., u p) are such that: u N ( 0, (I p p D B W ) 1), where B = {b jk } is a p p symmetric matrix. The joint distribution of φ is φ N ( 0, (A I n n ) (I p p D B W ) 1 (A I n n ) ). 19 Adapted from Sudipto Banerjee s notes SANOVA vs MCAR

39 MCAR models through latent spatial effects Basic idea Case 3: p dependent and not identical latent CAR effects Assume that the latent CAR effects u = (u 1,..., u p) are such that: u N ( 0, (I p p D B W ) 1), where B = {b jk } is a p p symmetric matrix. The joint distribution of φ is φ N ( 0, (A I n n ) (I p p D B W ) 1 (A I n n ) ). We call this the MCAR(B, Σ) distribution 19 Adapted from Sudipto Banerjee s notes SANOVA vs MCAR

40 MCAR models through latent spatial effects Basic idea Method behind this madness? The M CAR(B, Σ) distribution can be expressed (e.g. p = 2 diseases) as ( ( ) ) 1 (D α11 W )Λ φ N 0, 11 (D α 12 W )Λ 12. (D α 12 W )Λ 12 (D α 22 W )Λ 22 Result holds for general p, i.e. matrices of the form {(D α ij W )Λ ij } There exists a bijection: ({a jk }, {b jk }) ({Λ ij }, {α ij }) 20 Adapted from Sudipto Banerjee s notes SANOVA vs MCAR

41 MCAR models through latent spatial effects Basic idea Computation Notes Restriction on B: ξ i = eigenvalues of D 1/2 W D 1/2, i = 1,..., n; (ξ min < 0) ζ j = eigenvalues of B, j = 1,, p ξ i ζ j = eigenvalues for B (D 1/2 W D 1/2 ) I p D B W is p.d. iff ζ j (1/ξ min, 1). 21 Adapted from Sudipto Banerjee s notes SANOVA vs MCAR

42 MCAR models through latent spatial effects Basic idea Computation Notes Restriction on B: ξ i = eigenvalues of D 1/2 W D 1/2, i = 1,..., n; (ξ min < 0) ζ j = eigenvalues of B, j = 1,, p ξ i ζ j = eigenvalues for B (D 1/2 W D 1/2 ) I p D B W is p.d. iff ζ j (1/ξ min, 1). Update B: Spectral decomposition of B: B = P P ; P = p 1 i=1 p j=i+1 G ij(θ ij ) where G ij is I p with i-th and j-th diagonal elements replaced by cos(θ ij ) and (i, j)-th and (j, i)-th entry replaced by ± sin(θ ij ); priors: θ ij U( π/2, π/2), ζ j U(1/ξ min, 1) 21 Adapted from Sudipto Banerjee s notes SANOVA vs MCAR

43 Minnesota Cancer Example Minnesota Cancer Surveillance data used; Counties in Minnesota are treated as locations: j = 1,..., types of diseases considered: cancers of the lung, larynx and oesophagus: j = 1, 2, 3. E ij computed as the expected age adjusted number of deaths due to cancer j in county i. 22 Adapted from Sudipto Banerjee s notes SANOVA vs MCAR

44 Minnesota Cancer Example 23 Adapted from Sudipto Banerjee s notes SANOVA vs MCAR

45 Do we really need this madness? Do we really need these complex multivariate structures? Can the data really identify MCAR s elaborate dependence structures? What if we want to work with an improper MCAR? Can we have a simpler and more interpretable approach? 24 Adapted from Sudipto Banerjee s notes SANOVA vs MCAR

46 A new approach to multiple disease mapping Two-way ANOVA with one observation per cell County 1 County 2 County n Cancer 1 Cancer Cancer p 25 Adapted from Sudipto Banerjee s notes SANOVA vs MCAR

47 A new approach to multiple disease mapping Two-way ANOVA table Effects df Cancer p 1 County n 1 Error=Cancer County (n 1)(p 1) Total np 1 26 Adapted from Sudipto Banerjee s notes SANOVA vs MCAR

