Modern Bayesian Statistics Part III: high-dimensional modeling Example 2: Spatial hierarchical factor analysis

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1 Modern Bayesian Statistics Part III: high-dimensional modeling Example 2: Spatial hierarchical factor analysis Hedibert Freitas Lopes 1 hedibert.org 13 a amostra de Estatística IME-USP, October Professor of Statistics and Econometrics at Insper, São Paulo. 1

2 Example 2: Spatial hierarchical factor analysis Vulnerability index for Uruguay Uruguay has an area of 176,215 km 2 and roughly 3.3 million inhabitants, half of which live in the capital,. Around 93% of the population lives in urban areas. MELO Argentina BELLA_UNION ARTIGAS Brazil RIVERA SALTO TACUAREMBO PAYSANDU MELO FRAY_BENTOS MERCEDES DURAZNO TRINIDAD TREINTA_Y_TRES COLONIA FLORIDA SAN_JOSE MINAS ROCHA CANELONES MALDONADO MONTEVIDEO Atlantic Ocean 2

3 Census tracts per capital Capital Census tracts Capital Census tracts Bella Unión Tacuarembó Paysandú San José 34 3

4 Main goals To characterize the vulnerability of the population of Uruguay to diseases transmitted through vectors (e.g. Dengue Fever, Malaria, etc.); To help prioritizing the allocation of fundings; We have information on p = 11 variables per census tracts of the I = 19 Departamental Capitals of the country. Source: Census 1996 (latest Census in Uruguay) 4

5 Table: Description of the p = 11 variables, observed in the census tract level of the departmental capitals, to build the vulnerability index of the population of Uruguay to vector-borne diseases. Levels Personal characteristic Household characteristic Variables Illiteracy rate (ILL) Population with access to public health care (PHC) Male without formal jobs (UQW) Owed houses (OWH) Households headed by a woman (WHF) Households without sewage system (AHS) Average number of persons per household (APH) Households with more than two persons per room (OVC) Households without access to drinkable water (ADW) Households with air conditioner (ACO) Households poorly built (HOQ) 5

6 Sample correlations ILL PHC OVC UQW AHS ADW APH PHC 0.85 OVC UQW AHS ADW APH HOQ The sample correlations between OWH or WHF or ACO and any one of the attributes are below 18% (in absolute value). 6

7 Model structure Observational Level: y ijk = µ k + β k f ij + σ k ε ijk k = 1,, p, where µ k represents the overall grand mean. Modeling f ij : f ij = θ i + f ij + ω i u ij where θ i is the common factor for capital i. Spatial variation within capitals: f i N(0, τ 2 i P i ) where P i = (I ni + φm i ) 1, M i = D i W i, with w ijl, the (j, l) component of W i, given by w ijl = 1/d jl if j and l are neighbors (denoted here by j l) and zero otherwise, d jl = s j s l is the Euclidean distance between centroids of capitals j and l, D i = diag(w i1+,..., w ini +) and w ij+ = l j w ijl. 7

8 Model structure (cont.) Spatial variation between capitals: where θ = (θ 1,, θ I ). θ N ( 1 I θ 0, δ 2 H(λ) ), Although each capital i has its own vulnerability factor, the above model allows borrowing-strength across neighboring regions. 8

9 Table: Comparing SHFM and UHFM: Comparing the unstructured hierarchical factor (UHFM) and spatial hierarchical factor models (SHFM) for different values of φ. Best models appear in italic. DIC: deviance information criterion, EPD: expected posterior deviation, CRPS: continuous ranked probability score, MSE: mean square error and MAE: mean absolute error. CRPS are in tens of thousands. UHFM SHFM Criterion θ = 0 unknown θ φ = 1 φ = 5 φ = 7 DIC EPD CRPS MAE MSE

10 latitude Argentina SALTO PAYSANDU FRAY_BENTOS MERCEDES COLONIA BELLA_UNION ARTIGAS DURAZNO TRINIDAD SAN_JOSE FLORIDA RIVERA TACUAREMBO CANELONES MONTEVIDEO MINAS MELO TREINTA_Y_TRES ROCHA MALDONADO Brazil Atlantic Ocean latitude Argentina PAYSANDU FRAY_BENTOS MERCEDES BELLA_UNION SALTO COLONIA ARTIGAS DURAZNO TRINIDAD SAN_JOSE FLORIDA RIVERA TACUAREMBO CANELONES MONTEVIDEO MINAS MELO TREINTA_Y_TRES ROCHA MALDONADO Brazil Atlantic Ocean longitude longitude Figure: Posterior mean of θ i and standard deviations (second column) for observed and unobserved cities under the SHFM when φ = 5. 10

11 Posterior mean Posterior mean Posterior mean Posterior mean Figure: Posterior means of the θ i and 95% CI. Top row: SHFM with φ = 5 (left) and UHFM (right). Bottom row: ASFM (left) and AFM (right). 11

12 Rank Rank Rank Rank Figure: Posterior rankings of the capitals. Top row: SHFM with φ = 5 (left) and UHFM (right).bottom row: ASFM (left) and AFM (right). 12

13 Rank

14 (a) Bella Unión (b) under over 0.8 under over 0.8 meters meters (c) (d) meters under over 0.8 under over 0.8 meters Figure: Within-city posterior vulnerability index per census 14

15 under over 0.8 meters