Hierarchical Additive Modeling of Nonlinear Association with Spatial Correlations

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1 1 ENAR09 LSUHSC p. 1/18 Hierarchical Additive Modeling of Nonlinear Association with Spatial Correlations Qingzhao Yu, Bin Li, Richard Scribner Louisiana State University, School of Public Health Supported by: NIAAA-5 R01 AA (PI Richard Scribner)

2 1 ENAR09 LSUHSC p. 2/18 Objective Data and Challenges Multivariate Additive Regression Trees Conditional Autoregressive Model Hierarchical Additive Model Model Convergence Results

3 1 ENAR09 LSUHSC p. 3/18 The aim is to determine whether a change in the alcohol environment results in a change of assaultive violence rates in the neighborhood. Hypothesis: Neighborhood alcohol outlet density is positively associated with alcohol related crimes, such as the assaultive crime.

4 1 ENAR09 LSUHSC p. 4/18 In 1992, a civil unrest occurred over a large area of south central Los Angeles. During the civil unrest, a number of alcohol outlets were damaged, so that the alcohol density decreased in that area. We restrict our analysis to the areas affected by civil unrest thereby controlling for a possible global effect of the unrest on outcomes. We control for the other factors that are associated with both alcohol density and the related crime rates.

5 1 ENAR09 LSUHSC p. 5/18 There are a total of 290 census tracts. All data are collected at the census tract level. We collected 10 years data: 1990 to 1999.

6 1 ENAR09 LSUHSC p. 5/18 There are a total of 290 census tracts. All data are collected at the census tract level. We collected 10 years data: 1990 to Alcohol Environment: data are obtained from the PRC and California ABC.

7 1 ENAR09 LSUHSC p. 5/18 There are a total of 290 census tracts. All data are collected at the census tract level. We collected 10 years data: 1990 to Alcohol Environment: data are obtained from the PRC and California ABC. outlet license surrender data following the 1992 Civil Unrest: revealing 279 outlets suspended operation following the 1992 unrest, 252 of those outlets were off-sale type outlets. annual count of offsale and onsale alcohol outlets

8 1 ENAR09 LSUHSC p. 6/18 Sociodemographic Data. These data are collected through US Census and LA county department of health services for the years 1990 to The variables we used include: % Black, % White, % Hispanic, % Asian, % male, % adults and % households in poverty

9 1 ENAR09 LSUHSC p. 6/18 Sociodemographic Data. These data are collected through US Census and LA county department of health services for the years 1990 to The variables we used include: % Black, % White, % Hispanic, % Asian, % male, % adults and % households in poverty Outcomes. The crime rates are collected from the LAPD annual police reports for the years 1990 to The data collected include: Assaultive Violence Rate: homicide, rape, robbery and assault. Complicated interactions might exist among covariates.

10 1 ENAR09 LSUHSC p. 7/18 The association between the assault violence rates and other covariate maybe nonlinear.

11 1 ENAR09 LSUHSC p. 7/18 The association between the assault violence rates and other covariate maybe nonlinear. Complicated interactions might exist among covariates.

12 1 ENAR09 LSUHSC p. 7/18 The association between the assault violence rates and other covariate maybe nonlinear. Complicated interactions might exist among covariates. More than 7% of the observations are missing one or more values for some covariates.

13 1 ENAR09 LSUHSC p. 7/18 The association between the assault violence rates and other covariate maybe nonlinear. Complicated interactions might exist among covariates. More than 7% of the observations are missing one or more values for some covariates. we should take into account the spatial correlations among observations in adjacent tracts.

14 MART is a special case of the generic gradient boosting approach. 1 ENAR09 LSUHSC p. 8/18

15 1 ENAR09 LSUHSC p. 8/18 MART is a special case of the generic gradient boosting approach. MART approximates the target function F(x) by an additive expansion of trees ˆf(x) = M m=1 νb H (x;γ m ).

16 1 ENAR09 LSUHSC p. 8/18 MART is a special case of the generic gradient boosting approach. MART approximates the target function F(x) by an additive expansion of trees ˆf(x) = M m=1 νb H (x;γ m ). Friedman, 2001, Friedman and Meulman, 2003.

17 1 ENAR09 LSUHSC p. 9/18 MART is able to capitalize on the nonlinear relationships between the dependent and independent variables with no need for specifying the basis functions.

18 1 ENAR09 LSUHSC p. 9/18 MART is able to capitalize on the nonlinear relationships between the dependent and independent variables with no need for specifying the basis functions. MART is able to capture complex and/or high order interaction effects.