48 A new approach to multiple disease mapping Two-way ANOVA table Effects df Cancer p 1 County n 1 Error=Cancer County (n 1)(p 1) Total np 1 Saturated linear model: Grand Mean + Cancer Main Effects + County Main Effects + Cancer County Interaction Effects 26 Adapted from Sudipto Banerjee s notes SANOVA vs MCAR

49 A new approach to multiple disease mapping A new approach to multiple disease mapping? Smoothed ANOVA (Hodges et al, 2007): g(e[y]) = A 1 Θ 1 + A 2 Θ 2 ; M 1 and M 2 degrees of freedom for main effects and interactions respectively A 1 s columns correspond to the grand mean and main effects; A 1 A 1 = I M1. A 2 s columns correspond to interactions; A 2 A 2 = I M2. Rather than choosing interaction (and main effects) to include/exclude...include all these effects, but shrink toward zero. 27 Adapted from Sudipto Banerjee s notes SANOVA vs MCAR

50 A new approach to multiple disease mapping A 1 and A 2 are design matrices of contrasts for the unshrunk (overall & main) and shrunk (interaction) effects. Smoothing : Θ 2 N(0, diag( 1 η j )). Usually the interaction effects are shrunk towards zero. Θ 1 is assigned a flat prior; grand mean and main effects are not smoothed. 28 Adapted from Sudipto Banerjee s notes SANOVA vs MCAR

51 A new approach to multiple disease mapping A 1 and A 2 are design matrices of contrasts for the unshrunk (overall & main) and shrunk (interaction) effects. Smoothing : Θ 2 N(0, diag( 1 η j )). Usually the interaction effects are shrunk towards zero. Θ 1 is assigned a flat prior; grand mean and main effects are not smoothed. How does the improper CAR fit into SANOVA? 28 Adapted from Sudipto Banerjee s notes SANOVA vs MCAR

52 The intrinsic CAR Model Define Q = D W. The intrinsic CAR model (Besag et al 1991) with L 2 norm has the improper density ( p(φ τ) τ n G 2 exp τ ) 2 φ Qφ, 29 Adapted from Sudipto Banerjee s notes SANOVA vs MCAR

53 The intrinsic CAR Model Define Q = D W. The intrinsic CAR model (Besag et al 1991) with L 2 norm has the improper density ( p(φ τ) τ n G 2 exp τ ) 2 φ Qφ, τ is the spatial dispersion parameter. G is the number of islands" (disconnected parts). Assume that G = 1 (no major issues for general G) 29 Adapted from Sudipto Banerjee s notes SANOVA vs MCAR

54 Fit CAR into SANOVA Q1 = 0 and rank of Q is n Adapted from Sudipto Banerjee s notes SANOVA vs MCAR

55 Fit CAR into SANOVA Q1 = 0 and rank of Q is n 1. Let Q have spectral decomposition Q = V ΛV, Λ is diagonal and V is orthogonal with columns as eigenvectors. 30 Adapted from Sudipto Banerjee s notes SANOVA vs MCAR

56 Fit CAR into SANOVA Q1 = 0 and rank of Q is n 1. Let Q have spectral decomposition Q = V ΛV, Λ is diagonal and V is orthogonal with columns as eigenvectors. Eigenvector 1 n 1 n has eigenvalue Adapted from Sudipto Banerjee s notes SANOVA vs MCAR

57 Fit CAR into SANOVA Q1 = 0 and rank of Q is n 1. Let Q have spectral decomposition Q = V ΛV, Λ is diagonal and V is orthogonal with columns as eigenvectors. Eigenvector 1 n 1 n has eigenvalue 0. Let V s other columns are orthogonal to 1 n ; call them V 1 and their corresponding eigenvalue matrix as Λ Adapted from Sudipto Banerjee s notes SANOVA vs MCAR

58 Fit CAR into SANOVA Q1 = 0 and rank of Q is n 1. Let Q have spectral decomposition Q = V ΛV, Λ is diagonal and V is orthogonal with columns as eigenvectors. Eigenvector 1 n 1 n has eigenvalue 0. Let V s other columns are orthogonal to 1 n ; call them V 1 and their corresponding eigenvalue matrix as Λ 1. V 1 s n 1 columns also define contrasts in φ s: 1 V 1 = Adapted from Sudipto Banerjee s notes SANOVA vs MCAR