19 1 ENAR09 LSUHSC p. 9/18 MART is able to capitalize on the nonlinear relationships between the dependent and independent variables with no need for specifying the basis functions. MART is able to capture complex and/or high order interaction effects. MART can handle mixed-type predictors and missing values in covariates.

20 1 ENAR09 LSUHSC p. 10/18 We use the vector {φ Ti,C i } to capture spatial autocorrelations among areas C i at time T i. Assume that an area C i is correlated with only the areas that are adjacent to it. y with a hierarchical structure on its mean function: y i N(µ i, σ 2 ) and µ i = f(x i ) + φ Ti,C i ; (1) The hierarchical CAR structure for {φ Ti,C i } has the form φ Ti,C i =j φ Ti,C i j N 1 1 φ Ti,k,. n j n j τ Ti k j

21 1 ENAR09 LSUHSC p. 11/18 1. Let φ 0 T i,c i = 0 where C i {1,..., C}, T i {1,..., T }; q = 0, δ = 1000, µ 1i = 0 and i = 1,..., n. 2. If δ <, go to step 3, otherwise q=q+1 and (a) Let z i = y i φ [q 1] T i,c i. Fit MART f [q] (x) on z and the covariates x. (b) Let e i = y i f [q] (x i ), calculate the Moran s I of e i within time slots 1 to T. Let S be the collection of time slots in which the spatial correlation test show a p-value smaller than 0.01(this indicates a strong spatial correlation in e i ).

22 1 ENAR09 LSUHSC p. 12/18 (c) If S is empty, let φ [q] T i,c i = 0 and go to step 3; otherwise for the observations i {i : T i S}, let the f(x i ) in Equation (1) be f [q] (x i ) and calculate the generalized MLEs of ˆφ Ti,C i. Let φ [q] T i S,C i = ˆφ Ti,C i and φ [q] T i / S,C i = 0. (d) Let µ 0i = µ 1i, µ 1i = f [q] (x i ) + φ [q] T i,c i and let δ = go back to Output the results from step q. n i=1 (µ 1i µ 0i ) 2 n i=1 µ2 1i,

23 1 ENAR09 LSUHSC p. 13/18 Little is known of the convergence properties of the above procedure.

24 1 ENAR09 LSUHSC p. 13/18 Little is known of the convergence properties of the above procedure. ICM (Iterated Conditional Modes), Besag (1986).

25 1 ENAR09 LSUHSC p. 13/18 Little is known of the convergence properties of the above procedure. ICM (Iterated Conditional Modes), Besag (1986). Under certain conditions, iterative conditional maximization converges to local maximizers. Meng and Rubin (1993).

26 1 ENAR09 LSUHSC p. 13/18 Little is known of the convergence properties of the above procedure. ICM (Iterated Conditional Modes), Besag (1986). Under certain conditions, iterative conditional maximization converges to local maximizers. Meng and Rubin (1993). Backfitting process, Hastie and Tibshirani, 2000; Buja et al

27 1 ENAR09 LSUHSC p. 13/18 Little is known of the convergence properties of the above procedure. ICM (Iterated Conditional Modes), Besag (1986). Under certain conditions, iterative conditional maximization converges to local maximizers. Meng and Rubin (1993). Backfitting process, Hastie and Tibshirani, 2000; Buja et al Let = 10 7 in our analysis.

28 1 ENAR09 LSUHSC p. 14/18 pctoffsurryn2 pctonsurryn2 hispanic male_15_30 damage asian totaldensity year black white poverty onsurryn offsurryn hispanic offsale onsale male_15_30 damage asian year black white poverty Relative Variable Importance Relative Variable Importance Figure 1: Relative variable importance in MART-fitted models.

29 Partial dependence Partial dependence poverty black Partial dependence Partial dependence white year Partial dependence Partial dependence totaldensity male_15_30 Figure 2: Partial dependence plots. 1 ENAR09 LSUHSC p. 15/18

30 1 ENAR09 LSUHSC p. 16/18 Year Origin P-value Res1 P-value Res2 P-value < e < e < e < e < e < e < e < e < e < e

31 1 ENAR09 LSUHSC p. 17/18 values for loghatrate98 (4) < 0.0 values for logmartrate98 (2) < 0.0 N (20) N (117) (8) (113) (128) (165) (21) >= 3.0 (2) >= 3.0 values for spatialerror km (6) < values for randomerror km (7) < N (51) N (165) (67) (135) (65) (76) (3) >= 0.75 (5) >= 0.75

32 1 ENAR09 LSUHSC p. 18/18