59 Fit CAR into SANOVA Define a new parameter Θ = V φ, so ( p(θ τ) τ n G 2 exp τ ) 2 Θ ΛΘ, 31 Adapted from Sudipto Banerjee s notes SANOVA vs MCAR

60 Fit CAR into SANOVA Define a new parameter Θ = V φ, so ( p(θ τ) τ n G 2 exp τ ) 2 Θ ΛΘ, Setting a CAR prior on φ setting Θ s prior to N(0, (τλ) 1 ). 31 Adapted from Sudipto Banerjee s notes SANOVA vs MCAR

61 Fit CAR into SANOVA Define a new parameter Θ = V φ, so ( p(θ τ) τ n G 2 exp τ ) 2 Θ ΛΘ, Setting a CAR prior on φ setting Θ s prior to N(0, (τλ) 1 ). Let φ = (φ 1, φ 2,..., φ n ), then the SANOVA model φ = V Θ [ ] [ 1 = V 1 n Θ 1 Reg n Θ GM where Θ REG N(0, (τλ 1 ) 1 ) and the precision Λ nn = 0 for the overall level, i.e., a flat prior on Θ GM. ]. 31 Adapted from Sudipto Banerjee s notes SANOVA vs MCAR

62 Fit MCAR into SANOVA Suppose φ, the spatial effects for p cancers in n counties, arises from a saturated linear regression model: φ = [ φ 1, φ 2,..., φ n ] = [A1 A 2 ] Θ = 1 np 1 np }{{} 1 1 n H CA n }{{} V 1 1 p 1 p }{{} V 1 H CA (1)... V1 H CA (p 1) }{{} Θ GM Θ CA Θ CO Θ CO CA Cancer County Cancer County Grand mean main eect main eect interaction np 1 np (p 1) np (n 1) np (n 1)(p 1) Let Θ CO N n 1 (0, (τ 0 Λ 1 ) 1 ); Θ CO CA (j) N n 1 (0, (τ j Λ 1 ) 1 ), where H CA is p (p 1) matrix whose columns are contrasts for cancer, Λ 1 corresponds to V 1 ; τ 0, τ j > 0 are unknown and τ 0 Λ 1, τ j Λ 1 are precision matrices. 32 Adapted from Sudipto Banerjee s notes SANOVA vs MCAR

63 Fit MCAR into SANOVA We can prove that a priori, A 2 Θ CO Θ CO CA (1). Θ CO CA (p 1) has precision Q (H A (+) (+) diag(τ j )H A ), 33 Adapted from Sudipto Banerjee s notes SANOVA vs MCAR

64 Fit MCAR into SANOVA We can prove that a priori, A 2 Θ CO Θ CO CA (1). Θ CO CA (p 1) has precision Q (H A (+) (+) diag(τ j )H A ), A 2 refers to the columns of the design matrix corresponding to the county main effects and the cancer-by-county interactions 33 Adapted from Sudipto Banerjee s notes SANOVA vs MCAR

65 Fit MCAR into SANOVA We can prove that a priori, A 2 Θ CO Θ CO CA (1). Θ CO CA (p 1) has precision Q (H A (+) (+) diag(τ j )H A ), A 2 refers to the columns of the design matrix corresponding to the county main effects and the cancer-by-county interactions H A (+) = ( 1 p 1 p H CA ) is an orthogonal matrix 33 Adapted from Sudipto Banerjee s notes SANOVA vs MCAR

66 MCAR vs SANOVA The marginal precision matrix of [ φ 1, φ 2,..., φ N ] is: MCAR SANOVA 34 Adapted from Sudipto Banerjee s notes SANOVA vs MCAR

67 Simulation Experiment We simulated both normal and Poisson distributed data for different settings of precision parameters. When generating data, we set H (+) A = HA 1 = We fit three SANOVA models: SANOVA with the correct H A (+), HA 1 ; SANOVA with a somewhat incorrect H A (+), HA 2 ; a variant SANOVA with a very incorrect H A (+), H AM, where HA 2 = H AM = , Adapted from Sudipto Banerjee s notes SANOVA vs MCAR

68 Simulation Findings Six models were compared. Three flavours of SANOVA SANOVA Correct has the H (+) A used to generate the data. SANOVA Incorrect used a slightly wrong H (+) A matrix. SANOVA Variant data generated from an MCAR (very wrong H (+) A ). Compared to an MCAR model with varying values for Ω s hyper-prior matrix (i.e. the Wishart scale matrix) R. R = I 3 R = I 3 R = 200 I 3 36 Adapted from Sudipto Banerjee s notes SANOVA vs MCAR

69 Simulation Findings Metrics used: AMSE: Average (over L = 100 replications) MSE of the cell means plugging in the posterior medians MBIAS: 2.5, 50, & 97.5 percentiles of the average (over replications) bias of the cell means (plugging in the posterior median). PI Rate: average coverage of the 95% posterior intervals of the cell means. DIC scores 37 Adapted from Sudipto Banerjee s notes SANOVA vs MCAR

70 Simulation Findings Findings: SANOVA Correct did the best (smallest AMSE & MBIAS, nearly nominal PI Rate). For normal data: SANOVA mostly outperformed MCAR. SANOVA models were comparable but worse than MCAR for good choices of the Wishart s scale matrix; MCAR performed poorly for very bad choices on R Overall: SANOVA appears competitive than MCAR and is fairly robust to mis-specifying H (+) A, unlike MCAR which is relatively sensitive to the hyper-prior on Ω. 38 Adapted from Sudipto Banerjee s notes SANOVA vs MCAR

71 Maps of raw and fitted SMR s Lung Cancer 39 Adapted from Sudipto Banerjee s notes SANOVA vs MCAR

72 Maps of raw and fitted SMR s Larynx Cancer 40 Adapted from Sudipto Banerjee s notes SANOVA vs MCAR

73 Maps of raw and fitted SMR s Esophagus Cancer 41 Adapted from Sudipto Banerjee s notes SANOVA vs MCAR

74 Discussion and future work We extended SANOVA to smooth spatial random effects taking advantage of the spatial structure. For the cases considered, SANOVA does in fact, compete well with MCAR. To improve SANOVA in the future, Extend SANOVA to more general MCAR case (ours is intrinsic MCAR). SANOVA in spatiotemporal models. SANOVA in spatial survival models (e.g., Cox model with spatial effects). Further investigations on choice of priors and robustness issues. 42 Adapted from Sudipto Banerjee s notes SANOVA vs MCAR

75 References 1 Besag, J, York, JC, and Mollié, A (1991). Bayesian image restoration, with two applications in spatial statistics (with discussion). Annals of the Institute of Statistical Mathematics, 43: Gelfand AE, Vounatsou P (2003). Proper multivariate conditional autoregressive models for spatial data analysis. Biostatistics, 4: Hodges JS, Cui Y, Sargent DJ, Carlin BP (2007). Smoothing balanced single-error-term analysis of variance. Technometrics, 49: Jin X, Carlin BP, Banerjee S (2005) Generalized hierarchical multivariate CAR models for areal data. Biometrics, 61: Jin X, Banerjee S, Carlin BP (2007). Order-free coregionalized lattice models with application to multiple disease mapping. J. Roy. Stat. Soc., Ser. B, 69: Kim H, Sun D, Tsutakawa RK (2001). A bivariate Bayes method for improving the estimates of mortality rates with a twofold conditional autoregressive model. J. Amer. Stat. Assn., 96: Sain SR, Cressie N (2002). Multivariate lattice models for spatial environmental data. In Proc. A.S.A. Section on Statistics and the Environment. Alexandria, VA: American Statistical Association, pp D. Sun, R.K. Tsutakawa, H. Kim and Z. He (2000). Bayesian Analysis of Mortality Rates with Disease Maps. Statistics in Medicine, 19: Zhang, Y., Hodges, J.S. and Banerjee, S. (2009) Smoothed ANOVA with Spatial Effects as a Competitor to MCAR in Multivariate Spatial Smoothing. Annals of Applied Statistics, in press. 43 Adapted from Sudipto Banerjee s notes SANOVA vs MCAR

76 Questions Thank You! 44 Adapted from Sudipto Banerjee s notes SANOVA vs MCAR